1. Introduction to Logic for Artificial Intelligence Lecture 2
Erik Sandewall
2010

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 [all .x (or [= .x c1][= .x c2] ... [= .x cn])]
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.
[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.