Policy Iteration-Based Conditional Termination and Ranking Functions

Abstract

Termination analyzers generally synthesize ranking functions or relations, which represent checkable proofs of their results. In [23], we proposed an approach for conditional termination analysis based on abstract fixpoint computation by policy iteration. This method is not based on ranking functions and does not directly provide a ranking relation, which makes the comparison with existing approaches difficult. In this paper we study the relationships between our approach and ranking functions and relations, focusing on extensions of linear ranking functions. We show that it can work on programs admitting a specific kind of segmented ranking functions, and that the results can be checked by the construction of a disjunctive ranking relation. Experimental results show the interest of this approach.

Proving Termination Through Conditional Termination

“…A first technique combined constraint-based methods for finding potential ranking functions with quantifier elimination [15] to infer preconditions. More recently, policy iteration-based methods [34], backwards reasoning in the abstract interpretation framework [45] and an adaptation of conflict-driven learning from satisfiability solving [17] have been adapted to find conditions for termination. Our algorithm CondTerm differs in its relative simplicity (by delegating the majority of the work to a constraint solver), and our procedure Term could combine it with or replace it by other approaches.…”

Bit-Precise Procedure-Modular Termination Analysis

“…We consider our template-based abstract interpretation that automatically synthesises abstract transformers more amenable to refinement techniques than classical abstract interpretations where abstract transformers are implemented manually. Sufficient preconditions to termination Conditional termination has recently attracted increased interest [Cook et al 2008;Bozga et al 2012;Ganty and Genaim 2013;Massé 2012;Massé 2014;Urban and Miné 2014;Urban and Miné 2017]. In this paper, we compute sufficient preconditions, i.e., under-approximating preconditions to termination via computing over-approximating preconditions to potential non-termination.…”

“…While Gupta et al [2008] dynamically enumerate lasso-shaped candidate paths for counterexamples, and then statically prove their feasibility, prove non-termination via reduction to safety proving and Le et al [2015] use bi-abduction to construct summaries of terminating and non-terminating behaviours for each method. Massé [2014] uses policy iteration techniques in order to compute an over-approximation of the potentially non-terminating states. In order to prove both termination and non-termination, Harris et al [2010] compose several program analyses (termination provers for multi-path loops, non-termination provers for cycles, and safety provers).…”

Inference of ranking functions for proving temporal properties by abstract interpretation

“…In [39], preconditions are acquired in order to strengthen a termination argument, while our preconditions are inherently obtained from the inferred ranking functions as the set of program states for which the ranking function is defined. Thus, our preconditions are derived by under-approximation of the set of terminating states as opposed to the approaches presented in [40,41] where the preconditions are derived by (complementing an) over-approximation of the non-terminating states.…”