Logic + control: On program construction and verification

Abstract

This paper presents an example of formal reasoning about the semantics of a Prolog program of practical importance (the SAT solver of Howe and King). The program is treated as a definite clause logic program with added control. The logic program is constructed by means of stepwise refinement, hand in hand with its correctness and completeness proofs. The proofs are declarative -they do not refer to any operational semantics. Each step of the logic program construction follows a systematic approach to constructing programs which are provably correct and complete. We also prove that correctness and completeness of the logic program is preserved in the final Prolog program. Additionally, we prove termination, occur-check freedom and non-floundering.Our example shows how dealing with "logic" and with "control" can be separated. Most of the proofs can be done at the "logic" level, abstracting from any operational semantics.The example employs approximate specifications; they are crucial in simplifying reasoning about logic programs. It also shows that the paradigm of semantics-preserving program transformations may be not sufficient. We suggest considering transformations which preserve correctness and completeness with respect to an approximate specification.

Backjumping is Exception Handling

“…However, catch and throw provide a more general solution, not requiring search to be completely restarted, and there seems no reason why this strategy cannot extend to SMT solving (Robbins et al 2015) by another instantiation of the template. It has been recently shown (Drabent 2018) that the vanilla SAT solver of Fig. 4 can be understood as a logic program with added control, however reasoning about the correctness of learnt clauses is more challenging still, a research problem that is not exclusive to logic programming.…”

On Correctness and Completeness of an n Queens Program

“…In this paper, the treatment of specifications and reasoning about correctness and completeness follows that of Drabent (2016); missing proofs and further explanations can be found there. For further discussion, examples and references, see also Drabent (2018) and Drabent and Mi lkowska (2005).…”

“…Our construction of the approximate specification exemplifies a general pattern (Drabent 2016;2018). Some ground atoms may be irrelevant for the program properties On Correctness and Completeness of an n Queens Program…”

A Note on Occur-Check

“…This may be not sufficient when the occur-check depends on other features of the terms in argument positions. For instance, in the SAT-solver of Howe and King [HK12] an argument is a non-linear list of lists of pairs, and occur-check freeness depends on the first element of each pair being ground [Dra18]. In such cases our methods fail, and some analysis of the queries in SLD-trees is needed instead.…”