111

Artificial Intelligence

CS 165A

Tuesday, November 6, 2007

Inference in FOL (Ch 9)

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George Boole (1815-1864)

• More than 100 years later, he didn’t know about Leibniz, but proceeded to bring to life part of Leibniz’ dream

• His insight: Logical relationships are expressible as a kind of algebra– Letters represent classes (rather than numbers)

– So logic can be viewed as a form of mathematics

• Published The Laws of Thought

British

• He extended Aristotle's simple syllogisms to a broader range of reasoning– Syllogism: Premise_1, Premise_2 Conclusion– His logic: Propositional logic

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Gottlob Frege (1848-1925)

• He provided the first fully developed system of logic that encompassed all of the deductive reasoning in ordinary mathematics.

• He intended for logic to be the foundation of mathematics – all of mathematics could be based on, and derived from, logic

• In 1879 he published Begriffsschrift, subtitled “A formula language, modeled upon that of arithmetic, for pure thought”

– This can be considered the ancestor of all current computer programming languages

– Made the distinction between syntax and semantics critical

German

• He invented what we today call predicate calculus (or first-order logic)

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Notes

• HW#3 due Thursday

• Midterm exam one week from today– Short-answer conceptual questions and HW-like questions

– Topics: Reading, lectures, material through chapter 8 What AI is, main issues and areas in AI Problem solving and intelligent agents Blind search, informed search, adversarial search Logic and inference in propositional calculus Basics of first-order logic

– Closed book

– You may bring one 8.5”x11” piece of paper with your notes Some formulas, procedures will be given (posted in advance)

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Reminder

• Term– Constant, variable, function( )

• Atomic sentence– Predicate( ), term1 = term2

• Literal– An atomic sentence or a negated atomic sentence

• Sentence– Atomic sentence, sentences with quantifiers and/or connectives

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Simple example of inference in FOL

Bob is a buffalo

Pat is a pig

Buffaloes outrun pigs

Does Bob outrun Pat?

Buffalo(Bob)

Pig(Pat)

Buffalo(x) Pig(y) Outrun(x,y)

KB entails Outrun(Bob, Pat)?

KB0 |– Buffalo(Bob) Pig(Pat)

(And-Introduction)

KB1 |– Buffalo(Bob) Pig(Pat) Outrun(Bob, Pat)

(Universal Instantiation) [coming soon]

KB2 |– Outrun(Bob, Pat)

(Modus Ponens)

KB1

KB0

KB2

KB3

KB0

S

Review

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Using FOL to express knowledge

• One can express the knowledge of a particular domain in first-order logic

• Example: The “kinship domain”– Objects: people– Properties: gender, family relationships– Unary predicates: Male, Female, MotherOf, FatherOf– Binary predicates: Parent, Sibling, Brother, Sister, Son,

Daughter, Father, Mother, Uncle, Aunt, Grandparent, Grandfather, Grandmother, Husband, Wife, Spouse, Brother-in-law, Stepmother, etc….

– Functions: MotherOf, FatherOf…

• Note: There is usually (always?) more than one way to specify knowledge

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Kinship domain

• Write down what we know (what we want to be in the KB)– One’s mother is one’s female parent

m, c Mother(m, c) Female(m) Parent(m, c) m, c TheMotherOf(c) = m Female(m) Parent(m, c)

– One’s husband is one’s male spouse w, h Husband(h, w) Male(h) Spouse(h, w)

– One is either male or female x Male(x) Female(x)

– Parent-child relationship p, c Parent(p, c) Child(c, p)

– Grandparent-grandchild relationship g, c Grandparent(g, c) p Parent(g, p) Parent(p, c)

– Etc…

• Now we can reason about family relationships. (How?)

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Kinship domain (cont.)

Assertions (“Add this sentence to the KB”)

TELL( KB, m, c Mother(c) = m Female(m) Parent(m, c) )

TELL( KB, w, h Husband(h, w) Male(h) Spouse(h, w) )

TELL( KB, x Male(x) Female(x) )

TELL( KB, Female(Mary) Parent(Mary, Frank) Parent(Frank, Ann) )– Note: TELL( KB, S1 S2 ) TELL( KB, S1) and TELL( KB, S2)

(because of and-elimination and and-introduction)

Queries (“Does the KB entail this sentence?”)

ASK( KB, Grandparent(Mary, Ann) ) True

ASK( KB, x Child(x, Frank) ) True– But a better answer would be { x / Ann }– This returns a substitution or binding

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Implementing ASK: Inference

• We want a sound and complete inference algorithm so that we can produce (or confirm) entailed sentences from the KB

KB KB i

• The resolution rule, along with a complete search algorithm, provides a complete inference algorithm to confirm or refute a sentence in propositional logic(Sec. 7.5)

– Based on proof by contradiction (refutation)

• Refutation: To prove that the KB entails P, assume P and show a contradiction:

(KB P False) (KB P)Prove this!

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Inference in First-Order Logic

• Inference rules for propositional logic:– Modus ponens, and-elimination, and-introduction, or-introduction,

resolution, etc.

– These are valid for FOL also

• But since these don’t deal with quantifiers and variables, we need new rules, especially those that allow for substitution (binding) of variables to objects– These are called lifted inference rules

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Substitution and variable binding

• Notation for substitution:– SUBST ( Binding list, Sentence )

Binding list: { var / ground term, var / ground term, … } “ground term” = term with no variables

– SUBST( {var/gterm}, Func (var) ) = Func (gterm) SUBST (, p)

– Examples: SUBST ( {x/Mary}, FatherOf (x) ) = FatherOf (Mary) SUBST ( {x/Joe, y/Lisa}, Siblings (x,y) ) = Siblings (Joe, Lisa)

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Three new inference rules using SUBST(, p)

• Universal Instantiation

)},/({

gvSUBST

v

• Existential Instantiation

)},/({

kvSUBST

v k – constant that does not appear elsewhere in the knowledge base

g – ground term

• Existential Introduction

)},/({

vgSUBSTvv – variable not in g – ground term in

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To Add to These Rules

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Universal Instantiation – examples

)},/({

gvSUBST

v g – ground term

x Sleepy(x)– SUBST({x/Joe}, )

Sleepy(Joe)

x Mother(x) Female(x)– SUBST({x/Mary}, )

Mother(Mary) Female(Mary)– SUBST({x/Dad}, )

Mother(Dad) Female(Dad)

x, y Buffalo(x) Pig(y) Outrun(x,y)– SUBST({x/Bob}, )

y Buffalo(Bob) Pig(y) Outrun(Bob,y)

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Existential Instantiation – examples

)},/({

kvSUBST

v k – constant that does not appear elsewhere in the knowledge base

x BestAction(x)– SUBST({x/B_A}, )

BestAction(B_A)– “B_A” is a constant; it is not in our universe of actions

y Likes(y, Broccoli)– SUBST({y/Truman}, )

x Likes(Truman, Broccoli)– “Truman” is a constant; it is not in our universe of people

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Existential Introduction – examples

)},/({

vgSUBSTvv – variable not in g – ground term in

• Likes(Jim, Broccoli)– SUBST({Jim/x}, )

x Likes(x, Broccoli)

x Likes(x, Broccoli) Healthy(x)– SUBST({Broccoli/y}, )

y x Likes(x, y) Healthy(x)

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What’s our goal here?

• Formulate a search process:– Initial state

KB– Operators

Inference rules– Goal test

KB contains S

• What is a node?– KB + new sentences (generated by applying the inference rules)– In other words, the new state of the KB

• What kind of search to use?– I.e., which node to expand next?

• How to apply inference rules? – Need to match the premise pattern

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Applying inference rules

• FOL query– ASK(KB, Outruns(Bob, Pat))

– ASK(KB, x BestAction(x, t))

– ASK(KB, x Equals(x, AuntOf(HusbandOf(Mary))))

• Proof – From axioms (KB) to hypothesis (S)– Does the KB entail a sentence S – if so, what steps were taken?

• We need to apply inference rules and appropriate substitutions to reach the desired sentence– Answer: Yes, SUBST({x/Grab}, Action(x, t))

– Answer: Yes, the KB entails S (and here’s how…)

• Reasoning systems may require the solution path, not just the final answer

Path = proof

Path shows reasoning (here’s why I think you have cancer)

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Concerns

• Is the inference procedure sound and complete?

• Is it efficient?– We need to be concerned about the branching factor

– Generalized modus ponens (GMP) is one way of reducing branching factor

• How do we know which operators (inference rules) are valid at any particular time?– Inference rule: Pig(x) Cuddly(x)

– KB: Pig(Bob)

Pig(x) and Pig(Bob) are unified by the substitution = {x/Bob}

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Problem of branching factor

• 10 inference rules– Branching factor can be huge!

• Common inference pattern:– And-Introduction

– Universal Instantiation

– Modus ponens

• Let’s introduce a new inference rule which reduces the branching factor by combining these three steps into one– Generalized Modus Ponens

Buffaloes outrun pigs example

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Generalized Modus Ponens

• For atomic sentences pi , pi' , and q , where there is a substitution such that SUBST(, pi') = SUBST(, pi) for all i, then

),(

)(,,,, 2121

qSUBST

qpppppp nn

• GMP features:– Efficient – it combines several inferences into one

– Sensible – substitutions are purposeful (unification)

– Efficient – when sentences are in a certain canonical form (Horn clauses), this is the only inference rule necessary (!!!)

Specific facts General rule

Instantiated conclusion

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Generalized Modus Ponens Example

• Go from– Buffalo(Bob), Pig(Pat), (Buffalo(x) Pig(y) Outrun(x,y))

to

– Outrun(Bob, Pat)

• More generally, go from– p1', p2', (p1 p2

q)

to

– SUBST(, q)

where SUBST(, pi') = SUBST(, pi)

• Reminder: What is ?

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Unification

• Unification takes two atomic sentences p and q and returns a substitution that would make p and q look the same (or else it fails)– UNIFY(p, q) = where SUBST(, p) = SUBST(, q) is the unifier of the sentences p and q

• Examples– p = Knows(John, x) q = Knows(John, Jane)

= {x/Jane}

– p = Knows(John, x) q = Knows(y, Fred) = {x/Fred, y/John}

– p = Knows(John, x) q = Knows(y, MotherOf(y)) = {x/MotherOf(John), y/John}

– p = Knows(John, x) q = Knows(x, Mary) = fail

– p = Knows(John, x1) q = Knows(x2, Mary)

= {x1/Mary, x2/John}

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From GMP to GR

• GMP requires sentences in Horn clause

• Prolog is based on this: Horn clause form and GMP inference– Horn clause: an implication of positive literals

• It is “reasonably powerful,” but – Cannot handle disjunctions or negations

– Many legal sentences cannot be expressed in a Horn clause

– I.e., this form is incomplete

• It turns out we can do better by putting the sentences of the KB into a different format and using just one inference rule, which is both complete and sound!

– KB Format: Conjunctive normal form (CNF)

– Inference Rule: Generalized (first-order) resolution

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Note: Completeness

• An inference procedure i is complete iff– KB i whenever KB

– “The procedure finds the needle in the haystack when there is one”

• An inference procedure using GMP is incomplete– There are sentences entailed by the KB that the procedure cannot

infer

• A complete inference procedure exists: Generalized Resolution– Generalized resolution will produce if KB entails – However, what if is not entailed by the KB? Will resolution tell us

this? No, there exists no certain procedure to show this in general

– Related to the famous “Halting problem”

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Resolution

rp

rqqp

,

Simple resolution

Let’s rewrite it:rp

rqqp

,

ca

cbba

,or:

Remember: p q p q

So resolution is really the “chaining” of implications….

Review

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Generalized (first-order) resolution

• Generalized resolution is the basis for a complete inference procedure– Can derive new implications (P Q)

GMP can only derive atomic conclusions (Q)

– Any sentence can be put into its normal form

• Just like “regular” resolution except– Uses UNIFY/SUBST rather than =

– Deals with more than 2 disjuncts

• GR is complete, but semidecidable– Not always able to show that a sentence is not entailed by the KB

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Reminder: Atomic sentences vs. Literals

• Atomic sentences– Predicate and its arguments (terms)

P(x), Q(y), Siblings(Bill, z), Equal(x, y)

• Literals– Atomic sentences and negated atomic sentences

P(x), Q(y), Siblings(Bill, z), Equal(x, y)

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Two versions of Generalized Resolution

kj qandpexceptnm

nk

mj

qqppSUBST

qqq

ppp

),( 11

1

1

Disjunctions

For literals pi and qi , where UNIFY(pj , qk) =

kj qandpexceptnnnn

nkn

nnj

qqrrssppSUBST

qqqss

rrppp

),(4231

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1111

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Implications

For atomic sentences pi, qi, ri, si , where UNIFY(pj , qk) =