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Expressive Logics for Coalgebras via Terminal Sequence Induction
 Notre Dame J. Formal Logic
, 2002
"... This paper introduces the proof principle of terminal sequence induction and shows, how terminal sequence induction can be used to obtain expressiveness results for logics, interpreted over coalgebras. ..."
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Cited by 39 (12 self)
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This paper introduces the proof principle of terminal sequence induction and shows, how terminal sequence induction can be used to obtain expressiveness results for logics, interpreted over coalgebras.
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 37 (19 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.
Proof Methods for Corecursive Programs
 Fundamenta Informaticae Special Issue on Program Transformation
, 1999
"... This article is a tutorial on four methods for proving properties of corecursive programs: fixpoint induction, the approximation lemma, coinduction, and fusion. ..."
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Cited by 27 (8 self)
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This article is a tutorial on four methods for proving properties of corecursive programs: fixpoint induction, the approximation lemma, coinduction, and fusion.
GSOS for Probabilistic Transition Systems
, 2002
"... We introduce PGSOS, an operator specification format for (reactive) probabilistic transition systems which bears similarity to the known GSOS format for labelled (nondeterministic) transition systems. Like the standard one, the format is well behaved in the sense that on all models bisimilarity is a ..."
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Cited by 21 (1 self)
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We introduce PGSOS, an operator specification format for (reactive) probabilistic transition systems which bears similarity to the known GSOS format for labelled (nondeterministic) transition systems. Like the standard one, the format is well behaved in the sense that on all models bisimilarity is a congruence and the uptocontext proof principle is valid. Moreover, guarded recursive equations involving the specified operators have unique solutions up to bisimilarity. These results generalize wellbehavedness results given in the literature for specific operators that turn out to be definable by our format. PGSOS arose from the following procedure: Turi and Plotkin proposed to model specifications in the (standard) GSOS format as natural transformations of a type they call abstract GSOS. This formulation allows for simple proofs of several wellbehavedness properties, such as bisimilarity being a congruence on all models of such a specification. First, we give a full proof of Turi and Plotkin's claim about the correspondence of abstract GSOS and standard GSOS for labelled transition systems. Next, we instantiate their categorical framework to yield a specification format for probabilistic transition systems. The main contribution of the present paper is the derivation of the PGSOS format as a rulestyle representation of the natural transformations obtained this way. We benefit from the fact that some parts of our argument for the nondeterministic case can be reused. The wellbehavedness results for abstract GSOS immediately carry over to the new concrete format.
Beating the Productivity Checker Using Embedded Languages
"... Abstract. Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures th ..."
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Cited by 10 (3 self)
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Abstract. Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures that programs are productive, i.e. that every finite prefix of an infinite value can be computed in finite time. However, many productive programs are not guarded, and it can be nontrivial to put them in guarded form. This paper gives a method for turning a productive program into a guarded program. The method amounts to defining a problemspecific language as a data type, writing the program in the problemspecific language, and writing a guarded interpreter for this language. 1
The Dual of Substitution is Redecoration
, 2002
"... It is well known that type constructors of incomplete trees (trees with variables) carry the structure of a monad with substitution as the extension operation. Less known are the facts that the same is true of type constructors of incomplete cotrees (=nonwellfounded trees) and that the correspondin ..."
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Cited by 7 (3 self)
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It is well known that type constructors of incomplete trees (trees with variables) carry the structure of a monad with substitution as the extension operation. Less known are the facts that the same is true of type constructors of incomplete cotrees (=nonwellfounded trees) and that the corresponding monads exhibit a special structure. We wish to draw attention to the dual facts which are as meaningful for functional programming: type constructors of decorated cotrees carry the structure of a comonad with redecoration as the coextension operation, and so doeven more interestinglytype constructors of decorated trees.
Contributions to the Theory of Syntax with Bindings and to Process Algebra
, 2010
"... We develop a theory of syntax with bindings, focusing on: methodological issues concerning the convenient representation of syntax; techniques for recursive definitions and inductive reasoning. Our approach consists of a combination of FOAS (FirstOrder Abstract Syntax) and HOAS (HigherOrder Abst ..."
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Cited by 5 (4 self)
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We develop a theory of syntax with bindings, focusing on: methodological issues concerning the convenient representation of syntax; techniques for recursive definitions and inductive reasoning. Our approach consists of a combination of FOAS (FirstOrder Abstract Syntax) and HOAS (HigherOrder Abstract Syntax) and tries to take advantage of the best of both worlds. The connection between FOAS and HOAS follows some general patterns and is presented as a (formally certified) statement of adequacy. We also develop a general technique for proving bisimilarity in process algebra Our technique, presented as a formal proof system, is applicable to a wide range of process algebras. The proof system is incremental, in that it allows building incrementally an a priori unknown bisimulation, and patternbased, in that it works on equalities of process patterns (i.e., universally quantified equations of process terms containing process variables), thus taking advantage of equational reasoning in a “circular ” manner, inside coinductive proof loops. All the work presented here has been formalized in the Isabelle theorem prover. The formalization is performed in a general setting: arbitrary manysorted syntax with bindings and arbitrary SOSspecified process algebra in de Simone format. The usefulness of our techniques is illustrated by several formalized case studies: a development of callbyname and callbyvalue λcalculus with constants, including ChurchRosser theorems, connection with de Bruijn representation, connection with other Isabelle formalizations, HOAS representation, and contituationpassingstyle (CPS) transformation; a proof in HOAS of strong normalization for the polymorphic secondorder λcalculus (a.k.a. System F). We also indicate the outline and some details of the formal development. ii to Leili R. Marleene iii
Incremental patternbased coinduction for process algebra and its Isabelle formalization
"... Abstract. We present a coinductive proof system for bisimilarity in transition systems specifiable in the de Simone SOS format. Our coinduction is incremental, in that it allows building incrementally an a priori unknown bisimulation, and patternbased, in that it works on equalities of process patt ..."
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Abstract. We present a coinductive proof system for bisimilarity in transition systems specifiable in the de Simone SOS format. Our coinduction is incremental, in that it allows building incrementally an a priori unknown bisimulation, and patternbased, in that it works on equalities of process patterns (i.e., universally quantified equations of process terms containing process variables), thus taking advantage of equational reasoning in a “circular ” manner, inside coinductive proof loops. The proof system has been formalized and proved sound in Isabelle/HOL. 1
A coalgebraic approach to the semantics of the ambient calculus
 ALGEBRA AND COALGEBRA IN COMPUTER SCIENCE
, 2005
"... Recently, various process calculi have been introduced which are suited for the modelling of mobile computation and in particular the mobility of program code; a prominent example is the ambient calculus. Due to the complexity of the involved spatial reduction, there is — in contrast to the situatio ..."
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Cited by 5 (2 self)
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Recently, various process calculi have been introduced which are suited for the modelling of mobile computation and in particular the mobility of program code; a prominent example is the ambient calculus. Due to the complexity of the involved spatial reduction, there is — in contrast to the situation in standard process algebra — up to now no satisfying coalgebraic representation of a mobile process calculus. Here, we discuss a coalgebraic denotational semantics for the ambient calculus, viewed as a step towards a generic coalgebraic framework for modelling mobile systems. Crucial features of our modelling are a set of GSOS style transition rules for the ambient calculus, a hardwiring of the socalled hardening relation in the functorial signature, and a setbased treatment of hidden name sharing. The formal representation of this framework is cast in the algebraiccoalgebraic specification language CoCasl.
Unifying structured recursion schemes
 in International Conference on Functional Programming. ACM
"... Folds over inductive datatypes are well understood and widely used. In their plain form, they are quite restricted; but many disparate generalisations have been proposed that enjoy similar calculational benefits. There have also been attempts to unify the various generalisations: two prominent su ..."
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Cited by 3 (3 self)
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Folds over inductive datatypes are well understood and widely used. In their plain form, they are quite restricted; but many disparate generalisations have been proposed that enjoy similar calculational benefits. There have also been attempts to unify the various generalisations: two prominent such unifications are the ‘recursion schemes from comonads ’ of Uustalu, Vene and Pardo, and our own ‘adjoint folds’. Until now, these two unified schemes have appeared incompatible. We show that this appearance is illusory: in fact, adjoint folds subsume recursion schemes from comonads. The proof of this claim involves standard constructions in category theory that are nevertheless not well known in functional programming: EilenbergMoore categories and bialgebras.