Sergei Soloviev. A complete axiom system for isomorphism of types in closed categories. In A. Voronkov, editor, Proceedings 4th Int. Conf. on Logic Programming and Automated Reasoning, LPAR'93, St. Petersburg, Russia, 13--20 July 1993, volume 698, pages 360--371. Springer-Verlag, Berlin, 1993.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... details [10, 3] A characterisation of type isomorphisms has been obtained for monomorphic type systems with various combinations of the unit, product, and arrow type constructors [29, 5, 4] as well as for ML style [11] and second order polymorphism [14] and for linear lambda calculus [30] and multiplicative linear logic [2] Type isomorphism and category theory. Type isomorphism in foundational theories of functional programming languages, like typed lambda calculi, can be studied by their associated categorical models. From this perspective, our investigations fall in the ....

....and arrow by exponentiation) are equal in the standard model of natural numbers. In these cases, type isomorphism (and numerical equality) is finitely axiomatisable and decidable; hence so is the equational theory of isomorphisms in cartesian closed categories. In the same vein, Soloviev [30], gave a complete axiomatisation of isomorphisms in symmetric monoidal closed categories, and Dosen and Petric [16] provided the arithmetic structure that exactly corresponds to these isomorphisms. The question has been open as to whether such correspondence was limited to the case of the ....

S. V. Soloviev. A complete axiom system for isomorphism of types in closed categories. In A. Voronkov, editor, Logic Programming and Automated Reasoning, 4th International Conference, volume 698 of Lecture Notes in Artificial Intelligence (subseries of LNCS), pages 360--371, St. Petersburg, Russia, 1993. Springer-Verlag.

....isomorphism of types corresponds to the isomorphism of objects in free SMC category, and can also be described as the isomorphism of types in the system of lambda calculus which corresponds to intuitionistic multiplicative linear logic. A description of this system can be found in [5] 6] [7], 8] In [7] it was shown that the subsystem of the axiom system above, consisting of the axioms 1) 6) where is understood as times and as 3 linear implication) with the same rules, defines an equivalence relation on types that coincides with the relation of linear isomorphism of types. ....

....of types corresponds to the isomorphism of objects in free SMC category, and can also be described as the isomorphism of types in the system of lambda calculus which corresponds to intuitionistic multiplicative linear logic. A description of this system can be found in [5] 6] 7] 8] In [7] it was shown that the subsystem of the axiom system above, consisting of the axioms 1) 6) where is understood as times and as 3 linear implication) with the same rules, defines an equivalence relation on types that coincides with the relation of linear isomorphism of types. Of course, the ....

[Article contains additional citation context not shown here]

S.V. Soloviev. A complete axiom system for isomorphism of types in closed categories. Lecture Notes in Artificial Intelligence, v. 698 (1993), 380-392.

.... of isomorphism of types in Cartesian Closed categories [27, 8] If we exclude Rules (Distrib ) Ident ) then the remaining axiom system, denoted T SMC , gives a sound and complete axiomatization of isomorphism (called linear isomorphism) of types in Symmetric Monoidal Closed categories [26]. Rittri [23, 24, 25] used both kinds of isomorphism in his 10 work on using types as search keys. The following table summarizes some decidability results for TCC and T SMC . Axioms Word problem Matching problem Uni cation problem TCC n 2 log(n) 9] NP hard, decidable [19] Undecidable [19] ....

Sergei Soloviev. A complete axiom system for isomorphism of types in closed categories. pages 380-392. Springer-Verlag (LNAI 698), 1993.

....this is a stronger requirement than the linear logical equivalence ffi Gamma Gammaffi. It is far from obvious that the five linear axioms in table 1 form a complete equational axiomatization of linear isomorphism, but I had the good luck to be able to contact Sergei Soloviev, who found a proof [24]. Instead of generating the isomorphisms that hold in any Cartesian closed category, the five axioms generate those that hold in all symmetric monoidal closed (SMC) categories, sometimes just called closed categories [13, section VII.7] The equational axioms in table 1 are decorated with ....

....: f n ; f Gamma1 n ) A consequence of this inductive definition is that whenever a function is in Linb, then so is its inverse. The set Linb depends on which type operators exist in our language. If the only type operators are Theta and , then Soloviev s proof of equational completeness [24] says that Linb contains exactly the bijections that exist between types that are isomorphic in every SMC category. But when we search functional libraries, we must allow all type operators that occur in the library. 8 3 Experiments with equational unification When we retrieve library functions ....

S. V. Soloviev. A complete axiom system for isomorphism of types in closed categories. In A. Voronkov, editor, Logic Programming and Automated Reasoning, 4th Int. Conf., St. Petersburg, Russia. Volume 698 of Lecture Notes in Artificial Intelligence, pages 360--371, Springer-Verlag, 1993. 17

....Monoidal Categories, and can be also described as the isomorphism of types in the system of lambda calculus which corresponds to 1 Provided X is free for Y in A, and Y 62 FTV (A) 2 Provided X 62 FTV (A) intuitionistic multiplicative linear logic. A description of this system can be found in [22]) In [22] it was shown that the axiom system consisting of the axioms 1, 2, 3, 5, and 7 defines an equivalence relation on types that coincides with the relation of linear isomorphism of types, and in [1] a very efficient algorithm for deciding equality of linear isomorphisms is provided; also, ....

....Categories, and can be also described as the isomorphism of types in the system of lambda calculus which corresponds to 1 Provided X is free for Y in A, and Y 62 FTV (A) 2 Provided X 62 FTV (A) intuitionistic multiplicative linear logic. A description of this system can be found in [22] In [22] it was shown that the axiom system consisting of the axioms 1, 2, 3, 5, and 7 defines an equivalence relation on types that coincides with the relation of linear isomorphism of types, and in [1] a very efficient algorithm for deciding equality of linear isomorphisms is provided; also, 12] ....

[Article contains additional citation context not shown here]

S. V. Soloviev. A complete axiom system for isomorphism of types in closed categories. In A. Voronkov, editor, Logic Programming and Automated Reasoning, 4th International Conference, volume 698 of Lecture Notes in Artificial Intelligence (subseries of LNCS), pages 360--371, St. Petersburg, Russia, 1993. Springer-Verlag.

....isomorphism of types corresponds to the isomorphism of objects in free SMC category, and can be also described as the isomorphism of types in the system of lambda calculus which corresponds to intuitionistic multiplicative linear logic. A description of this system can be found in [5] 6] [7], 8] In [7] it was shown that the subsystem of the axiom system above, consisting of the axioms 1) 6) where is understood as times and as linear implication) with the same rules, defines an equivalence relation on types that coincides with the relation of linear isomorphism of types. Of ....

....of types corresponds to the isomorphism of objects in free SMC category, and can be also described as the isomorphism of types in the system of lambda calculus which corresponds to intuitionistic multiplicative linear logic. A description of this system can be found in [5] 6] 7] 8] In [7] it was shown that the subsystem of the axiom system above, consisting of the axioms 1) 6) where is understood as times and as linear implication) with the same rules, defines an equivalence relation on types that coincides with the relation of linear isomorphism of types. Of course, the use ....

[Article contains additional citation context not shown here]

S.V. Soloviev. A complete axiom system for isomorphism of types in closed categories. - Lecture Notes in Artificial Intelligence, 698 (1993), 380-392.

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Sergei Soloviev. A complete axiom system for isomorphism of types in closed categories. In A. Voronkov, editor, Proceedings 4th Int. Conf. on Logic Programming and Automated Reasoning, LPAR'93, St. Petersburg, Russia, 13--20 July 1993, volume 698, pages 360--371. Springer-Verlag, Berlin, 1993.

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