logic with quantifiers (informally)

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EDAA40 Discrete Structures in Computer Science 8: Quantificational logic Jörn W.

of Computer Science, Lund University logic with quantifiers (informally) Given a

notations represents there exists an x, such that Forall is universal

1 EDAA40 Discrete Structures in Computer Science 8: Quantificational logic Jörn W. Janneck, Dept. of Computer Science, Lund University logic with quantifiers (informally) Given a logical formula that depends on a variable x : represents forall x, alt. notations represents there exists an x, such that Forall is universal quantification, exists is existential. Examples: SLAM, p the language of quantificational logic terms also 1-place! formulae also other relation symbols SLAM, p

following equivalences hold for any formula : Remember de Morgan's laws? Explain.

2 more examples... Zermelo-Fraenkel Set Theory w/choice (ZFC) extensionality regularity specification union replacement infinity power set choice 4 equivalences: quantifier interchange The following equivalences hold for any formula : Remember de Morgan's laws? Explain. 5 finite transforms Suppose we quantify all variables over a finite set D, and we have constant symbols for each of its elements. A finite transform of a universally/existentially quantified formula removes the quantifier, and instantiates the body for each element of D in a chained conjunction/disjunction. Do this for the following formula, until all quantifiers are gone. Assume a domain with two values, with constant names a and b. 6

equivalences: distribution The following equivalences hold for any formula

x x z quantifier scopes, and the variables bound in/by them 8 free and

inside the scope of a quantifier that binds that variable.

3 equivalences: distribution The following equivalences hold for any formula : This should be easy to see if you think about what this would look like in a finite transform. 7 quantifier scopes variable uses, and the quantifier they are bound by Y x x z quantifier scopes, and the variables bound in/by them 8 free and bound variable occurrences A variable occurrence is bound iff it occurs inside the scope of a quantifier that binds that variable. It is free otherwise. A formula with no free variable occurrences is called closed. A closed formula is a sentence. bound occurrences free occurrences 9

the symbols in it. Also, we need to determine what values the quantified variables can assume. A domain or universe D is the set of values that quantified variables range over.

4 equivalences: vacuity, relettering Vacuity: If x does not occur free in, then Can you think of a case where this is not true? Relettering: If x does not occur at all in, and is the result of replacing every bound occurrence of some variable y in with x, then 10 interpretations The value of a formula depends on how you read the symbols in it. Also, we need to determine what values the quantified variables can assume. A domain or universe D is the set of values that quantified variables range over. An interpretation v is a function assigning mathematical objects to the symbols occurring in a formula. Specifically... - to each constant a - to each variable x - to each n-place function letter f - to each n-place relation letter P - to the identity symbol the identity over D 11 evaluating terms and formulae Given a domain D and an interpretation v, the value of a term t is defined as follows (reusing the v): With this, we can determine the truth value (0 or 1) of a formula as follows: We are overloading the name of the interpretation, like we did for assignments and valuations in propositional logic. 12

5 method 1: substitution Given a formula, a variable x, and a term t, is the formula resulting from substituting every free occurrence of x in with t. Now we make two assumptions: (1) There are enough constant letters so each value in D can have its own constant letter. (2) The interpretation from the constant letters is onto D, i.e. every value in D is represented by (at least) one constant letter under v. 13 method 1: substitution With this we can evaluate quantified formulae as follows: It sounds a bit circular, but it does give an algorithm for computing the truth value of a quantified formula. Also, larger domains require more constant letters. 14 method 2: x-variant To avoid having to tinker with the constant symbols, we can instead tinker with the interpretation. Given an interpretation v and a variable x, an x-variant of v (with value c) is another interpretation v' that agrees with v on all constants, function and predicate letters, and variables other than x, and that assigns x the value c. With this we can evaluate quantified formulae as follows: 15

logical implication Given a set of formulae and a formula, we say that A logically implies iff there is no interpretation v such that all the formulae in A are true under v, but is false: Also:

6 logical implication Given a set of formulae and a formula, we say that A logically implies iff there is no interpretation v such that all the formulae in A are true under v, but is false: Also: logical equivalence logical truth contradiction A formula that is neither logically true nor a contradiction is contingent. 16 some implications Why isn't this an equivalence? 17 clean substitution Given a formula, a variable x, and a term t, a substitution is clean iff no free occurrence of x is in the scope of a quantifier binding a free variable in t. y 18

7 instantiation & generalization The following are general rules regarding quantifiers: universal instantiation (UI) if substitution is clean universal generalization (UG) if x not free in existential generalization (EG) if substitution is clean not vacuity, as x could be free in! existential instantiation (EI) if x not free in Why does UG / EI work? EG involves reverse substitution 19 replacement Given a formula, and a term t, a occurrence of t in in a scope that binds a variable in t. is free iff it is not non-free occurrences free occurrences A replacement of a term t by t' in a formula free occurrences of t in by t'. is the result of replacing It is clean if the t' replaced for t are also free. 20 principle of replacement Principle of Replacement: If the replacement is clean. 21

Foundations of AI. 9. Predicate Logic. Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution

Foundations of AI. 9. Predicate Logic. Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution Foundations of AI 9. Predicate Logic Syntax and Semantics, Normal Forms, Herbrand Expansion, Resolution Wolfram Burgard, Andreas Karwath, Bernhard Nebel, and Martin Riedmiller 09/1 Contents Motivation