first order logic (syntax, semantics and inference)

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First Order Logic(Syntax, Semantics and Inference)

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Knowledge based agents can represent the world it is in and can deduce the actions to take

In most programming languages the data and operations are tied closely

Each update to a data structure is done via a domain-specific procedure whose details are derived by the programmer from his own knowledge of the domain

Is there a way to say “pit is in [1,2] or [2,1]” ? Propositional logic could represent the partial

information, with negation and disjunction

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Propositional Logic is “compositional” Propositional logic is less expressive when the world

consists many objects (why?) But english/.../.. are very much expressive

Why not use engilsh for our representation? Ambiguous & not compositional

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First-order logic

Propositional logic assumes the world contains facts, (ontological commitment)

First-order logic (like natural language) assumes the world contains– Objects: people, houses, numbers, colors, baseball

games, wars, … – Relations: red, round, prime, brother of, bigger than,

part of, comes between, …– Functions: father of, best friend, one more than, plus,

…

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Ontological Commitments Propositional Logic : facts which hold ot does not. FOL : Objects and relations

Epistemological Commitments (possible states of knowledge w.r.t each facts)

Propositional Logic : true/false FOL : true/false

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Syntax of FOL: Basic elements

Constants KingJohn, 2, ... Predicates Brother, >,... Functions Sqrt, LeftLegOf,... Variables x, y, a, b,... Connectives , , , , Equality = Quantifiers ,

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Atomic sentences

Sentence = AtomicSentence | (Sentence Connective Sentence) | Quantifier Variables,…. Sentence | Sentence

Atomic sentence = predicate (term1,...,termn) | term1 = term2

Term = function (term1,...,termn) | constant | variable

E.g., Brother(KingJohn,RichardTheLionheart) > (Length(LeftLegOf(Richard)), Length(LeftLegOf(KingJohn)))

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Truth in first-order logic

Sentences are true with respect to a model (possible world) and an interpretation

Model contains objects (domain elements) and relations among them

Interpretation specifies referents forconstant symbols → objectspredicate symbols → relationsfunction symbols → functional relations

An atomic sentence predicate(term1,...,termn) is true iff the objects referred to by term1,...,termn are in the relation referred to by predicate

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Universal quantification

<variables> <sentence>

Everyone at England is smart:x At(x, England) Smart(x)

x P is true in a model m iff P is true with x being each possible object in the model

Roughly speaking, equivalent to the conjunction of instantiations of PAt(KingJohn,England) Smart(KingJohn)

At(Richard, England) Smart(Richard) At(England, England) Smart(England) ...

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A common mistake to avoid

Typically, is the main connective with Common mistake: using as the main connective

with :x At(x, England) Smart(x)

means “Everyone is at England and everyone is smart”

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Existential quantification

<variables> <sentence>

Someone at England is smart:x At(x, England) Smart(x)

x P is true in a model m iff P is true with x being some possible object in the model

Roughly speaking, equivalent to the disjunction of instantiations of PAt(KingJohn, England) Smart(KingJohn)

At(Richard, England) Smart(Richard) At(England, England) Smart(England) ...

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Another common mistake to avoid

Typically, is the main connective with

Common mistake: using as the main connective with :

x At(x,England) Smart(x)

is true if there is anyone who is not at England!

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Properties of quantifiers

x y is the same as y x x y is the same as y x

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Properties of quantifiers

x y is the same as y xx y is the same as y x

x y is not the same as y xx 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”

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Properties of quantifiers

x y is the same as y xx y is the same as y x

x y is not the same as y xx 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”

Quantifier duality: each can be expressed using the otherx Likes(x,IceCream) x Likes(x,IceCream)x Likes(x,Broccoli) x Likes(x,Broccoli)

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Interacting with FOL KBs

Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t=5:Tell(KB,Percept([Smell,Breeze,None],5))Ask(KB,a BestAction(a,5))

i.e., does the KB entail some best action at t=5? Answer: Yes, {a/Shoot} ← substitution (binding list) Given a sentence S and a substitution σ, Sσ denotes the result of plugging σ into S; e.g.,

S = Smarter(x,y)σ = {x/Hillary,y/Bill}Sσ = Smarter(Hillary,Bill)

• Ask(KB,S) returns some/all σ such that KB╞ Sσ

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Universal instantiation (UI)

Every instantiation of a universally quantified sentence is entailed by it:

v αSubst({v/g}, α)

for any variable v and ground term g E.g.,

x King(x) Greedy(x) Evil(x) yields:King(John) Greedy(John) Evil(John)

King(Richard) Greedy(Richard) Evil(Richard)

King(Father(John)) Greedy(Father(John)) Evil(Father(John))

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. … substitutions are {x/Richard} , {x/John}, {x/ Father(John)} We can infer any sentence by replacing the variable with the ground term

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Existential instantiation (EI)

For any sentence α, variable v, and constant symbol k that does not appear elsewhere in the knowledge base:

v αSubst({v/k}, α)

E.g., x Crown(x) OnHead(x,John) yields:

Crown(C1) OnHead(C1,John)

provided C1 is a new constant symbol, called a Skolem constant

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Reduction to propositional inference

Suppose the KB contains just the following:x King(x) Greedy(x) Evil(x)King(John)Greedy(John)Brother(Richard,John)

Instantiating the universal sentence in all possible ways, we have:King(John) Greedy(John) Evil(John)King(Richard) Greedy(Richard) Evil(Richard)King(John)Greedy(John)Brother(Richard,John)

The new KB is propositionalized: proposition symbols are King(John), Greedy(John), Evil(John), King(Richard), etc.

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Reduction contd.

• Idea: propositionalize KB and query, apply resolution, return result

• Problem: with function symbols, there are infinitely many ground terms,

e.g., Father(Father(Father(John)))

• Theorem: Herbrand (1930). If a sentence α is entailed by an FOL KB, it is entailed by a finite subset of the propositionalized KB

• Idea: For n = 0 to ∞ do Create a propositional KB by instantiating with depth-n terms See if α is entailed by this KB.

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Reduction contd.

• Problem: Works if α is entailed, loops if α is not entailed

• Theorem: Turing (1936), Church (1936) Entailment for FOL is semidecidable

algorithms exist that say yes to every entailed sentence, but no algorithm exists that also says no to every nonentailed sentence.

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Problems with Propositionalization

Propositionalization seems to generate lots of irrelevant sentences.

E.g., from:x King(x) Greedy(x) Evil(x)King(John)y Greedy(y)Brother(Richard,John)

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Forward Chaining

Unification

We can get the inference immediately if we can find a substitution θ such that King(x) and Greedy(x) match King(John) and Greedy(y)θ = {x/John,y/John} worksUnify(α,β) = θ if αθ = βθ

p q θ Knows(John,x) Knows(John,Jane) {x/Jane}Knows(John,x) Knows(y,OJ) {x/OJ, y/John}Knows(John,x) Knows(y,Mother(y)) {y/Jane, x/Mother(y)}Knows(John,x) Knows(x,OJ) {Fail}

Standardizing apart eliminates overlap of variables, e.g., Knows(z17,OJ)

Unification

To unify Knows(John,x) and Knows(y,z),θ = {y/John, x/z } or θ = {y/John, x/John, z/John}

The first unifier is more general than the second, places fewer restriction on the values of the variables

There is a single most general unifier (MGU) that is unique up to renaming of variables.

MGU = { y/John, x/z }

First Order Definite Clauses

Exactly one positive literalAtomic literalImplication whose antecedent is conjunction of positive literal and a single positive literal as consequent

ExamplesKing(x) Λ Greedy(x) Evil(x)King(John)Greedy(y)

Generalized Modus Ponens

p1', p2', … , pn', ( p1 p2 … pn q) qθ

p1' is King(John) p1 is King(x)

p2' is Greedy(y) p2 is Greedy(x) θ is {x/John,y/John} q is Evil(x) q θ is Evil(John)

Lifted version of modus ponens GMP used with KB of definite clausesAll variables assumed universally quantified

An Example KB

Consider the passageThe law says that it is a crime for an American to sell weapons to hostile nations. The country Nono, an enemy of America, has some missiles, and all of its missiles were sold to it by Colonel West, who is American.

The KB should answer the queryIs Col. West a criminal?

An Example KB (contd…)

... it is a crime for an American to sell weapons to hostile nations:American(x) Weapon(y) Sells(x,y,z) Hostile(z) Criminal(x)

Nono … has some missiles, i.e., x Owns(Nono,x) Missile(x):Owns(Nono,M1) and Missile(M1)

… all of its missiles were sold to it by Colonel WestMissile(x) Owns(Nono,x) Sells(West,x,Nono)

Missiles are weapons:Missile(x) Weapon(x)

An enemy of America counts as "hostile“:Enemy(x,America) Hostile(x)

West, who is American …American(West)

The country Nono, an enemy of America …Enemy(Nono,America)

No Function Symbols in the KB>> Datalog KB

Forward Chaining Algorithm

Forward chaining - Example

Forward chaining - Example

Forward chaining - Example

Forward Chaining Algorithm - discussion

No new sentences can be added to the KB after it has generated criminal(west)Fixed point of inference process

SoundComplete (?)Datalog KB

• k- maximum arity of predicates, n – constant symbols, p- predicates pnk distinct ground facts

KB with function symbols• Use Herbrand’s theorem, if the query has answer.

Remember, entailment with FOL is semidecidable

Forward Chaining Algorithm - discussion

Efficiency concernsNotice the inner-loop generate all possible θ

• Expensive pattern matching

Algorithm rechecks every rule in every iteration to see if the premises are satisfiedGenerates many facts which are irrelevant to the goal

Addressing Efficiency ConcernsMatching against known rules only

To apply the rule, Missile(x) Weapon (x)Look for the rules that unify only with Missile(x)Use Indexed KB

Missile(x) Λ Owns(Nono,x) Sells(West, x, Nono)Owns(Nono,x)Nono may own thousands of objectsFind all missiles first, then see if these missiles are owned by Nono : conjunct orderingRemember heuristic : Most constrained variable

Addressing Efficiency ConcernsMatching against known rules only

Colorable() is inferred iff the CSP has a solutionCSPs include 3SAT as a special case, hence matching is NP-hardForward chaining has an NP Hard Matching Problem in its inner loop

Diff(wa,nt) Diff(wa,sa) Diff(nt,q) Diff(nt,sa) Diff(q,nsw) Diff(q,sa) Diff(nsw,v) Diff(nsw,sa) Diff(v,sa) Colorable()

Diff(Red,Blue) Diff (Red,Green) Diff(Green,Red) Diff(Green,Blue) Diff(Blue,Red) Diff(Blue,Green)

Addressing Efficiency ConcernsMatching against known rules only

> Most Rules in real world are small and simple, upper bounds on the rule size and arity.> Data complexity> Consider only subclasses of databases for which matching is efficient datalog KBUse better algorithm that avoids redundant matchings

Addressing Efficiency ConcernsIncremental Forward Chaining

Every fact inferred at iteration t will use at least one fact derived at iteration t-1At iteration t, consider only rules whose premise include conjuncts which unifies with a fact derived at iteration t-1With indexing, we may find out all rules which may trigger at iteration t

Rete Algorithm

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Backward Chaining

Backward chaining algorithm

SUBST(COMPOSE(θ1, θ2), p) = SUBST(θ2, SUBST(θ1, p))

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Backward chaining example

Properties of backward chaining

Depth-first recursive proof search: space is linear in size of proofIncomplete due to infinite loops fix by checking current goal against every goal on stack

Inefficient due to repeated subgoals (both success and failure) fix using caching of previous results (extra space)

Widely used for logic programming

Resolution

Converting sentences to CNF

1. Eliminate all ↔ connectives (P ↔ Q) ((P Q) ^ (Q P))

2. Eliminate all connectives (P Q) (P Q)

3. Reduce the scope of each negation symbol to a single predicate P P

(P Q) P Q

(P Q) P Q

(x)P (x)P

(x)P (x)P

4. Standardize variables: rename all variables so that each quantifier has its own unique variable name

Converting sentences to clausal form

5. Eliminate existential quantification by introducing Skolem constants/functions(x)P(x) P(c)

c is a Skolem constant (a brand-new constant symbol that is not used in any other sentence)(x)(y)P(x,y) (x)P(x, f(x))since is within the scope of a universally quantified variable, use a Skolem function f to construct a new value that depends on the universally quantified variable

f must be a brand-new function name not occurring in any other sentence in the KB.

E.g., (x)(y)loves(x,y) (x)loves(x,f(x)) In this case, f(x) specifies the person that x loves

Converting sentences to clausal form

6. Remove universal quantifiers by (1) moving them all to the left end; (2) making the scope of each the entire sentence; and (3) dropping the “prefix” partEx: (x)P(x) P(x)

7. Put into conjunctive normal form (conjunction of disjunctions) using distributive and associative laws(P Q) R (P R) (Q R)(P Q) R (P Q R)

8. Split conjuncts into separate clauses9. Standardize variables so each clause contains only variable names

that do not occur in any other clause

An example

(x)(P(x) ((y)(P(y) P(f(x,y))) (y)(Q(x,y) P(y)))) 2. Eliminate

(x)(P(x) ((y)(P(y) P(f(x,y))) (y)(Q(x,y) P(y))))

3. Reduce scope of negation(x)(P(x) ((y)(P(y) P(f(x,y))) (y)(Q(x,y) P(y))))

4. Standardize variables(x)(P(x) ((y)(P(y) P(f(x,y))) (z)(Q(x,z) P(z))))

5. Eliminate existential quantification(x)(P(x) ((y)(P(y) P(f(x,y))) (Q(x,g(x)) P(g(x)))))

6. Drop universal quantification symbols(P(x) ((P(y) P(f(x,y))) (Q(x,g(x)) P(g(x)))))

Example

7. Convert to conjunction of disjunctions(P(x) P(y) P(f(x,y))) (P(x) Q(x,g(x))) (P(x) P(g(x)))

8. Create separate clausesP(x) P(y) P(f(x,y))

P(x) Q(x,g(x))

P(x) P(g(x))

9. Standardize variablesP(x) P(y) P(f(x,y))

P(z) Q(z,g(z))

P(w) P(g(w))

Resolution: brief summary

Full first-order version:l1 ··· lk, m1 ··· mn

(l1 ··· li-1 li+1 ··· lk m1 ··· mj-1 mj+1 ··· mn)θwhere Unify(li, mj) = θ.

The two clauses are assumed to be standardized apart so that they share no variables.For example,

Rich(x) Unhappy(x) , Rich(Ken)Unhappy(Ken)

with θ = {x/Ken}

Apply resolution steps to CNF(KB α); complete for FOL

Resolution proof: definite clauses