Propositional Logic 2 Syntactic Elements Logic An atomic sentence consists of a single propositional symbol, representing a proposition that can be true or false. A literal is a propositional symbol or its negation. Complex sentences are constructed from simpler sentences using logical connectives: ¬ (not), ∧ (and), ∨ (or), → (implies) [I prefer → to ⇒], and ↔ (iff). CS 5233 Artificial Intelligence Logic – 2 Propositional Logic 2 SyntacticElements.................................. ...... 2 Syntax Syntax............................................. ... 3 A proposition or propositional sentence can be formed as follows: Semantics.......................................... .... 4 Every propositional symbol is a sentence. Deduction.......................................... .... 5 If A is a sentence, then ¬A is a sentence. Entailment ......................................... .... 6 If A1 and A2 are sentences, then so are: TheWumpusWorldExample.............................. ... 7 A1 ∧ A2, (and, conjunction) InferenceRules ..................................... ..... 8 A1 ∨ A2, (or, disjunction) InferenceProcedures................................ ....... 9 A1 → A2, and (implies, implication) Resolution Inference Procedure. ........ 10 A1 ↔ A2. (iff, biconditional) UsefulEquivalences ................................. ..... 11 CS 5233 Artificial Intelligence Logic – 3 First-Order Logic 12 SyntacticElements.................................. ..... 12 Semantics Syntax............................................. .. 13 Semantics.......................................... ... 14 A world or domain is the set of facts we want to represent. InferenceRules ..................................... .... 15 An interpretation maps each propositional symbol to the world. Example:Setup...................................... ... 16 A sentence is true if its interpretation in the world is true. Example,Proof,Part1................................ .... 17 A knowledge base is a set of sentences. Example,Proof,Part2................................ .... 18 A world is a model of a KB if the KB is true for that world. Unification ......................................... ... 19 For convenience, a model can be thought of as a truth assignment to the Unify-ListsProcedure............................... ...... 20 symbols. Unify-TermsProcedure............................... ..... 21 CS 5233 Artificial Intelligence Logic – 4 Resolve Procedure Preliminaries . ........ 22 ResolveProcedure ................................... .... 23 1 2 Deduction The Wumpus World Example A sentence S is satisfiable within a KB if S is true in some model of the KB, i.e., some truth assignment makes S and the KB true. A sentence S is entailed by a KB if S is true in all models of the KB (denoted as KB |= S), i.e., every truth assignment that makes KB true also makes S true. Logical inference or deduction is concerned with producing entailed sentences from KBs. CS 5233 Artificial Intelligence Logic – 5 Entailment Sentences (KB) Sentence Interpretation Models Entails True in all CS 5233 Artificial Intelligence Logic – 7 Possible Truth Possible Truth Assignments Assignments Inference Rules Interpretation Modus Ponens From A → B and A Infer B Unit Resolution From A ∨ B Models and ¬B True in all Infer A Possible Worlds Possible Worlds Resolution From A ∨ B CS 5233 Artificial Intelligence Logic – 6 and ¬B ∨ C Infer A ∨ C CS 5233 Artificial Intelligence Logic – 8 3 4 Inference Procedures First-Order Logic 12 An inference procedure uses inference rules to produce proofs. Denoted as Syntactic Elements KB ⊢ S. An inference procedure is sound if it can only prove entailed sentences, i.e., A symbol in first-order logic can be a predicate, a function, a constant every sentence it proves is entailed. term, or a variable term. An inference procedure is complete if it can prove any entailed sentence, First-order logic uses the connectives ¬ (not), ∧ (and), ∨ (or), → i.e., every entailed sentence can be proved. (implies), and ↔ (iff). CS 5233 Artificial Intelligence Logic – 9 First-order logic also uses the quantifiers ∀ (for all) and ∃ (there exists). CS 5233 Artificial Intelligence Logic – 12 Resolution Inference Procedure Syntax Resolution applies to conjunctive normal form. A term is a constant or variable, or a function applied to a sequence of – A KB is a conjunction of sentences. terms. – Each sentence is a disjunction of literals. A ground term is a term with no variables. Resolution is refutation complete. If a KB is in CNF, and if the KB is An atomic sentence or atom is a predicate applied to a sequence of terms. inconsistent, then resolution will infer inconsistent literals. A ground atom is an atom with no variables. To deduce a sentence S, first temporarily add ¬S to the KB. Then, if Connectives can be used to construct more complex sentences in the usual resolution infers inconsistent literals, then ¬S is not satisfiable within the way. KB, which means that S is entailed. If A is a sentence and x is a variable, then: CS 5233 Artificial Intelligence Logic – 10 – ∀x A is a sentence (universal quantifier). – ∃x A is a sentence (existential quantifier). Useful Equivalences for Converting to CNF A well-formed formula is a sentence in which all the variables are quantified. Several logical equivalences are handy for converting sentences to CNF CS 5233 Artificial Intelligence Logic – 13 (p → q) ≡ (¬p ∨ q) ((p ∧ r) → (q ∨ s)) ≡ (¬p ∨¬r ∨ q ∨ s) Semantics (p → (q ∧ r)) The connectives ∧, ∨, ¬, →, and ↔ are evaluated in the usual way. ≡ ((p → q) ∧ (p → r)) ∀x A is true if every substitution for x makes A true. ≡ ((¬p ∨ q) ∧ (¬p ∨ r)) ∃x A is true if at least one substitution for x makes A true. ((p ∨ q) → r) An interpretation consists of objects in the world and mappings for terms ≡ ((p → r) ∧ (q → r)) and predicates. ≡ ((¬p ∨ r) ∧ (¬q ∨ r)) Each ground term is mapped to an object. CS 5233 Artificial Intelligence Logic – 11 Each predicate is mapped to a relation (think relational database). A ground atom is true if the predicate’s relation holds between the terms’ objects. CS 5233 Artificial Intelligence Logic – 14 5 6 Inference Rules Example, Proof, Part 1 Universal From ∀x A(x) -parent(x,y) | -ancestor(y,z) | ancestor(x,z) Elimination Infer A(t) where t is any term -ancestor(Liz,Billy) --------------------------------------------- Existential From ∃x A(x) -parent(Liz,y) | -ancestor(y,Billy) Elimination Infer A(c) where c is a new constant Resolution From ∀x,y B(y) ∨ A(x) -mother(x,y) | parent(x,y) (disjunctive) and ∀x, z ¬A(x) ∨ C(z) -parent(Liz,y) | -ancestor(y,Billy) --------------------------------------------- Infer ∀y,z B(y) ∨ C(z) -mother(Liz,y) | -ancestor(y,Billy) Resolution From ∀x,y B(y) → A(x) (implicative) and ∀x,z A(x) → C(z) mother(Liz,Charley) Infer ∀y,z B(y) → C(z) -mother(Liz,y) | -ancestor(y,Billy) --------------------------------------------- CS 5233 Artificial Intelligence Logic – 15 -ancestor(Charley,Billy) CS 5233 Artificial Intelligence Logic – 17 Example: Setup Example, Proof, Part 2 Knowledge Base -parent(x,y) | ancestor(x,y) -parent(x,y) | -ancestor(y,z) | ancestor(x,z) -ancestor(Charley,Billy) --------------------------------------------- -parent(x,y) | ancestor(x,y) -parent(Charley,Billy) -mother(x,y) | parent(x,y) -father(x,y) | parent(x,y) -father(x,y) | parent(x,y) mother(Liz,Charley) -parent(Charley,Billy) father(Charley,Billy) --------------------------------------------- -father(Charley,Billy) To prove ancestor(Liz,Billy) father(Charley,Billy) Refute -ancestor(Liz,Billy) -father(Charley,Billy) CS 5233 Artificial Intelligence Logic – 16 --------------------------------------------- contradiction CS 5233 Artificial Intelligence Logic – 18 Unification To use the resolution inference rule, we need to be able to match atoms in sentences. This is called unification. If successful, unification returns a substitution. A substitution specifies values for variables that would make the two atoms identical. CS 5233 Artificial Intelligence Logic – 19 7 8 Unify-Lists Procedure Resolve Procedure Preliminaries The Resolve procedure assumes that sentences A and B are disjunctions function Unify-Lists(A, B, θ) represented as sets of literals. A literal is an atom or its negation. if A and B have different predicates/functions Resolve assumes that sentences A and B have no variables with the or have different numbers of arguments same names. then return failure Resolve returns all the sentences it infers. for i ← 1 to number of arguments do CS 5233 Artificial Intelligence Logic – 22 termA ← ith argument of A termB ← ith argument of B θ ← Unify-Terms(termA, termB, θ) Resolve Procedure if θ = failure then return failure end for function Resolve(A, B) return θ for each litA in A do for each litB in B do CS 5233 Artificial Intelligence Logic – 20 if one of litA and litB is negated, not both then θ ← Unify-Lists(litA, litB, empty substitution) Unify-Terms Procedure if θ =6 failure and has no recursive substs. then A′ ← apply substitution θ to A litA′ ← apply substitution θ to litA function Unify-Terms(A, B, θ) B′ ← apply substitution θ to B while A is a variable and θ[A] exists litB ′ ← apply substitution θ to litB do A ← θ[A] C ← (A′ −{litA′}) ∪ (B′ −{litB ′}) while B is a variable and θ[B] exists add C to formulas inferred do B ← θ[B] return formulas inferred if A = B then do nothing A A B else if is a variable then θ[ ] ← CS 5233 Artificial Intelligence Logic – 23 else if B is a variable then θ[B] ← A else if A or B is a constant then θ ← failure else θ ← Unify-Lists(A, B, θ) return θ CS 5233 Artificial Intelligence Logic – 21 9 10.