Termination of Context-Sensitive Rewriting with Built-In Numbers and Collection Data Structures

Abstract

Abstract. Context-sensitive rewriting is a restriction of rewriting that can be used to elegantly model declarative specification and programming languages such as Maude. Furthermore, it can be used to model lazy evaluation in functional languages such as Haskell. Building upon our previous work on an expressive and elegant class of rewrite systems (called CERSs) that contains built-in numbers and supports the use of collection data structures such as sets or multisets, we consider contextsensitive rewriting with CERSs in this paper. This integration results in a natural way for specifying algorithms in the rewriting framework. In order to prove termination of this kind of rewriting automatically, we develop a dependency pair framework for context-sensitive rewriting with CERSs, resulting in a flexible termination method that can be automated effectively. Several powerful termination techniques are developed within this framework. An implementation in the termination prover AProVE has been successfully evaluated on a large collection of examples.

Rewriting Induction + Linear Arithmetic = Decision Procedure

“…We are interested in developing an inductive proof method for more general classes of CERSs as defined in [14]. In contrast to Z-TRSs, these CERSs also support (nonfree) collection data structures such as sets or multisets and thus provide an even more expressive kind of term rewrite systems.…”

“…This class of TRSs can be seen as a simplified special case of the constrained equational rewrite systems (CERSs) from [14]. We assume familiarity with the notions and concepts from (many-sorted) term rewriting and refer to [5] for details.…”

“…While this is in general undecidable, the methods for proving termination of CERSs developed in [14] are applicable since Z-TRSs can be seen as a restricted kind of CERSs. These methods are based on the dependency pair approach and have been implemented in the termination tool AProVE.…”

“…In order to check for termination of R ∪ H as needed in the side condition of Expand, the implementation of the methods for proving termination of CERSs developed in [14] in the termination tool AProVE is used. Since that implementation is currently limited to Z-rules l → r ϕ with V(r) ∪ V(ϕ) ⊆ V(l), Sail2 imposes the same requirement on Z-TRSs and atomic conjectures.…”

“…While the time spent within the SMT solvers can probably be reduced by using the SMT solver via its API (since this would eliminates the overhead of calling an external tool), the remaining parts of the proof attempt still require much more time than checking whether a conjecture is a member of the class of conjectures whose inductive validity is decidable. A more promising direction for improving the performance of Sail2 is to shift the burden of proving termination from AProVE to a more optimized implementation of the termination methods developed in [14].…”