This is an advanced 2001 textbook on modal logic, a field which caught the attention of computer scientists in the late 1970s. Model theory of modal logic 3 over the given frame in e. A modal logic has the nite model property fmp if it is an intersection of tabular logics. Modal logic modal logic is the logic of possible worlds. It is essentially game theoretic and so we build on this view. An established modal logic due to milner, parrow, and walker characterises late bisimilarity, that is, two processes satisfy the same set of formulae if and only if they are bisimilar.
Well look at some more metatheory of propositional. Modal logic can be shown to be as expressive as the socalled bisimulation inarianvt part of rstorder logic. Since kripke models are a special case of labelled state transition systems, bisimulation is also a topic in modal logic. But what kind of structures can modal logic talk about. Ivano ciardelli institute for logic, language, and computation university of amsterdam i. Often in applications of modal logic, we want to reason about a restricted class of structures, e. In modal logic, bisimulation is the fundamental concept of equivalence between structures, originally introduced by van. Firstorder modal logic, kripke semantics, bisimulation, goldblattthomason theorem, lindstrom theorem. A modal logic determined by a single nite frame is called tabular.
Im trying to muddle my way through a nonclassical logic course, and im especially stuck on the notion of bisimulation. Modal logic is, strictly speaking, the study of the deductive behavior of the. A modala word that expresses a modalityqualifies a statement. We illustrate this by establishing the finite model property and proving invariance and definability results. Pdf a characterisation of open bisimulation using an. An n modal logic lis called locally tabular if for any nite kthere exist nitely many n modal kformulas up to equivalence in l. A characterisation of open bisimilarity using an intuitionistic modal. Notation for simulation, bisimulation and modal equivalence. Bisimulation and hennessymilner logic for generalized. Its origins can be found in the analysis of modal logic but it was independently rediscovered by. Bisimulation invariant monadicsecond order logic in the. In this paper we consider modal team logic, a generalization of classical modal logic in which it is possible to describe dependence phenomena between data. In particular, the bisimulation based analysis of modal languages has been extensively studied in the amsterdam school of modal logic dr93, ger99, mv03, and it has even been suggested that this notion is as important for modal logic as the notion of partial isomorphism has been for the model theory of classical logic dr93. In fact, modal logic is the fragment of firstorder logic invariant under bisimulation van benthems theorem.
Similar to the van benthemhennessymilner result for nondeterministic transition systems, but for probabilistic systems. But worse than that, by the early 1980s, modal logic had also acquired powerful enemies within philosophy, preaching its imminent demise. A semantic perspective 3 chapters in this handbook. Its origins can be found in the analysis of modal logic but it was independently rediscovered by computer sci. On the other hand, the passage from local to global semantics is achieved if one looks at truth in all states an abstraction through implicit universal. Modal logic can be shown to be as expressive as the socalled bisimulation inarianvt part of.
Inquisitive modal logic, inqml, is a generalisation of standard kripkestyle. From bisimulation quantifiers to classifying toposes. A second track, comprising sections 3 to 5, is primarily devoted to modal logic as a logic of kripke structures. This note sketches the extension of the basic characterisation theorems as the bisimulationinvariant fragment of firstorder logic to. So we also describe modal mucalculus, modal logic with. Furthermore, in 11 a general theory of games is introduced in order to characterize process equivalences of the linearbranching time spectrum. Uniform interpolation for propositional and modal team. In each round, traveler moves along a link to arrive at a goal node, while demon deletes one link to prevent traveler.
It can also be characterised in terms of modal logic hennessymilner logic. Computer science, philosophical logic precisely, modal logic, set theory. According to ghilardi and zawadowski 1995a, it is known that pitts quantifiers pitts 1992 are semantically characterizable through direct and dual direct image in the topos of sheaves over the site given by the opposite category of finite heyting algebras endowed with the canonical topology. Modal logics characterising late bisimilarity and coarser bisimulations were developed early in the literature on the. The study of modal logics and various bisimulation equivalences so far shows the following progression.
The good properties of the basic modal logic may or may not survive. Bisimulation quantiers are a natural extension of modal logics. Such formalization allows to import, for free, a wealth of new notions e. These notes are meant to present the basic facts about modal logic and so to provide a common. Bisimulation games and locally tabular modal logics. Bisimulations between pointed frames can be defined likewise, by omitting atom equivalence. This is a graduatelevel text for a first course in propositional modal logic.
A bisimulation is a counterpart relation between states of two such models. Lecture notes modal logic linguistics and philosophy. In section 3 bisimulation equivalence due to park and milner is described. Metric bisimulation games and realvalued modal logics. Pdf minimal models and bisimulation in modal logic. It is written from the semantical point of view rather than the more usual proof theoretic approach, and the book covers all. The van benthem characterization theorem monotonic bisimulation let m hw,n,vi and m0 hw0,n0,v0i be two monotonic neighborhood models. Just as it is common to use metalevel application to represent objectlevel ap.
Pdf on the logic of argumentation theory semantic scholar. Acknowledgements we thank paul wild, lutz schroder and dirk pattinson for inspiring discussions on fuzzy modal logic, and in particular on. Handbook of modal logic download ebook pdf, epub, tuebl. Let m be the following modal logic where a ranges over a. However, we also nd cases where such intuitions are not preserved. This logic driven analysis of argumentation allows. The modeltheoretic invariant for the basic modal logic presented above is modal bisimulation. We show this can be a natural extension of modal logic preserving the intuitions of both modal logic and propositional quanti cation. Logics for bisimulation and divergence springerlink. On the origins of bisimulation and coinduction davide sangiorgi university of bologna, italy the origins of bisimulation and bisimilarity are examined, in the three. Modal bisimulation a bisimulation is a binary relation ebetween the worlds of two pointed models m. Modal formulas are interpreted on the states of labelled transition systems. Jan 21, 20 a brief, intuitive introduction to the basic concepts of modal logic. They preserve the bisimulation invariance of modal logic, while allowing monadic secondorder expressivity.
We further investigate the operators and semantics to which these results apply. Expressiveness of probabilistic modal logics, revisited. Modality, bisimulation and interpolation in infinitary logic. Model constructions bounded morphism let f 1 hw 1,n 1i and f. We prove that most known fragment of full modal team logic allow the elimination of the so called existential bisimulation quantifiers, where the existence of a certain set is made modulo. Pdf a note on bisimulation and modal equivalence in. We shall do this by using the fo logic miller and tiu 2005. Bisimulations and the standard translation guest lecture. Bisimulation and modal logic since kripke models are a special case of labelled state transition systems, bisimulation is also a topic in modal logic. Modal logic polymodal logic is an extension of propositional logic with formulas \ a \. An introduction to modal logic 2009 formosan summer school on logic, language, and computation 29 june10 july, 2009. Abstract model theory for extensions of modal logic. A logical characterization of probabilistic bisimulation. Using these bisimulations the model theory of graded modal logic can be developed in a uniform manner.
We introduce a notion of bisimulation for graded modal logic. An nmodal logic lis called locally tabular if for any nite kthere exist nitely many nmodal kformulas up to equivalence in l. Pdf doing argumentation theory in modal logic semantic. The paper applies modal logic to formalize fragments of argumentation theory. Probably silly question about bisimulation in modal logic. We provide a highlevel introduction to this logic here before presenting more technical aspects of it in the next section.
In this way, one obtains a correspondence between a family of classical logics and a family of modal logics. Researchers in areas ranging from economics to computational linguistics have since realised its worth. For example, modal logic can be given an algebraic semantics, and under this interpretation modal logic is a tool for talking about what are known as boolean algebras with operators. Despite the seemingly dense air of those concepts, it turns out the logic formalising them is limited in its power. Bisimulation games the discovery of bisimulation in modal logic. The present paper applies wellinvestigated modal logics to provide formal foundations to specific fragments of argumentation theory. It is the key semantic in variance for the modal propositional language over ltss, which has the usual boolean. The handbook of modal logic contains 20 articles, which collectively introduce contemporary modal logic, survey current research, and indicate the way in which the field is developing. As it happens, this notion first emerged in modal logic. Notions of bisimulation for heytingvalued modal languages. Characterisation theorems of this type have a strong tradition in the. There is a simple modal logic which characterizes behavioural equivalence of states in a probabilistic transition system. Sabotage game and sabotage modal logic this work is inspired by the work on sabotage game sg 10 and sabotage modal logic sml 5. Bisimulation for conditional modalities springerlink.
We give a definition of bisimulation for conditional modalities interpreted on selection functions and prove the correspondence between bisimilarity and modal equivalence, generalizing the hennessymilner theorem to a wide class of conditional operators. Counterfactuals, neighborhood semantics, probability, predicative necessity, etc. Besides bisimulation invariance, there is the related notion of bisimulation safety that generalises the same. Metric bisimulation games and realvalued modal logics for. A note on bisimulation and modal equivalence in provability logic and interpretability logic. A characterisation of the bisimulation invariant fragment of a given classical logic relates this logic to a suitable modal logic. Modal reasoning university of california, berkeley. In this paper, we are interested in one important notion called bisimulation. 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. Well look at some more metatheory of propositional modal logic. Modal logics and bounded fragments of predicate logic. Section 2 provides a new abstract version of this result, and a characterization of bisimulation follows naturally. Proof theory of modal logic download ebook pdf, epub. Using this notion, the model theory of graded modal logic can be developed in a uniform manner.