An arithmetical hierarchy of the law of excluded middle and related principles

Akama, Yohji; Berardi, Stefano; Hayashi, Susumu; Kohlenbach, Ulrich

doi:10.1109/lics.2004.1319613

Cited by 47 publications

(209 citation statements)

References 13 publications

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“…We call this di culty global ilegibility. 1 . If proof animation is for nding useful information such as bounds for solutions and algorithms in classical proofs as proof mining in 14], global ilegibility is not a real obstacle.…”

Section: Limit Interpretationsmentioning

confidence: 99%

“…At the beginning, we temporarily assume that f(0) is the minimal value among f(0) f (1) : : : . Then, we start to compare the value of f (0) with the values f(1) f (2) : : : to con rm our hypothesis.…”

Section: Limit Interpretationsmentioning

confidence: 99%

“…Let p 0 beu 0 : : : u n . I f u 0 : : : u n u n+1 : : : u m is an extension of p 0 in P( ), then u n+1 : : : u m never contains backtracking moves to 9x 1 . Namely, p 0 is a play \stable" with respect to 9x 1 .…”

Section: -Backtracking Gamementioning

confidence: 99%

“…L e t v 0 be any p l a y i n S 1 . Since this does not satisfy the second condition for p 0 , there is an extension v 1 whose last move i s a backtrack t o 9x 1 . It again belongs to S 1 .…”

Section: -Backtracking Gamementioning

confidence: 99%

“…It maintains that there is some x such that f(x) is the minimum among f(0) f (1) : : : , namely, a minimum number principle for the ordering P m n .…”

mentioning

confidence: 99%

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Can Proofs Be Animated By Games?

Hayashi

2005

Lecture Notes in Computer Science

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Abstract. Proof animation is a way of executing proofs to nd errors in the formalization of proofs. It is intended to be \testing in proof engineering". Although the realizability i n terpretation as well as the functional interpretation based on limit-computations were introduced as means for proof animation, they were unrealistic as an architectural basis for actual proof animation tools. We h a ve found game theoretical semantics corresponding to these interpretations, which is likely to be the right architectural basis for proof animation.

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Section: Limit Interpretationsmentioning

confidence: 99%

Section: Limit Interpretationsmentioning

confidence: 99%

Section: -Backtracking Gamementioning

confidence: 99%

Section: -Backtracking Gamementioning

confidence: 99%

“…It maintains that there is some x such that f(x) is the minimum among f(0) f (1) : : : , namely, a minimum number principle for the ordering P m n .…”

mentioning

confidence: 99%

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Can Proofs Be Animated By Games?

Hayashi

2005

Lecture Notes in Computer Science

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Continuous homomorphisms of R onto a compact group

Bridges¹,

Hendtlass

2010

Math. Log. Quart.

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It is shown within Bishop's constructive mathematics that, under one extra, classically automatic, hypothesis, a continuous homomorphism from R onto a compact metric abelian group is periodic, but that the existence of the minimum value of the period is not derivable.In this paper, which is written within the framework of Bishop's constructive mathematics (BISH), 1) we consider a partial abstraction of the well-known classical result COPEvery compact orbit of a dynamical system is periodic [12].The abstraction is the following: Theorem 1 Let θ be a continuous homomorphism of R onto a compact (metric) abelian group G, such thatis open. Then θ is periodic, in the sense that there exists τ > 0 such that θ(τ) = 0.Here we use standard additive notation for abelian group operations. By a metric abelian group we mean an abelian group G equipped with a metric ρ, such that the mapping (x, y)y − x is pointwise continuous at (0, 0) ∈ G × G, and uniformly continuous on compact subsets of G × G. The mappings x −x and (x, y)x + y are then pointwise continuous throughout their domains, and uniformly continuous on compact subsets of their domains. Moreover, for each positive integer n, the mapping x nx is continuous at 0. Note that if G is locally compact -that is, every bounded subset of G is contained in a compact subset of G -then the pointwise continuity of the mapping (x, y)y − x is a consequence of its uniform continuity on compact sets. We say that a homomorphism θ of the abelian group R into a metric abelian group G is continuous if it is uniformly continuous on each compact (or, equivalently, on each bounded) subset of R.Although we know of no reference for Theorem 1 in the literature, we believe that the following argument, based on the standard classical one used to prove COP, would be the natural one for the classical mathematician to use. Under the hypotheses of Theorem 1, first observe that for each positive integer n, since θ is continuous, the closed subsetof G is compact. Since C 1 ⊃ C 2 ⊃ · · · , the set n 1 C n is nonempty; from which it is not hard to deduce that each C n is dense in G. On the other hand, by the continuity of θ, the sets θ[−n, n] are compact and hence closed. Since G is complete and equals the union of the sets θ[−n, n] (n 1), it follows from the Baire category theorem that the interior of θ[−N, N] is nonempty for some N. We can therefore find t 1 , t 2 such that |t 1 | N, t 2 N + 1, and θ(t 1 ) = θ(t 2 ). Setting τ = t 2 − t 1 , with very little more work we see that θ(τ) = 0.

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Classical provability of uniform versions and intuitionistic provability

Fujiwara

Kohlenbach

2015

Mathematical Logic Qtrly

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Along the line of Hirst-Mummert [9] and Dorais [4], we analyze the relationship between the classical provability of uniform versions Uni(S) of Π 2 -statements S with respect to higher order reverse mathematics and the intuitionistic provability of S. Our main theorem states that (in particular) for every Π 2 -statement S of some syntactical form, if its uniform version derives the uniform variant of ACA over a classical system of arithmetic in all finite types with weak extensionality, then S is not provable in strong semi-intuitionistic systems including bar induction BI in all finite types but also nonconstructive principles such as König's lemma KL and uniform weak König's lemma UWKL. Our result is applicable to many mathematical principles whose sequential versions imply ACA.

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An arithmetical hierarchy of the law of excluded middle and related principles

Cited by 47 publications

References 13 publications

Can Proofs Be Animated By Games?

Can Proofs Be Animated By Games?

Continuous homomorphisms of R onto a compact group

Classical provability of uniform versions and intuitionistic provability

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