From Big-Step to Small-Step Semantics and Back with Interpreter Specialisation

Abstract

We investigate representations of imperative programs as constrained Horn clauses. Starting from operational semantics transition rules, we proceed by writing interpreters as constrained Horn clause programs directly encoding the rules. We then specialise an interpreter with respect to a given source program to achieve a compilation of the source language to Horn clauses (an instance of the first Futamura projection). The process is described in detail for an interpreter for a subset of C, directly encoding the rules of big-step operational semantics for C. A similar translation based on small-step semantics could be carried out, but we show an approach to obtaining a small-step representation using a linear interpreter for big-step Horn clauses. This interpreter is again specialised to achieve the translation from big-step to small-step style. The linear small-step program can be transformed back to a big-step non-linear program using a third interpreter. A regular path expression is computed for the linear program using Tarjan's algorithm, and this regular expression then guides an interpreter to compute a program path. The transformation is realised by specialisation of the path interpreter. In all of the transformation phases, we use an established partial evaluator and exploit standard logic program transformation to remove redundant data structures and arguments in predicates and rename predicates to make clear their link to statements in the original source program.

Regular Path Clauses and Their Application in Solving Loops

“…As another limitation, our approach takes a multipath loop and turns it into nested single-path loops. The result of such a transformation using regular expressions is linear in the number of paths, since our approach produces one path predicate for each distinct subexpression of the regular path expression [12], and the number of distinct subexpressions in the transformed expressions is linear in the number of paths. However, transforming a given program into a multi-path loop expression of form (e 1 + .…”

“…A well known algorithm by Tarjan [33] computes a regular path expression describing all paths in the CFG from a designated entry node to an exit node. We previously developed an interpreter that is guided by a regular path expression [12], and partially evaluated it with respect to a set of linear CHCs P. Here, we outline the idea behind the interpreter, which is that given a regular expression e, each subexpression of e defines a set of subpaths, which are composed to give the overall paths. These path predicates are defined by non-linear CHCs even though the CFG is defined from a set of linear CHCs.…”

“…So is not the intention to analyse the computational cost of the path program or its termination behaviour. The only property of the path clauses that we exploit in Section 4 is that path predicates capture the relationship between the values of variables at the start and end of derivations in P. The notion of path clauses is related to the big-step semantics [12] and path predicaes are sometimes called summary predicates.…”

“…We present an approach to solving such loops formulated in terms of constrained Horn clauses (CHCs), which are capable of expressing the semantics of imperative programs [28,17,24,13,14,16,6,21,12]. Clauses are assumed to have numerical variables (of infinite precision); we assume that variables values have been abstracted with respect to their numerical sizes, by performing a program transformation that uses the desired metrics (such as list length, term depth, term size, etc.…”

Analysis and Transformation of Constrained Horn Clauses for Program Verification

“…Derivation of small-step CHCs from big-step CHCs, and vice versa, can also be defined Gallagher et al (2020). Big-step and small-step styles can be mixed; for example, a procedure call in the small-step style can be defined as a single step that completely executes the procedure body, such as is done in De Angelis et al (2017b).…”