logic review a summary of chapters 7, 8, 9 of artificial intelligence: a modern approach by russel...

Logic review

A summary of chapters 7, 8, 9 of

Artificial Intelligence: A modern approachby Russel and Norving

Outline

• Knowledge-based agents

• Logic in general - models and entailment• Propositional (Boolean) logic• Equivalence, validity, satisfiability• Inference rules– forward chaining– backward chaining– resolution

Knowledge bases

• Knowledge base = set of sentences in a formal language• Declarative approach to building an agent (or other system):

– Tell it what it needs to know

– Then it can Ask itself what to do - answers should follow from the KB

• Agents can be viewed at the knowledge leveli.e., what they know, regardless of how implemented

• Or at the implementation level– i.e., data structures in KB and algorithms that manipulate them

A simple knowledge-based agent

• The agent must be able to:– Represent states, actions, etc.– Incorporate new percepts– Update internal representations of the world– Deduce hidden properties of the world– Deduce appropriate actions

Logic in general

• Logics are formal languages for representing information such that conclusions can be drawn

• Syntax defines the sentences in the language• Semantics define the "meaning" of sentences;

– i.e., define truth of a sentence in a world

• E.g., the language of arithmetic– x+2 ≥ y is a sentence; x2+y > {} is not a sentence– x+2 ≥ y is true iff the number x+2 is no less than the number

y– x+2 ≥ y is true in a world where x = 7, y = 1– x+2 ≥ y is false in a world where x = 0, y = 6

Entailment

• Entailment means that one thing follows from another:

KB ╞ α• Knowledge base KB entails sentence α if and

only if α is true in all worlds where KB is true

– E.g., the KB containing “the Giants won” and “the Reds won” entails “Either the Giants won or the Reds won”

– E.g., x+y = 4 entails 4 = x+y– Entailment is a relationship between sentences

(i.e., syntax) that is based on semantics

Models

• Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated

• We say m is a model of a sentence α if α is true in m

• M(α) is the set of all models of α

• Then KB ╞ α iff M(KB) M(α)– E.g. KB = Giants won and Reds

won α = Giants won

Inference

• KB ├i α = sentence α can be derived from KB by procedure i

• Soundness: i is sound if whenever KB ├i α, it is also true that KB╞ α

• Completeness: i is complete if whenever KB╞ α, it is also true that KB ├i α

• Preview: we will define a logic (first-order logic) which is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure.

• That is, the procedure will answer any question whose answer follows from what is known by the KB.

Propositional logic: Syntax

• Propositional logic is the simplest logic – illustrates basic ideas

• The proposition symbols P1, P2 etc are sentences

– If S is a sentence, S is a sentence (negation)– If S1 and S2 are sentences, S1 S2 is a sentence (conjunction)– If S1 and S2 are sentences, S1 S2 is a sentence (disjunction)– If S1 and S2 are sentences, S1 S2 is a sentence

(implication)– If S1 and S2 are sentences, S1 S2 is a sentence

(biconditional)

Propositional logic: Semantics

Each model specifies true/false for each proposition symbolE.g. P1,2 P2,2 P3,1

false true false

With these symbols, 8 possible models, can be enumerated automatically.

Rules for evaluating truth with respect to a model m:S is true iff S is false S1 S2 is true iff S1 is true and S2 is trueS1 S2 is true iff S1is true or S2 is trueS1 S2 is true iff S1 is false orS2 is true i.e., is false iff S1 is true and S2 is falseS1 S2 is true iff S1S2 is true andS2S1 is true

Simple recursive process evaluates an arbitrary sentence, e.g.,

P1,2 (P2,2 P3,1) = true (true false) = true true = true

Truth tables for connectives

Inference by enumeration

• Depth-first enumeration of all models is sound and complete

• For n symbols, time complexity is O(2n), space complexity is O(n)

Logical equivalence

• Two sentences are logically equivalent} iff true in same models: α ≡ ß iff α╞ β and β╞ α

Validity and satisfiability

A sentence is valid if it is true in all models,e.g., True, A A, A A, (A (A B)) B

Validity is connected to inference via the Deduction Theorem:KB ╞ α if and only if (KB α) is valid

A sentence is satisfiable if it is true in some modele.g., A B, C

A sentence is unsatisfiable if it is true in no modelse.g., AA

Satisfiability is connected to inference via the following:KB ╞ α if and only if (KB α) is unsatisfiable

Proof methods

• Proof methods divide into (roughly) two kinds:

– Application of inference rules• Legitimate (sound) generation of new sentences from old• Proof = a sequence of inference rule applications

Can use inference rules as operators in a standard search algorithm

• Typically require transformation of sentences into a normal form

– Model checking• truth table enumeration (always exponential in n)• improved backtracking, e.g., Davis--Putnam-Logemann-

Loveland (DPLL)• heuristic search in model space (sound but incomplete)

e.g., min-conflicts-like hill-climbing algorithms

Resolution

Conjunctive Normal Form (CNF) conjunction of disjunctions of literals

clausesE.g., (A B) (B C D)

• Resolution inference rule (for CNF):l1 … lk, m1 … mn

l1 … li-1 li+1 … lk m1 … mj-1 mj+1 ... mn

where li and mj are complementary literals. E.g., P1,3 P2,2, P2,2

P1,3

• Resolution is sound and complete for propositional logic

Resolution

Soundness of resolution inference rule:

(li … li-1 li+1 … lk) li

mj (m1 … mj-1 mj+1 ... mn)

(li … li-1 li+1 … lk) (m1 … mj-1 mj+1 ... mn)

Conversion to CNF

B1,1 (P1,2 P2,1)

1. Eliminate , replacing α β with (α β)(β α).(B1,1 (P1,2 P2,1)) ((P1,2 P2,1) B1,1)

2. Eliminate , replacing α β with α β.(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)

3. Move inwards using de Morgan's rules and double-negation:(B1,1 P1,2 P2,1) ((P1,2 P2,1) B1,1)

4. Apply distributivity law ( over ) and flatten:(B1,1 P1,2 P2,1) (P1,2 B1,1) (P2,1 B1,1)

Resolution algorithm

• Proof by contradiction, i.e., show KBα unsatisfiable

Resolution example

• KB = (B1,1 (P1,2 P2,1)) B1,1 α = P1,2

Forward and backward chaining

• Horn Form (restricted)KB = conjunction of Horn clauses

– Horn clause = • proposition symbol; or• (conjunction of symbols) symbol

– E.g., C (B A) (C D B)• Modus Ponens (for Horn Form): complete for Horn KBs

α1, … ,αn, α1 … αn ββ

• Can be used with forward chaining or backward chaining.• These algorithms are very natural and run in linear time

Forward chaining

• Idea: fire any rule whose premises are satisfied in the KB,– add its conclusion to the KB, until query is found

Forward chaining algorithm

• Forward chaining is sound and complete for Horn KB

Forward chaining example

Proof of completeness

• FC derives every atomic sentence that is entailed by KB

1. FC reaches a fixed point where no new atomic sentences are derived

2. Consider the final state as a model m, assigning true/false to symbols

3. Every clause in the original KB is true in m a1 … ak b

4. Hence m is a model of KB5. If KB╞ q, q is true in every model of KB,

including m

Backward chaining

Idea: work backwards from the query q:to prove q by BC,

check if q is known already, orprove by BC all premises of some rule concluding q

Avoid loops: check if new subgoal is already on the goal stack

Avoid repeated work: check if new subgoal1. has already been proved true, or2. has already failed

Backward chaining example

Forward vs. backward chaining

• FC is data-driven, automatic, unconscious processing,– e.g., object recognition, routine decisions

• May do lots of work that is irrelevant to the goal

• BC is goal-driven, appropriate for problem-solving,– e.g., Where are my keys? How do I get into a PhD program?

• Complexity of BC can be much less than linear in size of KB

• KB contains "physics" sentences for every single square

• For every time t and every location [x,y],

Lx,y FacingRightt Forwardt Lx+1,y

• Rapid proliferation of clauses

Expressiveness limitation of propositional logic

tt

Summary of propositional logic

• Logical agents apply inference to a knowledge base to derive new information and make decisions

• Basic concepts of logic:– syntax: formal structure of sentences– semantics: truth of sentences wrt models– entailment: necessary truth of one sentence given another– inference: deriving sentences from other sentences– soundness: derivations produce only entailed sentences– completeness: derivations can produce all entailed sentences

• Resolution is complete for propositional logicForward, backward chaining are linear-time, complete for Horn clauses

• Propositional logic lacks expressive power

First-Order Logic

Outline

• Why FOL?

• Syntax and semantics of FOL

• Using FOL

• Knowledge engineering in FOL

Pros and cons of propositional logic

Propositional logic is declarative Propositional logic allows partial/disjunctive/negated

information– (unlike most data structures and databases)

Propositional logic is compositional:– meaning of B1,1 P1,2 is derived from meaning of B1,1 and of

P1,2

Meaning in propositional logic is context-independent– (unlike natural language, where meaning depends on

context)

Propositional logic has very limited expressive power– (unlike natural language)– E.g., cannot say "pits cause breezes in adjacent squares“

• except by writing one sentence for each square

First-order logic

• Whereas propositional logic assumes the world contains facts,

• 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, …

Syntax of FOL: Basic elements

• Constants KingJohn, 2, NUS,...

• Predicates Brother, >,...

• Functions Sqrt, LeftLegOf,...

• Variables x, y, a, b,...

• Connectives , , , , • Equality =

• Quantifiers ,

Atomic sentences

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

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

• E.g., – Brother(KingJohn,RichardTheLionheart)– Married(FatherOf(Richard), MotherOf(John))

Complex sentences

• Complex sentences are made from atomic sentences using connectives

S, S1 S2, S1 S2, S1 S2, S1 S2,

E.g. Sibling(KingJohn,Richard) Sibling(Richard,KingJohn)

>(1,2) ≤ (1,2)

>(1,2) >(1,2)

Truth in first-order logic

• Sentences are true with respect to a model 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 trueiff the objects referred to by term1,...,termn

are in the relation referred to by predicate

Models for FOL: Example

Universal quantification

<variables> <sentence>

Everyone at NUS is smart:x At(x,NUS) 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 P

At(KingJohn,NUS) Smart(KingJohn) At(Richard,NUS) Smart(Richard) At(NUS,NUS) Smart(NUS) ...

Existential quantification

<variables> <sentence>

• Someone at NUS is smart: x At(x,NUS) 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 P

At(KingJohn,NUS) Smart(KingJohn) At(Richard,NUS) Smart(Richard) At(NUS,NUS) Smart(NUS) ...

Properties of quantifiers

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

x y is not the same as y x 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”

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

Equality

• term1 = term2 is true under a given interpretation if and only if term1 and term2 refer to the same object

• E.g., definition of Sibling in terms of Parent:x,y Sibling(x,y) [(x = y) m,f (m = f)

Parent(m,x) Parent(f,x) Parent(m,y) Parent(f,y)]

Using FOL

The kinship domain:

• Brothers are siblingsx,y Brother(x,y) Sibling(x,y)

• One's mother is one's female parentm,c Mother(c) = m (Female(m) Parent(m,c))

• “Sibling” is symmetricx,y Sibling(x,y) Sibling(y,x)

Using FOL

The set domain: s Set(s) (s = {} ) (x,s2 Set(s2) s = {x|s2}) x,s {x|s} = {} x,s x s s = {x|s} x,s x s [ y,s2} (s = {y|s2} (x = y x s2))] s1,s2 s1 s2 (x x s1 x s2) s1,s2 (s1 = s2) (s1 s2 s2 s1) x,s1,s2 x (s1 s2) (x s1 x s2) x,s1,s2 x (s1 s2) (x s1 x s2)

Knowledge engineering in FOL1. Identify the task2. Assemble the relevant knowledge3. Decide on a vocabulary of predicates,

functions, and constants4. Encode general knowledge about the

domain5. Encode a description of the specific

problem instance6. Pose queries to the inference procedure

and get answers7. Debug the knowledge base

The electronic circuits domain

One-bit full adder


1. Identify the task– Does the circuit actually add properly? (circuit

verification)2. Assemble the relevant knowledge– Composed of wires and gates; Types of gates

(AND, OR, XOR, NOT)– Irrelevant: size, shape, color, cost of gates

3. Decide on a vocabulary– Alternatives:

Type(X1) = XORType(X1, XOR)XOR(X1)


1. Encode general knowledge of the domain t1,t2 Connected(t1, t2) Signal(t1) = Signal(t2) t Signal(t) = 1 Signal(t) = 0– 1 ≠ 0 t1,t2 Connected(t1, t2) Connected(t2, t1) g Type(g) = OR Signal(Out(1,g)) = 1 n

Signal(In(n,g)) = 1 g Type(g) = AND Signal(Out(1,g)) = 0 n

Signal(In(n,g)) = 0 g Type(g) = XOR Signal(Out(1,g)) = 1

Signal(In(1,g)) ≠ Signal(In(2,g)) g Type(g) = NOT Signal(Out(1,g)) ≠

Signal(In(1,g))


1. Encode the specific problem instanceType(X1) = XOR Type(X2) = XOR

Type(A1) = AND Type(A2) = AND

Type(O1) = OR

Connected(Out(1,X1),In(1,X2)) Connected(In(1,C1),In(1,X1))

Connected(Out(1,X1),In(2,A2)) Connected(In(1,C1),In(1,A1))

Connected(Out(1,A2),In(1,O1)) Connected(In(2,C1),In(2,X1))

Connected(Out(1,A1),In(2,O1)) Connected(In(2,C1),In(2,A1))

Connected(Out(1,X2),Out(1,C1)) Connected(In(3,C1),In(2,X2))

Connected(Out(1,O1),Out(2,C1)) Connected(In(3,C1),In(1,A2))


1. Pose queries to the inference procedure

What are the possible sets of values of all the terminals for the adder circuit?

i1,i2,i3,o1,o2 Signal(In(1,C_1)) = i1 Signal(In(2,C1)) = i2 Signal(In(3,C1)) = i3 Signal(Out(1,C1)) = o1 Signal(Out(2,C1)) = o2

1. Debug the knowledge base

May have omitted assertions like 1 ≠ 0

Inference in first-order logic

Outline

• Reducing first-order inference to propositional inference

• Unification

• Generalized Modus Ponens

• Forward chaining

• Backward chaining

• Resolution

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))

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

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.

Reduction contd.

• Every FOL KB can be propositionalized so as to preserve entailment

• (A ground sentence is entailed by new KB iff entailed by original KB)

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

Reduction contd.

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

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.)

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)

• it seems obvious that Evil(John), but propositionalization produces lots of facts such as Greedy(Richard) that are irrelevant

• With p k-ary predicates and n constants, there are p·nk instantiations.

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} works

• Unify(α,β) = θ if αθ = βθ

p q θ

Knows(John,x) Knows(John,Jane)

Knows(John,x) Knows(y,OJ)

Knows(John,x) Knows(y,Mother(y))

Knows(John,x) Knows(x,OJ)

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

Unification




p q θ

Knows(John,x) Knows(John,Jane) {x/Jane}}

Knows(John,x) Knows(y,OJ)




Unification




p q θ


Knows(John,x) Knows(y,OJ) {x/OJ,y/John}}




Unification




p q θ



Knows(John,x) Knows(y,Mother(y)) {y/John,x/Mother(John)}}



Unification




p q θ



Knows(John,x) Knows(y,Mother(y)) {y/John,x/Mother(John)}}

Knows(John,x) Knows(x,OJ) {fail}


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.

• There is a single most general unifier (MGU) that is unique up to renaming of variables.MGU = { y/John, x/z }

The unification algorithm

Generalized Modus Ponens (GMP)

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)

• GMP used with KB of definite clauses (exactly one positive literal)

• All variables assumed universally quantified

where pi'θ = pi θ for all i

Soundness of GMP

• Need to show that p1', …, pn', (p1 … pn q) ╞ qθ

provided that pi'θ = piθ for all I

• Lemma: For any sentence p, we have p ╞ pθ by UI

1. (p1 … pn q) ╞ (p1 … pn q)θ = (p1θ … pnθ qθ)2. p1', \; …, \;pn' ╞ p1' … pn' ╞ p1'θ … pn'θ 3. From 1 and 2, qθ follows by ordinary Modus Ponens

Example knowledge base

• The 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.

• Prove that Col. West is a criminal

Example knowledge base 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)

Forward chaining algorithm

Forward chaining proof

Properties of forward chaining

• Sound and complete for first-order definite clauses

• Datalog = first-order definite clauses + no functions• FC terminates for Datalog in finite number of

iterations

• May not terminate in general if α is not entailed

• This is unavoidable: entailment with definite clauses is semidecidable

Efficiency of forward chaining

Incremental forward chaining: no need to match a rule on iteration k if a premise wasn't added on iteration k-1 match each rule whose premise contains a newly added

positive literal

Matching itself can be expensive:Database indexing allows O(1) retrieval of known facts

– e.g., query Missile(x) retrieves Missile(M1)

Forward chaining is widely used in deductive databases

Hard matching example

• Colorable() is inferred iff the CSP has a solution

• CSPs include 3SAT as a special case, hence matching is NP-hard

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)

Backward chaining algorithm

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

Backward chaining example

Properties of backward chaining

• Depth-first recursive proof search: space is linear in size of proof

• Incomplete 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

Logic programming: Prolog

• Algorithm = Logic + Control

• Basis: backward chaining with Horn clauses + bells & whistlesWidely used in Europe, Japan (basis of 5th Generation project)Compilation techniques 60 million LIPS

• Program = set of clauses = head :- literal1, … literaln.criminal(X) :- american(X), weapon(Y), sells(X,Y,Z), hostile(Z).

• Depth-first, left-to-right backward chaining• Built-in predicates for arithmetic etc., e.g., X is Y*Z+3• Built-in predicates that have side effects (e.g., input and output • predicates, assert/retract predicates)• Closed-world assumption ("negation as failure")

– e.g., given alive(X) :- not dead(X).– alive(joe) succeeds if dead(joe) fails

Prolog

• Appending two lists to produce a third:

append([],Y,Y).

append([X|L],Y,[X|Z]) :- append(L,Y,Z).

• query: append(A,B,[1,2]) ?

• answers: A=[] B=[1,2]

A=[1] B=[2]

A=[1,2] B=[]

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

Conversion to CNF

• Everyone who loves all animals is loved by someone:x [y Animal(y) Loves(x,y)] [y Loves(y,x)]

• 1. Eliminate biconditionals and implicationsx [y Animal(y) Loves(x,y)] [y Loves(y,x)]

• 2. Move inwards: x p ≡ x p, x p ≡ x px [y (Animal(y) Loves(x,y))] [y Loves(y,x)] x [y Animal(y) Loves(x,y)] [y Loves(y,x)] x [y Animal(y) Loves(x,y)] [y Loves(y,x)]

Conversion to CNF contd.

1. Standardize variables: each quantifier should use a different one

x [y Animal(y) Loves(x,y)] [z Loves(z,x)]

2. Skolemize: a more general form of existential instantiation.Each existential variable is replaced by a Skolem function of the

enclosing universally quantified variables: x [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x)

3. Drop universal quantifiers: [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x)

4. Distribute over : [Animal(F(x)) Loves(G(x),x)] [Loves(x,F(x)) Loves(G(x),x)]

Resolution proof: definite clauses

logic review a summary of chapters 7, 8, 9 of artificial intelligence: a modern approach by russel...

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