Intransitive temporal multi-agent’s logic, knowledge and uncertainty, plausibility

© Springer International Publishing Switzerland 2016. We study intransitive temporal logic implementing multiagent’s approach and formalizing knowledge and uncertainty. An innovative point here is usage of non-transitive linear time and multi-valued models - the ones using separate valuations Vj for agent’s knowledge of facts and summarized (agreed) valuation together with rules for computation truth values for compound formulas. The basic mathematical problems we study here are - decidability and decidability w.r.t. admissible rules. First, we study general case - the logic with non-uniform intransitivity and solve its decidability problem. Also we consider a modification of this logic - temporal logic with uniform non-transitivity and solve problem of recognizing admissibility in this logic

Non-transitive linear temporal logic and logical knowledge operations

© 2015 The Author, 2015. Published by Oxford University Press. All rights reserved.We study a linear temporal logic LTLNT with non-transitive time (with NEXT and UNTIL) and possible interpretations for logical knowledge operations in this approach. We assume time to be non-transitive, linear and discrete, it is a major innovative part of our article. Motivation for our approach that time might be non-transitive and comments on possible interpretations of logical knowledge operations are given. The main result of Section 5 is a solution of the decidability problem for LTLNT, we find and describe in details the decision algorithm. In Section 6 we introduce non-transitive linear temporal logic LTLNT(m) with uniform bound (m) for non-transitivity. We compare it with standard linear temporal logic LTL and the logic LTLNT - where non-transitivity has no upper bound - and show that LTLNT may be approximated by logics LTLNT(m). Concluding part of the article contains a list of open interesting problems

Unification in Linear Modal Logic on Non-transitive Time with the Universal Modality

We investigate the question of unification in the linear modal logic on non-transitive time with the universal modality. The semantic construction of logic on linear non-transitive Kripke frames is proposed, effective definability and projectivity of the unifiable formulas are proved. An algorithm for construction the most general unifier is found

Multiagent Temporal Logics with Multivaluations

We study multiagent logics and use temporal relational models with multivaluations. The key distinction from the standard relational models is the introduction of a particular valuation for each agent and the computation of the global valuation using all agents’ valuations. We discuss this approach, illustrate it with examples, and demonstrate that this is not a mechanical combination of standard models, but a much more subtle and sophisticated modeling of the computation of truth values in multiagent environments. To express the properties of these models we define a logical language with temporal formulas and introduce the logics based at classes of such models. The main mathematical problem under study is the satisfiability problem. We solve it and find deciding algorithms. Also we discuss some interesting open problems and trends of possible further investigations

Inference Rules in some temporal multi-epistemic propositional logics

Multi-modal logics are among the best tools developed so far to analyse human reasoning and agents’ interactions. Recently multi-modal logics have found several applications in Artificial Intelligence (AI) and Computer Science (CS) in the attempt to formalise reasoning about the behavior of programs. Modal logics deal with sentences that are qualified by modalities. A modality is any word that could be added to a statement p to modify its mode of truth. Temporal logics are obtained by joining tense operators to the classical propositional calculus, giving rise to a language very effective to describe the flow of time. Epistemic logics are suitable to formalize reasoning about agents possessing a certain knowledge. Combinations of temporal and epistemic logics are particularly effective in describing the interaction of agents through the flow of time. Although not yet fully investigated, this approach has found many fruitful applications. These are concerned with the development of systems modelling reasoning about knowledge and space, reasoning under uncertainty, multi-agent reasoning et c. Despite their power, multi modal languages cannot handle a changing environment. But this is exactly what is required in the case of human reasoning, computation and multi-agent environment. For this purpose, inference rules are a core instrument. So far, the research in this field has investigated many modal and superintuitionistic logics. However, for the case of multi-modal logics, not much is known concerning admissible inference rules. In our research we extend the investigation to some multi-modal propositional logics which combine tense and knowledge modalities. As far as we are concerned, these systems have never been investigated before. In particular we start by defining our systems semantically; further we prove such systems to enjoy the effective finite model property and to be decidable with respect to their admissible inference rules. We turn then our attention to the syntactical side and we provide sound and complete axiomatic systems. We conclude our dissertation by introducing the reader to the piece of research we are currently working on. Our original results can be found in [9, 4, 11] (see Appendix A). They have also been presented by the author at some international conferences and schools (see [8, 10, 5, 7, 6] and refer to Appendix B for more details). Our project concerns philosophy, mathematics, AI and CS. Modern applications of logic in CS and AI often require languages able to represent knowledge about dynamic systems. Multi-modal logics serve these applications in a very efficient way, and we would absorb and develop some of these techniques to represent logical consequences in artificial intelligence and computation

Behavioural Economics: Classical and Modern

In this paper, the origins and development of behavioural economics, beginning with the pioneering works of Herbert Simon (1953) and Ward Edwards (1954), is traced, described and (critically) discussed, in some detail. Two kinds of behavioural economics – classical and modern – are attributed, respectively, to the two pioneers. The mathematical foundations of classical behavioural economics is identified, largely, to be in the theory of computation and computational complexity; the corresponding mathematical basis for modern behavioural economics is, on the other hand, claimed to be a notion of subjective probability (at least at its origins in the works of Ward Edwards). The economic theories of behavior, challenging various aspects of 'orthodox' theory, were decisively influenced by these two mathematical underpinnings of the two theoriesClassical Behavioural Economics, Modern Behavioural Economics, Subjective Probability, Model of Computation, Computational Complexity. Subjective Expected Utility

Information sharing among ideal agents

Multi-agent systems operating in complex domains crucially require agents to interact with each other. An important result of this interaction is that some of the private knowledge of the agents is being shared in the group of agents. This thesis investigates the theme of knowledge sharing from a theoretical point of view by means of the formal tools provided by modal logic. More specifically this thesis addresses the following three points. First, the case of hypercube systems, a special class of interpreted systems as defined by Halpern and colleagues, is analysed in full detail. It is here proven that the logic S5WDn constitutes a sound and complete axiomatisation for hypercube systems. This logic, an extension of the modal system S5n commonly used to represent knowledge of a multi-agent system, regulates how knowledge is being shared among agents modelled by hypercube systems. The logic S5WDn is proven to be decidable. Hypercube systems are proven to be synchronous agents with perfect recall that communicate only by broadcasting, in separate work jointly with Ron van der Meyden not fully reported in this thesis. Second, it is argued that a full spectrum of degrees of knowledge sharing can be present in any multi-agent system, with no sharing and full sharing at the extremes. This theme is investigated axiomatically and a range of logics representing a particular class of knowledge sharing between two agents is presented. All the logics but two in this spectrum are proven complete by standard canonicity proofs. We conjecture that these two remaining logics are not canonical and it is an open problem whether or not they are complete. Third, following a influential position paper by Halpern and Moses, the idea of refining and checking of knowledge structures in multi-agent systems is investigated. It is shown that, Kripke models, the standard semantic tools for this analysis are not adequate and an alternative notion, Kripke trees, is put forward. An algorithm for refining and checking Kripke trees is presented and its major properties investigated. The algorithm succeeds in solving the famous muddy-children puzzle, in which agents communicate and reason about each other's knowledge. The thesis concludes by discussing the extent to which combining logics, a promising new area in pure logic, can provide a significant boost in research for epistemic and other theories for multi-agent systems