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19
Induction and coinduction in sequent calculus
 Postproceedings of TYPES 2003, number 3085 in LNCS
, 2003
"... Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than sett ..."
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Abstract. Proof search has been used to specify a wide range of computation systems. In order to build a framework for reasoning about such specifications, we make use of a sequent calculus involving induction and coinduction. These proof principles are based on a proof theoretic (rather than settheoretic) notion of definition [13, 20, 25, 51]. Definitions are akin to (stratified) logic programs, where the left and right rules for defined atoms allow one to view theories as “closed ” or defining fixed points. The use of definitions makes it possible to reason intensionally about syntax, in particular enforcing free equality via unification. We add in a consistent way rules for pre and post fixed points, thus allowing the user to reason inductively and coinductively about properties of computational system making full use of higherorder abstract syntax. Consistency is guaranteed via cutelimination, where we give the first, to our knowledge, cutelimination procedure in the presence of general inductive and coinductive definitions. 1
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 21 (8 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.
On the proof theory of modal mucalculus
 Studia Logica
, 2008
"... We study the prooftheoretic relationship between two deductive systems for the modal mucalculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a ..."
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Cited by 7 (2 self)
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We study the prooftheoretic relationship between two deductive systems for the modal mucalculus. First we recall an infinitary system which contains an omega rule allowing to derive the truth of a greatest fixed point from the truth of each of its (infinitely many) approximations. Then we recall a second infinitary calculus which is based on nonwellfounded trees. In this system proofs are finitely branching but may contain infinite branches as long as some greatest fixed point is unfolded infinitely often along every branch. The main contribution of our paper is a translation from proofs in the first system to proofs in the second system. Completeness of the second system then follows from completeness of the first, and a new proof of the finite model property also follows as corollary. 1
Canonical completeness of infinitary µ
 Submitted. Address Thomas Studer Institut für Informatik und angewandte Mathematik, Universität Bern Neubrückstrasse 10, CH3012
"... This paper presents a new model construction for a natural cutfree infinitary version K + ω (µ) of the propositional modal µcalculus. Based on that the completeness of K + ω (µ) and the related system Kω(µ) can be established directly – no detour, for example through automata theory, is needed. As ..."
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This paper presents a new model construction for a natural cutfree infinitary version K + ω (µ) of the propositional modal µcalculus. Based on that the completeness of K + ω (µ) and the related system Kω(µ) can be established directly – no detour, for example through automata theory, is needed. As a side result we also obtain a finite, cutfree sound and complete system for the propositional modal µcalculus. 1
Cut elimination for a logic with induction and coinduction
 JOURNAL OF APPLIED LOGIC
, 2012
"... ..."
On the proof theory of regular fixed points
"... Abstract. We consider encoding finite automata as least fixed points in a prooftheoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to prooftheory. This leads ..."
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Abstract. We consider encoding finite automata as least fixed points in a prooftheoretical framework equipped with a general induction scheme, and study automata inclusion in that setting. We provide a coinductive characterization of inclusion that yields a natural bridge to prooftheory. This leads us to generalize these observations to regular formulas, obtaining new insights about inductive theorem proving and cyclic proofs in particular. 1
Ambiguous Classes in the Games µCalculus Hierarchy
"... Every parity game is a combinatorial representation of a closed Boolean µterm. When interpreted in a distributive lattice every Boolean µterm is equivalent to a fixedpoint free term. The alternationdepth hierarchy is therefore trivial in this case. This is not the case for non distributive la ..."
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Every parity game is a combinatorial representation of a closed Boolean µterm. When interpreted in a distributive lattice every Boolean µterm is equivalent to a fixedpoint free term. The alternationdepth hierarchy is therefore trivial in this case. This is not the case for non distributive lattices, as the second author has shown that the alternationdepth hierarchy is infinite. In this paper
The Variable Hierarchy for the Games µCalculus
, 2007
"... 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 pa ..."
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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.