Robert L. Constable and Karl Crary. Computational complexity and induction for partial computable functions in type theory. In Wilfried Sieg, Richard Sommer, and Carolyn Talcott, editors, Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman, Lecture Notes in Logic, pages 166--183. Association for Symbolic Logic, 2001.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... :A] always holds. Note that we can not prove that A :A is valid even using a classical meta theory, because there is no uniform witness for for A :A. Corollary 5.2. The rule (4. 3) and its equivalents) is consistent with the NuPRL type theory containing the theory of partial functions [7]. Note however that the rule of excluded middle A :A is known to be inconsistent with the theory of [7] In particular, in that theory we can prove that there exists an undecidable proposition. That is, for some P the following is provable: 8n : N:P (n) P (n) 5.1) Therefore even using ....

....because there is no uniform witness for for A :A. Corollary 5.2. The rule (4. 3) and its equivalents) is consistent with the NuPRL type theory containing the theory of partial functions [7] Note however that the rule of excluded middle A :A is known to be inconsistent with the theory of [7]. In particular, in that theory we can prove that there exists an undecidable proposition. That is, for some P the following is provable: 8n : N:P (n) P (n) 5.1) Therefore even using rule (4.3) we can not prove that [8n : N:P (n) P (n) which would contradict (5.1) But we can prove ....

Robert L. Constable and Karl Crary. Computational complexity and induction for partial computable functions in type theory. In Preprint, 1998.

.... system [deB70] and by Scott s Constructive Validity paper [Sco70] The extensions include the quotient type [CZ84] the set type [Con83, Con85] and the inductive type [CM85, Men88] 3 Recently, rules for the intersection type have been added, as have rules for a newly developed partial type [CC98] which allows for reasoning about partial functions in type theory. In France, Coquand and Huet designed the Calculus of Constructions [CH85, CH88a] a higher order impredicative constructive type system based on Girard s system F [Gir86] The Coq system [DFH 93a, CCF 95] provides a ....

Robert L. Constable and Karl Crary. Computational complexity and induction for partial computable functions in type theory. In Preprint, 1998.

No context found.

Robert L. Constable and Karl Crary. Computational complexity and induction for partial computable functions in type theory. In Preprint, 1998.

....computation is another active area related to this article [12, 69, 72, 73, 74] These topics are covered also in Schwichtenberg [13] and in the articles of Jones [70] Schwichtenberg [98] and Wainer [87] in this book. The work reported here is new and based largely on Constable and Crary [38] and Benzinger [14] as well as examples from Kreitz and Pientka [90] One interesting application of the resource indexed types is to define types like Parikh s feasible numbers [89] numbers that may be computed in a reasonable time. Benzinger [14] shows another application. Acknowledgements ....

Robert L. Constable and Karl Crary. Computational complexity and induction for partial computable functions in type theory. In W. Sieg, R. Sommer, and C. Talcott, editors, Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman, Lecture Notes in Logic, pages 166--183. Association for Symbolic Logic,

....do this by adding to the type theory representations for all terms, and by adding rules for reasoning intensionally about those representations and for linking those representations back to the terms they represent. Mechanisms for accomplishing this are discussed in detail in Constable and Crary [23]. 4.3.5 Other Contributions The version of the Nuprl type theory presented in this chapter makes a few minor contributions other than those contributions relating to partial types: ffl By adding a primitive subtyping assertion, subtyping becomes negatable (recall Section 4.1.4) This makes it ....

....from discoveries outside the logic. Of the Nuprl proof rules of Appendix B, 84 rules (not quite half) deal with admissibility. It would be preferable to deal with admissibility within the logic. A theory with intensional reasoning principles, such as the one proposed in Constable and Crary [23], would allow reasoning about computation internally. Then these results could be proved within the theory and the only extra rule that would be required would be a single rule relating admissibility to the the fixpoint principle. However they are placed into the logic, these results allow for ....

Robert L. Constable and Karl Crary. Computational complexity and induction for partial computable functions in type theory. Technical report, Department of Computer Science, Cornell University, 1997.

....but it is unpleasant in that it leads to a proliferation of new types and inference rules stemming from discoveries outside the logic. It would be preferable to deal with admissibility within the logic. A theory with intensional reasoning principles, such as the one proposed in Constable and Crary [7], would allow reasoning about computation internally. Then these results could be proved within the theory and the only extra rule that would be required would be a single rule relating admissibility to the the fixpoint principle. However they are placed into the logic, these results allow for ....

Robert L. Constable and Karl Crary. Computational complexity and induction for partial computable functions in type theory. Technical report, Department of Computer Science, Cornell University, 1997.

....partial types (Constable and Smith, 1987; Smith, 1989; Crary, 1998c) For any type T , the partial type T is a supertype of T that contains all the elements of T and also all divergent terms. A total type is one that contains only convergent terms. The induction principles on T (Smith, 1989; Constable and Crary, 1997) are different than those on T , so we can safely type fix with the rule: H e 2 T T H T admissible H fix (e) 2 T (8) We use partial types to interpret the possibly non terminating computations of K . When (in K ) a term e has type , the embedded term [ e] will have type [ ....

Constable, R. L. and Crary, K. (1997) Computational complexity and induction for partial computable functions in type theory. Technical report, Department of Computer Science, Cornell University.

.... This theory is a subset of the type theory of Crary [11] and is similar to Smith s theory [24] The major difference between the theory used here and Smith s is that the latter does not provide a notion of equality; the ramifications of handling equality are discussed in Constable and Crary [6] and at greater length in Crary [11] 2.1 Preliminaries As data types, the theory contains natural numbers (denoted by N) disjoint unions (denoted by T 1 T 2 ) dependent products 1 (denoted by Sigma x:T 1 :T 2 ) and dependent function spaces (denoted by Pi x:T 1 :T 2 ) When x does not ....

....but it is unpleasant in that it leads to a proliferation of new types and inference rules stemming from discoveries outside the logic. It would be preferable to deal with admissibility within the logic. A theory with intensional reasoning principles, such as the one proposed in Constable and Crary [6], would allow reasoning about computation internally. Then these results could be proved within the theory and the only extra rule that would be required would be a single rule relating admissibility to the the fixpoint principle. However they are placed into the logic, these results allow for ....

R. L. Constable and K. Crary. Computational complexity and induction for partial computable functions in type theory. Technical report, Department of Computer Science, Cornell University, 1997.

....more work) for almost every type, including function types (to which the fix rule of K is restricted) It is clear, then, that fix cannot be used to define new members of the basic types. How then can recursive functions be typed The solution is to add a new type constructor for partial types [10, 11, 51, 9, 17]. For any type T , the partial type T is a supertype of T that contains all the elements of T and also all divergent terms. A total type is one that contains only convergent terms. The induction principles on T [51, 9] are different than those on T , so we can safely type fix with the rule: 2 ....

....The solution is to add a new type constructor for partial types [10, 11, 51, 9, 17] For any type T , the partial type T is a supertype of T that contains all the elements of T and also all divergent terms. A total type is one that contains only convergent terms. The induction principles on T [51, 9] are different than those on T , so we can safely type fix with the rule: 2 H e 2 T T H T admissible H fix (e) 2 T We use partial types to interpret the possibly non terminating computations of K . When (in K ) a term e has type , the embedded term [ e] will have type [ ....

Robert L. Constable and Karl Crary. Computational complexity and induction for partial computable functions in type theory. Technical report, Department of Computer Science, Cornell University, 1997.

No context found.

Robert L. Constable and Karl Crary. Computational complexity and induction for partial computable functions in type theory. In Wilfried Sieg, Richard Sommer, and Carolyn Talcott, editors, Reflections on the Foundations of Mathematics: Essays in Honor of Solomon Feferman, Lecture Notes in Logic, pages 166--183. Association for Symbolic Logic, 2001.

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