Multiphase-Linear Ranking Functions and Their Relation to Recurrent Sets

Abstract

Multiphase ranking functions (MΦRFs) are tuples f 1 , . . . , f d of linear functions that are often used to prove termination of loops in which the computation progresses through a number of "phases". Our work provides new insights regarding such functions for loops described by a conjunction of linear constraints (Single-Path Constraint loops). The decision problem existence of a MΦRF asks to determine whether a given SLC loop admits a MΦRF; and the corresponding bounded decision problem restricts the search to MΦRFs of depth d, where the parameter d is part of the input. The algorithmic and complexity aspects of the bounded problem have been completely settled in a recent work. In this paper we make progress regarding the existence problem, without a given depth bound. In particular, we present an approach that reveals some important insights into the structure of these functions. Interestingly, it relates the problem of seeking MΦRFs to that of seeking recurrent sets (used to prove non-termination). It also helps in identifying classes of loops for which MΦRFs are sufficient. Our research has led to a new representation for single-path loops, the difference polyhedron replacing the customary transition polyhedron. This representation yields new insights on MΦRFs and SLC loops in general. For example, a result on bounded SLC loops becomes straightforward.

Proving Non-Termination via Loop Acceleration

“…We also tested with our tool AProVE [22], but excluded it as it uses a similar approach like T2, but finds fewer non-termination proofs. The remaining participants of the respective category of TermComp, Ctrl [29] and iRankFinder [2,16], cannot prove non-termination. 7 We used the TermComp '19 version of VeryMax and the TermComp '17 version of T2 (as T2 has not been developed further since 2017).…”

“…A term f(n) where n ⊂ Z is a configuration. An integer program T induces a relation → T on configurations: We have s − → T t if there is an α ∈ T and a model σ of guard α such that V(α) ⊆ dom(σ), σ(lhs α ) = s, and σ(rhs α ) = t. 2 Then we say that s evaluates to t. As usual, − → * T is the transitive-reflexive closure of − → T .…”

“…Hence, a non-empty recurrent set that contains an initial configuration witnesses non-termination. There are many techniques to find recurrent sets for simple loops [2], lassos [7,13,26,40], or more complex sub-programs [32]. Essentially, our invariant inference technique of Sect.…”

“…iRankFinder can prove non-termination of simple loops[2], but according to its authors it cannot yet check reachability of diverging configurations 8. Ultimate and AProVE were also the two most powerful tools in the "termination" category for C programs at SV-COMP '19[3].…”

Decision Tree Learning in CEGIS-Based Termination Analysis

“…In CEGIS, synthesizers receive a set of individual examples that synthesizers can use in various creative and speculative manners (such as machine learning). In contrast, in other methods such as [5][6][7][8]24,27], synthesizers receive logical constraints that are much more binding.…”