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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.
A Constructive Interpretation of Ramsey’s Theorem via the Product of Selection Functions
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A metastable dominated convergence theorem
, 2012
"... The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise a.e., then ( fn) converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: ..."
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The dominated convergence theorem implies that if (fn) is a sequence of functions on a probability space taking values in the interval [0, 1], and (fn) converges pointwise a.e., then ( fn) converges to the integral of the pointwise limit. Tao [26] has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion that is independent of the sequence (fn) and the underlying space. We prove a slight strengthening of Tao’s theorem which, moreover, provides an explicit description of the second bound in terms of the first. Specifically, we show that when the first bound is given by a continuous functional, the bound in the conclusion can be computed by a recursion along the tree of unsecured sequences. We also establish a quantitative version of Egorov’s theorem, and introduce a new mode of convergence related to these notions.
A gametheoretic computational interpretation of proofs in classical analysis. Preprint, available online at http://arxiv.org/abs/1204.5244
, 2012
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The peirce translation
, 2010
"... 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, as it eliminates uses of Peirce’s ..."
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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, as it eliminates uses of Peirce’s law. When combined with modified realizability this translation applies to full classical analysis, i.e. Peano arithmetic in the language of finite types extended with countable choice and dependent choice. A fundamental step in the interpretation is the realizability of a strengthening of the doublenegation shift via the iterated product of selection functions. In a separate paper we have shown that such a product of selection functions is equivalent, over system T, to modified bar recursion. Keywords: Peirce’s law, negative translation, countable choice, dependent choice 1.
HigherOrder Game Theory
"... I wish to report here on a novel formalisation of Game Theory based on higher order functionals. The starting point is the modelling of players via socalled selection functions, i.e. functionals of type (X → R) → X. Here X is the type of moves and R is the type of outcomes. If one thinks of X → R a ..."
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I wish to report here on a novel formalisation of Game Theory based on higher order functionals. The starting point is the modelling of players via socalled selection functions, i.e. functionals of type (X → R) → X. Here X is the type of moves and R is the type of outcomes. If one thinks of X → R as the type of possible game contexts, a selection function describes the optimal moves in each given game context. Some of the main results so far include: – The type constructor JRX ≡ (X → R) → X is a strong monad, and as such supports an operation JRX × JRY → JR(X × Y). This can be understood as a “merging ” of players. The new selection function JR(X × Y) is a new single player that captures the goals of the two given players JRX and JRY, see [3, 4]. – With an appropriate definition of equilibrium one can show that in sequential games the operation ⊗ calculates optimal strategies. Moreover, with argmax: (X → Rn) → X as selection functions, as in standard Game Theory, this construction coincides with backward induction [6]. – The binary operation ⊗ can be iterated not only finitely many times, but also a countable number of times, i.e. Πi∈NJRXi → JRΠi∈NXi is welldefined (assuming R a discrete type, and continuity of functionals) and in fact has been shown to be equivalent to bar recursion, a prooftheoretic construction used to give computational meaning to the countable axiom of choice [3]. Selection functions were first introduced in [1, 2] with R = B, and later generalised in [3–6].
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
To appear in EPTCS. Monad Transformers for Backtracking Search
"... This paper extends Escardo ́ and Oliva’s selection monad to the selection monad transformer, a general monadic framework for expressing backtracking search algorithms in Haskell. The use of the closely related continuation monad transformer for similar purposes is also discussed, including an imple ..."
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This paper extends Escardo ́ and Oliva’s selection monad to the selection monad transformer, a general monadic framework for expressing backtracking search algorithms in Haskell. The use of the closely related continuation monad transformer for similar purposes is also discussed, including an implementation of a DPLLlike SAT solver with no explicit recursion. Continuing a line of work exploring connections between selection functions and game theory, we use the selection monad transformer with the nondeterminism monad to obtain an intuitive notion of backward induction for a certain class of nondeterministic games. 1