John C. Shepherdson

Partial evaluation in logic programming

Abstract This paper gives a theoretical foundation for partial evaluation in logic programming. Let P be a normal program, G a normal goal, A a finite set of atoms, and P ′ a partial evaluation of P wrt A . We study, for both the declarative and procedural semantics, conditions under which P ′ is sound and complete wrt P for the goal G . We identify two relevant conditions, those of closedness and independence. For the procedural semantics, we show that, if P ′ ∪ { G } is A -closed and A is independent, then P ′ is sound and complete wrt P for the goal G . For the declarative semantics, we show that, if P ′ ∪ { G } is A -closed, then P ′ is sound wrt P for the goal G . However, we show that, unless strong conditions are imposed, we do not have completeness for the declarative semantics. A practical consequence of our results is that partial evaluators should enforce the closedness and independence conditions.

The reduction of two-way automata to one-way automata

Rabin has proved 1,2 that two-way finite automata, which are allowed to move in both directions along their input tape, are equivalent to one-way automata as far as the classification of input tapes is concerned. Rabins proof is rather complicated and consists in giving a method for the successive elimination of loops in the motion of the machine. The purposeo f this note is to give a short, direct proof of the result.

Negation as failure: a comparison of Clark's completed data base and Reiter's closed world assumption

Abstract Clark shows that a query evaluation procedure for data base deductions using a “negation as failure” inference rule can be regarded as making deductions from the “completed data base” (CDB) obtained by replacing the “if” clauses of the data base by “iff” clauses. We show they can also be regarded as deductions from the “closed world assumption” (CWA) of Reiter. Usually these deduction systems are incomplete and the CDB and CWA differ; one may be consistent and the other not, and when both are separately consistent they may be incompatible. However when the data base is Horn and definite they are compatible and when Clarks query evaluation procedure is complete for ground literals they are essentially equivalent. When the query evaluation procedure is not complete it may lack some basic closure properties. The conditions given by Clark for the completeness of his query evaluation procedure are not quite sufficient, and when made so are rather restrictive.

Negation as failure. II

Abstract The use of the negation as failure rule in logic programming is often considered to be tantamount to reasoning from Clarks “completed data base” [2]. Continuing the investigations of Clark and Shepherdson [2,7], we show that this is not fully equivalent to negation as failure either using classical logic or the more appropriate intuitionistic logic. We doubt whether there is any simple and useful logical meaning of negation as failure in the general case, and study in detail some special kinds of data base where the relationship of the completed data base to negation as failure is closer, e.g. where the data base is definite Horn or hierarchic.

The liar paradox and fuzzy logic

Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr( x ) saying “ x is true” and satisfying the “dequotation schema” for all sentences φ? This problem is investigated in the frame of Łukasiewicz infinitely valued logic.

A sound and complete semantics for a version of negation as failure

Abstract Negation as failure is sound both for the closed world assumption and the completed database or completion, comp ( P ) of a program P . In general it is not complete for either of these declarative semantics. Indeed there can be no semantics for which it is both sound and complete, for all programs and queries, because non-ground negative literals cannot be dealt with, and cause floundering. By extending the negation as failure rule we exclude floundering and we give a semantics T ω ( P ) for which the extended rule is both sound and complete. T ω ( P ) is a weak version of comp ( P ) based on an iterative construction. We show that the soundness and completeness results still hold if the classical consequence relation ⊢ is replaced by a weaker relation ⊢ 31 which is sound for both 3-valued logic and intuitionistic logic.

Unfold/fold transformations of logic programs

Unfold/fold transformations have been used in logic programming for some years to transform programs into more efficient ones. We describe recent work on the extent to which these transformations produce programs which are equivalent to the original one. Various notions of equivalence are considered: same success set; finite failure set; least Herbrand model; completion. This is used to illustrate the rather unsatisfactory relationship between logic programming and logic shown by the wide variety of different declarative semantics proposed for logic programs.

Unsolvable problems for SLDF resolution

Abstract Shepherdson showed that there are maximal safe computation rules R m for SLDNF resolution, such that if a query succeeds under any safe rule it succeeds under R m , and if it fails under any safe rule it fails under R m . Later he showed that such maximal rules also get all possible answers, i.e., if a query succeeds with answer θ under any safe rule, then it succeeds with answer equivalent to θ under R m . The question was raised there, whether there were always recursive maximal rules. We answer this negatively. We also show the unsolvability of the problems of deciding whether a query leads to a dead end, whether the Clark completion comp( P ) of a program P is consistent, and whether a query which is known to be a logical consequence of the program succeeds in PROLOG.

Godel's second incompleteness theorem for Q

For the first Godel incompleteness theorem, the existence in a formal system of arithmetic L of a sentence which is neither provable nor refutable, all that is required of the formula Th( x ) of L used to express the notion ‘ x is the g.n. (godel number) of a theorem of L ’ is mere numeralwise correctness, i.e. that for a numeral n , Th( n ) is provable in L iff n is the g.n. of a theorem of L . It is well known that much more is needed for the second Godel incompleteness theorem, the unprovability in L of the formula Con = df ¬(∃ y, z )(Th( y ) ∧ Th( z ) ∧ neg( z, y )), which (if neg expresses negation) expresses the consistency of L . Conditions sufficient for this second theorem, more or less as stated by Hilbert-Bernays [1, p. 286] and elegantly formulated by Lob [2] may with a cavalier disregard for the distinction between use and mention be stated thus: The result of the first incompleteness theorem: there is a sentence G such that ⊢ G ↔ ¬Th G ), together with, if ⊢ A then ⊢Th A , if ⊢( A → B ) then ⊢(Th A → Th B ), ⊢(Th A → Th Th A ). On the other hand Feferman [3], Kreisel [4, p. 154] and Jeroslow [9] have given examples of systems and consistency formulae, based on numeralwise correct formulae Th( x ), which are provable within the system.

A Semantically Meaningful Characterization of Reducible Flowchart Schemes

A “scalar” flowchart scheme, i.e. one with a siugle begin “instruction” is reducible iff its underlying flowgraph is reducible in the sense of Cocke and Allen tir Hecht and IJllman. We characterize the class of reducible scalar flowchart schemes as the smallest class containing certain members and closed under certain operations (on and to flowchart schemes). These operations are “semantically meaningful*’ in the sense tha operations of the same form are meaningful for “the” functions (or partial functions) computed by interpreted flowchart schemes; moreover, the schemes and the functions “are related by a homomorphism.” By appropriately generalizing “flowgraph” to (possibly) several begins (i.e. entries) we obtain a class of reducible “vector” flowchart schemes which can be characterized in a manner analogous to the scalar case but involving simpler more basic operations (which are also semantically meaningful). A significant side effect of this semantic viewpoint is the treatment of multi-exit flowchart schemes on an equal footing with single exit ones.