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24
PSPACE bounds for rank 1 modal logics
 IN LICS’06
, 2006
"... For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a sh ..."
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Cited by 25 (15 self)
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For lack of general algorithmic methods that apply to wide classes of logics, establishing a complexity bound for a given modal logic is often a laborious task. The present work is a step towards a general theory of the complexity of modal logics. Our main result is that all rank1 logics enjoy a shallow model property and thus are, under mild assumptions on the format of their axiomatisation, in PSPACE. This leads to a unified derivation of tight PSPACEbounds for a number of logics including K, KD, coalition logic, graded modal logic, majority logic, and probabilistic modal logic. Our generic algorithm moreover finds tableau proofs that witness pleasant prooftheoretic properties including a weak subformula property. This generality is made possible by a coalgebraic semantics, which conveniently abstracts from the details of a given model class and thus allows covering a broad range of logics in a uniform way.
A Finite Model Construction For Coalgebraic Modal Logic
"... In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness result ..."
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Cited by 24 (16 self)
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In recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. Here, we prove that (finitary) coalgebraic modal logic has the finite model property. This fact not only reproves known completeness results for coalgebraic modal logic, which we push further by establishing that every coalgebraic modal logic admits a complete axiomatization of rank 1; it also enables us to establish a generic decidability result and a first complexity bound. Examples covered by these general results include, besides standard HennessyMilner logic, graded modal logic and probabilistic modal logic.
A Study of Categories of Algebras and Coalgebras
, 2001
"... This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and ..."
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Cited by 13 (5 self)
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This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and the dual notion of coalgebras. Following this, we discuss (co)algebras for a (co)monad and develop a theory of regular subcoalgebras which will be used in the internal logic. We also prove that categories of coalgebras are complete, under reasonably weak conditions, and simultaneously prove the wellknown dual result for categories of algebras. We close the second chapter with a discussion of bisimulations in which we introduce a weaker notion of bisimulation than is current in the literature, but which is wellbehaved and reduces to the standard definition under the assumption of choice. The third chapter is a detailed look at three theorem's of G. Birkho# [Bir35, Bir44], presenting categorical proofs of the theorems which generalize the classical results and which can be easily dualized to apply to categories of coalgebras. The theorems of interest are the variety theorem, the equational completeness theorem and the subdirect product representation theorem. The duals of each of these theorems is discussed in detail, and the dual notion of "coequation" is introduced and several examples given. In the final chapter, we show that first order logic can be interpreted in categories of coalgebras and introduce two modal operators to first order logic to allow reasoning about "endomorphisminvariant" coequations and bisimulations internally. We also develop a translation of terms and formulas into the internal language of the base category, which preserves and reflects truth. La...
From Algebras and Coalgebras to Dialgebras
, 2001
"... This paper investigates the notion of dialgebra, which generalises the notions of algebra and coalgebra. We show that many (co)algebraic notions and results can be generalised to dialgebras, and investigate the essential dierences between (co)algebras and arbitrary dialgebras. ..."
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Cited by 10 (0 self)
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This paper investigates the notion of dialgebra, which generalises the notions of algebra and coalgebra. We show that many (co)algebraic notions and results can be generalised to dialgebras, and investigate the essential dierences between (co)algebras and arbitrary dialgebras.
Coalgebras For Binary Methods: Properties Of Bisimulations And Invariants
, 2001
"... Coalgebras for endofunctors C > C can be used to model classes of objectoriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors C^op x C > C . This extension ..."
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Cited by 9 (3 self)
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Coalgebras for endofunctors C > C can be used to model classes of objectoriented languages. However, binary methods do not fit directly into this approach. This paper proposes an extension of the coalgebraic framework, namely the use of extended polynomial functors C^op x C > C . This extension allows the incorporation of binary methods into coalgebraic class specifications. The paper also discusses how to define bisimulation and invariants for coalgebras of extended polynomial functors and proves many standard results. 1991 Mathematics Subject Classification. 03E20, 03G30, 68Q55, 68Q65.
Towards Weak Bisimulation For Coalgebras
, 2002
"... This report contains a novel approach to observation equivalence for coalgebras. ..."
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Cited by 9 (1 self)
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This report contains a novel approach to observation equivalence for coalgebras.
Dialgebraic Specification and Modeling
"... corecursive functions COALGEBRA state model constructors destructors data model recursive functions reachable hidden abstraction observable hidden restriction congruences invariants visible abstraction ALGEBRA visible restriction!e Swinging Cube ..."
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Cited by 4 (4 self)
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corecursive functions COALGEBRA state model constructors destructors data model recursive functions reachable hidden abstraction observable hidden restriction congruences invariants visible abstraction ALGEBRA visible restriction!e Swinging Cube
Greatest Bisimulations for Binary Methods
 In Proceedings of CMCS’02, volume 65(1) of ENTCS
, 2002
"... In previous work I introduced a generalised notion of coalgebra that is capable of modelling binary methods as they occur in objectoriented programming. An important problem with this generalisation is that bisimulations are not closed under union and that a greatest bisimulation does not exists in ..."
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Cited by 2 (0 self)
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In previous work I introduced a generalised notion of coalgebra that is capable of modelling binary methods as they occur in objectoriented programming. An important problem with this generalisation is that bisimulations are not closed under union and that a greatest bisimulation does not exists in general. There are two possible approaches to improve this situation: First, to strengthen the definition of bisimulation, and second, to place constraints on the coalgebras (i.e., on the behaviour of the binary methods). In this paper I combine both approaches to show that (under reasonable assumptions) the greatest bisimulation does exist for all coalgebras of extended polynomial functors.
Imperative Objectbased Calculi In (Co)Inductive Type Theories
 In Barendregt and Nipkow [2
, 2003
"... We discuss the formalization of Abadi and Cardelli's imp#, a paradigmatic objectbased calculus with types and side e#ects, in (Co)Inductive Type Theories. ..."
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Cited by 2 (0 self)
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We discuss the formalization of Abadi and Cardelli's imp#, a paradigmatic objectbased calculus with types and side e#ects, in (Co)Inductive Type Theories.
Coinductive Verification of Program Optimizations using Similarity Relations
"... Formal verification methods have gained increased importance due to their ability to guarantee system correctness and improve reliability. Nevertheless, the question how proofs are to be formalized in theorem provers is far from being trivial, yet very important as one needs to spend much more time ..."
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Cited by 2 (0 self)
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Formal verification methods have gained increased importance due to their ability to guarantee system correctness and improve reliability. Nevertheless, the question how proofs are to be formalized in theorem provers is far from being trivial, yet very important as one needs to spend much more time on verification if the formalization was not cleverly chosen. In this paper, we develop and compare two different possibilities to express coinductive proofs in the theorem prover Isabelle/HOL. Coinduction is a proof method that allows for the verification of properties of also nonterminating statetransition systems. Since coinduction is not as widely used as other proof techniques as e.g. induction, there are much fewer “recipes ” available how to formalize corresponding proofs and there are also fewer proof strategies implemented in theorem provers for coinduction. In this paper, we investigate formalizations for coinductive proofs of properties on state transition sequences. In particular, we compare two different possibilities for their formalization and show their equivalence. The first of these two formalizations captures the mathematical intuition, while the second can be used more easily in a theorem prover. We have formally verified the equivalence of these criteria in Isabelle/HOL, thus establishing a coalgebraic verification framework. To demonstrate that our verification framework is suitable for the verification of compiler optimizations, we have introduced three different, rather simple transformations that capture typical problems in the verification of optimizing compilers, even for nonterminating source programs.