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Complete sequent calculi for induction and infinite descent
 Proceedings of LICS22
, 2007
"... This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing induct ..."
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Cited by 18 (6 self)
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This paper compares two different styles of reasoning with inductively defined predicates, each style being encapsulated by a corresponding sequent calculus proof system. The first system supports traditional proof by induction, with induction rules formulated as sequent rules for introducing inductively defined predicates on the left of sequents. We show this system to be cutfree complete with respect to a natural class of Henkin models; the eliminability of cut follows as a corollary. The second system uses infinite (nonwellfounded) proofs to represent arguments by infinite descent. In this system, the left rules for inductively defined predicates are simple casesplit rules, and an infinitary, global condition on proof trees is required to ensure soundness. We show this system to be cutfree complete with respect to standard models, and again infer the eliminability of cut. The second infinitary system is unsuitable for formal reasoning. However, it has a natural restriction to proofs given by regular trees, i.e. to those proofs representable by finite graphs. This restricted “cyclic ” system subsumes the first system for proof by induction. We conjecture that the two systems are in fact equivalent, i.e., that proof by induction is equivalent to regular proof by infinite descent.
Conway Games, coalgebraically ⋆
"... Abstract. Using coalgebraic methods, we extend Conway’s original theory of games to include infinite games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on nonlosing strategies. Infinite games are a fruitf ..."
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Abstract. Using coalgebraic methods, we extend Conway’s original theory of games to include infinite games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on nonlosing strategies. Infinite games are a fruitful metaphor for nonterminating processes, Conway’s sum of games being similar to shuffling. Hypergames have a rather interesting theory, already in the case of generalized Nim. The theory of hypergames generalizes Conway’s theory rather smoothly, but significantly. We indicate a number of intriguing directions for future work. We briefly compare infinite games with other notions of games used in computer science.
CONWAY GAMES, ALGEBRAICALLY AND COALGEBRAICALLY
"... Abstract. First, we present Conway’s theory of games under an algebraic perspective. Then, using coalgebraic methods, we extend Conway’s theory to infinite games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focu ..."
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Abstract. First, we present Conway’s theory of games under an algebraic perspective. Then, using coalgebraic methods, we extend Conway’s theory to infinite games (hypergames). We take the view that a play which goes on forever is a draw, and hence rather than focussing on winning strategies, we focus on nonlosing strategies. Infinite games are a fruitful metaphor for nonterminating processes, Conway’s sum of games being similar to shuffling. Hypergames have a rather interesting theory, already in the case of impartial hypergames, for which we give a compositional semantics, in terms of a generalized GrundySprague function and a system of generalized Nim games. Equivalences and congruences on games and hypergames are discussed. The theory of hypergames generalizes Conway’s theory rather smoothly, but significantly. We indicate a number of intriguing directions for future work. We briefly compare infinite games with other notions of games used in computer science. 1.
The Variable Hierarchy for the Games µCalculus
, 710
"... Abstract. Parity games are combinatorial representations of closed Boolean µterms. By adding to them draw positions, they have been organized by Arnold and one of the authors [1, 2] into a µcalculus [3]. As done by Berwanger et al. [4, 5] for the propositional modal µcalculus, it is possible to c ..."
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Abstract. Parity games are combinatorial representations of closed Boolean µterms. By adding to them draw positions, they have been organized by Arnold and one of the authors [1, 2] into a µcalculus [3]. As done by Berwanger et al. [4, 5] for the propositional modal µcalculus, it is possible to classify parity games into levels of hierarchy according to the number of fixedpoint variables. We ask whether this hierarchy collapses w.r.t. the standard interpretation of the games µcalculus into the class of all complete lattices. We answer this question negatively by providing, for each n ≥ 1, a parity game Gn with these properties: it unravels to a µterm built up with n fixedpoint variables, it is semantically equivalent to no game with strictly less than n − 2 fixedpoint variables. 1
EQUIVALENCES AND CONGRUENCES ON INFINITE Conway Games
 THEORETICAL INFORMATICS AND APPLICATIONS
, 1999
"... Taking the view that infinite plays are draws, we study Conway nonterminating games and nonlosing strategies. These admit a sharp coalgebraic presentation, where nonterminating games are seen as a final coalgebra and game contructors, such as disjunctive sum, as final morphisms. We have shown, in ..."
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Taking the view that infinite plays are draws, we study Conway nonterminating games and nonlosing strategies. These admit a sharp coalgebraic presentation, where nonterminating games are seen as a final coalgebra and game contructors, such as disjunctive sum, as final morphisms. We have shown, in a previous paper, that Conway’s theory of terminating games can be rephrased naturally in terms of game (pre)congruences. Namely, various conceptually independent notions of equivalence can be defined and shown to coincide on Conway’s terminating games. These are the equivalence induced by the ordering on surreal numbers, the contextual equivalence determined by observing what player has a winning strategy, Joyal’s categorical equivalence, and, for impartial games, the denotational equivalence induced by Grundy semantics. In this paper, we discuss generalizations of such equivalences to nonterminating games and nonlosing strategies. The scenario is even more rich and intriguing in this case. In particular, we investigate efficient characterizations of the contextual equivalence,
Cuts for circular proofs: semantics and cutelimination
"... One of the authors introduced in [16] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products, finite coproducts, initial algebras, final coalgebras. The calculus presented [16] is cutfree; even if sound and complete for provab ..."
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One of the authors introduced in [16] a calculus of circular proofs for studying the computability arising from the following categorical operations: finite products, finite coproducts, initial algebras, final coalgebras. The calculus presented [16] is cutfree; even if sound and complete for provability, it lacked an important property for the semantics of proofs, namely fullness w.r.t. the class of intended categorical models (called µbicomplete categories in [18]). In this paper we fix this problem by adding the cut rule to the calculus and by modifying accordingly the syntactical constraint ensuring soundness of proofs. The enhanced proof system fully represents arrows of the canonical model (a free µbicomplete category). We also describe a cutelimination procedure as a a model of computation arising from the above mentioned categorical operations. The procedure constructs a cutfree prooftree with possibly infinite branches out of a finite circular proof with cuts.
DOI: 10.1007/9783642005961_3 Least and Greatest Fixpoints in Game Semantics
, 2009
"... We show how solutions to many recursive arena equations can be computed in a natural way by allowing loops in arenas. We then equip arenas with winning functions and total winning strategies. We present two natural winning conditions compatible with the loop construction which respectively provide i ..."
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We show how solutions to many recursive arena equations can be computed in a natural way by allowing loops in arenas. We then equip arenas with winning functions and total winning strategies. We present two natural winning conditions compatible with the loop construction which respectively provide initial algebras and terminal coalgebras for a large class of continuous functors. Finally, we introduce an intuitionistic sequent calculus, extended with syntactic constructions for least and greatest fixed points, and prove it has a sound and (in a certain weak sense) complete interpretation in our game model. 1