Outline. Why FOL? Syntax and semantics of FOL Using FOL. Knowledge engineering in FOL

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1 First-Order Logic

2 Outline Why FOL? Syntax and semantics of FOL Using FOL Wumpus world in FOL Knowledge engineering in FOL

3 Outline Knowledge-based agents Wumpus world Logic in general - models and entailment Propositional (Boolean) logic Equivalence, validity, satisfiability Inference rules and theorem proving forward chaining backward chaining resolution

4 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

5 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,

6 Syntax of FOL: Basic elements Constants KingJohn, 2, NUS,... Predicates Brother, >,... Functions Sqrt, LeftLegOf,... Variables x, y, a, b,... Connectives,,,, Equality = Quantifiers,

7 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) > (Length(LeftLegOf(Richard)), Length(LeftLegOf(KingJohn)))

8 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)

9 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

10 Models for FOL: Example

11 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)

12 A common mistake to avoid Typically, is the main connective with Common mistake: using as the main connective with : x At(x,NUS) Smart(x) means Everyone is at NUS and everyone is smart

13 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)...

14 Another common mistake to avoid Typically, is the main connective with Common mistake: using as the main connective with : x At(x,NUS) Smart(x) is true if there is anyone who is not at NUS!

15 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)

16 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)]

17 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

18 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 )

19 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 σ

20 Knowledge base for the wumpus world Perception t,s,b Percept([s,b,Glitter],t) Glitter(t) Reflex t Glitter(t) BestAction(Grab,t)

21 Deducing hidden properties x,y,a,b Adjacent([x,y],[a,b]) [a,b] {[x+1,y], [x-1,y],[x,y+1],[x,y-1]} Properties of squares: s,t At(Agent,s,t) Breeze(t) Breezy(s) Squares are breezy near a pit: Diagnostic rule---infer cause from effect s Breezy(s) \Exi{r} Adjacent(r,s) Pit(r)$

22 Knowledge engineering in FOL 1. Identify the task Assemble the relevant knowledge Decide on a vocabulary of predicates, functions, and constants Encode general knowledge about the domain Encode a description of the specific problem instance 6.

23 The electronic circuits domain One-bit full adder

24 The electronic circuits domain 1. Identify the task 2. Does the circuit actually add properly? (circuit verification) 2. Assemble the relevant knowledge 3. Composed of wires and gates; Types of gates (AND, OR, XOR, NOT) Irrelevant: size, shape, color, cost of gates 3. Decide on a vocabulary 4.

25 The electronic circuits domain 4. Encode general knowledge of the domain 5. t 1,t 2 Connected(t 1, t 2 ) Signal(t 1 ) = Signal(t 2 ) t Signal(t) = 1 Signal(t) = 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

26 The electronic circuits domain 5. Encode the specific problem instance 6. Type(X 1 ) = XOR Type(A 1 ) = AND Type(O 1 ) = OR Type(X 2 ) = XOR Type(A 2 ) = AND 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 ))

27 The electronic circuits domain 6. Pose queries to the inference procedure 7. 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 7. Debug the knowledge base

28 Summary First-order logic: objects and relations are semantic primitives syntax: constants, functions, predicates, equality, quantifiers Increased expressive power: sufficient to define wumpus world

29 Logical Agents

30 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

31 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

32 Wumpus World PEAS description Performance measure gold +1000, death per step, -10 for using the arrow Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter iff gold is in the same square Shooting kills wumpus if you are facing it Shooting uses up the only arrow

33 Wumpus world characterization Fully Observable No only local perception Deterministic Yes outcomes exactly specified Episodic No sequential at the level of actions Static Yes Wumpus and Pits do not move Discrete Yes

34 Exploring a wumpus world

35 Exploring a wumpus world

36 Exploring a wumpus world

37 Exploring a wumpus world

38 Exploring a wumpus world

39 Exploring a wumpus world

40 Exploring a wumpus world

41 Exploring a wumpus world

42 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

43 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.,

44 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

45 Entailment in the wumpus world Situation after detecting nothing in [1,1], moving right, breeze in [2,1] Consider possible models for KB assuming only pits 3 Boolean choices 8 possible models

46 Wumpus models

47 Wumpus models KB = wumpus-world rules + observations

48 Wumpus models KB = wumpus-world rules + observations α 1 = "[1,2] is safe", KB α 1, proved by model checking

49 Wumpus models KB = wumpus-world rules + observations

50 Wumpus models KB = wumpus-world rules + observations α 2 = "[2,2] is safe", KB α 2

51 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.

52 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)

53 Propositional logic: Semantics Each model specifies true/false for each proposition symbol E.g. P 1,2 P 2,2 P 3,1 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 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 iff S 1 is false or S 2 is true i.e., is false iff S 1 is true and S 2 is false S 1 S 2 is true iff S 1 S 2 is true ands 2 S 1 is true Simple recursive process evaluates an arbitrary sentence, e.g.,

54 Truth tables for connectives

55 Wumpus world sentences Let P i,j be true if there is a pit in [i, j]. Let B i,j be true if there is a breeze in [i, j]. P 1,1 B 1,1 B 2,1 "Pits cause breezes in adjacent squares" B 1,1 (P 1,2 P 2,1 ) B 2,1 (P 1,1 P 2,2 P 3,1 )

56 Truth tables for inference

57 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)

58 Logical equivalence Two sentences are logically equivalent} iff true in same models: α ß iff α β and β α

59 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

60 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

61 Resolution Conjunctive Normal Form (CNF) conjunction of disjunctions of literals clauses E.g., (A B) (B C D) Resolution inference rule (for CNF): l i l k, m 1 m n l i 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

62 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 )

63 Conversion to CNF B 1,1 (P 1,2 P 2,1 )β 1. Eliminate, replacing α β with (α β) (β α). 2. (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 doublenegation:

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

65 Resolution example KB = (B 1,1 (P 1,2 P 2,1 )) B 1,1 α = P 1,2

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

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

68 Forward chaining algorithm Forward chaining is sound and complete for Horn KB

69 Forward chaining example

70 Forward chaining example

71 Forward chaining example

72 Forward chaining example

73 Forward chaining example

74 Forward chaining example

75 Forward chaining example

76 Forward chaining example

77 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 Consider the final state as a model m, assigning true/false to symbols Every clause in the original KB is true in m 4. a 1 a k b 4. Hence m is a model of KB

78 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

79 Backward chaining example

80 Backward chaining example

81 Backward chaining example

82 Backward chaining example

83 Backward chaining example

84 Backward chaining example

85 Backward chaining example

86 Backward chaining example

87 Backward chaining example

88 Backward chaining example

89 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

90 Efficient propositional inference Two families of efficient algorithms for propositional inference: Complete backtracking search algorithms DPLL algorithm (Davis, Putnam, Logemann, Loveland) Incomplete local search algorithms WalkSAT algorithm

91 The DPLL algorithm Determine if an input propositional logic sentence (in CNF) is satisfiable. Improvements over truth table enumeration: 1. Early termination A clause is true if any literal is true. A sentence is false if any clause is false. 2. Pure symbol heuristic Pure symbol: always appears with the same "sign" in all clauses. e.g., In the three clauses (A B), ( B C), (C A), A and B are pure, C is impure. Make a pure symbol literal true. 3. Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true.

92 The DPLL algorithm

93 The WalkSAT algorithm Incomplete, local search algorithm Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses Balance between greediness and randomness

94 The WalkSAT algorithm

95 Hard satisfiability problems Consider random 3-CNF sentences. e.g., ( D B C) (B A C) ( C B E) (E D B) (B E C) m = number of clauses n = number of symbols

96 Hard satisfiability problems

97 Hard satisfiability problems Median runtime for 100 satisfiable random 3- CNF sentences, n = 50

98 Inference-based agents in the wumpus world A wumpus-world agent using propositional logic: P 1,1 W 1,1 B x,y (P x,y+1 P x,y-1 P x+1,y P x-1,y ) S x,y (W x,y+1 W x,y-1 W x+1,y W x-1,y ) W 1,1 W 1,2 W 4,4 W 1,1 W 1,2 W 1,1 W 1,3 64 distinct proposition symbols, 155 sentences

99

100 Expressiveness limitation of propositional logic KB contains "physics" sentences for every single square For every time t and every location [x,y], t t L x,y FacingRight t Forward t L x+1,y Rapid proliferation of clauses

101 Summary 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 Wumpus world requires the ability to represent partial and negated information, reason by cases, etc.

102 Inference in first-order logic

103 Outline Reducing first-order inference to propositional inference Unification Generalized Modus Ponens Forward chaining Backward chaining Resolution

104 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)

105 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

106 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.

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

108 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.)

109 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.

110 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)

111 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)

112 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.,

113 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.,

114 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}

115 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 }

116 The unification algorithm

117 The unification algorithm

118 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) where p i 'θ = p i θ for all i GMP used with KB of definite clauses (exactly one positive literal) All variables assumed universally quantified

119 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) 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 4.

120 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

121 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)

122 Forward chaining algorithm

123 Forward chaining proof

124 Forward chaining proof

125 Forward chaining proof

126 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

127 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

128 Hard matching example 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) Colorable() is inferred iff the CSP has a solution CSPs include 3SAT as a special case, hence matching is NP-hard

129 Backward chaining algorithm SUBST(COMPOSE(θ 1, θ 2 ), p) = SUBST(θ 2, SUBST(θ 1, p))

130 Backward chaining example

131 Backward chaining example

132 Backward chaining example

133 Backward chaining example

134 Backward chaining example

135 Backward chaining example

136 Backward chaining example

137 Backward chaining example

138 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)

139 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).

140 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=[]

141 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}

142 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)]

143 Conversion to CNF contd. 3. Standardize variables: each quantifier should use a different one 4. x [ y Animal(y) Loves(x,y)] [ z Loves(z,x)] 4. 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) 5. Drop universal quantifiers: [Animal(F(x)) Loves(x,F(x))] Loves(G(x),x) 6. Distribute over :

144 Resolution proof: definite clauses