Notation systems for infinitary derivations

Buchholz, Wilfried

doi:10.1007/bf01621472

Cited by 68 publications

(77 citation statements)

References 12 publications

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“…These address the two orthogonal concerns of §3.8: pattern-compilation produces terms that are finitely wide albeit infinitely deep, while defunctionalization gives these infinitely deep terms a finite, cyclic representation. In proof-theoretic terms, this attempt to relate infinitary and finitary derivations is very similar in spirit to the work of Mints (1978) and Buchholz (1991). These particular transformations seem to be closely related to the sequent calculus for infinite descent and cyclic proofs of Brotherston and Simpson (2007).…”

Section: A Shallow Dis-functional Syntaxmentioning

confidence: 83%

Refinement types and computational duality

Zeilberger

2009

Proceedings of the 3rd Workshop on Programming Languages Meets Program Verification

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One lesson learned painfully over the past twenty years is the perilous interaction of Curry-style typing with evaluation order and side-effects. This led eventually to the value restriction on polymorphism in ML, as well as, more recently, to similar artifacts in type systems for ML with intersection and union refinement types. For example, some of the traditional subtyping laws for unions and intersections are unsound in the presence of effects, while union-elimination requires an evaluation context restriction in addition to the value restriction on intersection-introduction.Our aim is to show that rather than being ad hoc artifacts, phenomena such as the value and evaluation context restrictions arise naturally in type systems for effectful languages, out of principles of duality. Beginning with a review of recent work on the CurryHoward interpretation of focusing proofs as pattern-matching programs, we explain how to interpret intersection and union refinements on these programs, and how to logically derive the subtyping relationship via an identity coercion interpretation. The value restriction, etc., emerge out of this analysis. However, this abstract account does not immediately yield a decidable type system, essentially because the syntax is infinitary-both "infinitely wide" and "infinitely deep". We show how to mechanically construct a finitary syntax by applying two well-known PL techniques-patterncompilation and defunctionalization-and conclude by giving this finitary syntax an algorithmic refinement type system. Parallel to the text, we describe an embedding in the dependently-typed functional language Agda, both for the sake of clarifying these ideas, and also because formalization was an important guide in developing them. As one example, the Agda encoding split very naturally into an intrinsic ("Church") view of well-typed programs, and an extrinsic ("Curry") view of refinement typing for those programs.

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Section: A Shallow Dis-functional Syntaxmentioning

confidence: 83%

Refinement types and computational duality

Zeilberger

2009

Proceedings of the 3rd Workshop on Programming Languages Meets Program Verification

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“…In [4,5], Buchholz presents a finitary reduction procedure for classical arithmetic, obtained using notations for infinitary derivations and Mints' continuous cut-elimination operators. In particular, in [5] one finds another method of extracting finitary reduction sequences from infinitary cut-elimination arguments, and an analysis of the relationship between these reductions and the ones used in Gentzen's original procedure.…”

Section: Logic Colloquium '98mentioning

confidence: 99%

“…In connection with these rules I will refer to the formulae in as the side formulae of the inference, and the other formulae in the hypotheses and conclusions as the main premises and principal formulae respectively. As in [4,5] The equality rules consist of quantifier-free sequents asserting the reflexivity, symmetry, and transitivity of equality, and the fact that equality acts as a congruence relation relative to all the functions and relations in the language.…”

Section: Preliminariesmentioning

confidence: 99%

A Realizability Interpretation for Classical Arithmetic

Avigad¹

2017

Logic Colloquium '98

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Summary.A constructive realizablity interpretation for classical arithmetic is presented, enabling one to extract witnessing terms from proofs of 1 sentences. The interpretation is shown to coincide with modified realizability, under a novel translation of classical logic to intuitionistic logic, followed by the Friedman-Dragalin translation. On the other hand, a natural set of reductions for classical arithmetic is shown to be compatible with the normalization of the realizing term, implying that certain strategies for eliminating cuts and extracting a witness from the proof of a 1 sentence are insensitive to the order in which reductions are applied.

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“…This inference rule (E) is called the height rule in [1], and its meaning is explained in Definition 4.4 as in [14].…”

Section: Finitary Analysis Of Fix I (T )mentioning

confidence: 99%

Intuitionistic fixed point theories over set theories

Arai

2015

Arch. Math. Logic

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In this paper we show that the intuitionistic fixed point theory FiX i (T ) over set theories T is a conservative extension of T if T can manipulate finite sequences and has the full foundation schema.1 Intuitionistic fixed point theory over set theories T For a theory T in a laguage L, let Q(X, x) be an X-positive formula in the language L ∪ {X} with an extra unary predicate symbol X. Introduce a fresh unary predicate symbol Q together with the axiom stating that Q is a fixed point of Q(X, x):By the completeness theorem, it is obvious that the resulting extension of T is conservative over T , though it has a non-elementary speed-up over T when T is a recursive theory containing the elementary recursive arithmetic EA, cf.[3]. When T has an axiom schema, e.g., T = PA, the Peano arithmetic with the complete induction schema, let us define the fixed point extension FiX(PA) to have the induction schema for any formula with the fixed point predicate Q. Then FiX(PA) is stronger than PA, e.g., FiX(PA) proves the consistency of PA. For the proof-theretic strength of the fixed point theory FiX(PA), see [2,10,16].On the other side, W. Buchholz [15] shows that an intuitionistic fixed point theory over the intuitionistic (Heyting) arithmetic HA for strongly positive formulae Q(X, x) is proof-theoretically reducible to HA. In a language of arithmetic strongly positive formulae with respect to X are generated from arithmetic formulae and atomic ones X(t) by means of positive connectives ∨, ∧, ∃, ∀. Then Rüede and Strahm [18] extends the result to the intuitionistic fixed point theory FiX i (HA) for strictly positive formulae Q(X, x), in which the predicate symbol X does not occur in the antecedent ϕ of implications ϕ → ψ nor in the scope 1

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Notation systems for infinitary derivations

Cited by 68 publications

References 12 publications

Refinement types and computational duality

Refinement types and computational duality

A Realizability Interpretation for Classical Arithmetic

Intuitionistic fixed point theories over set theories

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