A coalgebraic view on positive modal logic alessandra palmigiano. Overview 1 a brief introduction to category 2 coalgebra 3 logical languages and semantics coalgebraic logics via predicate liftings cover modality 4 summary wang yunsong sms coalgebraic modal logic may 28th, 2019235. Coalgebraic modal logic is a generic framework that captures a wide range of modal logics from the literature, e. Introduction logics of all colours, shapes, and sizes have traditionally. T2 applications in automata theory and modal logic. A finite model construction for coalgebraic modal logic lutz schr oder dfkilab bremen and department of computer science, university of bremen, germany abstract in recent years, a tight connection has emerged between modal logic on the one hand and coalgebras, understood as generic transition systems, on the other hand. It is the aim of coalgebraic modal logic to create a general framework for modal logics to reason. Coalgebra example 10 to capture labelled transition systems in the coalgebraic framework, we consider the functor pawhere ais a setof actions, or labels and pxais the set of all functions of type a. Coalgebraic semantics of modal logics anu college of. These systems will be described as coalgebras of socalled polynomial. Our main contribution then is a novel coalgebraic logic, which can be seen as an equational axiomatisation of mosss logic. Open access publications 51688 freely accessible full text publications.
Coalgebras can be seen as a natural abstraction of kripke frames. Coalgebraic trace semantics via forgetful logics bartekklin1,andjurriaanrot2 1 universityofwarsaw 2 leidenuniversity,cwi abstract. Abstract a category colog of distributive laws is introduced to unify di erent approaches to modal logic for coalgebras, based merely on the presence of a contravariant functor p. Modal logic which is usually interpreted over relation structures or kripke frames can also be seen as a boolean algebra with operators baos. Reversing the viewpoint that modal logic is a suitable speci cation language for coalgebra, coalgebraic semantics captures, in a uniform way, the structural similarities of a large variety of di erent modal logics. Finitary coalgebraic logics institute for logic, language.
Algebraic logic is a discipline which uses tools and techniques from universal algebra to study logic. Coalgebraic modal logic beyond sets bartek klin1,2 warsaw university, edinburgh university abstract polyadic coalgebraic modal logic is studied in the setting of locally presentable categories. We use modal logic as a framework for coalgebraic trace. Semantics of graded modal logic there are many choices of semantics. A \full logic for each type of coalgebras, using the basic properties of adjunction, is identi ed. Modal logic is a textbook on modal logic, intended for readers already acquainted with the elements of formal logic. Pdf coalgebraic modal logic in cocasl researchgate. Apr 04, 2011 first, we have the coalgebraic approach to modal logic, where we build on the duality between stone spaces and boolean algebras.
This paper substantiates the authors firm belief that the systematic exploitation of this coalgebraic nature will not only have impact on the field of modal logic itself. A finite model construction for coalgebraic modal logic. Modal logic has a good claim to being the logic of choice for describing the reactive behaviour of systems modeled as coalgebras. Coalgebraic semantics for positive modal logic article pdf available in electronic notes in theoretical computer science 821. It is shown that under certain assumptions, accessible functors admit expressive logics for their coalgebras. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability. Two syntaxoriented approaches to coalgebraic modal logic moss cover modality. Reversing the viewpoint that modal logic is a suitable speci. A finite model construction for coalgebraic modal logic 159 state x in every tcoalgebra, x. They will be used to reprove completeness results, and hennesseymilner style characterisation results for the modal logic, first established by ro. Apr 15, 2002 in this setting we will see lindenbaum mbao models as initial objects, and canonical coalgebraic models of maximally consistent sets of formulas as final objects. In this setting we will see lindenbaum mbao models as initial objects, and canonical coalgebraic models of maximally consistent sets of formulas as final objects. A new textbook on modal logic providing a modern introduction to modal logic. This type of semantics also provides an excellent motivation and.
A category colog of distributive laws is introduced to unify different approaches to modal logic for coalgebras, based merely on the presence of a contravariant functor p that maps a state space to its collection of predicates. In the same sense, coalgebraic logics are generalised modal logics. For more information and background on coalgebraic modal logic, we refer to 12. It is shown that under certain assumptions, accessible functors admit expressive logics for. This paper substantiates the authors firm belief that the systematic exploitation of this coalgebraic nature will not only have impact on the field of modal logic itself but also lead to significant progress in a number of areas within computer science, such as knowledge representation and concurrencymobility. We focus on three 1 standard framesemantics onkripke frames 2 multiframe semantics onmultiframes, a type of weighted frames 3 predicatelifting coalgebraic semantics oncoalgebrasthat are equivalent to multiframes 353.
The geometric relational kripke semantics of modal logics are instances of coalgebraic semantics. I was trying to describe a multiagent system and as david pointed out you can do that coalgebraically if the number of agents is fixed a a is the set of agent labels. Applications of modal logics are abundant in computer science, and a large number of structurally different modal logics have been successfully employed in a diverse spectrum of application contexts. This paper gives a semantical underpinning for a manysorted modal logic associated with certain dynamical systems, like transition systems, automata or classes in objectoriented languages. Bisimulation for weakly expressive coalgebraic modal logics. Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness.
This item appears in the following collections academic publications 176654 academic output radboud university. Coalgebraic modal logic, as in 21, is a framework in which modal logics for specifying coalgebras can be developed parametric in the signature of the modal. Coalgebraic semantics for positive modal logic sciencedirect. One can also concentrate on a fragment of the language for. Coalgebras and modal logic functionallogic development and.
Lncs 3921 a finite model construction for coalgebraic. As described in stone coalgebras we can derive an endofunctor on the category of boolean algebras, ba ba, from a modal operator, algebras for which are modal algebras think lindenbaum algebra of a propositional logic having the necessary operator \box applied. Modal logic is a type of formal logic primarily developed in the 1960s that extends classical propositional and predicate logic to include operators expressing modality. First we approach modal logic with the methodology of algebraic logic, a discipline which aims at study. Here we will outline the algebraic and coalgebraic environments of the theory of modal logic. Lncs 3921 a finite model construction for coalgebraic modal. The chellas text in uenced me the most, though the order of presentation is inspired more by goldblatt. Marcus kracht, tools and techniques in modal logic, studies in logic and the foundation of mathematics, 142, elsevier, 1999.
A modala word that expresses a modalityqualifies a statement. We propose to extend the algebraic coalgebraic specification language cocasl by full coalgebraic modal logic based on predicate liftings for functors. The usual algebraic semantics of modal logic is in terms of boolean algebras with operators and is described in the entry algebraic models for modal logic. Modal logics are coalgebraic probabilistic information of economic agents 19, whereas e.
The algebraic perspective adds a new dimension to the theory of modal logic. A coalgebraic approach to graded modal logic and graded. Basic concepts in modal logic1 stanford university. An introduction to modal logic geared towards philosophy students. Van benthemrosen theorem for coalgebraic predicate logic. Wang yunsong sms coalgebraic modal logic may 28th, 20191035. While the canonical model construction is well understood for kripke semantics, nonnormal modal logics often present subtle difficulties up to the point that canonical models may fail to exist, as is the case e. We propose to extend the algebraiccoalgebraic specification language cocasl by full coalgebraic modal logic based on predicate liftings for functors.
On a purely categorical framework for coalgebraic modal logic. Pdf applications of modal logics are abundant in computer science, and a large number of structurally dierent modal logics have been successfully. The class of positive modal algebras is the one canonically associated with pml. Modularity of coalgebraic modal logics are recognised as standard categorical constructions in this category, avoiding any syntactic bookkeeping. This logic is more general than the modal logic previously used in cocasl and supports the specification of a variety of modal logics, such as graded modal logic, majority logic, and probabilistic modal logic. As described in stone coalgebras we can derive an endofunctor on the category of boolean algebras, ba ba, from a modal operator, algebras for which are modal algebras think lindenbaum algebra of a propositional logic having the necessary operator \box. General books on modal logics that include information on algebraic models include. On a categorical framework for coalgebraic modal logic. Relation liftings in coalgebraic modal logic msc thesis afstudeerscriptie written by johannes marti born december 19, 1986 in basel, switzerland under the supervision of prof dr yde venema, and submitted to the board of examiners in partial ful. A coalgebraic view on positive modal logic article in theoretical computer science 32712. Coalgebraic semantics, on the other hand, provides a uniform and encompassing view on the large variety of specific logics used in particular domains. Coalgebraic modal logic i was unable to continue this yesterday, so let me give a bit more precision to my ideas. Simulations and bisimulations for coalgebraic modal logics.
A coalgebraic view on positive modal logic request pdf. It is the aim of coalgebraic modal logic to create a general framework for modal logics to reason about a variety of semantic domains in a uniform way. This framework is parametric in syntax and semantics. We start by recalling basic notions from coalgebraic modal logic, and. Polyadic coalgebraic modal logic is studied in the setting of locally presentable categories. The framework of coalgebraic modal logic 9 covers a broad range of modalities including modal operators with nonrelational semantics, such as probabilistic and gametheoretic phenomena as well as neighbourhood semantics and nonmaterial conditionals. Venema, modal logic, cambridge tracts in theoretical computer science, vol.
387 768 776 1288 373 1512 992 880 1205 1378 645 1159 828 541 107 1060 181 506 748 681 1019 928 1509 514 925 1078 680 1345 441 1304 191 630 423 825 895 162 1090 635 46 180 536 1097 1396 597 1231 831 971 1324