Computational Paths. The Calculus of Equality
Language: English
Published by College Publications, 2026
- Softcover
- New
Condition: New
US$ 38.61
Quantity: 1 available
Add to basketSeller Inventory # KALAMO-9781848905153
- Title
- Computational Paths. The Calculus of Equality
- Author
- Freitas Ramos, Arthur; De Queiroz, Ruy J G B; Grisi De Oliveira, Anjolina
- Publisher
- College Publications
- Publication year
- 2026
- Condition
- New
- Binding
- Soft cover
- Language
- English
- ISBN 10
- 1848905157
- ISBN 13
- 9781848905153
When 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.
Part 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.
The 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.
Aimed 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.
"Synopsis" may belong to another edition of this title.
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