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The Peirce Translation and the Double Negation Shift
"... We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of classical minimal logic into minimal logic, which we refer to as the Peirce translation, and which we apply to interpret b ..."
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We develop applications of selection functions to proof theory and computational extraction of witnesses from proofs in classical analysis. The main novelty is a translation of classical minimal logic into minimal logic, which we refer to as the Peirce translation, and which we apply to interpret both a strengthening of the doublenegation shift and the axioms of countable and dependent choice, via infinite products of selection functions.
Sequential Games and Optimal Strategies
"... This article gives an overview of recent work on the theory of selection functions. We explain the intuition behind these highertype objects, and define a general notion of sequential game whose optimal strategies can be computed via a certain product of selection functions. Several instances of th ..."
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Cited by 3 (3 self)
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This article gives an overview of recent work on the theory of selection functions. We explain the intuition behind these highertype objects, and define a general notion of sequential game whose optimal strategies can be computed via a certain product of selection functions. Several instances of this game are considered in a variety of areas such as fixed point theory, topology, game theory, highertype computability, and proof theory. These examples are intended to illustrate how the fundamental construction of optimal strategies based on products of selection functions permeates several research areas.
Searchable Sets, DubucPenon Compactness, Omniscience Principles, and the Drinker Paradox
"... We show that a number of contenders for an abstract and general notion of compactness, applicable in particular to computability theory and constructive mathematics, coincide in some well known frameworks. We consider compactness of sets rather than of spaces, where we replace topologies by the res ..."
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We show that a number of contenders for an abstract and general notion of compactness, applicable in particular to computability theory and constructive mathematics, coincide in some well known frameworks. We consider compactness of sets rather than of spaces, where we replace topologies by the restriction to constructive reasoning, as in the work by a number of authors, including Penon, Dubuc, Taylor and Escardó. Sets here are conceived in a very liberal way, including types of HA ω and Martin Löf type theory, and objects of toposes, among others. Some of the equivalences require instances of the axiom of choice, which are available in some of the above frameworks but not all, as is well known. We relate the instances of the axiom of choice applied in the above equivalences to the topological notion of total separatedness.
niques]: functional programming
, 2010
"... This is a tutorial for mathematically inclined functional programmers, based on previously published, peered reviewed theoretical work. We discuss a highertype functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a compu ..."
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This is a tutorial for mathematically inclined functional programmers, based on previously published, peered reviewed theoretical work. We discuss a highertype functional, written here in the functional programming language Haskell, which (1) optimally plays sequential games, (2) implements a computational version of the Tychonoff Theorem from topology, and (3) realizes the DoubleNegation Shift from logic and proof theory. The functional makes sense for finite and infinite (lazy) lists, and in the binary case it amounts to an operation that is available in any (strong) monad. In fact, once we define this monad in Haskell, it turns out that this amazingly versatile functional is already available in Haskell, in the standard prelude, called sequence, which iterates this binary operation. Therefore Haskell proves that this functional is even more versatile than anticipated, as the function sequence was introduced for other purposes by the language designers, in particular the iteration of a list of monadic effects (but effects are not what we discuss here). D.1.1 [Programming tech
BAR RECURSION AND PRODUCTS OF SELECTION FUNCTIONS
"... Abstract. We show how two iterated products of selection functions can both be used in conjunction with system T to interpret, via the dialectica interpretation and modified realizability, full classical analysis. We also show that one iterated product is equivalent over system T to Spector’s bar re ..."
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Abstract. We show how two iterated products of selection functions can both be used in conjunction with system T to interpret, via the dialectica interpretation and modified realizability, full classical analysis. We also show that one iterated product is equivalent over system T to Spector’s bar recursion, whereas the other is Tequivalent to modified bar recursion. Modified bar recursion itself is shown to arise directly from the iteration of a different binary product of ‘skewed ’ selection functions. Iterations of the dependent binary products are also considered but in all cases are shown to be Tequivalent to the iteration of the simple products. §1. Introduction. Gödel’s [13] socalled dialectica interpretation reduces the consistency of Peano arithmetic to the consistency of a quantifierfree calculus of functionals T. In order to extend Gödel’s interpretation to full classical analysis PA ω + CA, Spector [18] made use of the fact that PA ω + CA can be embedded, via the negative translation, into HA ω + ACN + DNS. Here PA ω
System T and the Product of Selection Functions
"... We show that the finite product of selection functions (for all finite types) is primitive recursively equivalent to Gödel’s highertype recursor (for all finite types). The correspondence is shown to hold for similar restricted fragments of both systems: The recursor for type level n 1 is primitive ..."
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We show that the finite product of selection functions (for all finite types) is primitive recursively equivalent to Gödel’s highertype recursor (for all finite types). The correspondence is shown to hold for similar restricted fragments of both systems: The recursor for type level n 1 is primitive recursively equivalent to the finite product of selection functions of type level n. Whereas the recursor directly interprets induction, we show that other classical arithmetical principles such as bounded collection and finite choice are more naturally interpreted via the product of selection functions.
Computing Nash Equilibria of Unbounded
"... Using techniques from highertype computability theory and proof theory we extend the wellknown gametheoretic technique of backward induction to certain general classes of unbounded games. The main application is a closed formula for calculating strategy profiles in Nash equilibrium and subgame pe ..."
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Using techniques from highertype computability theory and proof theory we extend the wellknown gametheoretic technique of backward induction to certain general classes of unbounded games. The main application is a closed formula for calculating strategy profiles in Nash equilibrium and subgame perfect equilibrium even in the case of games where the length of play is not apriori fixed. 1
A Constructive Proof of Dependent Choice, Compatible with Classical Logic
, 2012
"... Abstract—MartinLöf’s type theory has strong existential elimination (dependent sum type) that allows to prove the full axiom of choice. However the theory is intuitionistic. We give a condition on strong existential elimination that makes it computationally compatible with classical logic. With thi ..."
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Abstract—MartinLöf’s type theory has strong existential elimination (dependent sum type) that allows to prove the full axiom of choice. However the theory is intuitionistic. We give a condition on strong existential elimination that makes it computationally compatible with classical logic. With this restriction, we lose the full axiom of choice but, thanks to a lazilyevaluated coinductive representation of quantification, we are still able to constructively prove the axiom of countable choice, the axiom of dependent choice, and a form of bar induction in ways that make each of them computationally compatible with classical logic. KeywordsDependent choice; classical logic; constructive logic; strong existential I.