Models for Logics and Conditional Constraints in Automated Proofs of Termination

Abstract

Abstract. Reasoning about termination of declarative programs, which are described by means of a computational logic, requires the definition of appropriate abstractions as semantic models of the logic, and also handling the conditional constraints which are often obtained. The formal treatment of such constraints in automated proofs, often using numeric interpretations and (arithmetic) constraint solving, can greatly benefit from appropriate techniques to deal with the conditional (in)equations at stake. Existing results from linear algebra or real algebraic geometry are useful to deal with them but have received only scant attention to date. We investigate the definition and use of numeric models for logics and the resolution of linear and algebraic conditional constraints as unifying techniques for proving termination of declarative programs.

Use of Logical Models for Proving Operational Termination in General Logics

“…This observation was a main motivation to develop the idea of convex polytopic domain [12] as a sufficiently simple but flexible approach to obtain a variety of domains that can be used in proofs of termination and which are amenable for automation [10]. The research in this paper closes some gaps left during these developments and provides a basis for the implementation of P RP in the OT Framework by means of the automatic generation of logical models for order-sorted first-order theories.…”

“…In [10] we have shown that convex domains [12] provide an appropriate basis to the automatic definition of algebras and structures that can be used in program analysis with order-sorted first-order specifications. In the following definition, vectors x, y ∈ R n are compared using the coordinate-wise extension of the ordering ≥ among numbers which, by abuse, we denote using ≥ as well:…”

“…In [6] Order-Sorted First-Order Logic (OS-FOL) is proposed as a sufficiently general and expressive framework to represent declarative programs, semantics of programming languages, and program properties (see Section 3). In [10] we show how to systematically generate models for OS-FOL theories by using the convex polytopic domains introduced in [12]. In Section 4 we extend the work in [10] to generate appropriate interpretations of predicate symbols that can be then used to synthesize a model for a given OS-FOL theory S.…”

Synthesis of models for order-sorted first-order theories using linear algebra and constraint solving

“…The resolution of our running example (Example 6) shows that flexibility in the definition of domains A s for sorts s ∈ S is an asset: we have simultaneously used (due to the presence of sorts) an infinite domain like N (which is typical in termination proofs) and the finite domain {0}. In order to provide an appropriate computational basis to the automatic definition of algebras and structures that can be used in program analysis with order-sorted first-order specifications, we follow [11] and focus on domains that are obtained as the solution of polynomial and specially linear constraints. In Definition 3, vectors x, y ∈ R n are compared using the coordinate-wise extension of the ordering ≥ among numbers (by abuse, we use the same symbol): x = (x 1 , .…”

“…We also show how to transform an OS-FOL theory S into a derived parametric theory S ♯ of linear arithmetic where appropriate algorithms and constraint solving techniques can be used to give value to the parameters thus synthesizing a model of S . The convex domains introduced in [11] provide appropriate means for this. They can be used to define bounded and unbounded domains for the sorts in the OS signature.…”