Resolution for Predicate Logic The connection between general satisfiability and Her-brand satisfiability provides the basis for a refutational approach to first-order theorem proving. Validity of a first-order sentence φ can be checked as follows. 1. First convert the negated formula ¬φinto a prenex
nFin Skolem form we perform resolution on the clauses in the matrix Fwith the goal of deriving the empty clause. Although quantifiers do not explicitly appear in resolution proofs, we can see the variables in such a proof as being implicitly universally quantified. This is made more formal when we formulate the Resolution Lemma in the next section.
Some tautologies of predicate logic are analogs of tautologies for propo-sitional logic (Section 14.6), while others are not (Section 14.7). Proofs in predicate logic can be carried out in a manner similar to proofs in propositional logic (Sections 14.8 and 14.9). In Section 14.10 we discuss some of the implications of predicate logic as to our
Predicate logic admits the formulation of abstract, schematic assertions. (Object) variables are the technical tool for schematization. We assume that X is a given countably infinite set of symbols which we use for (the denotation of) variables. Ruzica Piskac First-Order Logic - Syntax, Semantics, Resolution 6 / 125
Refinements of resolution Problems of predicate resolution: • Branching degree of the search space too large • Too many dead ends • Combinatorial explosion of the search space Solution: Strategies and heuristics: forbid certain resolution steps, which narrows the search space. But: Completeness must be preserved! 26
Resolution For Predicates In predicate logic, a literal is a (possibly negated) predicate: not p and ¬p, but P(arg1) and ¬P(arg2) (different args) To derive a new clause from A ∨P(arg1) and B ∨¬P(arg2) must bring args to common form. Variables in clauses will be (implicitly) universally quantified
First-Order Logic • First-order logic is another way of knowledge representation in artificial intelligence. It is an extension to propositional logic. • FOL is sufficiently expressive to represent the natural language statements in a concise way. • First-order logic is also known as Predicate logic or First-order predicate logic. First ...
Resolution for predicate logic 12.1 Lecture 12 Resolution for predicate logic Unification, resolution Dr Christoph Haase University of Oxford (with small changes by Javier Esparza) 1/16. Resolution for predicate logic 12.2 Drawbacks of ground resolution Ground resolution good for showing semi-decidability, bad for
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 from S. A ...
Resolution for rst-order logic As for propositional logic, F ‘ Res C means that clause C can be derived from a set of clauses F by a sequence of resolution steps, i.e. that there is a sequence of clauses C 1;:::;C m = C such that for every C i Ieither C i 2F Ior C i is the resolvent of C a and C b where a;b <i. Questions: CorrectnessDoes F ‘
Translating English to Logic Translate the following sentence into predicate logic: “Every student in this class has taken a course in Java.” Solution: First decide on the domain U. Solution 1:If U is all students in this class, define a propositional function J(x) denoting “x has taken a course in Java” and translate as 8x J(x).
Resolution method in predicate logic Substitutions Substitutions A substitution is a (finite) setσ= {x1/t 1,...,x n/t n}, where x i’s are distinctvariables, t i’s are terms, and the term t i is not x i. If all t i’s are ground terms, then σis a ground substitution. If all t i’s are distinct variables, then σis a renaming of variables. An expression is a literal or a term.
• Resolution is refutation complete: it can establish that a given sentence Q is entailed by KB, but can’t (in general) generate all logical consequences of a set of sentences • Also, it cannot be used to prove that Q is not entailed by KB • Resolution won’t always give an answer since entailment is only semi-decidable
Predicate Logic Resolution Algorithm • While no empty clause exists and there are clauses that can be resolved: – select 2 clauses that can be resolved. – resolve the clauses (after unification), apply the unification substitution to the result and store in the knowledge base.
A predicate-logic resolution derivation of a clause Cfrom a set of clauses Fis a sequence of clauses C 1;:::;C m, with C m = Csuch that each C i is either a clause of F(possibly with the variables renamed) or follows by a resolution step from two preceding clauses C j;C k, with j;k<i. We write Res (F) for the set of clauses C
IA008 Computational logic Revision II 1 Resolution in predicate logic Exercise 1.1: Find all possible resolvents of the following pairs of clauses: a) C1 = {P(x)},C2 = {¬P(f(x))} ... IA008 Computational logic Revision II c) LD resolution: a step towards implementation. It is defined only for Horn
(c) Paul Fodor (CS Stony Brook) and Elsevier Interpretation An interpretation I of an alphabet is: a non-empty domain D, and a mapping that associates: each constant c∈A with an element c I∈D each n-ary functor f∈A with an function f I: Dn →D each n-ary predicate symbol p∈A with an relation p I⊆Dn For instance, one interpretation of the symbols in our “relations”
USING PREDICATE LOGIC Representation of Simple Facts in Logic Propositional logic is useful because it is simple to deal with and a decision procedure for it exists. ... Propositional Resolution 1. Convert all the propositions of F to clause form. 2. Negate P and convert the result to clause form. Add it to the set of clauses obtained in step 1.
