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27
A Coalgebraic Perspective on Linear Weighted Automata
, 2011
"... Weighted automata are a generalization of nondeterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for nondeterministic automata, their behaviours can be expressed in terms of either ( ..."
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Cited by 11 (6 self)
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Weighted automata are a generalization of nondeterministic automata where each transition, in addition to an input letter, has also a quantity expressing the weight (e.g. cost or probability) of its execution. As for nondeterministic automata, their behaviours can be expressed in terms of either (weighted) bisimilarity or (weighted) language equivalence. Coalgebras provide a categorical framework for the uniform study of statebased systems and their behaviours. In this work, we show that coalgebras can suitably model weighted automata in two different ways: coalgebras on
ContextFree Languages, Coalgebraically
, 2011
"... We give a coalgebraic account of contextfree languages using the functor D(X) =2 × X A for deterministic automata over an alphabet A, in three different but equivalent ways: (i) by viewing contextfree grammars as Dcoalgebras; (ii) by defining a format for behavioural differential equations (w.r. ..."
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Cited by 10 (8 self)
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We give a coalgebraic account of contextfree languages using the functor D(X) =2 × X A for deterministic automata over an alphabet A, in three different but equivalent ways: (i) by viewing contextfree grammars as Dcoalgebras; (ii) by defining a format for behavioural differential equations (w.r.t. D) for which the unique solutions are precisely the contextfree languages; and (iii) as the Dcoalgebra of generalized regular expressions in which the Kleene star is replaced by a unique fixed point operator. In all cases, semantics is defined by the unique homomorphism into the final coalgebra of all languages, paving the way for coinductive proofs of contextfree language equivalence. Furthermore, the three characterizations can serve as the basis for the definition of a general coalgebraic notion of contextfreeness, which we see as the ultimate longterm goal of the present study.
A coalgebraic decision procedure for NetKAT
, 2014
"... Program equivalence is a fundamental problem that has practical applications across a variety of areas of computing including compilation, optimization, software synthesis, formal verification, and many others. Equivalence is undecidable in general, but in certain settings it is possible to develop ..."
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Cited by 7 (4 self)
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Program equivalence is a fundamental problem that has practical applications across a variety of areas of computing including compilation, optimization, software synthesis, formal verification, and many others. Equivalence is undecidable in general, but in certain settings it is possible to develop domainspecific languages that are expressive enough to be practical and yet sufficiently restricted so that equivalence remains decidable. In previous work we introduced NetKAT, a domainspecific language for specifying and verifying network packetprocessing functions. NetKAT provides familiar constructs such as tests, assignments, union, sequential composition, and iteration as well as custom primitives for modifying packet headers and encoding network topologies. Semantically, NetKAT is based on Kleene algebra with tests (KAT) and comes equipped with a sound and complete equational theory. Although NetKAT equivalence is decidable, the best known algorithm is hardly practical—it uses Savitch’s theorem to determinize a PSPACE algorithm and requires quadratic space. This paper presents a new algorithm for deciding NetKAT equivalence. This algorithm is based on finding bisimulations between finite automata constructed from NetKAT programs. We investigate the coalgebraic theory of NetKAT, generalize the notion of Brzozowski derivatives to NetKAT, develop efficient representations of NetKAT automata in terms of spines and sparse matrices, and discuss the highlights of our prototype implementation. 1.
A Probabilistic Kleene Theorem
"... We provide a Kleene Theorem for (Rabin) probabilistic automata over finite words. Probabilistic automata generalize deterministic finite automata and assign to a word an acceptance probability. We provide probabilistic expressions with probabilistic choice, guarded choice, concatenation, and a sta ..."
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Cited by 2 (1 self)
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We provide a Kleene Theorem for (Rabin) probabilistic automata over finite words. Probabilistic automata generalize deterministic finite automata and assign to a word an acceptance probability. We provide probabilistic expressions with probabilistic choice, guarded choice, concatenation, and a star operator. We prove that probabilistic expressions and probabilistic automata are expressively equivalent. Our result actually extends to twoway probabilistic automata with pebbles and corresponding expressions.
Realization of Coinductive Types
, 2011
"... We give an explicit combinatorial construction of final coalgebras for a modest generalization of polynomial functors on Set. Type signatures are modeled as directed multigraphs instead of endofunctors. The final coalgebra for a type signature F involves the notion of Brzozowski derivative on sets o ..."
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We give an explicit combinatorial construction of final coalgebras for a modest generalization of polynomial functors on Set. Type signatures are modeled as directed multigraphs instead of endofunctors. The final coalgebra for a type signature F involves the notion of Brzozowski derivative on sets of paths in F. 1
Coalgebraic characterizations of contextfree languages
 Logical Methods in Computer Science
"... Abstract. In this article, we provide three coalgebraic characterizations of the class of contextfree languages, each based on the idea of adding coalgebraic structure to an existing algebraic structure by specifying outputderivative pairs. Final coalgebra semantics then gives an interpretation fu ..."
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Abstract. In this article, we provide three coalgebraic characterizations of the class of contextfree languages, each based on the idea of adding coalgebraic structure to an existing algebraic structure by specifying outputderivative pairs. Final coalgebra semantics then gives an interpretation function into the final coalgebra of all languages with the usual output and derivative operations. The first characterization is based on systems, where each derivative is given as a finite language over the set of nonterminals; the second characterization on systems where derivatives are given as elements of a termalgebra; and the third characterization is based on adding coalgebraic structure to a class of closed (unique) fixed point expressions. We prove equivalences between these characterizations, discuss the generalization from languages to formal power series, as well as the relationship to the generalized powerset construction. 1.
LeftHanded Completeness
, 2011
"... We give a new, significantly shorter proof of the completeness of the lefthanded star rule of Kleene algebra. The proof reveals the rich interaction of algebra and coalgebra in the theory. 1 ..."
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We give a new, significantly shorter proof of the completeness of the lefthanded star rule of Kleene algebra. The proof reveals the rich interaction of algebra and coalgebra in the theory. 1
Completeness and Incompleteness in Nominal Kleene Algebra
, 2014
"... Gabbay and Ciancia (2011) presented a nominal extension of Kleene algebra as a framework for trace semantics with dynamic allocation of resources, along with a semantics consisting of nominal languages. They also provided an axiomatization that captures the behavior of the scoping operator and its ..."
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Gabbay and Ciancia (2011) presented a nominal extension of Kleene algebra as a framework for trace semantics with dynamic allocation of resources, along with a semantics consisting of nominal languages. They also provided an axiomatization that captures the behavior of the scoping operator and its interaction with the Kleene algebra operators and proved soundness over nominal languages. In this paper we show that the axioms are complete and describe the free language models. 1
Towards a Theory of Glue
"... We propose and study the notions of behaviour type and composition operator making a first step towards the definition of a formal framework for studying behaviour composition in a setting sufficiently general to provide insight into how the componentbased systems should be modelled and compared. W ..."
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We propose and study the notions of behaviour type and composition operator making a first step towards the definition of a formal framework for studying behaviour composition in a setting sufficiently general to provide insight into how the componentbased systems should be modelled and compared. We illustrate the proposed notions on classical examples (Traces, Labelled Transition Systems and Coalgebras). Finally, the definition of memoryless glue operators, takes us one step closer to a formal understanding of the separation of concerns principle stipulating that computational aspects of a system should be localised within its atomic components, whereas coordination layer responsible for managing concurrency should be realised by memoryless glue operators. 1