Howard, W.A., Functional interpretation of bar induction by bar recursion. Compositio Mathematica 20, pp. 107-124 (1968). Home/Search Document Not in Database Summary Related Articles Check |
This paper is cited in the following contexts:
On the No-Counterexample Interpretation - Kohlenbach (1997) (2 citations) (Correct)
.... relativizes uniformly to the case where t is allowed to contain number and function parameters yielding a primitive recursive functional (in the sense of T ) in these parameters and z; u; n; y (to see this one could also use the technique of elimination of free variables from section 5 of [9]) 2 0 1 : m ae 1 : ae l ffi 1 : ffi k a (closed) term of T 1 BR 0;1 , where ffi 1 = ffi k = 0; deg(ae 1 ) deg(ae l ) 1; deg( 1 ) deg( m ) 3. Let Phi m be closed terms in T . Then s : x :t(x; h; Phi 1 ; Phi m ) is definable as a closed ....
Howard, W.A., Functional interpretation of bar induction by bar recursion. Compositio Mathematica 20, pp. 107-124 (1968).
Proof Interpretations and the Computational Content of Proofs - Kohlenbach (2002) (1 citation) (Correct)
....over PRA. Suggested further reading: The combination of negative translation and functional interpretation has been studied as a single interpretation in [113] An extension of functional interpretation to classical analysis was given by C. Spector using so called bar recursive functionals ( 118] [57], 101] 2 In [38] Spector s approach is 2 For an interesting alternative to functional interpretation in this context see [8] CHAPTER 7. NEGATIVE AND FUNCTIONAL INTERPRETATION 62 extended to still stronger systems using bar recursive functionals of infinite types. A di#erent extension of ....
Howard, W.A., Functional interpretation of bar induction by bar recursion. Compositio Mathematica 20, pp. 107-124 (1968).
Fragments of Kripke-Platek Set Theory with Infinity - Rathjen (1992) (1 citation) (Correct)
....and Foundation for Definable Classes. So KP arises from ZF by completely omitting Power Set and restricting Separation and Collection to Delta 0 formulas. These alterations are suggested by the informal notion of predicative . KP is an impredicative theory, notwithstanding. It is known from [Ho1], Ho2] and [J] that KP proves the same arithmetical sentences as Feferman s system ID 1 of positive inductive definitions (cf. Fe] Its proof theoretic ordinal is the Howard ordinal Omega Gamma 0: This article deals with fragments resulting from KP by restricting the amount of ....
W.A. HOWARD, Functional interpretation of bar induction by bar recursion, Comp. Math. 20 (1968), 107-124.
Inductive Definitions and Type Theory: An Introduction - Coquand, Dybjer (Correct)
....Spector extended this to a consistency proof of second order arithmetic, by introducing the notion of bar recursion. The analysis of these systems seen as functional systems, and in particular, a meta theoretical proof that all the functions defined in these systems are total, by Tait and Howard [34, 14], has been historically an important step in the discovery of the deep analogy between functional systems and proof systems. One goal was to see if it was possible to understand Spector s consistency proof using only inductive definitions (with intuitionistic logic) These attempts are described ....
W. Howard. Functional interpretation of bar induction by bar recursion. Compositio Mathematica, 20:107 -- 124.
Inductive Definitions and Type Theory: An Introduction - Coquand, Dybjer (Correct)
....Spector extended this to a consistency proof of second order arithmetic, by introducing the notion of bar recursion. The analysis of these systems seen as functional systems, and in particular, a meta theoretical proof that all the functions defined in these systems are total, by Tait and Howard [28, 12], has been historically an important step in the discovery of the deep analogy between functional systems and proof systems. 2.4.2 Natural Deduction Martin Lof s type theory is formulated in the natural deduction style introduced by Gentzen. Reduction rules for natural deduction proofs were ....
W. Howard. Functional interpretation of bar induction by bar recursion. Compositio Mathematica, 20:107 -- 124.
On the computational content of the Axiom of Choice - Berardi, Bezem, Coquand (1995) (8 citations) (Correct)
....y Thierry Coquand Chalmers University of Gothenburg z Abstract We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel s Dialectica interpretation [10, 18]. Interestingly, this interpretation uses a refinement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We also show how to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and ....
....can be generalized to the case of the axiom of Dependent Choice, and in this case, reduce intuitionistically its correctness to a principle of bar induction. We also give an interpretation of the Double Negation Shift and try a comparison with Spector s bar recursive interpretation of DNS [18, 10, 11], which suggests a computational content of the negative interpretation of Axiom of Choice based on Godel s Dialectica translation. We end by an heuristic explanation of our realizability interpretation, based on a game theoretical analysis of proofs. Dip. Informatica, C.so Svizzera 185, 10149 ....
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W.A. Howard. Functional interpretation of bar induction by bar recursion. Compos. Mathematica 20 (1968), 107-124.
Gödel's Dialectica interpretation and its two-way stretch - Feferman (1997) (Correct)
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W. Howard [1968]. Functional interpretation of bar induction by bar recursion, Compositio Mathematica 20, 107--124.
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