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