Lecture Slides for MAT-60556 PART IV: First Order Logic: - PowerPoint PPT Presentation

Lecture Slides for MAT-60556 PART IV: First Order Logic: Resolution, Logic programming and Model theory Henri Hansen October 1, 2013 1

Resolution • Resolution is a sound and complete algorithm for propositional logic: A formula in clausal form is un- sat if and only if the algorithm reports unsat, and in propositional logic the procedure is guaranteed to terminate • The generalization to FOL is sound and complete as well, but the algorithm may fail to terminate • The generalization is done in two steps: First, we present ground resolution , which works on ground literals, and then general resolution , wchih uses uni- fication , to enable resolution on non-ground literals 2

Ground Resolution • Let C 1 and C 2 be ground clauses such that l ∈ C 1 annd l c ∈ C 2 . We say that C 1 and C 2 are clashing clauses and they clash on the ground literals l and l c . The resolvent of C 1 and C 2 is the clause C = ( C 1 − { l } ) ∪ ( C 2 − { l c } ) C 1 and C 2 are the parent clauses of C • Theorem: The resolvent is satisfiable if and only if its parents are both satisfiable • Remember: A ground literal is a predicate where the parameters are ground terms (only constants and functions!) 3

Substitution • A substitution of terms for variables is the set { x 1 ← t 1 , . . . , x n ← t n } where each x i is a distinct variable and t i is not identical to the corresponding variable. The set may be empty. • An expression is a term, a literal, a clause, or a set of clauses. If E is an expression and θ = { x 1 ← t 1 , . . . , x n ← t n } is a substitution, then an instance Eθ of E is obtained by simultaneously replacing x i with t i in E 4

Unification • Let U = { A 1 , . . . , A n } be a set of atoms. A unifier of U is a substitution θ such that A 1 θ = · · · = A n θ • A most general unifier (mgu) is a unifier µ such that any unifier θ of U can be expressed as θ = µλ for some substitution λ • Note that not all atoms are unifiable, for instance, it is impossible to unify atoms when predicate sym- bols are different, as well as atoms whose outer functions are different, or, for instance the atoms p ( x ) , p ( f ( x )), which cannot be unified. 5

Unification algorithm • A set of term equations is in solved form iff all equations are of the form x i = t i , where x i is a variable, and each variable that appears in the left- hand side of an equation, never appears anywhere else. • A set of equations in solved form defines a substi- tution, where x i ← t i for each i • It will turn out later, that the set in solved form is the mgu of the original set of term equations, 6

and thus for the set of atoms from which the terms were taken • The following algorithm transforms a set of term equations into solved form, or reports if it cannot be done 1. Transform t = x when t is not a variable, into x = t 2. Erase the equation x = x 3. Let t ′ = t ′′ be an equation where t ′ and t ′′ are not variables – If the outermost functions are not identical, terminate, not unifiable

– Otherwise, replace f ( t ′ 1 , . . . , t ′ k ) = f ( t ′′ 1 , . . . t ′′ k ) with the equations t ′ i = t ′′ i 4. Let x = t be an exuation such that x occurs elsewhere in the set – If x occurs in t and differs from t , terminate, not unifiable – Otherwise, transform the set by replacing oc- currences of x in other equations by t • note: Algorithms for unification may be extremely inefficient

General resolution step • The general resolution rule: Let C 1 and C 2 be clauses such that some predicate appears in C 1 and its negation appears in C 2 . Let l 1 and l 2 be the corresponding literals, such that l 1 σ and ( l 2 σ ) c are unified, by an mgu σ . The resolvent of C 1 and C 2 is defined C = ( C 1 σ − { l 1 σ } ) ∪ ( C 2 σ − { l 2 σ } ) • The resolvent is satisfiable if and only if C 1 and C 2 are satisfiable. 7

Resolution algorithm • Let S = S 0 be a set of clauses • S i +1 is constructed from S i by choosing a pair of clashing clauses C 1 , C 2 ∈ S i , and calculating the re- solvent C • If C = � , then terminate and S is not satisfiable. Otherwise S i +1 = S i ∪ { C } • If S i +1 = S i for all possible pairs of clashing clauses, then S is satisfiable 8

Soundness of resolution • Theorem: If the empty clause � is derived from S , then S is unsatisfiable – Idea of proof: If the parent clauses are simul- taneously satisfiable, then the resolvent is also; Because � is unsat, then the parent clauses must also be unsat – If parent clauses are sat, then there is a Herbrand interpretation for them. The elements of the Herbrand base that satisfy C 1 and C 2 have the same form as ground atoms, so there is a sub- stitution that unifies C i with some ground clause C ′ i 9

– We can infer that the Herbrand model for C 1 and C 2 is also a Herbrand model for C

Completeness of resolution • Lifting Lemma: Let C ′ 1 and C ′ 2 be ground instances of C 1 and C 2 . Let C ′ be a ground resolvent for C ′ 1 and C ′ 2 Then there exists a resolvent C of C 1 and C 2 such that C ′ is a ground instance of C • Theorem: If a set of clauses is unsat, the empty clause can be derived by resolution • Resolution has an important application in Logic programming , which we shall discuss forthwith 10

Logic Programming • Resolution was devoleped for automatic theorem proving • However, a restricted form of resolution can be used to carry out computation as well • Making use of this fact, logic programming has been developed • In logic programming, a program is a set of clauses, and and a query which is a negation of the results of the program. 11

From formulas to logic programming • Consider a deductive system with two kinds of ax- ioms: 1. Universally closed predicates, like ∀ x ( x + 0 = x ) 2. Universally closed implication, like ∀ x ∀ y ∀ z ( x ≤ y ∧ y ≤ z → x ≤ z ) • In clausal form, the first is a literal x + 0 = x , and the second one negates the left hand side, in the example ¬ ( x ≤ y ) ∨¬ ( y ≤ z ) ∨ ( x ≤ z ), so that there is exactly one positive literal. These are the so-called program clauses 12

From formulas to logic programming (contd) • Suppose that we ghave a set of program clauses and we want to prove G 1 ∧ · · · ∧ G n is a logical con- sequence of the set • The formula is negated to ¬ G 1 ∨ · · · ∨ ¬ G n , and this is refuted by resolution with program clauses • ¬ G 1 ∨ · · · ∨ ¬ G n is called the goal clause , and it can only clash on program clauses’ positive literal, as it consists of only negative literals 13

From formulas to logic programming (contd II) • The sequence of resolution produces a sequence of unifying substitutions, which then become the answer to the query • The computation is implicitly carried out as part of the substitutions • Given a gaol clause, it is possible that several literals clash with a positive literal of a program clause, which means the computation is nondeterministic 14

• A computation rule must be used to specify, how a literal in the goal is chosen • Once a literal is chosen, the next program rule must be chosen, so a search rule is needed

Horn Clauses and SLD-resolution • A Horn clause is a clause of the form A ← B 1 ∧ · · · ∧ B n , which is equivalent to A ∨ ¬ B 1 ∨ · · · ∨ B n , and it has at most one positive literal – A fact is a positive unit Horn clause, i.e. A ← – A goal clause is a Horn clause with no positive literals, i.e., ← B 1 · · · B n – A program clause is a Horn clause with one pos- itive, and one or more negative literals • ← means reverse implication, with the meaning “To prove A , prove B 1 , . . . B n ” 15

Correct Answer Substitution • Let P be a program and G be a goal clause. a sub- stitution θ for the variables in G is a correct answer substitution if P | = ∀ ( ¬ Gθ ), with quantification over all free variables in ¬ Gθ • Given a program, a goal clause G = ¬ G 1 ∨ · · · ¬ G n , and a correct substitution θ , P | = ∀ ( G 1 ∧ · · · G n ) θ , and therefore for any substitution σ that makes this conjunction into a ground formula ( G 1 ∧ · · · ∧ G n ) θσ is true for any model of P 16

SLD-resolution • Let P be a set of program clauses, R a computation rule, and G a goal clause. A derivation by SLD- resolution is a sequence of resolution steps between goals and program clauses. 1. The first goal clause G 0 = G 2. The clause G i +1 is derived from G i by selecting a literal A j i ∈ G i and choosing a clause C i ∈ P such that the head of C i unifies with A j i by mgu θ i , and resolving • An SLD-refutation is an SLD-derivation that results in � . 17

Excursion: Prolog • Prolog was the first logic programming language • The computation rule for Prolog is to choose the leftmost literal in the goal • The search rule is to choose clauses from top to bottom in the list of program clauses • In addition, Prolog implements some non-logic pred- icates whose purpose is to produce side-effects, such as I/O- functions 18

Prolog (example) • Consider the following program in Prolog ancestor(X,Y) :- parent(X,Y). ancestor(X,Y) :- parent(X,Z), ancestor(Z,Y). • in addition we should have a database of facts, like parent(bob,allen) and parent(dave,bob) • the goal :- ancestor(Y,bob), ancestor(bob,Z) will succeed and the correct substitution Y = dave, X = allen will be returned 19