Abstract This paper explores how using commutativity can improve the efficiency and efficacy of algorithmic termination checking for concurrent programs. If a program run is terminating, one can conclude that all other runs equivalent to it up-to-commutativity are also terminating. Since reasoning about termination involves reasoning about infinite behaviours of the program, the equivalence class for a program run may include infinite words with lengths strictly larger than $$\omega $$ ω that capture the intuitive notion that some actions may soundly be postponed indefinitely. We propose a sound proof rule which exploits these as well as classic bounded commutativity in reasoning about termination, and devise a way of algorithmically implementing this sound proof rule. We present experimental results that demonstrate the effectiveness of this method in improving automated termination checking for concurrent programs.

We propose SE2GIS, a novel inductive recursion synthesis approach with the ability to both synthesize code and declare a problem unsolvable. SE2GIS combines a symbolic variant of counterexample-guided inductive synthesis (CEGIS) with a new dual inductive procedure, which focuses on proving a synthesis problem unsolvable rather than finding a solution for it. A vital component of this procedure is a new algorithm that produces a witness, a set of concrete assignments to relevant variables, as a proof that the synthesis instance is not solvable. Witnesses in the dual inductive procedure play the same role that solutions do in classic CEGIS; that is, they ensure progress. Given a reference function, invariants on the input recursive data types, and a target family of recursive functions, SE2GIS synthesizes an implementation in this family that is equivalent to the reference implementation, or declares the problem unsolvable and produces a witness for it. We demonstrate that SE2GIS is effective in both cases; that is, for interesting data types with complex invariants, it can synthesize non-trivial recursive functions or output witnesses that contain useful feedback for the user.