Computing Truth of Logical Statements in Multi-Agents’ Environment

Thispaperdescribeslogical models and computational algorithmsforlogical statements(specs) including various versions ofChanceDiscovery(CD).The approachisbased attemporal multi-agentlogic. Prime question is how to express most essential properties of CD in terms of temporal logic (branching time multi-agents’ logic or a linear one), how to define CD by formulas in logical language. We, as an example, introduce several formulas in the language of temporal multi-agent logic which may express essential properties of CD. Then we study computational questions (in particular, using some light modification of the standard filtration technique we show that the constructed logic has the finite-model property with effectively computable upper bound; this proves that the logic is decidable and provides a decision algorithm). At the final part of the paper we consider interpretation of CD via uncertainty and plausibility in an extension ofthelineartemporallogicLTL and computationfortruth values(satisfiability) ofits formulas.Представленная статья посвящена построению логических моделей различных версий теории случайных открытий (СО) и описанию вычислительных алгоритмов для логических высказываний. Предлагаемый нами подход основывается на многоагентной временной логике. Главный вопрос состоит в том, как можно было бы выразить самые существенные свойства СО в терминах временной логики, многоагентной логики с ветвящимся временем или линейной логики и вообще как определить СО с помощью формул языка логики. Нами в статье введено несколько формул на языке многоагентной временной логики, которые способны выразить существенные свойства СО. Используя некоторую модифицированную стандартную технику фильтрации, мы показали, что сконструированная таким образом логика имеет свойство финитной аппроксимируемости с эффективно вычислимой верхней границей. Это доказывает, что такая логика разрешима и нами предъявлен алгоритм разрешения. В заключительной части статьи мы рассматриваем интерпретацию СО посредством неопределённости и вероятности в расширении временной линейной логики и вычисление истинностных значений её формул

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

Instability of coherent states of a real scalar field

We investigate stability of both localized time-periodic coherent states (pulsons) and uniformly distributed coherent states (oscillating condensate) of a real scalar field satisfying the Klein-Gordon equation with a logarithmic nonlinearity. The linear analysis of time-dependent parts of perturbations leads to the Hill equation with a singular coefficient. To evaluate the characteristic exponent we extend the Lindemann-Stieltjes method, usually applied to the Mathieu and Lame equations, to the case that the periodic coefficient in the general Hill equation is an unbounded function of time. As a result, we derive the formula for the characteristic exponent and calculate the stability-instability chart. Then we analyze the spatial structure of the perturbations. Using these results we show that the pulsons of any amplitudes, remaining well-localized objects, lose their coherence with time. This means that, strictly speaking, all pulsons of the model considered are unstable. Nevertheless, for the nodeless pulsons the rate of the coherence breaking in narrow ranges of amplitudes is found to be very small, so that such pulsons can be long-lived. Further, we use the obtaned stability-instability chart to examine the Affleck-Dine type condensate. We conclude the oscillating condensate can decay into an ensemble of the nodeless pulsons.Comment: 11 pages, 8 figures, submitted to Physical Review

A Hybrid of Tense Logic S4T and Multi-Agent Logic with Interacting Agents

In this paper we introduce a temporal multi-agent logic S4IA T , which implements interacting agents. Logic S4IA T is defined semantically as the set of all formulas of the appropriate propositional language that are valid in special Kripke models. The models are based on S4-like time frames, i.e., with reflexive and transitive time-accessibility relations. Agents knowledge-accessibility relations Ri, defined independently for each individual agent, are S5-relations on R-time clusters, and interaction of the agents consists of passing knowledge along arbitrary paths of such relations. The key result of the paper is an algorithm for checking satisfiability and recognizing theorems of S4IA T . We also prove the effective finite model property for the logic S4IA T

Multi-agent logics with interacting agents based on linear temporal logic: deciding algorithms

We introduce a multi-agent logic – a variant of the linear temporal logic LTL with embedded multi-agent knowledge with interacting agents. The logic is motivated by semantics based on potentially infinite runs with time points represented by clusters of states with distributed knowledge of the agents. We address properties of local and global knowledge modeled in this framework, consider modeling of interaction between agents by possibility to puss information from one agent to others via possible transitions within time clusters of states. Main question we are focused on is the satisfiability problem and decidability of the logic . Key result is proposed algorithm which recognizes theorems of (so we show that is decidable). It is based on verification of validity for special normal reduced forms of rules in models with at most triple exponential size in the testing rules. In the final part we discuss possible variations of the proposed logic

Temporal logic with interacting agents

The paper deals with a temporal multi-agent logic T MAZ , which imitates taking of decisions based on agents' access to knowledge by their interaction. The interaction is modeled by possible communication channels between agents in special temporal Kripke/Hintikka-like models. The logic T MAZ distinguishes local and global decisions-making. T MAZ is based on temporal Kripke/Hintikka models with agents' accessibility relations defined on states of all possible time clusters C(i) (where indexes i range over all integer numbers Z). The main result provides a decision algorithm for T MAZ (so, we prove that T MAZ is decidable). This algorithm also solves the satisfiability problem. In the final part of the paper, we consider the admissibility problem for inference rules in T MAZ , and show that this problem is decidable for T MAZ as well

Logics with the universal modality and admissible consecutions

In this paper we study admissible consecutions (inference rules) in multi-modal logics with the universal modality. We consider extensions of multi-modal logic S4n augmented with the universal modality. Admissible consecutions form the largest class of rules, under which a logic (as a set of theorems) is closed. We propose an approach based on the context effective finite model property. Theorem 7, the main result of the paper, gives sufficient conditions for decidability of admissible consecutions in our logics. This theorem also provides an explicit algorithm for recognizing such consecutions. Some applications to particular logics with the universal modality are given

Linear temporal logic LTLK extended by multi-agent logic Kn with interacting agents

We study an extension LTLK of the linear temporal logic LTL by implementing multi-agent knowledge logic KD45m (which is often referred as multi-modal logic S5m). The temporal language of our logic adapts the operations U (until) and N (next) and uses new temporal operations: Uw—weak until, and Us—strong until. We also employ the standard agents’ knowledge operations Ki from the multi-agent logic KD45m and extend them with an operation IntK responsible for knowledge obtained via interaction of agents. The semantic models for LTLK are Kripke/Hintikka-like structures NC based on the linear time. Structures NC use i∈N as indexes for time, and the base set of any NC consists of clusters C(i) (for all i∈N) containing all possible states at the time i. Agents’ knowledge is modelled in time clusters C(i) via agents’ knowledge accessibility relations Rj . The logic LTLK is the set of all formulas which are valid (true) in all such models NC w.r.t. all possible valuations.We prove that LTLK is decidable: we reduce the decidability problem to verification of validity for special normal reduced forms of rules in specific models (not LTLK models) of size single-exponential in size of the rules. Furthermore, we extend these results to a linear temporal logic LTLK (Z) based on the time flow indexed by all integer numbers (with additional operations Since and Previous). Also we show that LTLK has the finite model property (fmp) while LTLK (Z) has no standard fmp

Logic of discovery in uncertain situations – deciding algorithms

We study a logic LDU (logic of Discovery in Uncertain Situations) generated in a semantic way as the set of all formulas valid in Kripke/Hintikka models, which are models of linear discrete time with time clusters imitating possible uncertain states. The possibility of discovery and uncertain necessity of discovery are modeled by modal operations. The logic LDU differs from all standard normal and non-normal modal logics because the modalities ate not mutually expressible in standard way. We discuss properties of this logic, i.e. study its fragments, compare LDU with well known modal logics and study the main question about decidability of this logic. We propose an algorithm recognizing theorems of LDU (so we show that LDU is decidable), which is based on verification of validity of special normal reduced forms of rules in models of quadratic polynomial size in the testing rules