361 research outputs found
Strongly Complete Logics for Coalgebras
Coalgebras for a functor model different types of transition systems in a
uniform way. This paper focuses on a uniform account of finitary logics for
set-based coalgebras. In particular, a general construction of a logic from an
arbitrary set-functor is given and proven to be strongly complete under
additional assumptions. We proceed in three parts. Part I argues that sifted
colimit preserving functors are those functors that preserve universal
algebraic structure. Our main theorem here states that a functor preserves
sifted colimits if and only if it has a finitary presentation by operations and
equations. Moreover, the presentation of the category of algebras for the
functor is obtained compositionally from the presentations of the underlying
category and of the functor. Part II investigates algebras for a functor over
ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical
extensions of Boolean algebras with operators to this setting. Part III shows,
based on Part I, how to associate a finitary logic to any finite-sets
preserving functor T. Based on Part II we prove the logic to be strongly
complete under a reasonable condition on T
Semantics out of context: nominal absolute denotations for first-order logic and computation
Call a semantics for a language with variables absolute when variables map to
fixed entities in the denotation. That is, a semantics is absolute when the
denotation of a variable a is a copy of itself in the denotation. We give a
trio of lattice-based, sets-based, and algebraic absolute semantics to
first-order logic. Possibly open predicates are directly interpreted as lattice
elements / sets / algebra elements, subject to suitable interpretations of the
connectives and quantifiers. In particular, universal quantification "forall
a.phi" is interpreted using a new notion of "fresh-finite" limit and using a
novel dual to substitution.
The interest of this semantics is partly in the non-trivial and beautiful
technical details, which also offer certain advantages over existing
semantics---but also the fact that such semantics exist at all suggests a new
way of looking at variables and the foundations of logic and computation, which
may be well-suited to the demands of modern computer science
Nominal Topology for Data Languages
We propose a novel topological perspective on data languages recognizable by
orbit-finite nominal monoids. For this purpose, we introduce pro-orbit-finite
nominal topological spaces. Assuming globally bounded support sizes, they
coincide with nominal Stone spaces and are shown to be dually equivalent to a
subcategory of nominal boolean algebras. Recognizable data languages are
characterized as topologically clopen sets of pro-orbit-finite words. In
addition, we explore the expressive power of pro-orbit-finite equations by
establishing a nominal version of Reiterman's pseudovariety theorem.Comment: Extended version of the corresponding paper accepted for ICALP 202
Varieties of Data Languages
We establish an Eilenberg-type correspondence for data languages, i.e.
languages over an infinite alphabet. More precisely, we prove that there is a
bijective correspondence between varieties of languages recognized by
orbit-finite nominal monoids and pseudovarieties of such monoids. This is the
first result of this kind for data languages. Our approach makes use of nominal
Stone duality and a recent category theoretic generalization of Birkhoff-type
HSP theorems that we instantiate here for the category of nominal sets. In
addition, we prove an axiomatic characterization of weak pseudovarieties as
those classes of orbit-finite monoids that can be specified by sequences of
nominal equations, which provides a nominal version of a classical theorem of
Eilenberg and Sch\"utzenberger
Varieties of Data Languages
We establish an Eilenberg-type correspondence for data languages, i.e.
languages over an infinite alphabet. More precisely, we prove that there is a
bijective correspondence between varieties of languages recognized by
orbit-finite nominal monoids and pseudovarieties of such monoids. This is the
first result of this kind for data languages. Our approach makes use of nominal
Stone duality and a recent category theoretic generalization of Birkhoff-type
HSP theorems that we instantiate here for the category of nominal sets. In
addition, we prove an axiomatic characterization of weak pseudovarieties as
those classes of orbit-finite monoids that can be specified by sequences of
nominal equations, which provides a nominal version of a classical theorem of
Eilenberg and Sch\"utzenberger
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words
Coalgebras and Their Logics
Transition systems pervade much of computer science. This article outlines the beginnings of a general theory of specification languages for transition systems. More specifically, transition systems are generalised to coalgebras. Specification languages together with their proof systems, in the following called (logical or modal) calculi, are presented by the associated classes of algebras (e.g., classical propositional logic by Boolean algebras). Stone duality will be used to relate the logics and their coalgebraic semantics
Complete Additivity and Modal Incompleteness
In this paper, we tell a story about incompleteness in modal logic. The story
weaves together a paper of van Benthem, `Syntactic aspects of modal
incompleteness theorems,' and a longstanding open question: whether every
normal modal logic can be characterized by a class of completely additive modal
algebras, or as we call them, V-BAOs. Using a first-order reformulation of the
property of complete additivity, we prove that the modal logic that starred in
van Benthem's paper resolves the open question in the negative. In addition,
for the case of bimodal logic, we show that there is a naturally occurring
logic that is incomplete with respect to V-BAOs, namely the provability logic
GLB. We also show that even logics that are unsound with respect to such
algebras do not have to be more complex than the classical propositional
calculus. On the other hand, we observe that it is undecidable whether a
syntactically defined logic is V-complete. After these results, we generalize
the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van
Benthem's theme of syntactic aspects of modal incompleteness
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