{"product_id":"computational-paths-the-calculus-of-equality-9781848905153","title":"Computational Paths. The Calculus of Equality","description":"\u003cp\u003e • Author(s): Arthur Freitas Ramos | Ruy J. G. B. de Queiroz | Anjolina Grisi de Oliveira\u003cbr\u003e • Publisher: College Publications\u003cbr\u003e • Publisher Imprint: College Publications\u003cbr\u003e • BISAC: Logic\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e\u003cp\u003eWhen are two proofs of the same proposition equal? This book answers through computational paths - explicit syntactic witnesses that record how one proof-term rewrites into another. Continuing the programme begun in The Functional Interpretation of Logical Deduction (2011), it extends the Curry-Howard correspondence and the tradition of labelled deductive systems into a rewrite calculus of proof equalities.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003ePart I develops a 77-rule rewrite system on paths, proves it confluent and terminating, and derives decidable path equivalence from unique normal forms. Part II uncovers the higher-dimensional structure latent in this calculus: paths assemble into a weak ω-groupoid, with 2-cells as derivations between paths, 3-cells as coherences between derivations, and the Eckmann-Hilton argument recovered directly from rewrite combinatorics. Part III turns outward - to path induction and the J-eliminator, the failure of UIP and the resulting proof-relevance, fundamental groupoids of combinatorial spaces, and the framework's relationship to Homotopy Type Theory, category theory, and higher algebra.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003eThe approach shares conceptual ground with HoTT but diverges in its commitments: where HoTT posits univalence and higher inductive types, computational paths offer explicit rewrite witnesses and effective normalisation, with path equivalence rendered decidable rather than postulated. A companion Lean 4 formalization machine-checks the core constructions and major theorems.\u003c\/p\u003e\u003cp\u003e \u003c\/p\u003e\u003cp\u003eAimed at logicians, type theorists, and mathematicians concerned with the computational content of equality, the volume is self-contained - appendices cover term rewriting and higher groupoids - while charting a research programme that reaches toward dependent types, directed rewriting, and homotopical semantics.\u003c\/p\u003e\u003cp\u003e\u003c\/p\u003e","brand":"College Publications","offers":[{"title":"Paperback","offer_id":47892787888279,"sku":"9781848905153","price":3450.0,"currency_code":"INR","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9781848905153.webp?v=1781188961","url":"https:\/\/atlanticbooks.com\/products\/computational-paths-the-calculus-of-equality-9781848905153","provider":"Atlantic Books","version":"1.0","type":"link"}