Where l i and mj are complementary literals, there is a resolution in FOL. Because it only resolves perfectly, this rule is also known as the binary resolution rule. Example: We can determine two clauses which are given below: [Animal (g(x) V Loves (f(x), x)] and [￢ Loves(a, b) V ￢Kills(a, b)]
Valid Formula Schema H is valid iff valid for any FOL formula Fi obeying the side conditions Example: H1 and H2 are valid. 2- 17 Substitution σ of H σ : {F1 → ,...,Fn → } mapping place holders Fi of H to FOL formulae, (obeying the side conditions of H) Proposition (Formula Schema) If H is valid formula schema and
Lemma 8.3.1 (Completeness of ground resolution) The ground resolution method is complete for ground clauses. Proof: Observe that the systems G! and GCNF! are complete for quantifier-free formulae of a first-order language without equality. Hence, by theorem 4.3.1, the resolution method is also complete for sets of ground clauses.
Resolution in FOL Ralitsa Dardjonova June 2020 Abstract In this paper we consider resolution as a procedure for deciding unsatis ability of a formula. We start with presentation of resolution in propositional logic and we show it is a valid calculus which always terminates. We extend the resolution technique to predicate logic. We again show
The conjunction of CNF formulas is also a CNF formula. If all formulas are in CNF, we mayassume Σ as a set of clauses. cbna CS228 Logic for Computer Science 2023 Instructor: Ashutosh Gupta IITB, India 4 ... Applying resolution in FOL We apply resolution whenan atom and its negationare in two clauses. Resolution F ∨C ¬F ∨D
Resolution Procedure… • Resolution Rule for FOL: – From sentences P1 v ... v Pn and Q1 v ... v Qm where each Pi and Qi is a literal, – If P1 and ~Q1 unify with substitution Theta, i.e., if P1[Theta] = ~Q1[Theta] – Then infer the resolvent sentence: (P2 v … v Pn v Q2 v … v Qm)[Theta] • Example
Resolution • Basic algorithm for resolution for FOL: – Determine set of statements to be tested for consistency (e.g. if argument: negate conclusion) – Put all statements into PNF, and body into CNF – Skolemize all existentials – Drop universals and put into clauses – Resolve … using unification to match atomic formulas
A formula which is equivalent to a given formula and which consists of a product of elementary products is called a conjunctive normal form of given formula. Example : (P~ ∨ Q) ∧ (Q ∨ R) ∧ (~ P ∨ Q ∨ ~ R) If every elementary sum in CNF is tautology, then given formula is also tautology.
Resolution in Predicate Logic Resolution method is used to test unsatisfiability of a set S of clauses in Predicate Logic. – It is an extension of resolution method for propositional logic. The resolution principle basically checks whether empty clause is contained or derived ... Standard form of any formula α in FOL is of
The conjunction of CNF formulas is also a CNF formula. If all formulas are in CNF, we mayassume as a set of clauses. CS228 Logic for Computer Science 2021 Instructor: Ashutosh Gupta IITB, India 4 ... Applying resolution in FOL We apply resolution whenan atom and its negationare in two clauses. Resolution F _C :F _D C _D
Resolution Theorem Proving Properties of Resolution Theorem Proving: – sound (for propositional and FOL) – (refutation) complete (for propositional and FOL) Procedure may seem cumbersome but note that can be easily automated. Just “smash” clauses until empty clause or no more new clauses.
The last formula, respectively, defines s. Eliminating the existential quantifiers by Skolemization (cf. below) reintroduces the “redundant” symbols. Consequently there is a trade-off between the complexity of the quantification structure and the complexity of the signature. Ruzica Piskac First-Order Logic - Syntax, Semantics, Resolution ...
Correctness of Resolution for First-Order Logic De nition Theuniversal closureof a formula H with free variables x 1;:::;x n: 8H = 8x 18x 2:::8x nH Theorem Let F be a closed formula in Skolem form with matrix F in CNF. If F ‘ Res then F is unsatis able. 25
FOL resolution. A main limitation of the generalized MP rule is that it works only with universal conditions of the form we discussed above: $$\forall x\forall y\dots((P_1(x,y,\dots)\land P_2(x,y,\dots))\land \dots \to C(x,y,\dots))$$ ... Crucially, however, the formula is not necessarily equivalent to the original formula in FOL. It is what ...
Completeness of Resolution for First-Order Logic Theorem Let F be a closed formula in Skolem form with matrix F in CNF. If F is unsatis able then F ‘ Res. Proof If F is unsatis able, there is a ground resolution proof C0 1;:::;C n 0= . We transform this step by step into a resolution proof C 1;:::;C n = such that C0 i is a ground instance of ...
