Linear Ranking for Linear Lasso Programs

Abstract

The general setting of this work is the constraint-based synthesis of termination arguments. We consider a restricted class of programs called lasso programs. The termination argument for a lasso program is a pair of a ranking function and an invariant. We present theto the best of our knowledge-first method to synthesize termination arguments for lasso programs that uses linear arithmetic. We prove a completeness theorem. The completeness theorem establishes that, even though we use only linear (as opposed to non-linear) constraint solving, we are able to compute termination arguments in several interesting cases. The key to our method lies in a constraint transformation that replaces a disjunction by a sum.⋆ The final publication is available at link.springer.com.

Bit-Precise Procedure-Modular Termination Analysis

“…As opposed to the synthesis of termination arguments for linear programs over integers (rationals) Lee et al 2012;Ben-Amram and Genaim 2013;Podelski and Rybalchenko 2004;Heizmann et al 2013;Bradley et al 2005a;Cook et al 2013;Ben-Amram and Genaim 2014;Gonnord et al 2015], this subclass of termination analyses is substantially less covered. Moreover, construct a termination argument as a disjunction of ranking functions for paths, which has the drawback of having to consider the transitive closure of the transition relation when checking for disjunctive well-foundedness of the termination argument.…”

Ranking Templates for Linear Loops

“…See Figure 3 for a specification of our method in pseudocode. Following related approaches [2,5,6,7,14,20,24,25], we transform the ∃∀-constraint (2) into an ∃-constraint. This transformation makes the constraint more easily solvable because it reduces the number of non-linear operations in the constraint: every application of an affine-linear function symbol f corresponds to a non-linear term s T f x + t f .…”

“…Related approaches invoke Farkas' Lemma for the transformation into ∃-constraints [2,5,6,7,14,20,24,25]. The piecewise and the lexicographic ranking template contain both strict and non-strict inequalities, yet only non-strict inequalities can be transformed using Farkas' Lemma.…”

“…In contrast to some related methods [14,20] the constraint we generate is not linear, but rather a nonlinear algebraic constraint. Theoretically, this constraint can be decided in exponential time [10].…”

“…These consist of a linear loop program and a program stem, both of which are specified by boolean combinations of affine-linear inequalities over the program variables. Our method can be extended to linear lasso programs through the addition of affine-linear inductive invariants, analogously to related approaches [5,7,14,25].…”