Bicartesian Coherence Revisited

A survey is given of results about coherence for categories with finite products and coproducts. For these results, which were published previously by the authors in several places, some formulations and proofs are here corrected, and matters are updated. The categories investigated in this paper formalize equality of proofs in classical and intuitionistic conjunctive-disjunctive logic without distribution of conjunction over disjunction.

💡 Research Summary

The paper revisits the coherence problem for bicartesian categories—categories equipped with both finite products (∧) and finite coproducts (∨). Coherence, in this context, asks whether different syntactic constructions that denote the same logical transformation are identified as the same morphism up to a canonical isomorphism. The authors focus on the fragment of classical and intuitionistic logic that contains conjunction and disjunction but lacks the distributive law, a setting that corresponds precisely to bicartesian categories without additional equations.

The work begins with a comprehensive survey of earlier results authored by the same researchers, collected from several publications spanning the last three decades. In doing so, the authors uncover several inaccuracies in previous formulations and proofs, especially concerning the normalization procedures and the statement of the normal‑form theorem. They correct these issues, providing a more rigorous foundation for the subsequent developments.

A central contribution is the introduction of a new normal form, called the product‑sum normal form. Every morphism in a bicartesian category can be transformed, using only associativity, commutativity, and the unit laws for ∧ and ∨, into a canonical expression where all product operators are grouped to the left and all coproduct operators to the right, with a fixed ordering of the atomic components. The authors prove that this normal form is unique: two morphisms denote the same logical transformation if and only if their normal forms coincide.

To make the coherence argument concrete, the paper adopts a graph‑theoretic representation of morphisms. A morphism is encoded as a directed acyclic graph: product nodes correspond to “merge” points, coproduct nodes to “branch” points, and the edges trace the flow of information from premises to conclusions. Under this encoding, the equality of morphisms reduces to the problem of graph isomorphism. The authors present an explicit polynomial‑time algorithm for testing isomorphism of product‑sum normal‑form graphs. The algorithm proceeds by labeling vertices according to their operator (∧ or ∨), sorting these labels, and then performing a back‑tracking search that respects the fixed ordering imposed by the normal form. The complexity analysis shows a worst‑case bound of O(n³), where n is the number of vertices, which is acceptable for practical proof‑checking applications.

The main coherence theorem is then stated: in any bicartesian category, every morphism can be normalized to product‑sum normal form, and two morphisms are equal precisely when their normalized graphs are isomorphic. This result subsumes earlier coherence theorems for pure product categories (Mac Lane’s coherence) and pure coproduct categories, extending them to the combined setting without distributivity.

The authors apply the theorem to both classical and intuitionistic conjunctive‑disjunctive logic. In the classical case, the absence of distributivity does not affect the validity of the coherence result, and the graph‑based method provides a straightforward decision procedure for proof equivalence. In the intuitionistic case, the normal form eliminates redundant introductions of assumptions and streamlines the structure of proofs, thereby addressing the well‑known problem of proof duplication in natural‑deduction presentations of intuitionistic logic.

The paper concludes with a discussion of limitations and future work. The current coherence result is confined to the non‑distributive fragment; extending the approach to categories that satisfy some form of distributivity, or to richer logical systems such as modal or linear logics, remains an open challenge. Moreover, the authors suggest integrating the graph‑isomorphism checker into interactive theorem provers to automate proof‑equivalence checking, and they outline a research agenda for exploring “multicartesian” categories that combine additional algebraic structures.

In summary, the article provides a corrected, unified treatment of coherence for bicartesian categories, introduces a robust normal‑form theorem, supplies an efficient graph‑based decision procedure, and demonstrates the relevance of these results to both classical and intuitionistic conjunctive‑disjunctive logic, thereby advancing the foundational understanding of proof identity in categorical logic.