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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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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.
Proof systems for the coalgebraic cover modality
 Same volume. Clemens Kupke, Alexander Kurz and Yde Venema
, 2008
"... abstract. We investigate an alternative presentation of classical and positive modal logic where the coalgebraic cover modality is taken as primitive. For each logic, we present a sound and complete Hilbertstyle axiomatization. Moreover, we give a twosided sound and complete sequent calculus for t ..."
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Cited by 9 (4 self)
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abstract. We investigate an alternative presentation of classical and positive modal logic where the coalgebraic cover modality is taken as primitive. For each logic, we present a sound and complete Hilbertstyle axiomatization. Moreover, we give a twosided sound and complete sequent calculus for the negationfree language, and for the language with negation we provide a onesided sequent calculus which is sound, complete and cutfree.
SAHLQVIST THEOREM FOR MODAL FIXED POINT LOGIC
"... Abstract. We define Sahlqvist fixed point formulas. By extending the technique of Sambin and Vaccaro we show that (1) for each Sahlqvist fixed point formula ϕ there exists an LFPformula χ(ϕ), with no free firstorder variable or predicate symbol, such that a descriptive µframe (an ordertopological ..."
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Cited by 8 (3 self)
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Abstract. We define Sahlqvist fixed point formulas. By extending the technique of Sambin and Vaccaro we show that (1) for each Sahlqvist fixed point formula ϕ there exists an LFPformula χ(ϕ), with no free firstorder variable or predicate symbol, such that a descriptive µframe (an ordertopological structure that admits topological interpretations of least fixed point operators as intersections of clopen prefixed points) validates ϕ iff χ(ϕ) is true in this structure, and (2) every modal fixed point logic axiomatized by a set Φ of Sahlqvist fixed point formulas is sound and complete with respect to the class of descriptive µframes satisfying {χ(ϕ) : ϕ ∈ Φ}. We also give some concrete examples of Sahlqvist fixed point logics and classes of descriptive µframes for which these logics are sound and complete. 1.
Completeness for flat modal fixpoint logics
 Annals of Pure and Applied Logic, 162(1):55 – 82
, 2010
"... This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x, p1,..., pn), where x occurs only positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic a connective ♯γ for ..."
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Cited by 5 (2 self)
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This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics. Given a set Γ of modal formulas of the form γ(x, p1,..., pn), where x occurs only positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic a connective ♯γ for each γ ∈ Γ. The term ♯γ(ϕ1,..., ϕn) is meant to be interpreted as the least fixed point of the functional interpretation of the term γ(x, ϕ1,..., ϕn). We consider the following problem: given Γ, construct an axiom system which is sound and complete with respect to the concrete interpretation of the language L♯(Γ) on Kripke frames. We prove two results that solve this problem. First, let K♯(Γ) be the logic obtained from the basic polymodal K by adding a KozenPark style fixpoint axiom and a least fixpoint rule, for each fixpoint connective ♯γ. Provided that each indexing formula γ satisfies the syntactic criterion of being untied in x, we prove this axiom system to be complete. Second, addressing the general case, we prove the soundness and completeness of an extension K + ♯ (Γ) of K♯(Γ). This extension is obtained via an effective procedure that, given an indexing formula γ as input, returns a finite set of axioms and derivation rules for ♯γ, of size bounded by the length of γ. Thus the axiom system K + (Γ) is finite whenever Γ is finite.
Completeness for Flat Modal Fixpoint Logics (Extended Abstract)
"... Abstract. Given a set Γ of modal formulas of the form γ(x, p), where x occurs positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic K connectives ♯γ, γ ∈ Γ. Each term ♯γ is meant to be interpreted as the parametrized least fixed point of the functional interpre ..."
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Abstract. Given a set Γ of modal formulas of the form γ(x, p), where x occurs positively in γ, the language L♯(Γ) is obtained by adding to the language of polymodal logic K connectives ♯γ, γ ∈ Γ. Each term ♯γ is meant to be interpreted as the parametrized least fixed point of the functional interpretation of the term γ(x). Given such a Γ, we construct an axiom system K♯(Γ) which is sound and complete w.r.t. the concrete interpretation of the language L♯(Γ) on Kripke frames. If Γ is finite, then K♯(Γ) is a finite set of axioms and inference rules.
Flat Coalgebraic Fixed Point Logics
"... Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the µcalculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and in fact ..."
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Fixed point logics are widely used in computer science, in particular in artificial intelligence and concurrency. The most expressive logics of this type are the µcalculus and its relatives. However, popular fixed point logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the µcalculus. The family of such flat fixed point logics includes, e.g., CTL, the ∗nestingfree fragment of PDL, and the logic of common knowledge. Here, we extend this notion to the generic semantic framework of coalgebraic logic, thus covering a wide range of logics beyond the standard µcalculus including, e.g., flat fragments of the graded µcalculus and the alternatingtime µcalculus (such as ATL), as well as probabilistic and monotone fixed point logics. Our main results are completeness of the KozenPark axiomatization and a timedout tableaux method that matches EXPTIME upper bounds inherited from the coalgebraic µcalculus but avoids using automata.