Synthesis for Polynomial Lasso Programs

Abstract

1 This is an error-corrected version of the original thesis, updated last on July 11, 2018. AbstractThe scope of this work is the constraint-based synthesis of termination arguments for the restricted class of programs called linear lasso programs. A termination argument consists of a ranking function as well as a set of supporting invariants.We extend existing methods in several ways. First, we use Motzkin's Transposition Theorem instead of Farkas' Lemma. This allows us to consider linear lasso programs that can additionally contain strict inequalities. Existing methods are restricted to non-strict inequalities and equalities.Second, we consider several kinds of ranking functions: affine-linear, piecewise and lexicographic ranking functions. Moreover, we present a novel kind of ranking function called multiphase ranking function which proceeds through a fixed number of phases such that for each phase, there is an affine-linear ranking function. As an abstraction to the synthesis of specific ranking functions, we introduce the notion ranking function template. This enables us to handle all ranking functions in a unified way.Our method relies on non-linear algebraic constraint solving as a subroutine which is known to scale poorly to large problems. As a mitigation we formalize an assessment of the difficulty of our constraints and present an argument why they are of an easier kind than general non-linear constraints.We prove our method to be complete: if there is a termination argument of the form specified by the given ranking function template with a fixed number of affine-linear supporting invariants, then our method will find a termination argument.To our knowledge, the approach we propose is the most powerful technique of synthesis-based discovery of termination arguments for linear lasso programs and encompasses and enhances several methods having been proposed thus far [4,18,27].

A Nonlinear Real Arithmetic Fragment

“…Our procedure also works well on problems coming from template-based verification and synthesis [23,15] (see some nonlinear benchmark examples in [14]) and hybrid systems [21,27,11,22,25]. In fact, using a preliminary (and rather naive) implementation of our procedure (in Python, using floats and not using algebraic numbers) with some heuristics for handling cases where eigen-condition fails, we were able to solve all the nonlinear examples in [14] in time competitive with Z3 [13,18] (and faster than Z3 on a couple of problems). On the nonlinear encodings of SAT benchmarks, we are competitive with Z3's nonlinear solver on small problems, but much worse when problem size is larger -this is perhaps because our implementation does not learn from conflicts.…”

Ranking Templates for Linear Loops

“…For simplicity of presentation, we assume the loop program LOOP does not contain any strict inequalities, and the ranking template T does not contain any non-strict inequalities; however, recall that we are using Motzkin's theorem instead of Farkas' lemma precisely to lift this restriction. For the fully general constraints, see [Lei13,Ch. 5].…”

“…Acknowledgements. We wish to thank Samir Genaim for pointing our an error in the conference version of Theorem 4.5, and Amir M. Ben-Amram for detailed comments on the Master's thesis [Lei13] from which this paper was derived. Moreover, we thank Andreas Podelski for his detailed feedback and helpful suggestions.…”

Ranking Templates for Linear Loops

“…Now, Motzkin's Transposition Theorem will transform the constraint (19) into an equivalent existentially quantified constraint. This ∃-constraint is then checked for satisfiability.…”