Abstract
The theory of one-step rewriting for a given rewrite system R and signature ε is the first-order theory of the following structure: Its universe consists of all ε-ground terms, and its only predicate is the relation “x rewrites to y in one step by R”. The structure contains no function symbols and no equality. We show that there is no algorithm deciding the ∃*∀*-fragment of this theory for an arbitrary rewrite system. The proof uses both non-linear and non-shallow rewrite rules.
As a refinement of the proof, we show that the ∃*∀*-fragment of the first-order theory of encompassment (reducibility by rewrite rules) together with one-step rewriting by the rule f(x) → g(x) is undecidable.
Supported by the Human Capital and Mobility Programme of the European Union, under the contracts SOL (CHRX-CT92-0053) and CONSOLE (CHRX-CT94-0495).
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Treinen, R. (1996). The first-order theory of one-step rewriting is undecidable. In: Ganzinger, H. (eds) Rewriting Techniques and Applications. RTA 1996. Lecture Notes in Computer Science, vol 1103. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61464-8_59
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DOI: https://doi.org/10.1007/3-540-61464-8_59
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