Logic review A summary of chapters 7, 8, 9 of Artificial Intelligence: A modern approach by 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 level i.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 P 1, P 2 etc are sentences –If S is a sentence,  S is a sentence (negation) –If S 1 and S 2 are sentences, S 1  S 2 is a sentence (conjunction) –If S 1 and S 2 are sentences, S 1  S 2 is a sentence (disjunction) –If S 1 and S 2 are sentences, S 1  S 2 is a sentence (implication) –If S 1 and S 2 are sentences, S 1  S 2 is a sentence (biconditional)

Propositional logic: Semantics Each model specifies true/false for each proposition symbol E.g. P 1,2 P 2,2 P 3,1 falsetruefalse With these symbols, 8 possible models, can be enumerated automatically. Rules for evaluating truth with respect to a model m:  Sis true iff S is false S 1  S 2 is true iff S 1 is true and S 2 is true S 1  S 2 is true iff S 1 is true or S 2 is true S 1  S 2 is true iffS 1 is false orS 2 is true i.e., is false iffS 1 is true andS 2 is false S 1  S 2 is true iffS 1  S 2 is true andS 2  S 1 is true Simple recursive process evaluates an arbitrary sentence, e.g.,  P 1,2  (P 2,2  P 3,1 ) = 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(2 n ), 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 model e.g., A  B, C A sentence is unsatisfiable if it is true in no models e.g., A  A 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 clauses E.g., (A   B)  (B   C   D) Resolution inference rule (for CNF): l 1  …  l k, m 1  …  m n l 1  …  l i-1  l i+1  …  l k  m 1  …  m j-1  m j+1 ...  m n where l i and m j are complementary literals. E.g., P 1,3  P 2,2,  P 2,2 P 1,3 Resolution is sound and complete for propositional logic

Resolution Soundness of resolution inference rule:  ( l i  …  l i-1  l i+1  …  l k )  l i  m j  ( m 1  …  m j-1  m j+1 ...  m n )  ( l i  …  l i-1  l i+1  …  l k )  ( m 1  …  m j-1  m j+1 ...  m n )

Conversion to CNF B 1,1  (P 1,2  P 2,1 ) 1.Eliminate , replacing α  β with (α  β)  (β  α). (B 1,1  (P 1,2  P 2,1 ))  ((P 1,2  P 2,1 )  B 1,1 ) 2. Eliminate , replacing α  β with  α  β. (  B 1,1  P 1,2  P 2,1 )  (  (P 1,2  P 2,1 )  B 1,1 ) 3. Move  inwards using de Morgan's rules and double- negation: (  B 1,1  P 1,2  P 2,1 )  ((  P 1,2   P 2,1 )  B 1,1 ) 4. Apply distributivity law (  over  ) and flatten: (  B 1,1  P 1,2  P 2,1 )  (  P 1,2  B 1,1 )  (  P 2,1  B 1,1 )

Resolution algorithm Proof by contradiction, i.e., show KB  α unsatisfiable

Resolution example KB = (B 1,1  (P 1,2  P 2,1 ))  B 1,1 α =  P 1,2

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 α 1, …,α n,α 1  …  α n  β β 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 a 1  …  a k  b 4.Hence m is a model of KB 5.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, or prove 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 subgoal 1.has already been proved true, or 2.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], L x,y  FacingRight t  Forward t  L x+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 logic Forward, 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 B 1,1  P 1,2 is derived from meaning of B 1,1 and of P 1,2 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 ConstantsKingJohn, 2, NUS,... PredicatesBrother, >,... FunctionsSqrt, LeftLegOf,... Variablesx, y, a, b,... Connectives , , , ,  Equality= Quantifiers , 

Atomic sentences Atomic sentence =predicate (term 1,...,term n ) or term 1 = term 2 Term =function (term 1,...,term n ) 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, S 1  S 2, S 1  S 2, S 1  S 2, S 1  S 2, 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 for constant symbols → objects predicate symbols → relations function symbols →functional relations An atomic sentence predicate(term 1,...,term n ) is true iff the objects referred to by term 1,...,term n are in the relation referred to by predicate

Models for FOL: Example

Universal quantification  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  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 term 1 = term 2 is true under a given interpretation if and only if term 1 and term 2 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 siblings  x,y Brother(x,y)  Sibling(x,y) One's mother is one's female parent  m,c Mother(c) = m  (Female(m)  Parent(m,c)) “Sibling” is symmetric  x,y Sibling(x,y)  Sibling(y,x)

Using FOL The set domain:  s Set(s)  (s = {} )  (  x,s 2 Set(s 2 )  s = {x|s 2 })  x,s {x|s} = {}  x,s x  s  s = {x|s}  x,s x  s  [  y,s 2 } (s = {y|s 2 }  (x = y  x  s 2 ))]  s 1,s 2 s 1  s 2  (  x x  s 1  x  s 2 )  s 1,s 2 (s 1 = s 2 )  (s 1  s 2  s 2  s 1 )  x,s 1,s 2 x  (s 1  s 2 )  (x  s 1  x  s 2 )  x,s 1,s 2 x  (s 1  s 2 )  (x  s 1  x  s 2 )

Knowledge engineering in FOL 1.Identify the task 2.Assemble the relevant knowledge 3.Decide on a vocabulary of predicates, functions, and constants 4.Encode general knowledge about the domain 5.Encode a description of the specific problem instance 6.Pose queries to the inference procedure and get answers 7.Debug the knowledge base

The electronic circuits domain One-bit full adder

The electronic circuits domain 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(X 1 ) = XOR Type(X 1, XOR) XOR(X 1 )

The electronic circuits domain 1.Encode general knowledge of the domain  t 1,t 2 Connected(t 1, t 2 )  Signal(t 1 ) = Signal(t 2 )  t Signal(t) = 1  Signal(t) = 0 –1 ≠ 0  t 1,t 2 Connected(t 1, t 2 )  Connected(t 2, t 1 )  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))

The electronic circuits domain 1.Encode the specific problem instance Type(X 1 ) = XOR Type(X 2 ) = XOR Type(A 1 ) = AND Type(A 2 ) = AND Type(O 1 ) = OR Connected(Out(1,X 1 ),In(1,X 2 ))Connected(In(1,C 1 ),In(1,X 1 )) Connected(Out(1,X 1 ),In(2,A 2 ))Connected(In(1,C 1 ),In(1,A 1 )) Connected(Out(1,A 2 ),In(1,O 1 )) Connected(In(2,C 1 ),In(2,X 1 )) Connected(Out(1,A 1 ),In(2,O 1 )) Connected(In(2,C 1 ),In(2,A 1 )) Connected(Out(1,X 2 ),Out(1,C 1 )) Connected(In(3,C 1 ),In(2,X 2 )) Connected(Out(1,O 1 ),Out(2,C 1 )) Connected(In(3,C 1 ),In(1,A 2 ))

The electronic circuits domain 1.Pose queries to the inference procedure What are the possible sets of values of all the terminals for the adder circuit?  i 1,i 2,i 3,o 1,o 2 Signal(In(1,C_1)) = i 1  Signal(In(2,C 1 )) = i 2  Signal(In(3,C 1 )) = i 3  Signal(Out(1,C 1 )) = o 1  Signal(Out(2,C 1 )) = o 2 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 α Subst({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 α Subst({v/k}, α) E.g.,  x Crown(x)  OnHead(x,John) yields: Crown(C 1 )  OnHead(C 1,John) provided C 1 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·n k 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(z 17,OJ)

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) {x/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(z 17,OJ)

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) {x/Jane}} Knows(John,x) Knows(y,OJ) {x/OJ,y/John}} Knows(John,x) Knows(y,Mother(y)) Knows(John,x) Knows(x,OJ) Standardizing apart eliminates overlap of variables, e.g., Knows(z 17,OJ)

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) {x/Jane}} Knows(John,x)Knows(y,OJ) {x/OJ,y/John}} Knows(John,x) Knows(y,Mother(y)) {y/John,x/Mother(John)}} Knows(John,x)Knows(x,OJ) Standardizing apart eliminates overlap of variables, e.g., Knows(z 17,OJ)

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) {x/Jane}} Knows(John,x)Knows(y,OJ) {x/OJ,y/John}} Knows(John,x) Knows(y,Mother(y)) {y/John,x/Mother(John)}} Knows(John,x)Knows(x,OJ) {fail} Standardizing apart eliminates overlap of variables, e.g., Knows(z 17,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. 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) p 1 ', p 2 ', …, p n ', ( p 1  p 2  …  p n  q) qθ p 1 ' is King(John) p 1 is King(x) p 2 ' is Greedy(y) p 2 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 p i 'θ = p i θ for all i

Soundness of GMP Need to show that p 1 ', …, p n ', (p 1  …  p n  q) ╞ qθ provided that p i 'θ = p i θ for all I Lemma: For any sentence p, we have p ╞ pθ by UI 1.(p 1  …  p n  q) ╞ (p 1  …  p n  q)θ = (p 1 θ  …  p n θ  qθ) 2.p 1 ', \; …, \;p n ' ╞ p 1 '  …  p n ' ╞ p 1 'θ  …  p n '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,M 1 ) and Missile(M 1 ) … all of its missiles were sold to it by Colonel West Missile(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(M 1 ) 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(θ 1, θ 2 ), p) = SUBST(θ 2, SUBST(θ 1, 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 & whistles Widely used in Europe, Japan (basis of 5th Generation project) Compilation techniques  60 million LIPS Program = set of clauses = head :- literal 1, … literal n. 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: l 1  ···  l k, m 1  ···  m n ( l 1  ···  l i-1  l i+1  ···  l k  m 1  ···  m j-1  m j+1  ···  m n )θ where Unify ( l i,  m j ) = 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 implications  x [  y  Animal(y)  Loves(x,y)]  [  y Loves(y,x)] 2. Move  inwards:  x p ≡  x  p,   x p ≡  x  p  x [  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