Pattern matching as cut elimination

• Published in: Theoretical computer science Vol. 323; no. 1; pp. 71 - 127
• Main Authors: Cerrito, Serenella, Kesner, Delia
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 14.09.2004; Elsevier
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Artificial intelligence; Computer Science; Computer science; control theory; systems; Exact sciences and technology; General logic; Logic and foundations; Mathematical logic, foundations, set theory; Mathematics; Pattern recognition. Digital image processing. Computational geometry; Programming Languages; Proof theory and constructive mathematics; Sciences and techniques of general use; Theoretical computing; Cut elimination; Intuitionistic logic; Computer theory; Substitution; Confluence; Isomorphism; Pattern matching
• ISSN: 0304-3975; 1879-2294
• DOI: 10.1016/j.tcs.2004.03.032

We present a typed pattern calculus with explicit pattern matching and explicit substitutions, where both the typing rules and the reduction rules are modeled on the same logical proof system, namely Gentzen sequent calculus for intuitionistic minimal logic. Our calculus is inspired by the Curry–Howard Isomorphism, in the sense that types, both for patterns and terms, correspond to propositions, terms correspond to proofs, and term reduction corresponds to sequences of sequent proof normalization steps performed by cut elimination. The calculus enjoys subject reduction, confluence, preservation of strong normalization w.r.t a system with meta-level substitutions and strong normalization for well-typed terms. As a consequence, it can be seen as an implementation calculus for functional formalisms defined with meta-level operations for pattern matching and substitutions. This work is a revised and extended version of Cerrito and Kesner (14th Annual IEEE Symposium on Logic in Computer Science (LICS), IEEE Computer Society Press, Silver Spring, MD, 1999, pp. 98–108).