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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.



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Abramsky, S. and Jung, A. (1994) Domain theory. In: Abramsky, S., Gabbay, D. and Maibaum, T. (eds.) Handbook of Logic in Computer Science 3, Oxford science publications 1168.
Avigad, J. and Feferman, S. (1998) Gödel's functional (“Dialectica”) interpretation. In: Buss, S. R. (ed.) Handbook of proof theory, Studies in Logic and the Foundations of Mathematics 137, North-Holland337405.
Battenfeld, I., Schröder, M. and Simpson, A. (2006) Compactly generated domain theory. Mathematical Structures in Computer Science 16 (2)141161.
Battenfeld, I., Schröder, M. and Simpson, A. (2007) A convenient category of domains. In: Computation, meaning, and logic: articles dedicated to Gordon Plotkin. Electronic Notes in Theoretical Computer Science 172 6999.
Bauer, A. (2002) A relationship between equilogical spaces and type two effectivity. MLQ Math. Log. Q. 48 (suppl. 1)115.
Bekič, H. (1984) Programming languages and their definition – H. Bekič (1936–1982) Selected Papers edited by C. B. Jones. Springer-Verlag Lecture Notes in Computer Science 177.
Berardi, S., Bezem, M. and Coquand, T. (1998) On the computational content of the axiom of choice. The Journal of Symbolic Logic 63 (2)600622.
Berger, U. and Oliva, P. (2005) Modified bar recursion and classical dependent choice. In: Baaz, M., Friedman, S. D. and Kraijcek, J. (eds.) Logic Colloquium '01. Springer-Verlag Lecture Notes in Logic 20 89107.
Berger, U. and Oliva, P. (2006) Modified bar recursion. Mathematical Structures in Computer Science 16 (2)163183.
Bezem, M. (1985) Strongly majorizable functionals of finite type: a model for bar recursion containing discontinuous functionals. The Journal of Symbolic Logic 50 652660.
Bove, A. and Dybjer, P. (2008) Dependent types at work. Lecture notes from the LerNET Summer School, Piriapolis, available at the authors' web pages.
Escardó, M. (2007) Infinite sets that admit fast exhaustive search. In: Proceedings of the 22nd Annual IEEE Symposium on Logic In Computer Science, IEEE Computer Society 443–452.
Escardó, M. (2008) Exhaustible sets in higher-type computation. Logical Methods in Computer Science 4 (3:3)137.
Escardó, M., Lawson, J. and Simpson, A. (2004) Comparing Cartesian closed categories of (core) compactly generated spaces. Topology Appl. 143 (1–3)105145.
Hutton, G. (2007) Programming in Haskell, Cambridge University Press.
Johnstone, P. (2002) Sketches of an Elephant: a Topos Theory Compendium, Oxford University Press.
Kock, A. (1970a) Monads on symmetric monoidal closed categories. Arch. Math. (Basel) 21 110.
Kock, A. (1970b) On double dualization monads. Math. Scand. 27 151165.
Kock, A. (1972) Strong functors and monoidal monads. Arch. Math. (Basel) 23 113120.
Lambek, J. and Scott, P. (1986) Introduction to Higher Order Categorical Logic, Cambridge University Press.
Mac Lane, S. (1971) Categories for the Working Mathematician, Springer-Verlag.
Moggi, E. (1990) An abstract view of programming languages. Technical Report ECS-LFCS-90-113, Laboratory for Foundations of Computer Science, University of Edinburgh.
Moggi, E. (1991) Notions of computation and monads. Information and Computation 93 (1)5592.
Nisan, N. et al. (2007) Algorithmic Game Theory, Cambridge University Press.
Normann, D. (1980) Recursion on the countable functionals. Springer-Verlag Lecture Notes in Mathematics 811.
Normann, D. (1999) The continuous functionals. In: Griffor, E. R. (ed.) Handbook of Computability Theory, Chapter 8, North-Holland251275.
Oliva, P. (2006) Understanding and using Spector's bar recursive interpretation of classical analysis. In: Beckmann, A., Berger, U., Löwe, B. and Tucker, J. V. (eds.) Logical approaches to computational barriers. Proceedings second conference on computability in Europe, CiE 2006, Swansea. Springer-Verlag Lecture Notes in Computer Science 3988 423434.
Schröder, M. (2002) Extended admissibility. Theoretical Computer Science 284 (2)519538.
Smyth, M. (1977) Effectively given domains. Theoretical Computer Science 5 (1)256274.
Spector, C. (1962) Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles in current intuitionistic mathematics. In: Dekker, F. D. E. (ed.) Recursive Function Theory: Proc. Symposia in Pure Mathematics 5, American Mathematical Society 127.
Valiente, G. (2002) Algorithms on Trees and Graphs, Springer-Verlag.

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Selection functions, bar recursion and backward induction



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