The Resolution Principle for First Order Logic Resolution for FOL The Resolution Principle for First Order Logic Completeness of Resolution Examples of Resolution Deletion Strategy Summary The Resolution Principle for First Order Logic Resolution Resolution for FOL [Chang-Lee Ch. 5.5] for FOL Completeness Completeness of the resolution principle [Chang-Lee Ch. of Resolution 5.6] Examples of Resolution Examples of resolution [Chang-Lee Ch. 5.7] Deletion Strategy Deletion Strategy [Chang-Lee Ch. 5.8] Resolution Principle for FOL The Resolution Principle for First Order Logic Brief Recap. Resolution for FOL We introduced resolution as a refutation procedure for Completeness prop. logic of Resolution Examples of We know how to match literals containing variables using Resolution unication and substitutions Deletion Strategy We will see how to use these concepts to obtain a refutation procedure for FOL Factor The Resolution Principle for First Order Logic Resolution for FOL Denition (Factor) Completeness of Resolution If two ore more literals (with the same sign) in a clause C have Examples of a most general unier , then C is called a factor for C. If C Resolution σ σ σ Deletion is a unit clause then it is called a unit factor. Strategy Example The Resolution Principle for First Order Logic Resolution for FOL Example (unit factor) Completeness of Resolution Consider C = P(x) _ P(a). Examples of a x is a MGU for P x and P a . Resolution σ = f = g ( ) ( ) Deletion Cσ = P(a) is a unit factor of C Strategy Example II The Resolution Principle for First Order Logic Resolution for FOL Example (factor) Completeness of Resolution Consider C = P(x) _ P(f (y)) _:Q(x). Examples of f y x is a MGU for P x and P f y . Resolution σ = f ( )= g ( ) ( ( )) Deletion Cσ = P(f (y)) _:Q(f (y)) is a factor of C Strategy Binary Resolvent The Resolution Principle for First Order Logic Denition (Binary Resolvent) Resolution Given two clauses C and C (called parent clauses) with no for FOL 1 2 Completeness variables in common. Let L1 and L2 be two literals in C1 and of Resolution C2 respectively. If L1 and :L2 have a MGU σ then the clause Examples of Resolution C L C L Deletion ( 1σ − 1σ) [ ( 2σ − 2σ) Strategy is a binary resolvent of C1 and C2. L1 and L2 are the literals solved upon. Example: Binary Resolvent The Resolution Example (Binary Resolvent) Principle for First Order Consider the two clauses C P x Q x and Logic 1 = ( ) _ ( ) C2 = :P(a) _ R(x). Resolution Since x appears in both we will rename x with y in for FOL C2 = P(a) _ R(y) Completeness of Resolution Choose L1 = P(x) and L2 = :P(a). Examples of Resolution L1 and :L2 = P(a) have the MGU σ = a=x Deletion Strategy (C1σ − L1σ) [ (C2σ − L2σ) = (fP(a); Q(a)g − fP(a)g) [ ((:P(a); R(y)) − f:P(a)g) = (fQ(a)g [ fR(y)g = fQ(a); R(y)g = Q(a) _ R(y) Q(a) _ R(y) is the binary resolvent and P(x), :P(a) are the literals resolved upon Resolvent The Resolution Principle for First Order Logic Denition (Resolvent) Resolution for FOL Given two clauses C1 and C2 (parent clauses) a resolvent is one Completeness of the following binary resolvents: of Resolution Examples of a binary resolvent of C1 and C2 Resolution a binary resolvent of C1 and a factor of C2 Deletion Strategy a binary resolvent of a factor of C1 and C2 a binary resolvent of a factor of C1 and a factor of C2 Example: Resolvent The Resolution Principle for First Order Logic Example (Resolvent) Resolution for FOL Consider the two clauses C1 = P(x) _ P(f (y)) _ R(g(y)) and Completeness of Resolution C2 = :P(f (g(a))) _ Q(b). Examples of C 0 P f y R g y is a factor of C Resolution 1 = ( ( )) _ ( ( )) 1 0 Deletion Cr = R(g(g(a))) _ Q(b) is a binary resolvent of C1 and C2 Strategy Therefore Cr is a resolvent of C1 and C2 Completeness of Resolution The Resolution Principle for First Order Logic Completeness of resolution Resolution Resolution is an inference rule that produce resolvents from for FOL sets of clauses Completeness of Resolution It is more ecient than previous proof procedure (e.g. Examples of Resolution Gilmore + DPLL) Deletion Resolution is complete: if the set S of clauses is Strategy unsatisable using resolution we will always manage to obtain Example The Resolution Principle for Example (Trapezoid) First Order Logic Show that alternate interior angles formed by a diagonal of a trapezoid are equal. Resolution T (x; y; z; w) is true i xyzw are the vertices of a trapezoid. for FOL P(x; y; u; v) is true i line segment xy is parallel to line segment uv. Completeness of Resolution E(x; y; z; u; v; w) is true i the angle xyz is equal to uvw. Examples of Resolution Axioms: Deletion A1 (8x)(8y)(8u)(8v)(T (x; y; u; v) ! P(x; y; u; v)) Strategy , A2 , (8x)(8y)(8u)(8v)(P(x; y; u; v) ! E(x; y; v; u; v; y)). A3 , T (a; b; c; d). We want to proove that G , E(a; b; d; c; d; b) holds, given A1; A2; A3. Show that, by using resolution we can refute A1 ^ A2 ^ A3 ^ :G Resolution and Semantic trees The Resolution Principle for First Order Logic Resolution and Semantic trees Resolution for FOL Resolution is deeply related to semantic trees Completeness of Resolution Resolution generates clauses that can be used to prune Examples of branches of semantic trees Resolution Deletion Semantic trees can be used to prove completeness of Strategy resolution Example The Resolution Principle for Example (resolution and semantic trees) First Order Logic Consider the set of clauses S = fP; :P _ Q; :P _:Qg. We can nd a closed semantic tree with 5 nodes. Using resolution Resolution for FOL we can obtain: Completeness P Q P Q of Resolution : _ : _: Examples of :P Resolution Consider the set S0 S C, we can nd a closed semantic tree Deletion = [ Strategy with 3 nodes. Using resolution we can obtain: :PP 00 0 Consider the set S = S [ we can nd a closed semantic tree with one node. Semantic tree and completeness of Resolution The Resolution Principle for First Order Logic Resolution Semantic trees and Resolution for FOL A similar reasoning can be used to prove the completeness Completeness of Resolution of Resoluton Examples of Given a set of unsatisable clauses: Resolution Deletion 1 Construct a closed semantic tree Strategy 2 Force the tree to collapse while building a resolution proof. Lifting lemma The Resolution Theorem Principle for 0 0 First Order Lifting Lemma If C1 and C2 are instances of C1 and C2 Logic 0 0 0 0 respectively, and if C is a resolvent of C1 and C2, then C is an instance of C (resolvent of C1 and C2). Resolution for FOL Completeness Example of Resolution Consider C P x Q x and Examples of 1 = ( ) _ ( ) Resolution C2 = :P(f (y)) _:P(z) _ R(y). Deletion 0 Strategy C1 = P(f (a)) _ Q(f (a)) is an instance of C1 0 C2 = :P(f (a)) _ R(a) is an instance of C2 0 0 0 C3 = Q(f (a)) _ R(a) is a resolvent for C1 and C2 0 Lifting Lemma ) 9 C3 such that C3 is an instance of C3. For example, C3 = Q(f (y)) _ R(y) is a resolvent for C1 0 and C2 and C3 is an instance of C3 Lifting lemma: proof The Resolution Principle for First Order Logic Lifting Lemma If necessary we rename variables in C1 or C2 so that Resolution for FOL variables in C1 are all dierent from variables in C2. Completeness Let L0 and L0 be the literals resolved upon of Resolution 1 2 0 0 0 0 0 0 0 Examples of C = (C1γ − L1γ) [ (C2γ − L2γ), γ MGU for L1; L2. Resolution Since C 0 and C 0 are instances of C 0 and C 0 we can write Deletion 1 2 1 2 Strategy 0 0 C1 = C1θ and C2 = C2θ where θ is one substitution. 1 Ri Let Li ; ··· ; Li denote the literals in Ci corresponding to 0 1 Ri 0 Li (i.e. Li θ; ··· ; Li θ = Li ) Lifting lemma: proof II The Resolution Principle for First Order Logic Lifting Lemma 1 Ri assume i > 1 obtain a MGU λi for Li ; ··· ; Li . and let Resolution 1 for FOL Li = Li λi for i = 1; 2. Completeness then L is a literal in factor C of C . of Resolution i i λi i 1 Examples of assume i = 1 then λi = and Li = Li λi . Resolution Let Deletion λ = λ1 [ λ2 Strategy 0 Then Li is an instance of Li 0 0 Since L1 and L2 are uniable then L1 and L2 are uniable. Let σ be a MGU of L1 and L2 Lifting lemma: proof III The Resolution Principle for First Order Logic Proof. Resolution (Lifting Lemma) for FOL 1 R1 Completeness Let C = (C1(λ ◦ σ) − (fL1; ··· ; L1 g)(λ ◦ σ)) [ ((C2(λ ◦ of Resolution 1 R2 σ) − (fL2; ··· ; L2 g)(λ ◦ σ))) Examples of Resolution 0 1 R1 Then C = (C1(θ ◦ γ) − (fL1; ··· ; L1 g)(θ ◦ γ)) [ ((C2(θ ◦ Deletion 1 R2 Strategy γ) − (fL2; ··· ; L2 g)(θ ◦ γ))) is an instance of C as λ ◦ σ is a more general unier than θ ◦ γ Completeness of Resolution The Resolution Principle for First Order Logic Resolution for FOL Theorem (Completeness of Resolution) Completeness of Resolution A set S of clauses is unsatisable i there is a resolution Examples of Resolution deduction of the empty clause from S Deletion Strategy Completeness of Resolution: proof ( The Resolution .