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GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS
"... The powerset construction is a standard method for converting a nondeterministic automaton into a deterministic one recognizing the same language. In this paper, we lift the powerset construction from automata to the more general framework of coalgebras with structured state spaces. Coalgebra is an ..."
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The powerset construction is a standard method for converting a nondeterministic automaton into a deterministic one recognizing the same language. In this paper, we lift the powerset construction from automata to the more general framework of coalgebras with structured state spaces. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the type of systems (Fcoalgebras) and a notion of behavioural equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for nondeterministic automata it is ordinary bisimilarity. We give several examples of applications of our generalized determinization construction, including partial Mealy machines, (structured) Moore automata, Rabin probabilistic automata, and, somewhat surprisingly, even pushdown automata. To further witness the generality of the approach we show how to characterize coalgebraically several equivalences which have been object of interest in the concurrency community, such as failure or ready
Borrowing Interpolation
, 2011
"... We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a ..."
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We present a generic method for establishing interpolation properties by ‘borrowing ’ across logical systems. The framework used is that of the socaled ‘institution theory’ which is a categorical abstract model theory providing a formal definition for the informal concept of ‘logical system’ and a mathematical concept of ‘homomorphism’ between logical systems. We develop three different styles or patterns to apply the proposed borrowing interpolation method. These three ways are illustrated by the development of a series of concrete interpolation results for logical systems that are used in mathematical logic or in computing science, most of these interpolation properties apparently being new results. These logical systems include fragments of (classical many sorted) first order logic with equality, preordered algebra and its Horn fragment, partial algebra, higher order logic. Applications are also expected for many other logical systems, including membership algebra, various types of order sorted algebra, the logic of predefined types, etc., and various combinations of the logical systems discussed here.
Revisiting Bisimilarity and its Modal Logic for Nondeterministic and Probabilistic Processes
 ACTA INFORMATICA
"... The logic PML is a probabilistic version of HennessyMilner logic introduced by Larsen and Skou to characterize bisimilarity over probabilistic processes without internal nondeterminism. In this paper, two alternative interpretations of PML over nondeterministic and probabilistic processes as mod ..."
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The logic PML is a probabilistic version of HennessyMilner logic introduced by Larsen and Skou to characterize bisimilarity over probabilistic processes without internal nondeterminism. In this paper, two alternative interpretations of PML over nondeterministic and probabilistic processes as models are considered, and two new bisimulationbased equivalences that are in full agreement with those interpretations are provided. The new equivalences include as coarsest congruences the two bisimilarities for nondeterministic and probabilistic processes proposed by Segala and Lynch. The latter equivalences are instead known to agree with two versions of HennessyMilner logic extended with an additional probabilistic operator interpreted over state distributions in place of individual states. The new interpretations of PML and the corresponding new bisimilarities are thus the first ones to offer a uniform framework for reasoning on processes that are purely nondeterministic or reactive probabilistic or that mix nondeterminism and probability in an alternating/nonalternating way.
Compositional reasoning for weighted Markov decision processes
, 2013
"... Weighted Markov decision processes (MDPs) have long been used to model quantitative aspects of systems in the presence of uncertainty. However, much of the literature on such MDPs takes a monolithic approach, by modelling a system as a particular MDP; properties of the system are then inferred by an ..."
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Weighted Markov decision processes (MDPs) have long been used to model quantitative aspects of systems in the presence of uncertainty. However, much of the literature on such MDPs takes a monolithic approach, by modelling a system as a particular MDP; properties of the system are then inferred by analysis of that particular MDP. In contrast in this paper we develop compositional methods for reasoning about weighted MDPs, as a possible basis for compositional reasoning about their quantitative behaviour. In particular we approach these systems from a process algebraic point of view. For these we define a coinductive simulationbased behavioural preorder which is compositional in the sense that it is preserved by structural operators for constructing weighted MDPs from components. For finitary convergent processes, which are finitestate and finitely branching systems without divergence, we provide two characterisations of the behavioural preorder. The first uses a novel quantitative probabilistic logic, while the second is in terms of a novel form of testing, in which benefits are accrued during the execution of tests. 1
Coinduction UpTo in a Fibrational Setting ∗
"... Bisimulation upto enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows ..."
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Bisimulation upto enhances the coinductive proof method for bisimilarity, providing efficient proof techniques for checking properties of different kinds of systems. We prove the soundness of such techniques in a fibrational setting, building on the seminal work of Hermida and Jacobs. This allows us to systematically obtain upto techniques not only for bisimilarity but for a large class of coinductive predicates modelled as coalgebras. By tuning the parameters of our framework, we obtain novel techniques for unary predicates and nominal automata, a variant of the GSOS rule format for similarity, and a new categorical treatment of weak bisimilarity. Categories and Subject Descriptors F.3 [Logics and meanings of programs]; F.4 [Mathematical logic and formal languages]
Algebraic Enriched Coalgebras?
"... Abstract. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the types of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be ca ..."
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Abstract. Coalgebra is an abstract framework for the uniform study of different kinds of dynamical systems. An endofunctor F determines both the types of systems (Fcoalgebras) and a notion of behavioral equivalence (∼F) amongst them. Many types of transition systems and their equivalences can be captured by a functor F. For example, for deterministic automata the derived equivalence is language equivalence, while for nondeterministic automata is ordinary bisimilarity. The powerset construction is a standard method for converting a nondeterministic automaton into an equivalent deterministic one as far as language is concerned. In this paper, we lift the powerset construction on automata to the more general framework of coalgebras with enriched state spaces. Examples of application include partial Mealy machines, (enriched) Moore automata, and Rabin probabilistic automata.
Beyond Regularity for Presburger Modal Logics
"... Satisfiability problem for modal logic K with quantifierfree Presburger and regularity constraints (EML) is known to be pspacecomplete. In this paper, we consider its extension with nonregular constraints, and more specifically those expressed by visibly pushdown languages (VPL). This class of lan ..."
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Satisfiability problem for modal logic K with quantifierfree Presburger and regularity constraints (EML) is known to be pspacecomplete. In this paper, we consider its extension with nonregular constraints, and more specifically those expressed by visibly pushdown languages (VPL). This class of languages behaves nicely, in particular when combined with Propositional Dynamic Logic (PDL). By extending EML, we show that decidability is preserved if we allow at most one positive VPLconstraint at each modal depth. However, the presence of two VPLcontraints or the presence of a negative occurrence of a single VPLconstraint leads to undecidability. These results contrast with the decidability of PDL augmented with VPLconstraints. Keywords: Presburger constraint, contextfree constraint, decidability
GENERALIZING DETERMINIZATION FROM AUTOMATA TO COALGEBRAS
, 2011
"... Vol. 9(1:09)2013, pp. 1–27 www.lmcsonline.org ..."
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Strong Completeness for IterationFree Coalgebraic Dynamic Logics
"... Abstract. We present a (co)algebraic treatment of iterationfree dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL), both without star. The main observation is that the program/game constructs of PDL/GL arise from monad structure, and the axioms of these logics corre ..."
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Abstract. We present a (co)algebraic treatment of iterationfree dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL), both without star. The main observation is that the program/game constructs of PDL/GL arise from monad structure, and the axioms of these logics correspond to certain compatibilty requirements between the modalities and this monad structure. Our main contribution is a general soundness and strong completeness result for PDLlike logics for Tcoalgebras where T is a monad and the ”program” constructs are given by sequential composition, test, and pointwise extensions of operations of T. 1