145 research outputs found
Coalgebraic completeness-via-canonicity for distributive substructural logics
We prove strong completeness of a range of substructural logics with respect
to a natural poset-based relational semantics using a coalgebraic version of
completeness-via-canonicity. By formalizing the problem in the language of
coalgebraic logics, we develop a modular theory which covers a wide variety of
different logics under a single framework, and lends itself to further
extensions. Moreover, we believe that the coalgebraic framework provides a
systematic and principled way to study the relationship between resource models
on the semantics side, and substructural logics on the syntactic side.Comment: 36 page
Bialgebraic Semantics for Logic Programming
Bialgebrae provide an abstract framework encompassing the semantics of
different kinds of computational models. In this paper we propose a bialgebraic
approach to the semantics of logic programming. Our methodology is to study
logic programs as reactive systems and exploit abstract techniques developed in
that setting. First we use saturation to model the operational semantics of
logic programs as coalgebrae on presheaves. Then, we make explicit the
underlying algebraic structure by using bialgebrae on presheaves. The resulting
semantics turns out to be compositional with respect to conjunction and term
substitution. Also, it encodes a parallel model of computation, whose soundness
is guaranteed by a built-in notion of synchronisation between different
threads
Generic Trace Semantics and Graded Monads
Models of concurrent systems employ a wide variety of semantics inducing various notions of process equivalence, ranging from linear-time semantics such as trace equivalence to branching-time semantics such as strong bisimilarity. Many of these generalize to system types beyond standard transition systems, featuring, for example, weighted, probabilistic, or game-based transitions; this motivates the search for suitable coalgebraic abstractions of process equivalence that cover these orthogonal dimensions of generality, i.e. are generic both in the system type and in the notion of system equivalence. In recent joint work with Kurz, we have proposed a parametrization of system equivalence over an embedding of the coalgebraic type functor into a monad. In the present paper, we refine this abstraction to use graded monads, which come with a notion of depth that corresponds, e.g., to trace length or bisimulation depth. We introduce a notion of graded algebras and show how they play the role of formulas in trace logics
Graded Monads and Graded Logics for the Linear Time - Branching Time Spectrum
State-based models of concurrent systems are traditionally considered under a variety of notions of process equivalence. In the case of labelled transition systems, these equivalences range from trace equivalence to (strong) bisimilarity, and are organized in what is known as the linear time - branching time spectrum. A combination of universal coalgebra and graded monads provides a generic framework in which the semantics of concurrency can be parametrized both over the branching type of the underlying transition systems and over the granularity of process equivalence. We show in the present paper that this framework of graded semantics does subsume the most important equivalences from the linear time - branching time spectrum. An important feature of graded semantics is that it allows for the principled extraction of characteristic modal logics. We have established invariance of these graded logics under the given graded semantics in earlier work; in the present paper, we extend the logical framework with an explicit propositional layer and provide a generic expressiveness criterion that generalizes the classical Hennessy-Milner theorem to coarser notions of process equivalence. We extract graded logics for a range of graded semantics on labelled transition systems and probabilistic systems, and give exemplary proofs of their expressiveness based on our generic criterion
Universal Constructions for (Co)Relations: categories, monoidal categories, and props
Calculi of string diagrams are increasingly used to present the syntax and
algebraic structure of various families of circuits, including signal flow
graphs, electrical circuits and quantum processes. In many such approaches, the
semantic interpretation for diagrams is given in terms of relations or
corelations (generalised equivalence relations) of some kind. In this paper we
show how semantic categories of both relations and corelations can be
characterised as colimits of simpler categories. This modular perspective is
important as it simplifies the task of giving a complete axiomatisation for
semantic equivalence of string diagrams. Moreover, our general result unifies
various theorems that are independently found in literature and are relevant
for program semantics, quantum computation and control theory.Comment: 22 pages + 3 page appendix, extended version of arXiv:1703.0824
Towards a Uniform Theory of Effectful State Machines
Using recent developments in coalgebraic and monad-based semantics, we
present a uniform study of various notions of machines, e.g. finite state
machines, multi-stack machines, Turing machines, valence automata, and weighted
automata. They are instances of Jacobs' notion of a T-automaton, where T is a
monad. We show that the generic language semantics for T-automata correctly
instantiates the usual language semantics for a number of known classes of
machines/languages, including regular, context-free, recursively-enumerable and
various subclasses of context free languages (e.g. deterministic and real-time
ones). Moreover, our approach provides new generic techniques for studying the
expressivity power of various machine-based models.Comment: final version accepted by TOC
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