Multi-dimensional Rankings, Program Termination, and Complexity Bounds of Flowchart Programs

Abstract

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Better Termination Proving through Cooperation

“…in Cooperating-T2 vs. AProVE we see that many results cannot be obtained with the invariant true, even though AProVE also uses modern rank function synthesis algorithms (e.g. [2]). Furthermore, the result of AProVE+Interproc indicates that an eager search for invariants in a preprocessing step is not a suitable solution to this problem, as this leads to more timeouts.…”

“…Based on experience with the two tools we expect that ARMC and HSF will have comparable results when proving termination [38]. -AProVE [21], a termination prover based on the dependency pair framework [3,22,27] and including an implementation of the rank function synthesisà la Alias et al [2]. AProVE does not generate invariants on demand and hence always uses the supporting invariant true.…”

“…In our setting, we need to find rank functions for the SCC contexts of counterexamples in the termination subgraph, which might involve transitions over several program points. For this purpose we use the lexicographic rank function synthesis due to Alias et al [2] to find linear rank functions for a set of transitions using possibly several program locations.…”

“…and calls to a rank function synthesis tool (e.g. [2], [7], [8], [35], etc.). In this setting a candidate termination argument is iteratively constructed.…”

“…For example, we adapt a program simplification strategy from the dependency pair framework [3,22,27] to our shared graph as a way of recording lemmas during the proof search. We use a recently developed tech-i := 0; while i < n do j := 0; while j ≤ i do j := j + 1; done i := i + 1; done nique for efficient rank function synthesis with multiple control-flow locations [2]. Finally, we build upon a recently developed iterative method for finding lexicographic rank functions [16].…”