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150
A Tutorial on (Co)Algebras and (Co)Induction
 EATCS Bulletin
, 1997
"... . Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition pr ..."
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Cited by 228 (34 self)
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. Algebraic structures which are generated by a collection of constructors like natural numbers (generated by a zero and a successor) or finite lists and trees are of wellestablished importance in computer science. Formally, they are initial algebras. Induction is used both as a definition principle, and as a proof principle for such structures. But there are also important dual "coalgebraic" structures, which do not come equipped with constructor operations but with what are sometimes called "destructor" operations (also called observers, accessors, transition maps, or mutators). Spaces of infinite data (including, for example, infinite lists, and nonwellfounded sets) are generally of this kind. In general, dynamical systems with a hidden, blackbox state space, to which a user only has limited access via specified (observer or mutator) operations, are coalgebras of various kinds. Such coalgebraic systems are common in computer science. And "coinduction" is the appropriate te...
Structural Operational Semantics
 Handbook of Process Algebra
, 1999
"... Structural Operational Semantics (SOS) provides a framework to give an operational semantics to programming and specification languages, which, because of its intuitive appeal and flexibility, has found considerable application in the theory of concurrent processes. Even though SOS is widely use ..."
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Cited by 125 (19 self)
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Structural Operational Semantics (SOS) provides a framework to give an operational semantics to programming and specification languages, which, because of its intuitive appeal and flexibility, has found considerable application in the theory of concurrent processes. Even though SOS is widely used in programming language semantics at large, some of its most interesting theoretical developments have taken place within concurrency theory. In particular, SOS has been successfully applied as a formal tool to establish results that hold for whole classes of process description languages. The concept of rule format has played a major role in the development of this general theory of process description languages, and several such formats have been proposed in the research literature. This chapter presents an exposition of existing rule formats, and of the rich body of results that are guaranteed to hold for any process description language whose SOS is within one of these formats. As far as possible, the theory is developed for SOS with features like predicates and negative premises.
Reasoning about Information Change
, 1997
"... In this paper, we have combined techniques from epistemic and dynamic logic to arrive at a logic for describing multiagent information change. The key concept of dynamic semantics is that the meaning of an assertion is the way in which the assertion changes the information of the hearer. Thus a dyn ..."
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Cited by 95 (4 self)
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In this paper, we have combined techniques from epistemic and dynamic logic to arrive at a logic for describing multiagent information change. The key concept of dynamic semantics is that the meaning of an assertion is the way in which the assertion changes the information of the hearer. Thus a dynamic epistemic semantics consist in a explicit formal definition of the information change potential of a sentence. We used these ideas to arrive at the system of Dynamic Epistemic Semantics, which is semantics for a language describing information change in a multiagent setting. This semantics proved useful for analyzing the Muddy Children paradox, and also for giving a semantics for knowledge programs, since it enabled us to model knowledge change by giving an explicit semantics to the triggers of the information change (the latter being the assertions made, or the messages sent). We feel that this is an important extension, since standard approaches to for example the Muddy Children (e.g. Fagin et al. 1995) generally use static epistemic logics like S5 to describe the situation before and after a certain epistemic event, leaving the transition between `before' and `after' to considerations in the metalanguage.
Coalgebraic Logic
 Annals of Pure and Applied Logic
, 1999
"... We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The ..."
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Cited by 89 (0 self)
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We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every modelworld pair is characterized up to bisimulation by an infinitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a frangment of infinitary modal logic which contains the characterizing formulas and is closed under infinitary conjunction and an operation called 4. This fragment generalizes to a wide range of coalgebraic logics. We then apply the characterization result to get representation theorems for final coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of the final coalgebra. Keywords: infinitary modal logic, characterization theorem, functor on sets, coalgebra, greatest fixed point. 1 Intr...
Models of Sharing Graphs: A Categorical Semantics of let and letrec
, 1997
"... To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sha ..."
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Cited by 62 (10 self)
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To my parents A general abstract theory for computation involving shared resources is presented. We develop the models of sharing graphs, also known as term graphs, in terms of both syntax and semantics. According to the complexity of the permitted form of sharing, we consider four situations of sharing graphs. The simplest is firstorder acyclic sharing graphs represented by letsyntax, and others are extensions with higherorder constructs (lambda calculi) and/or cyclic sharing (recursive letrec binding). For each of four settings, we provide the equational theory for representing the sharing graphs, and identify the class of categorical models which are shown to be sound and complete for the theory. The emphasis is put on the algebraic nature of sharing graphs, which leads us to the semantic account of them. We describe the models in terms of the notions of symmetric monoidal categories and functors, additionally with symmetric monoidal adjunctions and traced
Semantics of Name and Value Passing
, 2001
"... We provide a semantic framework for (first order) messagepassing process calculi by combining categorical theories of abstract syntax with binding and operational semantics. In particular, we obtain abstract rule formats for name and value passing with both late and early interpretations. These for ..."
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Cited by 46 (4 self)
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We provide a semantic framework for (first order) messagepassing process calculi by combining categorical theories of abstract syntax with binding and operational semantics. In particular, we obtain abstract rule formats for name and value passing with both late and early interpretations. These formats induce an initialalgebra/finalcoalgebra semantics that is compositional, respects substitution, and is fully abstract for late and early congruence. We exemplify the theory with the #calculus and valuepassing CCS.
SetTheoretical and Other Elementary Models of the lambdacalculus
 Theoretical Computer Science
, 1993
"... Part 1 of this paper is the previously unpublished 1972 memorandum [43], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary settheoretical model of the ficalculus. F ..."
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Cited by 40 (0 self)
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Part 1 of this paper is the previously unpublished 1972 memorandum [43], with editorial changes and some minor corrections. Part 2 presents what happened next, together with some further development of the material. The first part begins with an elementary settheoretical model of the ficalculus. Functions are modeled in a similar way to that normally employed in set theory, by their graphs; difficulties are caused in this enterprise by the axiom of foundation. Next, based on that model, a model of the fijcalculus is constructed by means of a natural deduction method. Finally, a theorem is proved giving some general properties of those nontrivial models of the fijcalculus which are continuous complete lattices. The second part begins with a brief discussion of models of the calculus in set theories with antifoundation axioms. Next the model of the fi calculus of Part 1 and also the closely relatedbut different!models of Scott [53, 54] and of Engeler [21, 22] are reviewed....
A Coinduction Principle for Recursive Data Types Based on Bisimulation
, 1996
"... This paper provides foundations for a reasoning principle (coinduction) for establishing the equality of potentially infinite elements of selfreferencing (or circular) data types. As it is wellknown, such data types not only form the core of the denotational approach to the semantics of programmin ..."
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Cited by 37 (3 self)
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This paper provides foundations for a reasoning principle (coinduction) for establishing the equality of potentially infinite elements of selfreferencing (or circular) data types. As it is wellknown, such data types not only form the core of the denotational approach to the semantics of programming languages [SS71], but also arise explicitly as recursive data types in functional programming languages like Standard ML [MTH90] or Haskell [HPJW92]. In the latter context, the coinduction principle provides a powerful technique for establishing the equality of programs with values in recursive data types (see examples herein and in [Pit94]).
Dynamic Epistemic Logic
 Logic, Language, and Information 2, Stanford University, CSLI Publication
, 1997
"... This paper is the result of combining two traditions in formal logic: epistemic logic and dynamic semantics. Dynamic semantics is a branch of formal semantics that is concerned with ..."
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Cited by 36 (1 self)
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This paper is the result of combining two traditions in formal logic: epistemic logic and dynamic semantics. Dynamic semantics is a branch of formal semantics that is concerned with
DYNAMIC CONGRUENCE vs. PROGRESSING BISIMULATION for CCS
 Fundamenta Informaticae
, 1992
"... Weak Observational Congruence (woc) defined on CCS agents is not a bisimulation since it does not require two states reached by bisimilar computations of woc agents to be still woc, e.g. ff:ø:fi:nil and ff:fi:nil are woc but ø:fi:nil and fi:nil are not. This fact prevent us from characterizing CCS s ..."
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Cited by 32 (12 self)
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Weak Observational Congruence (woc) defined on CCS agents is not a bisimulation since it does not require two states reached by bisimilar computations of woc agents to be still woc, e.g. ff:ø:fi:nil and ff:fi:nil are woc but ø:fi:nil and fi:nil are not. This fact prevent us from characterizing CCS semantics (when ø is considered invisible) as a final algebra, since the semantic function would induce an equivalence over the agents that is both a congruence and a bisimulation. In the paper we introduce a new behavioural equivalence for CCS agents, which is the coarsest among those bisimulations which are also congruences. We call it Dynamic Observational Congruence because it expresses a natural notion of equivalence for concurrent systems required to simulate each other in the presence of dynamic, i.e. run time, (re)configurations. We provide an algebraic characterization of Dynamic Congruence in terms of a universal property of finality. Furthermore we introduce Progressing Bisimulatio...