Introduction to Logic for Artificial Intelligence Lecture 2

1. Erik Sandewall 2010 Introduction to Logic for Artificial IntelligenceLecture 2

2. Relational Logic, Datalog(a Restricted Predicate Logic) • Use the operators not, and, or, etc like for propositional logic, but: • The elements of the logic expressions can consist of a predicate with its arguments, and are not restricted to proposition symbols • Proposition symbols are considered as predicates with zero arguments • The arguments of predicates may be constant symbols or variables • Examples (in CKL): constant symbol may be London, apple-3, variable may be written .a3 or a3

3. Relational Logic: Quantifiers • Use the operators not, and, or, etc like for propositional logic, but also: • [all .v P] where .v is a variable and P is a formula • [exists .v P] where .v is a variable and P is a formula • Evaluation rule: consider for example evaluation in an episode, i.e. an instance of a microworld that has a specific set of entities associated with it. • [all .v P] is true in that episode if P is true for all values of .v among the associated entities • [exists .v P] is true in that episode if P is true for some value of .v among the associated entities • Compare the operator 'some' that is used in the introductory example in lab2b.

4. Equivalence rules for quantifiers • The rules from propositional logic continue to hold • Choice of variable may be changed, by substitution throughout • [all x [all y P]] == [all y [all x P]] and same for exists • [not [all x P]] == [exists x [not P]] and similar for exists/all • [and [all x P][all x Q]] == [all x [and P Q]] • [or [exists x P][exists x Q]] == [exists x [or P Q]] • [or [all x P] Q] == [all x [or P Q]] if no x in Q • [and [exists x P] Q] == [exists x [and P Q]] if no x in Q • [exists x P] == P == [all x P] if no x in P • Actually, 'no x in Q' means 'no free x in Q' • A quantified expression encapsulates its variable so that it is not 'free' outside. Substitution within that scope is OK.

5. Some terms • Vocabulary for a logic formula: set of predicate symbols containing all those that occur in the formula (and maybe some more], with an indication of arity (i.e. its number of arguments] for each of them, even 0 arity • Domain: non-empty set of objects • Interpretation for a predicate: mapping from the set of all argument sequences over the domain, to T or F • Interpretation for a logic formula: an assignment of an interpretation to each predicate in its vocabulary • Model for a logic formula: an interpretation where the value of the formula is T • Joint vocabulary: like before • p == q holds if they have the same value for all interpretations in all their joint vocabularies

6. Relevance • Interpretation for a logic formula: an assignment of an interpretation to each predicate in its vocabulary • Model for a logic formula: an interpretation where the value of the formula is T • Joint vocabulary: like before • p == q holds if they have the same value for all interpretations in all their joint vocabularies • This definition for == is what one shall use in order to validate the correctness of the equivalence rules for the quantifiers.

7. Domains • Vocabulary for a logic formula: set of predicate symbols containing all those that occur in the formula (and maybe some more], with an indication of arity (i.e. its number of arguments] for each of them, even 0 arity • Domain: non-empty set of objects • Interpretation for a predicate: mapping from the set of all argument sequences over the domain, to T or F • Recall e.g.: [not [all x P]] == [exists x [not P]] • This rule is sound since the two sides obtain the same truth-value regardless of the choice of domain and the choice of the expression P.

8. Resolution for Relational Clause Logic • Convert given formulas to clause form like for propositional logic, making sure that quantifiers are outermost • The following applies if the resulting clauses only use 'all' and none of them uses 'exists' • Remove all quantifiers • Use a modified resolution rule where variable substitutions can be made. Examples: • [P .x .y] unifies with [-P a b] with .x = a, .y = b • [P .x b] unifies with [-P a .y] with the same bindings • [P .x] unifies with [-P .y] with .x = .y • [P a] does not unify with [-P b] • (continued)

9. Resolution for Relational Clause Logic • Use a modified resolution rule where variable substitutions can be made. Examples: • [P .x .y] unifies with [-P a b] with .x = a, .y = b • [P .x b] unifies with [-P a .y] with the same bindings • [P .x] unifies with [-P .y] with .x = .y • [P a] does not unify with [-P b] • To resolve two clauses, one must find substitutions whereby one pair of literals unify, and the same substitutions are applied to the rest of the clauses. Examples: • {[P a]}, {[-P .x], [Q .x]} resolve to {[Q a]} • {[P a .x], [R .x c]}, {[-P .y b], [-R d .y]} resolve to {[R b c], [-R d a]} • Notice that the same also resolve to {[P a d], [-P c b]}

10. Standard Predicate Logic(No types, no equality) • Arguments of predicates may be arbitrary terms, formed from variables and constant symbols by composition using function symbols. • Vocabulary for a logic formula: set of predicate symbols and function symbols containing all those that occur in the formula (and maybe some more], with an indication of arity (i.e. its number of arguments] for each of them, even 0 arity • Domain: non-empty set of objects • Interpretation for a predicate: mapping from the set of all argument sequences over the domain, to T or F • Interpretation for a function symbol: mapping from the set of all argument sequences over the domain, to the a member of the domain • Interpretation for a logic formula: an assignment of an interpretation to each predicate and function symbol in its vocabulary • Model for a logic formula: an interpretation where the value of the formula is T • Constant symbols may be considered as functions of arity zero

11. Equivalence rules • The same rules apply as for the case of relational predicate logic. No additions needed.

12. Resolution: Examples of Unification • [P .x .y] unifies with [-P (f a) b] with .x = (f a), .y = b • [P .x (f .y)] unifies with [-P (f a) .z] with .x = (f a), .y =.y,.z = (f .y) • but also with e.g. .x = (f a), .y = .w, .z = (f .w) • [P .x (f .y)] unifies with [-P (f .x) .y], but only using different substitutions in the two given literals, for example: • .x = (f .z), .y = w in the first one, • .x = .z, .y = (f .w) in the second one • But also e.g. .x = (f .x), .y = .y resp. .x = .x, .y = (f .y) • [P (f .x)] does not unify with [-P (g .y)], nor with [-P (g .x)]

13. Resolution: Examples • Consider (imp [P .x] [P (+ 1 .x)]) and [P 0] • Rewrite these as {[-P .x] [P (+ 1 .x)]} and {[P 0]} • Resolution gives {[P (+ 1 0)]} • How can this be simplified? Use additional clauses • {[= (+ .a 0) .a]} • {[-P .a] [/= .a .b] [P .b]} where /= is negation of = • The first of these gives {[= (+ 1 0) 1]} by substitution • Two resolution steps give {[P 1]} • Therefore, there is a need for two inference rules: resolution and substitution • Resolution is always done with built-in substitution (i.e. unification), but in principle it's sufficient to use primitive resolution + substitution.

14. Equality • Recall the axiom that was used above: • {[-P .a] [/= .a .b] [P .b]} • In order to use equality throughout, one needs one axiom similar to this for each predicate of one argument, • One needs two for each predicate of two arguments (one for each arg) • This is called an axiom schema. Notice that for a given vocabulary there is a fixed number of instances of the axiom schema. • Alternative 1: Second-order logic • [all .P (or (not [.P .a]) [/= .a .b] [.P .b])] i.e. • [-.P .a] [/= .a .b] [.P .b] • Alternative 2: Consider = as a part of the logic, just like quantifiers. Advantage: avoid messing up proofs with trivial uses of equality.

15. Equality • Important: equality is a predicate like all the others (unless one extends the logic considerably) • Equality axioms: • [all .x [= .x .x]] • [all .x [all .y (imp [= .x .y][= .y .x])]] • [all .x [all .y [all .z (imp (and [= .x .y][= .y .z]) [= .x .z])]]] • Schema for argument equality like in the example above • Also, if a and b are two different constant symbols, this does not in itself mean that [/= a b]. Some interpretations can assign the same value to them. • Applications may make separate statement of unique names assumption: different constant symbols have different values • Applications may also make separate statement of closed world assumption: each member of the domain is the value of some constant symbol.

16. Equality • Applications may make separate statement of unique names assumption: different constant symbols have different values • Expressed as a schema • [all .x (not (and [= .x a][= .x b]))] • that is instantiated for all combinations of different constant symbols a and b in the chosen vocabulary. • Applications may also make separate statement of closed world assumption: each member of the domain is the value of some constant symbol. • Expressed as a schema • [all .x (or [= .x c1][= .x c2] ... [= .x cn])] • listing all the constant symbols in the minimal vocabulary of the given premises.

17. Skolemization • In order to use the resolution method for a given set of premises, they shall be converted to conjunctive normal form preceded only by universal quantifiers (all), not by existential quantifiers. • What if the given set of premises do produce an expression with existential quantifiers as well?? • Method, example: [all .x [exists .y [P .x .y]]] is replaced by • [all .x [P .x (g .x)]] where g is a 'new' function symbol • i.e. one that is not already in the vocabulary • For [all .x [all .y [exists .z ----]]] replace .z by (g .x .y) • Intuitively reasonable • Formally nonobvious since the rewritten formula is not equivalent to the given one

18. Skolemization • In order to use the resolution method for a given set of premises, they shall be converted to conjunctive normal form preceded only by universal quantifiers (all), not by existential quantifiers. • What if the given set of premises do produce an expression with existential quantifiers as well?? • Method, example: [all .x [exists .y [P .x .y]]] is replaced by • [all .x [P .x (g .x)]] where g is a 'new' function symbol • i.e. one that is not already in the vocabulary • Intuitively reasonable • Formally problematic since the rewritten formula is not equivalent to the given one • But: the first formula is inconsistent if and only if the second one is. • Therefore, if resolution is used for proof by contradiction, this is a technique that works, but only then.