S P Vimal, Department of CSIS, BITS, Pilani

S P Vimal, Department of CSIS, BITS, Pilani

Agenda 2 • A quick review of syntax & semantics • Entailment in Propositional Logic • Discussion EA C461 Artificial Intelligence

Review: Propositional logic: Syntax 3 • Propositional logic is the simplest logic • Atomic sentence : A Propositional symbol (written in upper case, just for notational convenience) • W1,3 – Wumpus is in room (1,3) • True / False EA C461 Artificial Intelligence

Review: Propositional logic: Syntax 4 • 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) • Precedence  ,  ,  ,  ,  EA C461 Artificial Intelligence

Semantics • Defines the rules for defining the truth of a sentence w.r.t a particular model. • Let the sentence in the KB make use of the proposition symbols P12,P22,P31. • One possible model is m1 = {P12=true,P22=false,P31=true} 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 true S1 S2 is true iff S1is true or S2 is true S1 S2 is true iff S1 is false or S2 is true i.e., is false iff S1 is true and S2 is false S1 S2 is true iff S1S2 is true and S2S1 is true

Review: Truth tables for connectives 6 EA C461 Artificial Intelligence

Wumpus world sentences 7 Assume a wumpus world without wumpus (??) Let Pi,j be true if there is a pit in [i, j]. Let Bi,j be true if there is a breeze in [i, j]. R1:  P1,1 • "Pits cause breezes in adjacent squares" R2: B1,1 (P1,2 P2,1) R3: B2,1  (P1,1 P2,2  P3,1) R4: B1,1 R5: B2,1 EA C461 Artificial Intelligence

Truth tables for inference 8 EA C461 Artificial Intelligence

Inference by enumeration 9 • Depth-first enumeration of all models is sound and complete • For n symbols, time complexity is O(2n), space complexity is O(n) EA C461 Artificial Intelligence

Equivalence • Two sentences α and β are equivalent, if they are true in the same set of models, which is written as α <=> β • In other words, α ≡ β if and only if α ╞ β and β ╞ α

Validity • A sentence is valid, if it is true in all models • Example : P V ר P • Tautologies • Deduction theorem: For any sentence α and β, α ╞ β if and only if (α ╞ β ) valid • Now, What is inference? • Rephrase in terms of Deduction Theorem? • All you do is to check the validity of KB╞ α • Conversely??? Conversely, every valid implication represents an inference.

Satisfiablity • A sentence is satisfiable if it is true in some model. • If a sentence α is true in model m, then m satisfies α, or we say m is the model of α. • How do you verify if a sentence is satisfiable? • Enumerate the possible models, until one is found to satisfy. • Many problems in Computer Science can be reduced to a satisfiablity problems • Ex. CSP is all about verifying if the constraints are satisfiable by some assignments

Connection between Validity and Satisfiablity • α is valid iff רα is not satisfiable • How do we use this observation for ‘entailment’? • Let us phrase it this way α╞ β if and only if the sentence (αΛרβ ) is not satisfiable. • Reduction to an absurd thing / Proof by refutation / Proof by contradiction

Proof methods 14 • 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 EA C461 Artificial Intelligence

A property 15 • Monotonicity • Set of entailed sentences can only increase as information is added to the knowledge base. • If KB╞ β then KBΛ α╞ β • Inference rules can be applied whenever suitable premises are found in the Knowledge base. EA C461 Artificial Intelligence

Resolution 16 Conjunctive Normal Form (CNF) conjunction of disjunctions of literals clauses E.g., (A B)  (B C D) • Resolution inference rule (for CNF): li… lk, m1 … mn li … 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 EA C461 Artificial Intelligence

Resolution algorithm 17 • Proof by contradiction, i.e., show KBα unsatisfiable EA C461 Artificial Intelligence

Resolution example 18 • KB = (B1,1 (P1,2 P2,1))  B1,1 • α = P1,2 EA C461 Artificial Intelligence

Completeness of Resolution 19 • Resolution Closure (RC) • RC(S) – set of clauses derivable by repeated application of resolution rule to clauses in S or their derivatives • RC(S) must be finite • Completeness is entailed by “Ground Resolution Theorem” • “If a set of clauses is unsatisfiable, then the resolution closure of those clauses contains the empty clause ” EA C461 Artificial Intelligence

Contd… 20 Any complete search algorithm, applying only resolution can derive any conclusion entailed by any knowledge base in propositional logic • This is true only in restricted sense • Resolution can always be used to either confirm or refute a sentence • Resolution cannot be used to enumerate entailed sentences • Refutation completeness EA C461 Artificial Intelligence

Forward and backward chaining 21 • 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β β • Can be used with forward chaining or backward chaining. • These algorithms are very natural and run in linear time EA C461 Artificial Intelligence

Forward chaining 22 • Idea: fire any rule whose premises are satisfied in the KB, • add its conclusion to the KB, until query is found EA C461 Artificial Intelligence

Forward chaining algorithm 23 • Forward chaining is sound and complete for Horn KB EA C461 Artificial Intelligence

Forward chaining example 24 EA C461 Artificial Intelligence

Proof of completeness 32 • FC derives every atomic sentence that is entailed by KB • 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 a1  … ak b • Hence m is a model of KB • If KB╞ q, q is true in every model of KB, including m EA C461 Artificial Intelligence

Backward chaining 33 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 • has already been proved true, or • has already failed EA C461 Artificial Intelligence

Backward chaining example 34 EA C461 Artificial Intelligence

Forward vs. backward chaining 44 • 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 EA C461 Artificial Intelligence

Refining model checking Algorithms checks satisfiability The connection between finding a solution to a CSP and finding satisfying model for a logical sentence implies the existence of backtracking algorithm DPLL (Davis, Putnam, Logemann & Loveland)

The DPLL algorithm 46 Determine if an input propositional logic sentence (in CNF) is satisfiable. Improvements over truth table enumeration: • Early termination A clause is true if any literal is true. A sentence is false if any clause is false. • 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. • Unit clause heuristic Unit clause: only one literal in the clause The only literal in a unit clause must be true. EA C461 Artificial Intelligence

The DPLL algorithm 47 EA C461 Artificial Intelligence

The WalkSAT algorithm 48 • Incomplete, local search algorithm • Evaluation function: The min-conflict heuristic of minimizing the number of unsatisfied clauses • Balance between greediness and randomness EA C461 Artificial Intelligence

The WalkSAT algorithm 49 EA C461 Artificial Intelligence

Hard satisfiability problems 50 • Consider randomly generated 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 • This CNF has 16 models out of 32 possible assignments. • Quite easy to solve. • So where are hard problems? • Fix n and keep increasing m, EA C461 Artificial Intelligence

S P Vimal, Department of CSIS, BITS, Pilani