Well-Definedness of Streams by Termination

Abstract

Abstract. Streams are infinite sequences over a given data type. A stream specification is a set of equations intended to define a stream. We propose a transformation from such a stream specification to a TRS in such a way that termination of the resulting TRS implies that the stream specification admits a unique solution. As a consequence, proving such well-definedness of several interesting stream specifications can be done fully automatically using present powerful tools for proving TRS termination.

Lazy productivity via termination

“…In [52,53], Zantema defines 'proper' stream specifications. These are similar to shallow tree specifications, but differ in two aspects:…”

A Tool Proving Well-Definedness of Streams Using Termination Tools

“…From [5] we recall the two variants of the main theorem. In [5] it is shown by an example that this distinction is essential: uniqueness of stream functions does not hold if the stream specification is data dependent, that is, left hand sides contain symbols 0 and 1.…”

“…In [5] it is shown by an example that this distinction is essential: uniqueness of stream functions does not hold if the stream specification is data dependent, that is, left hand sides contain symbols 0 and 1. Moreover, in [5] it has been shown that the technique is not complete: some fix-point specifications are well-defined while the observational variant is non-terminating.…”

“…If so, we call the stream specification well-defined. For this question in [5] a technique has been proposed in which the stream specification S is transformed to its observational variant Obs(S), being a term rewrite system (TRS). The main result of [5] states that if the TRS Obs(S) is terminating, then the stream specification S admits a unique solution.…”

“…For this question in [5] a technique has been proposed in which the stream specification S is transformed to its observational variant Obs(S), being a term rewrite system (TRS). The main result of [5] states that if the TRS Obs(S) is terminating, then the stream specification S admits a unique solution. As for proving termination of TRSs automatically during the last ten years strong tools have been developed like AProVE [2] and TTT2 [3], our approach to prove welldefinedness of a stream specification S simply consists of proving termination of Obs(S) by a tool like AProVE or TTT2.…”