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Register a new userCredulous Acceptability, Poison Games and Modal Logic
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Added by
Simon Rey
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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The Poison Game is a two-player game played on a graph in which one player can influence which edges the other player is able to traverse. It operationalizes the notion of existence of credulously admissible sets in an argumentation framework or, in graph-theoretic terminology, the existence of non-trivial semi-kernels. We develop a modal logic (poison modal logic, PML) tailored to represent winning positions in such a game, thereby identifying the precise modal reasoning that underlies the notion of credulous admissibility in argumentation. We study model-theoretic and decidability properties of PML, and position it with respect to recently studied logics at the cross-road of modal logic, argumentation, and graph games.
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Weakly Aggregative Modal Logic (WAML) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. WAML has some interesting applications on epistemic logic and logic of games, so we study some basic model theoretical aspects of WAML in this paper. Specifically, we give a van Benthem-Rosen characterization theorem of WAML based on an intuitive notion of bisimulation and show that each basic WAML system K_n lacks Craig Interpolation.
Term modal logics (TML) are modal logics with unboundedly many modalities, with quantification over modal indices, so that we can have formulas of the form $exists y. forall x. (Box_x P(x,y) supsetDiamond_y P(y,x))$. Like First order modal logic, TML is also notoriously undecidable, in the sense that even very simple fragments are undecidable. In this paper, we show the decidability of one interesting fragment, that of two variable TML. This is in contrast to two-variable First order modal logic, which is undecidable.
Mereology is the study of parts and the relationships that hold between them. We introduce a behavioral approach to mereology, in which systems and their parts are known only by the types of behavior they can exhibit. Our discussion is formally topos-theoretic, and agnostic to the topos, providing maximal generality; however, by using only its internal logic we can hide the details and readers may assume a completely elementary set-theoretic discussion. We consider the relationship between various parts of a whole in terms of how behavioral constraints are passed between them, and give an inter-modal logic that generalizes the usual alethic modalities in the setting of symmetric accessibility.
Quantified modal logic provides a natural logical language for reasoning about modal attitudes even while retaining the richness of quantification for referring to predicates over domains. But then most fragments of the logic are undecidable, over many model classes. Over the years, only a few fragments (such as the monodic) have been shown to be decidable. In this paper, we study fragments that bundle quantifiers and modalities together, inspired by earlier work on epistemic logics of know-how/why/what. As always with quantified modal logics, it makes a significant difference whether the domain stays the same across worlds, or not. In particular, we show that the bundle $forall Box$ is undecidable over constant domain interpretations, even with only monadic predicates, whereas $exists Box$ bundle is decidable. On the other hand, over increasing domain interpretations, we get decidability with both $forall Box$ and $exists Box$ bundles with unrestricted predicates. In these cases, we also obtain tableau based procedures that run in PSPACE. We further show that the $exists Box$ bundle cannot distinguish between constant domain and increasing domain interpretations.
Open bisimilarity is defined for open process terms in which free variables may appear. The insight is, in order to characterise open bisimilarity, we move to the setting of intuitionistic modal logics. The intuitionistic modal logic introduced, called $mathcal{OM}$, is such that modalities are closed under substitutions, which induces a property known as intuitionistic hereditary. Intuitionistic hereditary reflects in logic the lazy instantiation of free variables performed when checking open bisimilarity. The soundness proof for open bisimilarity with respect to our intuitionistic modal logic is mechanised in Abella. The constructive content of the completeness proof provides an algorithm for generating distinguishing formulae, which we have implemented. We draw attention to the fact that there is a spectrum of bisimilarity congruences that can be characterised by intuitionistic modal logics.
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