first order logics as a modelling languageoutline introduction fol formalization first order logics...
TRANSCRIPT
![Page 1: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/1.jpg)
OutlineIntroduction
FOL Formalization
First Order Logics as a Modelling Language
Luciano Serafini
Fondazione Bruno Kessler, Trento, Italy
December 14, 2017
Luciano Serafini First Order Logics as a Modelling Language
![Page 2: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/2.jpg)
OutlineIntroduction
FOL Formalization
1 IntroductionWell formed formulasFree and bounded variables
2 FOL FormalizationSimple SentencesFOL InterpretationFormalizing Problems
Graph Coloring Problem
Luciano Serafini First Order Logics as a Modelling Language
![Page 3: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/3.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Alphabet and formation rules
Logical symbols:⊥,∧,∨,→,¬,∀,∃,=Non Logical symbols:a set c1, .., cn of constantsa set f1, .., fm of functional symbolsa set P1, ..,Pm of relational symbols
Terms T :T := ci |xi |fi (T , ..,T )
Well formed formulas W:W := T = T |Pi (T , . . . ,T )|⊥|W ∧W |W ∨W |
W →W |¬W |∀x .W |∃x .W
Luciano Serafini First Order Logics as a Modelling Language
![Page 4: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/4.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
q(a);
p(y);
p(g(b));
¬r(x , a);
q(x , p(a), b);
p(g(f (a), g(x , f (x))));
q(f (a), f (f (x)), f (g(f (z), g(a, b))));
r(a, r(a, a));
Luciano Serafini First Order Logics as a Modelling Language
![Page 5: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/5.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
q(a);
p(y);
p(g(b));
¬r(x , a);
q(x , p(a), b);
p(g(f (a), g(x , f (x))));
q(f (a), f (f (x)), f (g(f (z), g(a, b))));
r(a, r(a, a));
Luciano Serafini First Order Logics as a Modelling Language
![Page 6: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/6.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
q(a);
p(y);
p(g(b));
¬r(x , a);
q(x , p(a), b);
p(g(f (a), g(x , f (x))));
q(f (a), f (f (x)), f (g(f (z), g(a, b))));
r(a, r(a, a));
Luciano Serafini First Order Logics as a Modelling Language
![Page 7: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/7.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
q(a);
p(y);
p(g(b));
¬r(x , a);
q(x , p(a), b);
p(g(f (a), g(x , f (x))));
q(f (a), f (f (x)), f (g(f (z), g(a, b))));
r(a, r(a, a));
Luciano Serafini First Order Logics as a Modelling Language
![Page 8: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/8.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
q(a);
p(y);
p(g(b));
¬r(x , a);
q(x , p(a), b);
p(g(f (a), g(x , f (x))));
q(f (a), f (f (x)), f (g(f (z), g(a, b))));
r(a, r(a, a));
Luciano Serafini First Order Logics as a Modelling Language
![Page 9: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/9.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
q(a);
p(y);
p(g(b));
¬r(x , a);
q(x , p(a), b);
p(g(f (a), g(x , f (x))));
q(f (a), f (f (x)), f (g(f (z), g(a, b))));
r(a, r(a, a));
Luciano Serafini First Order Logics as a Modelling Language
![Page 10: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/10.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
q(a);
p(y);
p(g(b));
¬r(x , a);
q(x , p(a), b);
p(g(f (a), g(x , f (x))));
q(f (a), f (f (x)), f (g(f (z), g(a, b))));
r(a, r(a, a));
Luciano Serafini First Order Logics as a Modelling Language
![Page 11: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/11.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
q(a);
p(y);
p(g(b));
¬r(x , a);
q(x , p(a), b);
p(g(f (a), g(x , f (x))));
q(f (a), f (f (x)), f (g(f (z), g(a, b))));
r(a, r(a, a));
Luciano Serafini First Order Logics as a Modelling Language
![Page 12: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/12.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
q(a);
p(y);
p(g(b));
¬r(x , a);
q(x , p(a), b);
p(g(f (a), g(x , f (x))));
q(f (a), f (f (x)), f (g(f (z), g(a, b))));
r(a, r(a, a));Luciano Serafini First Order Logics as a Modelling Language
![Page 13: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/13.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
r(a, g(a, a));
g(a, g(a, a));
∀x .¬p(x);
¬r(p(a), x);
∃a.r(a, a);
∃x .q(x , f (x), b)→ ∀x .r(a, x);
∃x .p(r(a, x));
∀r(x , a);
Luciano Serafini First Order Logics as a Modelling Language
![Page 14: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/14.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
r(a, g(a, a));
g(a, g(a, a));
∀x .¬p(x);
¬r(p(a), x);
∃a.r(a, a);
∃x .q(x , f (x), b)→ ∀x .r(a, x);
∃x .p(r(a, x));
∀r(x , a);
Luciano Serafini First Order Logics as a Modelling Language
![Page 15: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/15.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
r(a, g(a, a));
g(a, g(a, a));
∀x .¬p(x);
¬r(p(a), x);
∃a.r(a, a);
∃x .q(x , f (x), b)→ ∀x .r(a, x);
∃x .p(r(a, x));
∀r(x , a);
Luciano Serafini First Order Logics as a Modelling Language
![Page 16: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/16.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
r(a, g(a, a));
g(a, g(a, a));
∀x .¬p(x);
¬r(p(a), x);
∃a.r(a, a);
∃x .q(x , f (x), b)→ ∀x .r(a, x);
∃x .p(r(a, x));
∀r(x , a);
Luciano Serafini First Order Logics as a Modelling Language
![Page 17: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/17.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
r(a, g(a, a));
g(a, g(a, a));
∀x .¬p(x);
¬r(p(a), x);
∃a.r(a, a);
∃x .q(x , f (x), b)→ ∀x .r(a, x);
∃x .p(r(a, x));
∀r(x , a);
Luciano Serafini First Order Logics as a Modelling Language
![Page 18: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/18.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
r(a, g(a, a));
g(a, g(a, a));
∀x .¬p(x);
¬r(p(a), x);
∃a.r(a, a);
∃x .q(x , f (x), b)→ ∀x .r(a, x);
∃x .p(r(a, x));
∀r(x , a);
Luciano Serafini First Order Logics as a Modelling Language
![Page 19: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/19.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
r(a, g(a, a));
g(a, g(a, a));
∀x .¬p(x);
¬r(p(a), x);
∃a.r(a, a);
∃x .q(x , f (x), b)→ ∀x .r(a, x);
∃x .p(r(a, x));
∀r(x , a);
Luciano Serafini First Order Logics as a Modelling Language
![Page 20: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/20.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Say whether the following strings of symbols are well formed formulas orterms:
r(a, g(a, a));
g(a, g(a, a));
∀x .¬p(x);
¬r(p(a), x);
∃a.r(a, a);
∃x .q(x , f (x), b)→ ∀x .r(a, x);
∃x .p(r(a, x));
∀r(x , a);Luciano Serafini First Order Logics as a Modelling Language
![Page 21: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/21.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
FOL Syntax
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Exercises
Say whether the following strings of symbols are well formed formulas orterms:
a→ p(b);
r(x , b)→ ∃y .q(y , y , y);
r(x , b) ∨ ¬∃y .g(y , b);
¬y ∨ p(y);
¬¬p(a);
¬∀x .¬p(x);
∀x∃y .(r(x , y)→ r(y , x));
∀x∃y .(r(x , y)→ (r(y , x) ∨ (f (a) = g(a, x))));Luciano Serafini First Order Logics as a Modelling Language
![Page 22: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/22.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
A free occurrence of a variable x is an occurrence of x which is notbounded by a ∀x or ∃x quantifier.
A variable x is free in a formula φ (denoted by φ(x)) if there is atleast a free occurrence of x in φ.
A variable x is bounded in a formula φ if it is not free.
Luciano Serafini First Order Logics as a Modelling Language
![Page 23: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/23.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
A free occurrence of a variable x is an occurrence of x which is notbounded by a ∀x or ∃x quantifier.
A variable x is free in a formula φ (denoted by φ(x)) if there is atleast a free occurrence of x in φ.
A variable x is bounded in a formula φ if it is not free.
Luciano Serafini First Order Logics as a Modelling Language
![Page 24: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/24.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
A free occurrence of a variable x is an occurrence of x which is notbounded by a ∀x or ∃x quantifier.
A variable x is free in a formula φ (denoted by φ(x)) if there is atleast a free occurrence of x in φ.
A variable x is bounded in a formula φ if it is not free.
Luciano Serafini First Order Logics as a Modelling Language
![Page 25: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/25.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Find free and bounded variables in the following formulas:
p(x) ∧ ¬r(y , a)
∃x .r(x , y)
∀x .p(x)→ ∃y .¬q(f (x), y , f (y))
∀x∃y .r(x , f (y))
∀x∃y .r(x , f (y))→ r(x , y)
Luciano Serafini First Order Logics as a Modelling Language
![Page 26: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/26.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Find free and bounded variables in the following formulas:
p(x) ∧ ¬r(y , a)
∃x .r(x , y)
∀x .p(x)→ ∃y .¬q(f (x), y , f (y))
∀x∃y .r(x , f (y))
∀x∃y .r(x , f (y))→ r(x , y)
Luciano Serafini First Order Logics as a Modelling Language
![Page 27: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/27.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Find free and bounded variables in the following formulas:
p(x) ∧ ¬r(y , a)
∃x .r(x , y)
∀x .p(x)→ ∃y .¬q(f (x), y , f (y))
∀x∃y .r(x , f (y))
∀x∃y .r(x , f (y))→ r(x , y)
Luciano Serafini First Order Logics as a Modelling Language
![Page 28: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/28.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Find free and bounded variables in the following formulas:
p(x) ∧ ¬r(y , a)
∃x .r(x , y)
∀x .p(x)→ ∃y .¬q(f (x), y , f (y))
∀x∃y .r(x , f (y))
∀x∃y .r(x , f (y))→ r(x , y)
Luciano Serafini First Order Logics as a Modelling Language
![Page 29: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/29.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Find free and bounded variables in the following formulas:
p(x) ∧ ¬r(y , a)
∃x .r(x , y)
∀x .p(x)→ ∃y .¬q(f (x), y , f (y))
∀x∃y .r(x , f (y))
∀x∃y .r(x , f (y))→ r(x , y)
Luciano Serafini First Order Logics as a Modelling Language
![Page 30: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/30.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Examples
Find free and bounded variables in the following formulas:
p(x) ∧ ¬r(y , a)
∃x .r(x , y)
∀x .p(x)→ ∃y .¬q(f (x), y , f (y))
∀x∃y .r(x , f (y))
∀x∃y .r(x , f (y))→ r(x , y)
Luciano Serafini First Order Logics as a Modelling Language
![Page 31: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/31.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Non Logical symbols
constants a, b; functions f 1, g2; predicates p1, r2, q3.
Exercises
Find free and bounded variables in the following formulas:
∀x .(p(x)→ ∃y .¬q(f (x), y , f (y)))
∀x(∃y .r(x , f (y))→ r(x , y))
∀z .(p(z)→ ∃y .(∃x .q(x , y , z) ∨ q(z , y , x)))
∀z∃u∃y .(q(z , u, g(u, y)) ∨ r(u, g(z , u)))
∀z∃x∃y(q(z , u, g(u, y)) ∨ r(u, g(z , u)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 32: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/32.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 33: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/33.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y)
y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 34: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/34.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 35: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/35.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y)
no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 36: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/36.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 37: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/37.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12
x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 38: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/38.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 39: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/39.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12)
no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 40: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/40.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 41: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/41.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12)
y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 42: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/42.jpg)
OutlineIntroduction
FOL Formalization
Well formed formulasFree and bounded variables
Free variables
Intuitively..
Free variables represents individuals which must be instantiated tomake the formula a meaningful proposition.
Friends(Bob, y) y free
∀y .Friends(Bob, y) no free variables
Sum(x , 3) = 12 x free
∃x .(Sum(x , 3) = 12) no free variables
∃x .(Sum(x , y) = 12) y free
Luciano Serafini First Order Logics as a Modelling Language
![Page 43: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/43.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)
”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 44: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/44.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 45: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/45.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)
”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 46: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/46.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 47: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/47.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))
”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 48: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/48.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 49: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/49.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)
”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 50: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/50.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 51: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/51.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)
”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 52: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/52.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 53: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/53.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)
”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 54: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/54.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Examples
bought(Frank , dvd)”Frank bought a dvd.”
∃x .bought(Frank , x)”Frank bought something.”
∀x .(bought(Frank , x)→ bought(Susan, x))”Susan bought everything that Frank bought.”
∀x .bought(Frank , x)→ ∀x .bought(Susan, x)”If Frank bought everything, so did Susan.”
∀x∃y .bought(x , y)”Everyone bought something.”
∃x∀y .bought(x , y)”Someone bought everything.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 55: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/55.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Example
Which of the following formulas is a formalization of the sentence:”There is a computer which is not used by any student”
∃x .(Computer(x) ∧ ∀y .(¬Student(y) ∧ ¬Uses(y , x)))
∃x .(Computer(x)→ ∀y .(Student(y)→ ¬Uses(y , x)))
∃x .(Computer(x) ∧ ∀y .(Student(y)→ ¬Uses(y , x)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 56: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/56.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL: Intuitive Meaning
Example
Which of the following formulas is a formalization of the sentence:”There is a computer which is not used by any student”
∃x .(Computer(x) ∧ ∀y .(¬Student(y) ∧ ¬Uses(y , x)))
∃x .(Computer(x)→ ∀y .(Student(y)→ ¬Uses(y , x)))
∃x .(Computer(x) ∧ ∀y .(Student(y)→ ¬Uses(y , x)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 57: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/57.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake..
”Everyone studying at DIT is smart.”∀x .(At(x ,DIT )→ Smart(x))
and NOT∀x .(At(x ,DIT ) ∧ Smart(x))
”Everyone studies at DIT and everyone is smart”
”Someone studying at DIT is smart.”∃x .(At(x ,DIT ) ∧ Smart(x))
and NOT∃x .(At(x ,DIT )→ Smart(x))
which is true if there is anyone who is not at DIT.
Luciano Serafini First Order Logics as a Modelling Language
![Page 58: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/58.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake..
”Everyone studying at DIT is smart.”∀x .(At(x ,DIT )→ Smart(x))
and NOT∀x .(At(x ,DIT ) ∧ Smart(x))
”Everyone studies at DIT and everyone is smart”
”Someone studying at DIT is smart.”∃x .(At(x ,DIT ) ∧ Smart(x))
and NOT∃x .(At(x ,DIT )→ Smart(x))
which is true if there is anyone who is not at DIT.
Luciano Serafini First Order Logics as a Modelling Language
![Page 59: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/59.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake..
”Everyone studying at DIT is smart.”∀x .(At(x ,DIT )→ Smart(x))
and NOT∀x .(At(x ,DIT ) ∧ Smart(x))
”Everyone studies at DIT and everyone is smart”
”Someone studying at DIT is smart.”∃x .(At(x ,DIT ) ∧ Smart(x))
and NOT∃x .(At(x ,DIT )→ Smart(x))
which is true if there is anyone who is not at DIT.
Luciano Serafini First Order Logics as a Modelling Language
![Page 60: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/60.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake..
”Everyone studying at DIT is smart.”∀x .(At(x ,DIT )→ Smart(x))
and NOT∀x .(At(x ,DIT ) ∧ Smart(x))
”Everyone studies at DIT and everyone is smart”
”Someone studying at DIT is smart.”∃x .(At(x ,DIT ) ∧ Smart(x))
and NOT∃x .(At(x ,DIT )→ Smart(x))
which is true if there is anyone who is not at DIT.
Luciano Serafini First Order Logics as a Modelling Language
![Page 61: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/61.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake..
”Everyone studying at DIT is smart.”∀x .(At(x ,DIT )→ Smart(x))
and NOT∀x .(At(x ,DIT ) ∧ Smart(x))
”Everyone studies at DIT and everyone is smart”
”Someone studying at DIT is smart.”∃x .(At(x ,DIT ) ∧ Smart(x))
and NOT∃x .(At(x ,DIT )→ Smart(x))
which is true if there is anyone who is not at DIT.
Luciano Serafini First Order Logics as a Modelling Language
![Page 62: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/62.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake..
”Everyone studying at DIT is smart.”∀x .(At(x ,DIT )→ Smart(x))
and NOT∀x .(At(x ,DIT ) ∧ Smart(x))
”Everyone studies at DIT and everyone is smart”
”Someone studying at DIT is smart.”∃x .(At(x ,DIT ) ∧ Smart(x))
and NOT∃x .(At(x ,DIT )→ Smart(x))
which is true if there is anyone who is not at DIT.
Luciano Serafini First Order Logics as a Modelling Language
![Page 63: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/63.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake.. (2)
Quantifiers of different type do NOT commute
∃x∀y .φ is not the same as ∀y∃x .φ
Example
∃x∀y .Loves(x , y)”There is a person who loves everyone in the world.”
∀y∃x .Loves(x , y)”Everyone in the world is loved by at least one person.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 64: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/64.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake.. (2)
Quantifiers of different type do NOT commute∃x∀y .φ is not the same as ∀y∃x .φ
Example
∃x∀y .Loves(x , y)”There is a person who loves everyone in the world.”
∀y∃x .Loves(x , y)”Everyone in the world is loved by at least one person.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 65: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/65.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake.. (2)
Quantifiers of different type do NOT commute∃x∀y .φ is not the same as ∀y∃x .φ
Example
∃x∀y .Loves(x , y)”There is a person who loves everyone in the world.”
∀y∃x .Loves(x , y)”Everyone in the world is loved by at least one person.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 66: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/66.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Common mistake.. (2)
Quantifiers of different type do NOT commute∃x∀y .φ is not the same as ∀y∃x .φ
Example
∃x∀y .Loves(x , y)”There is a person who loves everyone in the world.”
∀y∃x .Loves(x , y)”Everyone in the world is loved by at least one person.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 67: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/67.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.
∀x .(Student(x)→ Smart(x))
There exists a student.∃x .Student(x)
There exists a smart student∃x .(Student(x) ∧ Smart(x))
Every student loves some student∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 68: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/68.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.∀x .(Student(x)→ Smart(x))
There exists a student.∃x .Student(x)
There exists a smart student∃x .(Student(x) ∧ Smart(x))
Every student loves some student∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 69: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/69.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.∀x .(Student(x)→ Smart(x))
There exists a student.
∃x .Student(x)
There exists a smart student∃x .(Student(x) ∧ Smart(x))
Every student loves some student∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 70: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/70.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.∀x .(Student(x)→ Smart(x))
There exists a student.∃x .Student(x)
There exists a smart student∃x .(Student(x) ∧ Smart(x))
Every student loves some student∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 71: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/71.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.∀x .(Student(x)→ Smart(x))
There exists a student.∃x .Student(x)
There exists a smart student
∃x .(Student(x) ∧ Smart(x))
Every student loves some student∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 72: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/72.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.∀x .(Student(x)→ Smart(x))
There exists a student.∃x .Student(x)
There exists a smart student∃x .(Student(x) ∧ Smart(x))
Every student loves some student∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 73: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/73.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.∀x .(Student(x)→ Smart(x))
There exists a student.∃x .Student(x)
There exists a smart student∃x .(Student(x) ∧ Smart(x))
Every student loves some student
∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 74: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/74.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.∀x .(Student(x)→ Smart(x))
There exists a student.∃x .Student(x)
There exists a smart student∃x .(Student(x) ∧ Smart(x))
Every student loves some student∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 75: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/75.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.∀x .(Student(x)→ Smart(x))
There exists a student.∃x .Student(x)
There exists a smart student∃x .(Student(x) ∧ Smart(x))
Every student loves some student∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.
∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 76: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/76.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
All Students are smart.∀x .(Student(x)→ Smart(x))
There exists a student.∃x .Student(x)
There exists a smart student∃x .(Student(x) ∧ Smart(x))
Every student loves some student∀x .(Student(x)→ ∃y .(Student(y) ∧ Loves(x , y)))
Every student loves some other student.∀x .(Student(x)→ ∃y .(Student(y) ∧ ¬(x = y) ∧ Loves(x , y)))
Luciano Serafini First Order Logics as a Modelling Language
![Page 77: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/77.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.
∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.Student(Bill)
Bill takes either Analysis or Geometry (but not both).Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 78: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/78.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.Student(Bill)
Bill takes either Analysis or Geometry (but not both).Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 79: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/79.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.
Student(Bill)
Bill takes either Analysis or Geometry (but not both).Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 80: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/80.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.Student(Bill)
Bill takes either Analysis or Geometry (but not both).Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 81: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/81.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.Student(Bill)
Bill takes either Analysis or Geometry (but not both).
Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 82: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/82.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.Student(Bill)
Bill takes either Analysis or Geometry (but not both).Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 83: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/83.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.Student(Bill)
Bill takes either Analysis or Geometry (but not both).Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.
Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 84: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/84.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.Student(Bill)
Bill takes either Analysis or Geometry (but not both).Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 85: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/85.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.Student(Bill)
Bill takes either Analysis or Geometry (but not both).Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.
¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 86: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/86.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
There is a student who is loved by every other student.∃x .(Student(x) ∧ ∀y .(Student(y) ∧ ¬(x = y)→ Loves(y , x)))
Bill is a student.Student(Bill)
Bill takes either Analysis or Geometry (but not both).Takes(Bill ,Analysis)↔ ¬Takes(Bill ,Geometry)
Bill takes Analysis and Geometry.Takes(Bill ,Analysis) ∧ Takes(Bill ,Geometry)
Bill doesn’t take Analysis.¬Takes(Bill ,Analysis)
Luciano Serafini First Order Logics as a Modelling Language
![Page 87: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/87.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.
¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 88: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/88.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 89: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/89.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.
∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 90: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/90.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 91: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/91.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.
¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 92: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/92.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 93: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/93.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.
∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 94: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/94.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 95: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/95.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.
∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 96: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/96.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 97: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/97.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.
∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 98: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/98.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
No students love Bill.¬∃x .(Student(x) ∧ Loves(x ,Bill))
Bill has at least one sister.∃x .SisterOf (x ,Bill)
Bill has no sister.¬∃x .SisterOf (x ,Bill)
Bill has at most one sister.∀x∀y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill)→ x = y)
Bill has (exactly) one sister.∃x .(SisterOf (x ,Bill) ∧ ∀y .(SisterOf (y ,Bill)→ x = y))
Bill has at least two sisters.∃x∃y .(SisterOf (x ,Bill) ∧ SisterOf (y ,Bill) ∧ ¬(x = y))
Luciano Serafini First Order Logics as a Modelling Language
![Page 99: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/99.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
Every student takes at least one course.
∀x .(Student(x)→ ∃y .(Course(y) ∧ Takes(x , y)))
Only one student failed Geometry.∃x .(Student(x) ∧ Failed(x ,Geometry) ∧ ∀y .(Student(y) ∧Failed(y ,Geometry)→ x = y))
No student failed Geometry but at least one student failedAnalysis.¬∃x .(Student(x) ∧ Failed(x ,Geometry)) ∧ ∃x .(Student(x) ∧Failed(x ,Analysis))
Every student who takes Analysis also takes Geometry.∀x .(Student(x) ∧ Takes(x ,Analysis)→ Takes(x ,Geometry))
Luciano Serafini First Order Logics as a Modelling Language
![Page 100: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/100.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
Every student takes at least one course.∀x .(Student(x)→ ∃y .(Course(y) ∧ Takes(x , y)))
Only one student failed Geometry.∃x .(Student(x) ∧ Failed(x ,Geometry) ∧ ∀y .(Student(y) ∧Failed(y ,Geometry)→ x = y))
No student failed Geometry but at least one student failedAnalysis.¬∃x .(Student(x) ∧ Failed(x ,Geometry)) ∧ ∃x .(Student(x) ∧Failed(x ,Analysis))
Every student who takes Analysis also takes Geometry.∀x .(Student(x) ∧ Takes(x ,Analysis)→ Takes(x ,Geometry))
Luciano Serafini First Order Logics as a Modelling Language
![Page 101: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/101.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
Every student takes at least one course.∀x .(Student(x)→ ∃y .(Course(y) ∧ Takes(x , y)))
Only one student failed Geometry.
∃x .(Student(x) ∧ Failed(x ,Geometry) ∧ ∀y .(Student(y) ∧Failed(y ,Geometry)→ x = y))
No student failed Geometry but at least one student failedAnalysis.¬∃x .(Student(x) ∧ Failed(x ,Geometry)) ∧ ∃x .(Student(x) ∧Failed(x ,Analysis))
Every student who takes Analysis also takes Geometry.∀x .(Student(x) ∧ Takes(x ,Analysis)→ Takes(x ,Geometry))
Luciano Serafini First Order Logics as a Modelling Language
![Page 102: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/102.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
Every student takes at least one course.∀x .(Student(x)→ ∃y .(Course(y) ∧ Takes(x , y)))
Only one student failed Geometry.∃x .(Student(x) ∧ Failed(x ,Geometry) ∧ ∀y .(Student(y) ∧Failed(y ,Geometry)→ x = y))
No student failed Geometry but at least one student failedAnalysis.¬∃x .(Student(x) ∧ Failed(x ,Geometry)) ∧ ∃x .(Student(x) ∧Failed(x ,Analysis))
Every student who takes Analysis also takes Geometry.∀x .(Student(x) ∧ Takes(x ,Analysis)→ Takes(x ,Geometry))
Luciano Serafini First Order Logics as a Modelling Language
![Page 103: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/103.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
Every student takes at least one course.∀x .(Student(x)→ ∃y .(Course(y) ∧ Takes(x , y)))
Only one student failed Geometry.∃x .(Student(x) ∧ Failed(x ,Geometry) ∧ ∀y .(Student(y) ∧Failed(y ,Geometry)→ x = y))
No student failed Geometry but at least one student failedAnalysis.
¬∃x .(Student(x) ∧ Failed(x ,Geometry)) ∧ ∃x .(Student(x) ∧Failed(x ,Analysis))
Every student who takes Analysis also takes Geometry.∀x .(Student(x) ∧ Takes(x ,Analysis)→ Takes(x ,Geometry))
Luciano Serafini First Order Logics as a Modelling Language
![Page 104: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/104.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
Every student takes at least one course.∀x .(Student(x)→ ∃y .(Course(y) ∧ Takes(x , y)))
Only one student failed Geometry.∃x .(Student(x) ∧ Failed(x ,Geometry) ∧ ∀y .(Student(y) ∧Failed(y ,Geometry)→ x = y))
No student failed Geometry but at least one student failedAnalysis.¬∃x .(Student(x) ∧ Failed(x ,Geometry)) ∧ ∃x .(Student(x) ∧Failed(x ,Analysis))
Every student who takes Analysis also takes Geometry.∀x .(Student(x) ∧ Takes(x ,Analysis)→ Takes(x ,Geometry))
Luciano Serafini First Order Logics as a Modelling Language
![Page 105: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/105.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
Every student takes at least one course.∀x .(Student(x)→ ∃y .(Course(y) ∧ Takes(x , y)))
Only one student failed Geometry.∃x .(Student(x) ∧ Failed(x ,Geometry) ∧ ∀y .(Student(y) ∧Failed(y ,Geometry)→ x = y))
No student failed Geometry but at least one student failedAnalysis.¬∃x .(Student(x) ∧ Failed(x ,Geometry)) ∧ ∃x .(Student(x) ∧Failed(x ,Analysis))
Every student who takes Analysis also takes Geometry.
∀x .(Student(x) ∧ Takes(x ,Analysis)→ Takes(x ,Geometry))
Luciano Serafini First Order Logics as a Modelling Language
![Page 106: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/106.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Examples
Every student takes at least one course.∀x .(Student(x)→ ∃y .(Course(y) ∧ Takes(x , y)))
Only one student failed Geometry.∃x .(Student(x) ∧ Failed(x ,Geometry) ∧ ∀y .(Student(y) ∧Failed(y ,Geometry)→ x = y))
No student failed Geometry but at least one student failedAnalysis.¬∃x .(Student(x) ∧ Failed(x ,Geometry)) ∧ ∃x .(Student(x) ∧Failed(x ,Analysis))
Every student who takes Analysis also takes Geometry.∀x .(Student(x) ∧ Takes(x ,Analysis)→ Takes(x ,Geometry))
Luciano Serafini First Order Logics as a Modelling Language
![Page 107: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/107.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Exercises
Define an appropriate language and formalize the followingsentences in FOL:
someone likes Mary.
nobody likes Mary.
nobody loves Bob but Bob loves Mary.
if David loves someone, then he loves Mary.
if someone loves David, then he (someone) loves also Mary.
everybody loves David or Mary.
Luciano Serafini First Order Logics as a Modelling Language
![Page 108: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/108.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Exercises
Define an appropriate language and formalize the followingsentences in FOL:
there is at least one person who loves Mary.
there is at most one person who loves Mary.
there is exactly one person who loves Mary.
there are exactly two persons who love Mary.
if Bob loves everyone that Mary loves, and Bob loves David,then Mary doesn’t love David.
Only Mary loves Bob.
Luciano Serafini First Order Logics as a Modelling Language
![Page 109: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/109.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Example
Define an appropriate language and formalize the followingsentences in FOL:
”A is above C, D is on E and above F.”
”A is green while C is not.”
”Everything is on something.”
”Everything that has nothing on it, is free.”
”Everything that is green is free.”
”There is something that is red and is not free.”
”Everything that is not green and is above B, is red.”
Luciano Serafini First Order Logics as a Modelling Language
![Page 110: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/110.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Example
”A is above C, D is above F and on E.”φ1 : Above(A,C ) ∧ Above(E ,F ) ∧ On(D,E )
”A is green while C is not.”φ2 : Green(A) ∧ ¬Green(C )
”Everything is on something.”φ3 : ∀x∃y .On(x , y)
”Everything that has nothing on it, is free.”φ4 : ∀x .(¬∃y .On(y , x)→ Free(x))
Luciano Serafini First Order Logics as a Modelling Language
![Page 111: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/111.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Example
”A is above C, D is above F and on E.”φ1 : Above(A,C ) ∧ Above(E ,F ) ∧ On(D,E )
”A is green while C is not.”φ2 : Green(A) ∧ ¬Green(C )
”Everything is on something.”φ3 : ∀x∃y .On(x , y)
”Everything that has nothing on it, is free.”φ4 : ∀x .(¬∃y .On(y , x)→ Free(x))
Luciano Serafini First Order Logics as a Modelling Language
![Page 112: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/112.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Example
”A is above C, D is above F and on E.”φ1 : Above(A,C ) ∧ Above(E ,F ) ∧ On(D,E )
”A is green while C is not.”φ2 : Green(A) ∧ ¬Green(C )
”Everything is on something.”φ3 : ∀x∃y .On(x , y)
”Everything that has nothing on it, is free.”φ4 : ∀x .(¬∃y .On(y , x)→ Free(x))
Luciano Serafini First Order Logics as a Modelling Language
![Page 113: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/113.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Example
”A is above C, D is above F and on E.”φ1 : Above(A,C ) ∧ Above(E ,F ) ∧ On(D,E )
”A is green while C is not.”φ2 : Green(A) ∧ ¬Green(C )
”Everything is on something.”φ3 : ∀x∃y .On(x , y)
”Everything that has nothing on it, is free.”φ4 : ∀x .(¬∃y .On(y , x)→ Free(x))
Luciano Serafini First Order Logics as a Modelling Language
![Page 114: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/114.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Example
”A is above C, D is above F and on E.”φ1 : Above(A,C ) ∧ Above(E ,F ) ∧ On(D,E )
”A is green while C is not.”φ2 : Green(A) ∧ ¬Green(C )
”Everything is on something.”φ3 : ∀x∃y .On(x , y)
”Everything that has nothing on it, is free.”φ4 : ∀x .(¬∃y .On(y , x)→ Free(x))
Luciano Serafini First Order Logics as a Modelling Language
![Page 115: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/115.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Example
”Everything that is green is free.”φ5 : ∀x .(Green(x)→ Free(x))
”There is something that is red and is not free.”φ6 : ∃x .(Red(x) ∧ ¬Free(x))
”Everything that is not green and is above B, is red.”φ7 : ∀x .(¬Green(x) ∧ Above(x ,B)→ Red(x))
Luciano Serafini First Order Logics as a Modelling Language
![Page 116: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/116.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Example
”Everything that is green is free.”φ5 : ∀x .(Green(x)→ Free(x))
”There is something that is red and is not free.”φ6 : ∃x .(Red(x) ∧ ¬Free(x))
”Everything that is not green and is above B, is red.”φ7 : ∀x .(¬Green(x) ∧ Above(x ,B)→ Red(x))
Luciano Serafini First Order Logics as a Modelling Language
![Page 117: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/117.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Formalizing English Sentences in FOL
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Example
”Everything that is green is free.”φ5 : ∀x .(Green(x)→ Free(x))
”There is something that is red and is not free.”φ6 : ∃x .(Red(x) ∧ ¬Free(x))
”Everything that is not green and is above B, is red.”φ7 : ∀x .(¬Green(x) ∧ Above(x ,B)→ Red(x))
Luciano Serafini First Order Logics as a Modelling Language
![Page 118: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/118.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
An interpretation I1 in the Blocks World
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
b3 b2
table
b4
b1
b5
Interpretation I1
I1(A) = b1, I1(B) = b2, I1(C) = b3, I1(D) = b4, I1(E) = b5,I1(F ) = table
I1(On) = {〈b1, b4〉, 〈b4, b3〉, 〈b3, table〉, 〈b5, b2〉, 〈b2, table〉}I1(Above) = {〈b1, b4〉, 〈b1, b3〉, 〈b1, table〉, 〈b4, b3〉, 〈b4, table〉,
〈b3, table〉, 〈b5, b2〉, 〈b5, table〉, 〈b2, table〉}I1(Free) = {〈b1〉, 〈b5〉}, I1(Green) = {〈b4〉}, I1(Red) = {〈b1〉, 〈b5〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 119: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/119.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
An interpretation I1 in the Blocks World
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.b3 b2
table
b4
b1
b5
Interpretation I1
I1(A) = b1, I1(B) = b2, I1(C) = b3, I1(D) = b4, I1(E) = b5,I1(F ) = table
I1(On) = {〈b1, b4〉, 〈b4, b3〉, 〈b3, table〉, 〈b5, b2〉, 〈b2, table〉}I1(Above) = {〈b1, b4〉, 〈b1, b3〉, 〈b1, table〉, 〈b4, b3〉, 〈b4, table〉,
〈b3, table〉, 〈b5, b2〉, 〈b5, table〉, 〈b2, table〉}I1(Free) = {〈b1〉, 〈b5〉}, I1(Green) = {〈b4〉}, I1(Red) = {〈b1〉, 〈b5〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 120: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/120.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
An interpretation I1 in the Blocks World
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.b3 b2
table
b4
b1
b5
Interpretation I1
I1(A) = b1, I1(B) = b2, I1(C) = b3, I1(D) = b4, I1(E) = b5,I1(F ) = table
I1(On) = {〈b1, b4〉, 〈b4, b3〉, 〈b3, table〉, 〈b5, b2〉, 〈b2, table〉}I1(Above) = {〈b1, b4〉, 〈b1, b3〉, 〈b1, table〉, 〈b4, b3〉, 〈b4, table〉,
〈b3, table〉, 〈b5, b2〉, 〈b5, table〉, 〈b2, table〉}I1(Free) = {〈b1〉, 〈b5〉}, I1(Green) = {〈b4〉}, I1(Red) = {〈b1〉, 〈b5〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 121: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/121.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
An interpretation I1 in the Blocks World
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.b3 b2
table
b4
b1
b5
Interpretation I1
I1(A) = b1, I1(B) = b2, I1(C) = b3, I1(D) = b4, I1(E) = b5,I1(F ) = table
I1(On) = {〈b1, b4〉, 〈b4, b3〉, 〈b3, table〉, 〈b5, b2〉, 〈b2, table〉}
I1(Above) = {〈b1, b4〉, 〈b1, b3〉, 〈b1, table〉, 〈b4, b3〉, 〈b4, table〉,〈b3, table〉, 〈b5, b2〉, 〈b5, table〉, 〈b2, table〉}
I1(Free) = {〈b1〉, 〈b5〉}, I1(Green) = {〈b4〉}, I1(Red) = {〈b1〉, 〈b5〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 122: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/122.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
An interpretation I1 in the Blocks World
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.b3 b2
table
b4
b1
b5
Interpretation I1
I1(A) = b1, I1(B) = b2, I1(C) = b3, I1(D) = b4, I1(E) = b5,I1(F ) = table
I1(On) = {〈b1, b4〉, 〈b4, b3〉, 〈b3, table〉, 〈b5, b2〉, 〈b2, table〉}I1(Above) = {〈b1, b4〉, 〈b1, b3〉, 〈b1, table〉, 〈b4, b3〉, 〈b4, table〉,
〈b3, table〉, 〈b5, b2〉, 〈b5, table〉, 〈b2, table〉}
I1(Free) = {〈b1〉, 〈b5〉}, I1(Green) = {〈b4〉}, I1(Red) = {〈b1〉, 〈b5〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 123: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/123.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
An interpretation I1 in the Blocks World
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.b3 b2
table
b4
b1
b5
Interpretation I1
I1(A) = b1, I1(B) = b2, I1(C) = b3, I1(D) = b4, I1(E) = b5,I1(F ) = table
I1(On) = {〈b1, b4〉, 〈b4, b3〉, 〈b3, table〉, 〈b5, b2〉, 〈b2, table〉}I1(Above) = {〈b1, b4〉, 〈b1, b3〉, 〈b1, table〉, 〈b4, b3〉, 〈b4, table〉,
〈b3, table〉, 〈b5, b2〉, 〈b5, table〉, 〈b2, table〉}I1(Free) = {〈b1〉, 〈b5〉}, I1(Green) = {〈b4〉}, I1(Red) = {〈b1〉, 〈b5〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 124: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/124.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
A different interpretation I2
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Interpretation I2
I2(A) = hat, I2(B) = Joe, I2(C) = bike, I2(D) = Jill , I2(E) = case,I2(F ) = ground
I2(On) = {〈hat, Joe〉, 〈Joe, bike〉, 〈bike, ground〉, 〈Jill , case〉,〈case, ground〉}I2(Above) = {〈hat, Joe〉, 〈hat, bike〉, 〈hat, ground〉, 〈Joe, bike〉,〈Joe, ground〉, 〈bike, ground〉, 〈Jill , case〉, 〈Jill , ground〉, 〈case, ground〉}I2(Free) = {〈hat〉, 〈Jill〉}, I2(Green) = {〈hat〉, 〈ground〉},I2(Red) = {〈bike〉, 〈case〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 125: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/125.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
A different interpretation I2
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Interpretation I2
I2(A) = hat, I2(B) = Joe, I2(C) = bike, I2(D) = Jill , I2(E) = case,I2(F ) = ground
I2(On) = {〈hat, Joe〉, 〈Joe, bike〉, 〈bike, ground〉, 〈Jill , case〉,〈case, ground〉}I2(Above) = {〈hat, Joe〉, 〈hat, bike〉, 〈hat, ground〉, 〈Joe, bike〉,〈Joe, ground〉, 〈bike, ground〉, 〈Jill , case〉, 〈Jill , ground〉, 〈case, ground〉}I2(Free) = {〈hat〉, 〈Jill〉}, I2(Green) = {〈hat〉, 〈ground〉},I2(Red) = {〈bike〉, 〈case〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 126: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/126.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
A different interpretation I2
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Interpretation I2
I2(A) = hat, I2(B) = Joe, I2(C) = bike, I2(D) = Jill , I2(E) = case,I2(F ) = ground
I2(On) = {〈hat, Joe〉, 〈Joe, bike〉, 〈bike, ground〉, 〈Jill , case〉,〈case, ground〉}
I2(Above) = {〈hat, Joe〉, 〈hat, bike〉, 〈hat, ground〉, 〈Joe, bike〉,〈Joe, ground〉, 〈bike, ground〉, 〈Jill , case〉, 〈Jill , ground〉, 〈case, ground〉}I2(Free) = {〈hat〉, 〈Jill〉}, I2(Green) = {〈hat〉, 〈ground〉},I2(Red) = {〈bike〉, 〈case〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 127: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/127.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
A different interpretation I2
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Interpretation I2
I2(A) = hat, I2(B) = Joe, I2(C) = bike, I2(D) = Jill , I2(E) = case,I2(F ) = ground
I2(On) = {〈hat, Joe〉, 〈Joe, bike〉, 〈bike, ground〉, 〈Jill , case〉,〈case, ground〉}I2(Above) = {〈hat, Joe〉, 〈hat, bike〉, 〈hat, ground〉, 〈Joe, bike〉,〈Joe, ground〉, 〈bike, ground〉, 〈Jill , case〉, 〈Jill , ground〉, 〈case, ground〉}
I2(Free) = {〈hat〉, 〈Jill〉}, I2(Green) = {〈hat〉, 〈ground〉},I2(Red) = {〈bike〉, 〈case〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 128: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/128.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
A different interpretation I2
Non Logical symbols
Constants: A,B,C ,D,E ,F ;
Predicates: On2,Above2,Free1,Red1,Green1.
Interpretation I2
I2(A) = hat, I2(B) = Joe, I2(C) = bike, I2(D) = Jill , I2(E) = case,I2(F ) = ground
I2(On) = {〈hat, Joe〉, 〈Joe, bike〉, 〈bike, ground〉, 〈Jill , case〉,〈case, ground〉}I2(Above) = {〈hat, Joe〉, 〈hat, bike〉, 〈hat, ground〉, 〈Joe, bike〉,〈Joe, ground〉, 〈bike, ground〉, 〈Jill , case〉, 〈Jill , ground〉, 〈case, ground〉}I2(Free) = {〈hat〉, 〈Jill〉}, I2(Green) = {〈hat〉, 〈ground〉},I2(Red) = {〈bike〉, 〈case〉}
Luciano Serafini First Order Logics as a Modelling Language
![Page 129: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/129.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
For each of the following formulas, decide whether they aresatisfied by I1 and/or I2:
φ1 : Above(A,C ) ∧ Above(E ,F ) ∧ On(D,E )φ2 : Green(A) ∧ ¬Green(C )φ3 : ∀x∃y .On(x , y)φ4 : ∀x .(¬∃y .On(y , x)→ Free(x))φ5 : ∀x .(Green(x)→ Free(x))φ6 : ∃x .(Red(x) ∧ ¬Free(x))φ7 : ∀x .(¬Green(x) ∧ Above(x ,B)→ Red(x))
Sol.
I1 |= ¬φ1 ∧ ¬φ2 ∧ ¬φ3 ∧ φ4 ∧ ¬φ5 ∧ ¬φ6 ∧ φ7
I2 |= φ1 ∧ φ2 ∧ ¬φ3 ∧ φ4 ∧ ¬φ5 ∧ φ6 ∧ φ7
Luciano Serafini First Order Logics as a Modelling Language
![Page 130: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/130.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
For each of the following formulas, decide whether they aresatisfied by I1 and/or I2:
φ1 : Above(A,C ) ∧ Above(E ,F ) ∧ On(D,E )φ2 : Green(A) ∧ ¬Green(C )φ3 : ∀x∃y .On(x , y)φ4 : ∀x .(¬∃y .On(y , x)→ Free(x))φ5 : ∀x .(Green(x)→ Free(x))φ6 : ∃x .(Red(x) ∧ ¬Free(x))φ7 : ∀x .(¬Green(x) ∧ Above(x ,B)→ Red(x))
Sol.
I1 |= ¬φ1 ∧ ¬φ2 ∧ ¬φ3 ∧ φ4 ∧ ¬φ5 ∧ ¬φ6 ∧ φ7
I2 |= φ1 ∧ φ2 ∧ ¬φ3 ∧ φ4 ∧ ¬φ5 ∧ φ6 ∧ φ7
Luciano Serafini First Order Logics as a Modelling Language
![Page 131: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/131.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
Consider the following sentences:
(1) All actors and journalists invited to the party are late.
(2) There is at least a person who is on time.
(3) There is at least an invited person who is neither ajournalist nor an actor.
Formalize the sentences and prove that (3) is not a logicalconsequence of (1) and (2)
Luciano Serafini First Order Logics as a Modelling Language
![Page 132: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/132.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
Consider the following sentences:
All actors and journalists invited to the party are late.
(1) ∀x .((a(x) ∨ j(x)) ∧ i(x)→ l(x))
There is at least a person who is on time.(2) ∃x .¬l(x)
There is at least an invited person who is neither a journalist nor an actor.(3) ∃x .(i(x) ∧ ¬a(x) ∧ ¬j(x))
It’s sufficient to find an interpretation I for which the logical consequence doesnot hold:
l(x) a(x) j(x) i(x)
Bob F T F FTom T T F TMary T F T T
Luciano Serafini First Order Logics as a Modelling Language
![Page 133: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/133.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
Consider the following sentences:
All actors and journalists invited to the party are late.(1) ∀x .((a(x) ∨ j(x)) ∧ i(x)→ l(x))
There is at least a person who is on time.(2) ∃x .¬l(x)
There is at least an invited person who is neither a journalist nor an actor.(3) ∃x .(i(x) ∧ ¬a(x) ∧ ¬j(x))
It’s sufficient to find an interpretation I for which the logical consequence doesnot hold:
l(x) a(x) j(x) i(x)
Bob F T F FTom T T F TMary T F T T
Luciano Serafini First Order Logics as a Modelling Language
![Page 134: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/134.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
Consider the following sentences:
All actors and journalists invited to the party are late.(1) ∀x .((a(x) ∨ j(x)) ∧ i(x)→ l(x))
There is at least a person who is on time.
(2) ∃x .¬l(x)
There is at least an invited person who is neither a journalist nor an actor.(3) ∃x .(i(x) ∧ ¬a(x) ∧ ¬j(x))
It’s sufficient to find an interpretation I for which the logical consequence doesnot hold:
l(x) a(x) j(x) i(x)
Bob F T F FTom T T F TMary T F T T
Luciano Serafini First Order Logics as a Modelling Language
![Page 135: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/135.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
Consider the following sentences:
All actors and journalists invited to the party are late.(1) ∀x .((a(x) ∨ j(x)) ∧ i(x)→ l(x))
There is at least a person who is on time.(2) ∃x .¬l(x)
There is at least an invited person who is neither a journalist nor an actor.(3) ∃x .(i(x) ∧ ¬a(x) ∧ ¬j(x))
It’s sufficient to find an interpretation I for which the logical consequence doesnot hold:
l(x) a(x) j(x) i(x)
Bob F T F FTom T T F TMary T F T T
Luciano Serafini First Order Logics as a Modelling Language
![Page 136: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/136.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
Consider the following sentences:
All actors and journalists invited to the party are late.(1) ∀x .((a(x) ∨ j(x)) ∧ i(x)→ l(x))
There is at least a person who is on time.(2) ∃x .¬l(x)
There is at least an invited person who is neither a journalist nor an actor.
(3) ∃x .(i(x) ∧ ¬a(x) ∧ ¬j(x))
It’s sufficient to find an interpretation I for which the logical consequence doesnot hold:
l(x) a(x) j(x) i(x)
Bob F T F FTom T T F TMary T F T T
Luciano Serafini First Order Logics as a Modelling Language
![Page 137: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/137.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
Consider the following sentences:
All actors and journalists invited to the party are late.(1) ∀x .((a(x) ∨ j(x)) ∧ i(x)→ l(x))
There is at least a person who is on time.(2) ∃x .¬l(x)
There is at least an invited person who is neither a journalist nor an actor.(3) ∃x .(i(x) ∧ ¬a(x) ∧ ¬j(x))
It’s sufficient to find an interpretation I for which the logical consequence doesnot hold:
l(x) a(x) j(x) i(x)
Bob F T F FTom T T F TMary T F T T
Luciano Serafini First Order Logics as a Modelling Language
![Page 138: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/138.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
Consider the following sentences:
All actors and journalists invited to the party are late.(1) ∀x .((a(x) ∨ j(x)) ∧ i(x)→ l(x))
There is at least a person who is on time.(2) ∃x .¬l(x)
There is at least an invited person who is neither a journalist nor an actor.(3) ∃x .(i(x) ∧ ¬a(x) ∧ ¬j(x))
It’s sufficient to find an interpretation I for which the logical consequence doesnot hold:
l(x) a(x) j(x) i(x)
Bob F T F FTom T T F TMary T F T T
Luciano Serafini First Order Logics as a Modelling Language
![Page 139: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/139.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Example
Consider the following sentences:
All actors and journalists invited to the party are late.(1) ∀x .((a(x) ∨ j(x)) ∧ i(x)→ l(x))
There is at least a person who is on time.(2) ∃x .¬l(x)
There is at least an invited person who is neither a journalist nor an actor.(3) ∃x .(i(x) ∧ ¬a(x) ∧ ¬j(x))
It’s sufficient to find an interpretation I for which the logical consequence doesnot hold:
l(x) a(x) j(x) i(x)
Bob F T F FTom T T F TMary T F T T
Luciano Serafini First Order Logics as a Modelling Language
![Page 140: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/140.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
FOL Satisfiability
Exercise
Let ∆ = {1, 3, 5, 15} and I be an interpretation on ∆ interpreting thepredicate symbols E 1 as ’being even’, M2 as ’being a multiple of’ and L2 as’being less then’, and s.t. I(a) = 1, I(b) = 3, I(c) = 5, I(d) = 15.Determine whether I satisfies the following formulas:
∃y .E(y) ∀x .¬E(x) ∀x .M(x , a) ∀x .M(x , b) ∃x .M(x , d)
∃x .L(x , a) ∀x .(E(x)→ M(x , a)) ∀x∃y .L(x , y) ∀x∃y .M(x , y)
∀x .(M(x , b)→ L(x , c)) ∀x∀y .(L(x , y)→ ¬L(y , x))
∀x .(M(x , c) ∨ L(x , c))
Luciano Serafini First Order Logics as a Modelling Language
![Page 141: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/141.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Graph Coloring Problem
Provide a propositional language and a set of axioms that formalizethe graph coloring problem of a graph with at most n nodes, withconnection degree ≤ m, and with less then k + 1 colors.
node degree: number of adjacent nodes
connection degree of a graph: max among all the degree of itsnodes
Graph coloring problem: given a non-oriented graph, associatea color to each of its nodes in such a way that no pair ofadjacent nodes have the same color.
Luciano Serafini First Order Logics as a Modelling Language
![Page 142: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/142.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Graph Coloring: FOL Formalization
FOL Language
A unary function color, where color(x) is the color associated to thenode x
A unary predicate node, where node(x) means that x is a node
A binary predicate edge, where edge(x , y) means that x isconnected to y
FOL Axioms
Two connected node are not equally colored:
∀x∀y .(edge(x , y)→ (color(x) 6= color(y)) (1)
A node does not have more than k connected nodes:
∀x∀x1 . . . ∀xk+1.
k+1∧h=1
edge(x , xh)→k+1∨
i,j=1,j 6=i
xi = xj
(2)
Luciano Serafini First Order Logics as a Modelling Language
![Page 143: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/143.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Graph Coloring: FOL Formalization
FOL Language
A unary function color, where color(x) is the color associated to thenode x
A unary predicate node, where node(x) means that x is a node
A binary predicate edge, where edge(x , y) means that x isconnected to y
FOL Axioms
Two connected node are not equally colored:
∀x∀y .(edge(x , y)→ (color(x) 6= color(y)) (1)
A node does not have more than k connected nodes:
∀x∀x1 . . . ∀xk+1.
k+1∧h=1
edge(x , xh)→k+1∨
i,j=1,j 6=i
xi = xj
(2)
Luciano Serafini First Order Logics as a Modelling Language
![Page 144: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/144.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Graph Coloring: FOL Formalization
FOL Language
A unary function color, where color(x) is the color associated to thenode x
A unary predicate node, where node(x) means that x is a node
A binary predicate edge, where edge(x , y) means that x isconnected to y
FOL Axioms
Two connected node are not equally colored:
∀x∀y .(edge(x , y)→ (color(x) 6= color(y)) (1)
A node does not have more than k connected nodes:
∀x∀x1 . . . ∀xk+1.
k+1∧h=1
edge(x , xh)→k+1∨
i,j=1,j 6=i
xi = xj
(2)
Luciano Serafini First Order Logics as a Modelling Language
![Page 145: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/145.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Graph Coloring: FOL Formalization
FOL Language
A unary function color, where color(x) is the color associated to thenode x
A unary predicate node, where node(x) means that x is a node
A binary predicate edge, where edge(x , y) means that x isconnected to y
FOL Axioms
Two connected node are not equally colored:
∀x∀y .(edge(x , y)→ (color(x) 6= color(y)) (1)
A node does not have more than k connected nodes:
∀x∀x1 . . . ∀xk+1.
k+1∧h=1
edge(x , xh)→k+1∨
i,j=1,j 6=i
xi = xj
(2)
Luciano Serafini First Order Logics as a Modelling Language
![Page 146: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/146.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Graph Coloring: FOL Formalization
FOL Language
A unary function color, where color(x) is the color associated to thenode x
A unary predicate node, where node(x) means that x is a node
A binary predicate edge, where edge(x , y) means that x isconnected to y
FOL Axioms
Two connected node are not equally colored:
∀x∀y .(edge(x , y)→ (color(x) 6= color(y)) (1)
A node does not have more than k connected nodes:
∀x∀x1 . . . ∀xk+1.
k+1∧h=1
edge(x , xh)→k+1∨
i,j=1,j 6=i
xi = xj
(2)
Luciano Serafini First Order Logics as a Modelling Language
![Page 147: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/147.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Graph Coloring: FOL Formalization
FOL Language
A unary function color, where color(x) is the color associated to thenode x
A unary predicate node, where node(x) means that x is a node
A binary predicate edge, where edge(x , y) means that x isconnected to y
FOL Axioms
Two connected node are not equally colored:
∀x∀y .(edge(x , y)→ (color(x) 6= color(y)) (1)
A node does not have more than k connected nodes:
∀x∀x1 . . . ∀xk+1.
k+1∧h=1
edge(x , xh)→k+1∨
i,j=1,j 6=i
xi = xj
(2)
Luciano Serafini First Order Logics as a Modelling Language
![Page 148: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/148.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Graph Coloring: FOL Formalization
FOL Language
A unary function color, where color(x) is the color associated to thenode x
A unary predicate node, where node(x) means that x is a node
A binary predicate edge, where edge(x , y) means that x isconnected to y
FOL Axioms
Two connected node are not equally colored:
∀x∀y .(edge(x , y)→ (color(x) 6= color(y)) (1)
A node does not have more than k connected nodes:
∀x∀x1 . . . ∀xk+1.
k+1∧h=1
edge(x , xh)→k+1∨
i,j=1,j 6=i
xi = xj
(2)
Luciano Serafini First Order Logics as a Modelling Language
![Page 149: First Order Logics as a Modelling LanguageOutline Introduction FOL Formalization First Order Logics as a Modelling Language Luciano Sera ni Fondazione Bruno Kessler, Trento, Italy](https://reader033.vdocument.in/reader033/viewer/2022060603/605753c04b960445e353c65e/html5/thumbnails/149.jpg)
OutlineIntroduction
FOL Formalization
Simple SentencesFOL InterpretationFormalizing Problems
Graph Coloring: Propositional Formalization
Prop. Language
For each 1 ≤ i ≤ n and 1 ≤ c ≤ k, coloric is a proposition, whichintuitively means that ”the i-th node has the c color”
For each 1 ≤ i 6= j ≤ n, edgeij is a proposition, which intuitively meansthat ”the i-th node is connected with the j-th node”.
Prop. Axioms
for each 1 ≤ i ≤ n,∨k
c=1 coloric”each node has at least one color”
for each 1 ≤ i ≤ n and 1 ≤ c, c ′ ≤ k, coloric → ¬coloric′”every node has at most 1 color”
for each 1 ≤ i , j ≤ n and 1 ≤ c ≤ k, edgeij → ¬(coloric ∧ colorjc)”adjacent nodes do not have the same color”
for each 1 ≤ i ≤ n, and each J ⊆ {1..n}, where |J| = m,∧j∈J edgeij →
∧j 6∈J ¬edgeij
”every node has at most m connected nodes”
Luciano Serafini First Order Logics as a Modelling Language