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**On the unification problem for Cartesian closed categories.**
*(English)*
Zbl 0882.03044

Summary: Cartesian closed categories (CCCs) have played and continue to play an important role in the study of the semantics of programming languages. An axiomatization of the isomorphisms which hold in all Cartesian closed categories discovered independently by Soloviev and Bruce, Di Cosmo and Longo leads to seven equalities. We show that the unification problem for this theory is undecidable, thus settling an open question. We also show that an important subcase, namely unification modulo the linear isomorphisms, is NP-complete. Furthermore, the problem of matching in CCCs is NP-complete when the subject term is irreducible.

CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by Rittri. It also has potential applications to the problem of polymorphic type inference and polymorphic higher-order unification, which in turn is relevant to theorem proving and logic programming.

CCC-matching and unification form the basis for an elegant and practical solution to the problem of retrieving functions from a library indexed by types investigated by Rittri. It also has potential applications to the problem of polymorphic type inference and polymorphic higher-order unification, which in turn is relevant to theorem proving and logic programming.

### MSC:

03D35 | Undecidability and degrees of sets of sentences |

18D15 | Closed categories (closed monoidal and Cartesian closed categories, etc.) |

68Q55 | Semantics in the theory of computing |

68Q25 | Analysis of algorithms and problem complexity |

### Keywords:

undecidability; Cartesian closed categories; semantics of programming languages; unification problem; unification modulo the linear isomorphisms; NP-complete; matching
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\textit{P. Narendran} et al., J. Symb. Log. 62, No. 2, 636--647 (1997; Zbl 0882.03044)

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