Neighbourhood structures are the standard semantic tool used to reason about non-normal modal log... more Neighbourhood structures are the standard semantic tool used to reason about non-normal modal logics. The logic of all neighbourhood models is called classical modal logic. In coalgebraic terms, a neighbourhood frame is a coalgebra for the contravariant powerset functor composed with itself, denoted by 2 2 . We use this coalgebraic modelling to derive notions of equivalence between neighbourhood structures. 2 2 -bisimilarity and behavioural equivalence are well known coalgebraic concepts, and they are distinct, since 2 2 does not preserve weak pullbacks. We introduce a third, intermediate notion whose witnessing relations we call precocongruences (based on pushouts). We give back-and-forth style characterisations for 2 2 -bisimulations and precocongruences, we show that on a single coalgebra, precocongruences capture behavioural equivalence, and that between neighbourhood structures, precocongruences are a better approximation of behavioural equivalence than 2 2 -bisimulations. We also introduce a notion of modal saturation for neighbourhood models, and investigate its relationship with definability and image-finiteness. We prove a Hennessy-Milner theorem for modally saturated and for image-finite neighbourhood models. Our main results are an analogue of Van Benthem's characterisation theorem and a model-theoretic proof of Craig interpolation for classical modal logic.
Proceedings of The Royal Society B: Biological Sciences, 2006
We survey a number of decidablity and undecidablity results concerning epistemic temporal logic. ... more We survey a number of decidablity and undecidablity results concerning epistemic temporal logic. The goal is to provide a general picture which will facilitate the 'sharing of ideas' from a number of different areas concerned with modeling agents in interactive social situations.
Studia Logica - An International Journal for Symbolic Logic, 2006
The paper focuses on extending to the first order case the semantical program for modalities firs... more The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K) in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics.
Various combinations of temporal logics, epistemic and doxastic logics, and action logics have be... more Various combinations of temporal logics, epistemic and doxastic logics, and action logics have been used to reason about (groups of) agents in social situations. A key issue that has emerged is how best to represent and reason about the underlying protocol that governs the agents' interactions in a particular social situation. In this paper, we propose a PDL-style logic for reasoning about protocols under imperfect information.
Studia Logica - An International Journal for Symbolic Logic, 2007
Adam Brandenburger and H. Jerome Keisler have recently discovered a two person Russell-style para... more Adam Brandenburger and H. Jerome Keisler have recently discovered a two person Russell-style paradox. They show that the following configurations of beliefs is impossible: Ann believes that Bob assumes that Ann believes that Bob's assumption is wrong. In [7] a modal logic interpretation of this paradox is proposed. The idea is to introduce two modal operators intended to represent the agents' beliefs and assumptions. The goal of this paper is to take this analysis further and study this paradox from the point of view of a modal logician. In particular, we show that the paradox can be seen as a theorem of an appropriate hybrid logic.
Artemov introduced the Logic of Proofs (LP) as a logic of explicit proofs. We can also offer an e... more Artemov introduced the Logic of Proofs (LP) as a logic of explicit proofs. We can also offer an epistemic reading of this formula: "t is a possible justification of φ". Motivated, in part, by this epistemic reading, Fitting introduced a Kripke style semantics for LP in . In this note, we prove soundness and completeness of some axiom systems which are not covered in [8].
A recurring issue in any formal model representing agents’ (changing) informational attitudes is ... more A recurring issue in any formal model representing agents’ (changing) informational attitudes is how to account for the fact that the agents are limited in their access to the available inference steps, possible observations and available messages. This may be because the agents are not logically omniscient and so do not have unlimited reasoning ability. But it can also be because the agents are following a predefined protocol that explicitly limits statements available for observation and/or communication. Within the broad literature on epistemic logic, there are a variety of accounts that make precise a notion of an agent’s “limited access” (for example, Awareness Logics, Justification Logics, and Inference Logics). This paper interprets the agents’ access set of formulas as a constraint on the agents’ information gathering process limiting which formulas can be observed.
Proc. of the 12th International Conference on …, 2010
We present a formal semantical model to capture action, belief and intention, based on the "datab... more We present a formal semantical model to capture action, belief and intention, based on the "database perspective" (Shoham 2009). We then provide postulates for belief and intention revision, and state a representation theorem relating our postulates to the formal model. Our belief postulates are in the spirit of the AGM theory; the intention postulates stand in rough correspondence with the belief postulates.
We develop a dynamic modal logic that can be used to model scenarios where agents negotiate over ... more We develop a dynamic modal logic that can be used to model scenarios where agents negotiate over the allocation of a finite number of indivisible resources. The logic includes operators to speak about both preferences of individual agents and deals regarding the reallocation of certain resources. We reconstruct a known result regarding the convergence of sequences of mutually beneficial deals to a Pareto optimal allocation of resources, and discuss the relationship between reasoning tasks in our logic and problems in negotiation. For instance, checking whether a given restricted class of deals is sufficient to guarantee convergence to a Pareto optimal allocation for a specific negotiation scenario amounts to a model checking problem; and the problem of identifying conditions on preference relations that would guarantee convergence for a restricted class of deals under all circumstances can be cast as a question in modal logic correspondence theory.
Results in social choice theory such as the Arrow and Gibbard-Satterthwaite theorems constrain th... more Results in social choice theory such as the Arrow and Gibbard-Satterthwaite theorems constrain the existence of rational collective decision making procedures in groups of agents. The Gibbard-Satterthwaite theorem says that no voting procedure is strategy-proof. That is, there will always be situations in which it is in a voter's interest to misrepresent its true preferences i.e., vote strategically. We present some properties of strategic voting and then examine-via a bimodal logic utilizing epistemic and strategizing modalities-the knowledge-theoretic properties of voting situations and note that unless the voter knows that it should vote strategically, and how, i.e., knows what the other voters' preferences are and that it should vote a certain preference P , the voter will not strategize. Our results suggest that opinion polls in election situations effectively serve as the first n − 1 stages in an n stage election.
History based models, introduced by Parikh and Ramanujam, provide a natural mathematical model of... more History based models, introduced by Parikh and Ramanujam, provide a natural mathematical model of social interactive situations. These models offer a "low level" description of a social situation -describing the situation in terms of events, sequences of events, and the agents' view of these events. A multi-agent epistemic temporal modal logic can be used to reason about these structures. A number of other models have been proposed in the literature which can be used as a semantics for such a logical language. Most notably, the interpreted systems discussed by Fagin et al. In this paper, we will discuss the differences and similarities between these two mathematical models. In particular, it is shown that these two semantics are modally equivalent. We will conclude with a discussion of a number of questions that are raised when history based models are used to reason about game-theoretic situations.
… of Knowledge Representation and Reasoning, KR, 2004
Graded modal logic, as presented in , extends propositional modal systems with a set of modal ope... more Graded modal logic, as presented in , extends propositional modal systems with a set of modal operators 3 n (n ∈ N) that express "there are more than n accessible worlds such that...". We extend 1 GML with a modal operator W that can express "there are more than or equal to half of the accessible worlds such that...". The semantics of W is straightforward provided there are only finitely many accessible worlds; however if there are infinitely many accessible worlds the situation becomes much more complex. In order to deal with such situations, we introduce a majority space. A majority space is a set W together with a collection of subsets of W intended to be the weak majority (more than or equal to half) subsets of W . We then extend a standard Kripke structure with a function that assigns a majority space over the set of accessible states to each state. Given this extended Kripke semantics, majority logic is proved sound and complete.
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Papers by Eric Pacuit