Costas D. Koutras

Notions of Bisimulation for Heyting-Valued Modal Languages

We examine the notion of bisimulation and its ramifications, in the context of the family of Heyting-valued modal languages introduced by M. Fitting. Each modal language in this family is built on an underlying space of truth values, a Heyting algebra H. All the truth values are directly represented in the language, which is interpreted on relational frames with an H-valued accessibility relation. We define two notions of bisimulation that allow us to obtain truth invariance results. We provide game semantics and, for the more interesting and complicated notion, we are able to provide characteristic formulae and prove a Hennessy–Milner-type theorem. If the underlying algebra H is finite, Heyting-valued modal models can be equivalently reformulated to a form relevant to epistemic situations with many interrelated experts. Our definitions and results draw inspiration from this formulation, which is of independent interest to Knowledge Representation applications.

Prolegomena to Concise Theories of Action

A new methodology for developing theories of action has recently emerged which provides means for formally evaluating the correctness of such theories. Yet, for a theory of action to qualify as a solution to the frame problem, not only does it need to produce correct inferences, but moreover, it needs to derive these inferences from a concise representation of the domain at hand. The new methodology however offers no means for assessing conciseness. Such a formal account of conciseness is developed in this paper. Combined with the existing criterion for correctness, our account of conciseness offers a framework where proposed solutions to the frame problem can be formally evaluated.

Maps in Multiple Belief Change

Multiple Belief Change extends the classical AGM framework for Belief Revision introduced by Alchourron, Gardenfors, and Makinson in the early ’80s. The extended framework includes epistemic input represented as a (possibly infinite) set of sentences, as opposed to a single sentence assumed in the original framework. The transition from single to multiple epistemic input worked out well for the operation of belief revision. The AGM postulates and the system-of-spheres model were adequately generalized and so was the representation result connecting the two. In the case of belief contraction however, the transition was not as smooth. The generalized postulates for contraction, which were shown to correspond precisely to the generalized partial meet model, failed to match up to the generalized epistemic entrenchment model. The mismatch was fixed with the addition of an extra postulate, called the limit postulate, that relates contraction by multiple epistemic input to a series of contractions by single epistemic input. The new postulate however creates problems on other fronts. First, the limit postulate needs to be mapped into appropriate constraints in the partial meet model. Second, via the Levi and Harper Identities, the new postulate translates into an extra postulate for multiple revision, which in turn needs to be characterized in terms of systems of spheres. Both these open problems are addressed in this article. In addition, the limit postulate is compared with a similar condition in the literature, called (K*F), and is shown to be strictly weaker than it. An interesting aspect of our results is that they reveal a profound connection between rationality in multiple belief change and the notion of an elementary set of possible worlds (closely related to the notion of an elementary class of models from classical logic).

Knowledge means ‘all’, belief means ‘most’

We introduce a bimodal epistemic logic intended to capture knowledge as truth in all epistemically alternative states and belief as a generalised ‘majority’ quantifier, interpreted as truth in most (i.e. a ‘majority’) of the epistemically alternative states. This doxastic interpretation is of interest in knowledge-representation applications and it also holds an independent philosophical and technical appeal. The logic comprises an epistemic modal operator, a doxastic modal operator of consistent and complete belief and ‘bridge’ axioms which relate knowledge to belief. To capture the notion of a ‘majority’ we use the ‘large sets’ introduced independently by K. Schlechta and V. Jauregui, augmented with a requirement of completeness, which furnishes a ‘weak ultrafilter’ concept. We provide semantics in the form of possible-worlds frames, properly blending relational semantics with a version of general Scott–Montague (neighbourhood) frames and we obtain soundness and completeness results. We examine the validity of certain epistemic principles discussed in the literature, in particular some of the ‘bridge’ axioms discussed by W. Lenzen and R. Stalnaker, as well as the ‘paradox of the perfect believer’, which is not a theorem of .

In All, but Finitely Many, Possible Worlds: Model-Theoretic Investigations on ‘ Overwhelming Majority ’ Default Conditionals

Defeasible conditionals of the form ‘if A then normally B’ are usually interpreted with the aid of a ‘normality’ ordering between possible states of affairs: \(A\Rightarrow B\) is true if it happens that in the most ‘normal’ (least exceptional) A-worlds, B is also true. Another plausible interpretation of ‘normality’ introduced in nonmonotonic reasoning dictates that \(A\Rightarrow B\) is true iff B is true in ‘most’ A-worlds. A formal account of ‘most’ in this majority-based approach to default reasoning has been given through the usage of (weak) filters and (weak) ultrafilters, capturing at least, a basic core of a size-oriented approach to defeasible reasoning. In this paper, we investigate defeasible conditionals constructed upon a notion of ‘overwhelming majority’, defined as ‘truth in a cofinite subset of \(\omega \)’, the first infinite ordinal. One approach employs the modal logic of the frame \((\omega , <)\), used in the temporal logic of discrete linear time. We introduce and investigate conditionals, defined modally over \((\omega , <)\); several modal definitions of the conditional connective are examined, with an emphasis on the nonmonotonic ones. An alternative interpretation of ‘majority’ as sets cofinal (in \(\omega \)) rather than cofinite (subsets of \(\omega \)) is examined. For all these modal approaches over \((\omega , <)\), a decision procedure readily emerges, as the modal logic \(\mathbf {KD4LZ}\) of this frame is well-known and a translation of the conditional sentences can be mechanically checked for validity. A second approach employs the conditional version of Scott-Montague semantics, in the form of \(\omega \), endowed with neighborhoods populated by its cofinite subsets. Again, different conditionals are introduced and examined. Although it is not feasible to obtain a completeness theorem, since it is not easy to capture ‘cofiniteness-in-\(\omega \)’ syntactically, this research reveals the possible structure of ‘overwhelming majority’ conditionals, whose relative strength is compared to (the conditional logic ‘equivalent’ of) KLM logics and other conditional logics in the literature.

On the ‘in many cases’ Modality: Tableaux, Decidability, Complexity, Variants

The modality ‘true in many cases’ is used to handle non-classical patterns of reasoning, like ‘probably φ is the case’ or ‘normally φ holds’. It is of interest in Knowledge Representation as it has found interesting applications in Epistemic Logic, ‘Typicality’ logics, and it also provides a foundation for defining ‘normality’ conditionals in Non-Monotonic Reasoning. In this paper we contribute to the study of this modality, providing results on the ‘majority logic’ Θ of V. Jauregui. The logic Θ captures a simple notion of ‘a large number of cases’, which has been independently introduced by K. Schlechta and appeared implicitly in earlier attempts to axiomatize the modality ‘probably φ’. We provide a tableaux proof procedure for the logic Θ and prove its soundness and completeness with respect to the class of neighborhood semantics modelling ‘large’ sets of alternative situations. The tableaux-based decision procedure allows us to prove that the satisfiability problem for Θ is NP-complete. We discuss a more natural notion of ‘large’ sets which accurately captures ‘clear majority’ and we prove that it can be also used, at the high cost however of destroying the finite model property for the resulting logic. Then, we show how to extend our results in the logic of complete majority spaces, suited for applications where either a proposition or its negation (but not both) are to be considered ‘true in many cases’, a notion useful in epistemic logic.

On a Modal Epistemic Axiom Emerging from McDermott-Doyle Logics

An important question in modal nonmonotonic logics concerns the limits of propositional definability for logics of the McDermott-Doyle family. Inspired by this technical question we define a variant of autoepistemic logic which provably corresponds to the logic of the McDermott-Doyle family that is based on themodal axiom p5: ◊p⊃ (¬sp⊃s¬sp). This axiomis a naturalweakening of classical negative introspection restricting its scope to possible facts. It closely resembles the axiom w5: p⊃ (¬sp⊃s¬sp) which restricts the effect of negative introspection to true facts. We examine p5 in the context of classical possible-worlds Kripke models, providing results for correspondence, completeness and the finite model property. We also identify the corresponding condition for p5 in the context of neighbourhood semantics. Although rather natural epistemically, this axiom has not been investigated in classical modal epistemic reasoning, probably because its addition to S4 gives the well-known strong modal system S5.

Stathis Zachos at 70

This year we are celebrating the 70th birthday of Stathis! We take this chance to recall some of his remarkable contributions to Computer Science.

In All But Finitely Many Possible Worlds: Model-Theoretic Investigations on `Overwhelming Majority' Default Conditionals

Defeasible conditionals are statements of the form ‘ifAthen normallyB’. One plausible interpretation introduced in nonmonotonic reasoning dictates that (