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Formalizing the Confluence of Orthogonal Rewriting Systems

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 Added by EPTCS
 Publication date 2013
and research's language is English




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Orthogonality is a discipline of programming that in a syntactic manner guarantees determinism of functional specifications. Essentially, orthogonality avoids, on the one side, the inherent ambiguity of non determinism, prohibiting the existence of different rules that specify the same function and that may apply simultaneously (non-ambiguity), and, on the other side, it eliminates the possibility of occurrence of repetitions of variables in the left-hand side of these rules (left linearity). In the theory of term rewriting systems (TRSs) determinism is captured by the well-known property of confluence, that basically states that whenever different computations or simplifications from a term are possible, the computed answers should coincide. Although the proofs are technically elaborated, confluence is well-known to be a consequence of orthogonality. Thus, orthogonality is an important mathematical discipline intrinsic to the specification of recursive functions that is naturally applied in functional programming and specification. Starting from a formalization of the theory of TRSs in the proof assistant PVS, this work describes how confluence of orthogonal TRSs has been formalized, based on axiomatizations of properties of rules, positions and substitutions involved in parallel steps of reduction, in this proof assistant. Proofs for some similar but restricted properties such as the property of confluence of non-ambiguous and (left and right) linear TRSs have been fully formalized.



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Convergent rewriting systems on algebraic structures give methods to solve decision problems, to prove coherence results, and to compute homological invariants. These methods are based on higher-dimensional extensions of the critical branching lemma that characterizes local confluence from confluence of the critical branchings. The analysis of local confluence of rewriting systems on algebraic structures, such as groups or linear algebras, is complicated because of the underlying algebraic axioms, and in some situations, local confluence properties require additional termination conditions. This article introduces the structure of algebraic polygraph modulo that formalizes the interaction between the rules of an algebraic rewriting system and the inherent algebraic axioms, and we show a critical branching lemma for algebraic polygraphs. We deduce from this result a critical branching lemma for rewriting systems on algebraic objects whose axioms are specified by convergent modulo rewriting systems. We illustrate our constructions for string, linear, and group rewriting systems.
125 - Joerg Endrullis 2014
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104 - Moa Johansson 2021
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