Paulo Oliva

Selection functions, bar recursion and backward induction

Bar recursion arises in constructive mathematics, logic, proof theory and higher-type computability theory. We explain bar recursion in terms of sequential games, and show how it can be naturally understood as a generalisation of the principle of backward induction that arises in game theory. In summary, bar recursion calculates optimal plays and optimal strategies, which, for particular games of interest, amount to equilibria. We consider finite games and continuous countably infinite games, and relate the two. The above development is followed by a conceptual explanation of how the finite version of the main form of bar recursion considered here arises from a strong monad of selections functions that can be defined in any cartesian closed category. Finite bar recursion turns out to be a well-known morphism available in any strong monad, specialised to the selection monad.

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 higher type 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, higher type 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.

Understanding and using spector's bar recursive interpretation of classical analysis

This note reexamines Spectors remarkable computational interpretation of full classical analysis. Spectors interpretation makes use of a rather abstruse recursion schema, so-called bar recursion, used to interpret the double negation shift DNS. In this note bar recursion is presented as a generalisation of a simpler primitive recursive functional needed for the interpretation of a finite (intuitionistic) version of DNS. I will also present two concrete applications of bar recursion in the extraction of programs from proofs of ∀∃-theorems in classical analysis.

Modified bar recursion

This paper studies modified bar recursion, a higher type recursion scheme, which has been used in Berardi et al. (1998) and Berger and Oliva (2005) for a realisability interpretation of classical analysis. A complete clarification of its relation to Spectors and Kohlenbachs bar recursion, the fan functional, Gandys functional

Proof mining in L1-approximation

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Computational interpretations of analysis via products of selection functions

and Kleenes notion of S1–S9 computability is given.

Hybrid Functional Interpretations

Abstract In this paper, we present another case study in the general project of proof mining which means the logical analysis of prima facie non-effective proofs with the aim of extracting new computationally relevant data. We use techniques based on monotone functional interpretation developed in Kohlenbach (Logic: from Foundations to Applications, European Logic Colloquium (Keele, 1993), Oxford University Press, Oxford, 1996, pp. 225–260) to analyze Cheneys simplification (Math. Mag. 38 (1965) 189) of Jacksons original proof (Trans. Amer. Math. Soc. 22 (1921) 320) of the uniqueness of the best L 1 -approximation of continuous functions f ∈ C [0,1] by polynomials p ∈ P n of degree ⩽ n . Cheneys proof is non-effective in the sense that it is based on classical logic and on the non-computational principle WKL (binary Konigs lemma). The result of our analysis provides the first effective (in all parameters) uniform modulus of uniqueness (a concept which generalizes ‘strong uniqueness’ studied extensively in approximation theory). Moreover, the extracted modulus has the optimal ϵ -dependency as follows from Kroo (Acta Math. Acad. Sci. Hungar. 32 (1978) 331). The paper also describes how the uniform modulus of uniqueness can be used to compute the best L 1 -approximations of a fixed f ∈ C [0,1] with arbitrary precision. The second author uses this result to give a complexity upper bound on the computation of the best L 1 -approximation in Oliva (Math. Logic Quart., 48 (S1) (2002) 66–77).

Reporting Exact and Approximate Regular Expression Matches

We show that the computational interpretation of full comprehension via two well-known functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions.

On Various Negative Translations

We show how different functional interpretations can be combined via a multi-modal linear logic. A concrete hybrid of Kreisels modified realizability and Godels Dialectica is presented, and several small applications are given. We also discuss how the hybrid interpretation relates to variants of Dialectica and modified realizability with non-computational quantifiers.

Bounded Functional Interpretation and Feasible Analysis

While much work has been done on determining if a document or a line of a document contains an exact or approximate match to a regular expression, less effort has been expended in formulating and determining what to report as “the match” once such a “hit” is detected. For exact regular expression pattern matching, we give algorithms for finding a longest match, all symbols involved in some match, and finding optimal submatches to tagged parts of a pattern. For approximate regular expression matching, we develop notions of what constitutes a significant match, give algorithms for them, and also for finding a longest match and all symbols in a match.