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Countdown μ-calculus
[article]

2022
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arXiv
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pre-print

We introduce the countdown μ-calculus, an extension of the modal μ-calculus with ordinal approximations of fixpoint operators. In addition to properties definable in the classical calculus, it can express (un)boundedness properties such as the existence of arbitrarily long sequences of specific actions. The standard correspondence with parity games and automata extends to suitably defined countdown games and automata. However, unlike in the classical setting, the scalar fragment is provably

arXiv:2208.00536v1
fatcat:os6irkf3cngingzjgybueutneq
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... er than the full vectorial calculus and corresponds to automata satisfying a simple syntactic condition. We establish some facts, in particular decidability of the model checking problem and strictness of the hierarchy induced by the maximal allowed nesting of our new operators.##
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Monadic Monadic Second Order Logic
[article]

2022
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arXiv
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pre-print

One of the main reasons for the correspondence of regular languages and monadic second-order logic is that the class of regular languages is closed under images of surjective letter-to-letter homomorphisms. This closure property holds for structures such as finite words, finite trees, infinite words, infinite trees, elements of the free group, etc. Such structures can be modelled using monads. In this paper, we study which structures (understood via monads in the category of sets) are such that

arXiv:2201.09969v1
fatcat:eva2hakhinhfnl3pxsardzgljy
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... the class of regular languages (i.e. languages recognized by finite algebras) are closed under direct images of surjective letter-to-letter homomorphisms. We provide diverse sufficient conditions for a monad to satisfy this property. We also present numerous examples of monads, including positive examples that do not satisfy our sufficient conditions, and counterexamples where the closure property fails.##
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Towards nominal computation

2012
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SIGPLAN notices
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Nominal sets are a different kind of set theory, with a more relaxed notion of finiteness. They offer an elegant formalism for describing λ-terms modulo α-conversion, or automata on data words. This paper is an attempt at defining computation in nominal sets. We present a rudimentary programming language, called Nλ. The key idea is that it includes a native type for finite sets in the nominal sense. To illustrate the power of our language, we write short programs that process automata on data words.

doi:10.1145/2103621.2103704
fatcat:m6jahqo55jeqldapa6vj3js33a
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Syntactic Formats for Free
[chapter]

2003
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Lecture Notes in Computer Science
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A framework of Plotkin and Turi's, originally aimed at providing an abstract notion of bisimulation, is modified to cover other operational equivalences and preorders. Combined with bialgebraic methods, it yields a technique for the derivation of syntactic formats for transition system specifications which guarantee operational preorders to be precongruences. The technique is applied to the trace preorder, the completed trace preorder and the failures preorder. In the latter two cases, new

doi:10.1007/978-3-540-45187-7_5
fatcat:2q43rctgcfgs5e2oniteegetfu
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... ctic formats ensuring precongruence properties are introduced.##
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Learning nominal automata

2017
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SIGPLAN notices
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We present an Angluin-style algorithm to learn nominal automata, which are acceptors of languages over infinite (structured) alphabets. The abstract approach we take allows us to seamlessly extend known variations of the algorithm to this new setting. In particular we can learn a subclass of nominal non-deterministic automata. An implementation using a recently developed Haskell library for nominal computation is provided for preliminary experiments.

doi:10.1145/3093333.3009879
fatcat:i36tyni3ezbuxbztgp33hqzabi
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Institution Independent Static Analysis for Casl
[chapter]

2002
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Lecture Notes in Computer Science
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We describe a way to make the static analysis for the inthe-large part of the Common Algebraic Specification Language (Casl) independent of the underlying logic that is used for specification in-thesmall. The logic here is formalized as an institution with some extra components. Following the institution independent semantics of Casl in-the-large, we thus get an institution independent static analysis for Casl in-the-large. With this, it is possible to re-use the Casl static analysis for

doi:10.1007/3-540-45645-7_11
fatcat:6nlzh3v6frg3jktvmxk6scsxdy
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... ons of Casl, or even completely different logics. One only has to provide a static analysis for specifications in-the-small for the given logic. This then can be plugged into the generic static analysis for Casl in-the-large.##
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Coalgebraic Trace Semantics via Forgetful Logics
[chapter]

2015
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Lecture Notes in Computer Science
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We use modal logic as a framework for coalgebraic trace semantics, and show the flexibility of the approach with concrete examples such as the language semantics of weighted, alternating and tree automata. We provide a sufficient condition under which a logical semantics coincides with the trace semantics obtained via a given determinization construction. Finally, we consider a condition that guarantees the existence of a canonical determinization procedure that is correct with respect to a

doi:10.1007/978-3-662-46678-0_10
fatcat:6b25hn4mezhzljpc6rvn22loam
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... n logical semantics. That procedure is closely related to Brzozowski's minimization algorithm.##
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Structural Operational Semantics for Weighted Transition Systems
[chapter]

2009
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Lecture Notes in Computer Science
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Weighted transition systems are defined, parametrized by a commutative monoid of weights. These systems are further understood as coalgebras for functors of a specific form. A general rule format for the SOS specification of weighted systems is obtained via the coalgebraic approach of Turi and Plotkin. Previously known formats for labelled transition systems (GSOS) and stochastic systems (SGSOS) appear as special cases.

doi:10.1007/978-3-642-04164-8_7
fatcat:dq65lpqzerfnzl7br7cdmzormy
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Bialgebras for structural operational semantics: An introduction

2011
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Theoretical Computer Science
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Bialgebras and distributive laws are an abstract, categorical framework to study various flavors of structural operational semantics. This paper aims to introduce the reader to the basics of bialgebras for operational semantics, and to sketch the state of the art in this research area. P is the (covariant) powerset, and P ω the finite powerset functor on Set. P does not admit final coalgebras for cardinality reasons, but a final P ω -coalgebra exists. Two states in an LTS are observationally

doi:10.1016/j.tcs.2011.03.023
fatcat:zct3uyknbrfyhem36ygu7z5laq
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... ivalent if and only if they are bisimilar; see e.g. [45] for more details on the coalgebraic understanding of LTSs. Example 6. A finitely supported probability distribution on a set X is a function ν : X → [0, 1] such that ν(x) = 0 for all but finitely many x, and ∑ x∈X ν(x) = 1. A (reactive) probabilistic transition system (PTS) is a triple (X, L, µ), where X is a set of states, L a set of labels, and the transition function µ : X × L × X → [0, 1] is such that µ(x, a, −) is either the constantly zero function or a finitely supported probability distribution on X , for every x ∈ X and a ∈ L. PTSs are in one-to-one correspondence with coalgebras for the endofunctor (1 is the set of all finitely supported probability distributions on X . This functor admits a final coalgebra. Two states in a PTS are observationally equivalent if and only if they are related by a probabilistic bisimulation; see [9] for a detailed coalgebraic treatment of PTSs. Several other kinds of transition systems can be modeled as coalgebras for various behavior functors; see e.g. [45] for further examples.##
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Coalgebraic Modal Logic Beyond Sets

2007
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Electronical Notes in Theoretical Computer Science
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Polyadic coalgebraic modal logic is studied in the setting of locally presentable categories. It is shown that under certain assumptions, accessible functors admit expressive logics for their coalgebras. Examples include typical functors used to describe systems with name binding, interpreted in nominal sets. The first abstract approach to logics for coalgebras was coalgebraic logic of Moss [25] , providing expressive logics for essentially all functors on Set. However general, coalgebraic

doi:10.1016/j.entcs.2007.02.034
fatcat:ysh64itqcbcibl47fgwz3mosgm
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... is rather difficult to use in practice, as its syntax involve applications of the behaviour functor to formulas, and it does not provide simple and natural modalities like those known from Hennessy-Milner or similar logics. On the other hand, logics developed in [15, 19, 26, 28] are close to their usual concrete presentations, but their expressivity depends on some conditions imposed on the behaviour functor. For example, modalities in [26] are predicate liftings, which map predicates on X to predicates on BX, where B is the behaviour functor, and the resulting modal logic is expressive provided enough predicate liftings exist for B. This approach was analyzed and generalized by Schröder [30] , who noted that predicate liftings are equivalent to functions B2 → 2 and considered polyadic modal logic, where modalities of any arity, such as functions B(2 n ) → 2, are allowed. He then proved polyadic modal logics expressive for all accessible behaviour functors. All results mentioned above apply to functors on Set. In particular, Schröder's expressivity proof is set-theoretic in nature and it is not immediately clear how to translate it to other base categories. It is the purpose of this paper to generalize the definition of polyadic modal logic, and the proof of its expressivity, to accessible functors on locally presentable categories that satisfy some additional conditions. Our approach is inspired by recent work by Kurz and Bonsangue [6, 7, 20, 21] , who use Stone dualities to obtain logics for coalgebras on arbitrary categories, and by that by Pavlovic, Mislove and Worrell [27], who exploit logical connections between data and tests to develop an abstract theory of testing. In those works, as in the present paper, contravariant adjunctions provide the infrastructure for linking processes and formulas. In [6, 7, 20] , the adjunctions are assumed to be categorical dualities. This easily implies the existence of expressive logics for all functors, and the main effort is directed towards the nontrivial task of finding concrete presentations of those logics; to that end, in [21] adjunctions that are not dualities were used. In the present, more flexible approach, the duality assumption is not made. This often makes concrete presentations of expressive logics easier to find, and opens a possibility to treat various interesting, but non-expressive logics in a uniform fashion, but it comes for a price: the existence of expressive logics depends on certain conditions, as in [30] . On the other hand, in [27] the duality assumption is not made, and the adjunctions arise from certain cogenerators in the relevant categories. This does not apply to all examples of interest, and in the present paper we work with more general adjunctions. Also, in [27] the main focus is on non-expressive logics, and no expressivity results are provided there. The paper is structured as follows. After §2 of technical preliminaries, §3 presents a categorical generalization of Schröder's polyadic modal logic, which is proved expressive under some conditions in §4. In §5, a categorical notion of modality is suggested. Examples for functors on three different categories are studied in §6.##
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Definable isomorphism problem

2018
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Logical Methods in Computer Science
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We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying structure of atoms, we prove decidability of the problem. The core result is parameter-elimination: existence of an isomorphism definable with parameters implies existence of an isomorphism definable without parameters. Example 1.1. Let Atoms be a countable set {1,

doi:10.23638/lmcs-15(4:14)2019
fatcat:febvwu6pcnhj5inrt6zmtqqhoi
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... 2, 3, . . .} equipped with the equality relation only; we shall call this structure the pure set. Let Both V and E are definable sets (over Atoms), as they are constructed from Atoms using (possibly nested) set-builder expressions with first-order guards ranging over Atoms. In general, we allow finite unions in the definitions, and finite tuples (as above) are allowed for notational convenience. Precise definitions are given in Section 2. The pair G = (V, E) is also a definable set, in fact, a definable graph. It is an infinite Kneser graph (a generalization Key words and phrases: isomorphism problem, definable sets, ω-categoricity.##
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Definable isomorphism problem
[article]

2019
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arXiv
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pre-print

We investigate the isomorphism problem in the setting of definable sets (equivalent to sets with atoms): given two definable relational structures, are they related by a definable isomorphism? Under mild assumptions on the underlying structure of atoms, we prove decidability of the problem. The core result is parameter-elimination: existence of an isomorphism definable with parameters implies existence of an isomorphism definable without parameters.

arXiv:1802.08500v3
fatcat:wwsazugwmjerbbzfs6vcgyu3t4
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Automata with Group Actions

2011
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2011 IEEE 26th Annual Symposium on Logic in Computer Science
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Our motivating question is a Myhill-Nerode theorem for infinite alphabets. We consider several kinds of those: alphabets whose letters can be compared only for equality, but also ones with more structure, such as a total order or a partial order. We develop a framework for studying such alphabets, where the key role is played by the automorphism group of the alphabet. This framework builds on the idea of nominal sets of Gabbay and Pitts; nominal sets are the special case of our framework where

doi:10.1109/lics.2011.48
dblp:conf/lics/BojanczykKL11
fatcat:nrrsy7qxojflzf2jkm2aloesfe
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... etters can be only compared for equality. We use the framework to uniformly generalize to infinite alphabets parts of automata theory, including decidability results. In the case of letters compared for equality, we obtain automata equivalent in expressive power to finite memory automata, as defined by Francez and Kaminski.##
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Bialgebraic methods and modal logic in structural operational semantics

2009
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Information and Computation
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Bialgebraic semantics, invented a decade ago by Turi and Plotkin, is an approach to formal reasoning about well-behaved structural operational semantics (SOS). An extension of algebraic and coalgebraic methods, it abstracts from concrete notions of syntax and system behaviour, thus treating various kinds of operational descriptions in a uniform fashion. In this paper, bialgebraic semantics is combined with a coalgebraic approach to modal logic in a novel, general approach to proving the

doi:10.1016/j.ic.2007.10.006
fatcat:pkohrmweqneh3jfglugaaxieiy
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... ionality of process equivalences for languages defined by structural operational semantics. To prove compositionality, one provides a notion of behaviour for logical formulas, and defines an SOS-like specification of modal operators which reflects the original SOS specification of the language. This approach can be used to define SOS congruence formats as well as to prove compositionality for specific languages and equivalences.##
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From Bialgebraic Semantics to Congruence Formats

2005
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Electronical Notes in Theoretical Computer Science
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*Klin*/ Electronic Notes in Theoretical ComputerScience 128 (2005) 3-37 ...

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