Showing papers in "Journal of Applied Non-Classical Logics in 2010"

Modal logics with Belnapian truth values

Abstract: Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification (support of truth) and the other for falsification (support of falsity). Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to twist structures over modal algebras. Sound and complete tableau calculi are presented as well. Moreover, a number of constructive non-modal logics with strong negation are faithfully embedded into BS4, into its three-valued extension B3S4, or into temporal BS4, BtS4. These logics include David Nelson's three-valued logic N3, the four-valued logic N4 bottom, the connexive logic C, and several extensions of bi-intuitionistic logic by strong ...

Model-checking CTL* over flat Presburger counter systems

Abstract: This paper studies model-checking of fragments and extensi ons of CTL* on infinite- state counter systems, where the states are vectors of integ ers and the transitions are determined by means of relations definable within Presburger arithmeti c. In general, reachability properties of counter systems are undecidable, but we have identified a n atural class of admissible counter systems (ACS) for which we show that the quantification over p in CTL* can be simulated by quantification over tuples of natural numbers, eventuall y allowing translation of the whole Presburger-CTL* into Presburger arithmetic, thereby enab ling effective model checking. We provide evidence that our results are close to optimal with r espect to the class of counter systems described above.

Continuity, freeness, and filtrations

Abstract: The role played by continuous morphisms in propositional modal logic is investigated: it turns out that they are strictly related to filtrations and to suitable variants of the notion of a free alg...

Label-free natural deduction systems for intuitionistic and classical modal logics

Abstract: In this paper we study natural deduction for the intuitionistic and classical (normal) modal logics obtained from the combinations of the axioms T, B, 4 and 5. In this context we introduce a new multi-contextual structure, called T-sequent, that allows to design simple labelfree natural deduction systems for these logics. After proving that they are sound and complete we show that they satisfy the normalization property and consequently the subformula property in the intuitionistic case.

Logics for multi-subset spaces

Abstract: We generalize Moss and Parikh's logic of knowledge, effort, and topological reasoning, in two ways. We develop both a multi-agent and a multi-method setting for it. In each of these cases, we prove...

Some modal aspects of XPath

Abstract: This paper provides several examples of how known results in modal logic can be applied to the XML document navigation language XPath. The applications concern complete axiomatizations and expressive power. XML is a standard for storage and exchange of data on the Internet. It is tailored towards hierarchically structured data, and its basic data structure is that of a (finite) sibling-ordered tree: a tree in which the children of each node are linearly ordered. XPath is a language used for retrieving elements from XML documents. As has been observed by several people, XPath is closely related to modal logic. In this paper, we illustrate the connection by showing how a number of (old and new) results on XPath fragments can be derived from known results in modal logic. These results concern axiomatizations and expressivity of XPath fragments, both of which are important topics in database theory. For instance, complete axiomatizations for XPath fragments can be understood as complete sets of rewrite rules. Identifying e! ective rules for rewriting expressions to simpler but equivalent ones is important for query optimization.

Back from the future

Abstract: Until is a notoriously difficult temporal operator as it is both existential and universal at the same time: A∪B holds at the current time instant w iff either B holds at w or there exists a time i...

An efficient approach to nominal equalities in hybrid logic tableaux

Abstract: Basic hybrid logic extends modal logic with the possibility of naming worlds by means of a distinguished class of atoms (called nominals) and the so-called satisfaction operator, that allows one to...

Verifying one hundred prisoners and a lightbulb

Abstract: This is a case-study in knowledge representation and dynamic epistemic protocol verification. We analyze the ‘one hundred prisoners and a lightbulb’ puzzle. In this puzzle it is relevant what the agents (prisoners) know, how their knowledge changes due to observations, and how they affect the state of the world by changing facts, i.e., by their actions. These actions depend on the history of previous actions and observations. Part of its interest is that all actions are local, i.e. not publicly observable, and part of the problem is therefore how to disseminate local results to other agents, and make them global. The various solutions to the puzzle are presented as protocols (iterated functions from agent's local states, and histories of actions, to actions). The paper consists of three parts. First, we present different versions of the puzzle, and their solutions. This includes a probabilistic version, and a version assuming synchronicity (the interval between prisoners' interrogations is known). The latter is very informative for the prisoners, and allows different protocols (with faster expected termination). Then, we model the puzzle in an epistemic logic incorporating dynamic operators for the effects of information changing events. Such events include both informative actions, where agents become more informed about the non-changing state of the world, and factual changes, wherein the world and the facts describing it change themselves as well. Finally, we verify the basic protocol to solve the problem. Novel contributions in this paper are: Firstly, Protocol 2 and Protocol 4. Secondly, the modelling in dynamic epistemic logic in its entirety — we do not know of a case study that combines factual and informational dynamics in a setting of non-public events, or of a similar proposal to handle asynchronous behaviour in a dynamic epistemic logic. Thirdly, our method to verify dynamic epistemic protocols by reasoning over possibly infinite execution sequences of these protocols. A precursor of the present paper, entitled ‘One hundred prisoners and a lightbulb – logic and computation’ (van Ditmarsch et al., 2010), was presented at KR 2010, Toronto. The differences with the present contribution are as follows: the former contains a section with computational results (expected runtime of different protocols before termination), whereas the focus of the present paper is the verification of one of the presented protocols in the former.

Judgment and Consequence Relations

Abstract: In this paper I argue that a variety of consequence relations can be subsumed under a common core. The reduction proceeds by taking the unconditional consequence, or judgment, as basic and deriving the conditional consequence via a uniform abstraction scheme. A specific outcome is that it is better not to base such a scheme on the semantic notion of a matrix and valuation but rather on theories and substitutions. I will also briefly look at consequence relations that are not reducible in this way.

Violation games: a new foundation for deontic logic ★

Abstract: In this paper I propose violation games as the basis of formal logics to represent and reason about norms, i.e. as the foundation of deontic logic. Deontic logic is an applied non-classical logic r...

Metric temporal reasoning with less than two clocks

Abstract: We introduce a new way of defining metric temporal logic on a real-numbers flow of time. The idea is based on having semantics which allow us to refer to a single universal clock of arbitrary precision in order to impose metric constraints. This gives us a new metric temporal logic which is very expressive, is natural to use, can be applied in very general situations, affords a wide range of useful abbreviations and operators, has a PSPACE decision procedure, and has the promise of being amenable to standard reasoning techniques. Thus it has many advantages over the existing metric temporal logics. We provide a decision procedure via conversion to a non-metric temporal logic, hence doing metric reasoning with no clocks.

An alert correlation approach based on security operator's knowledge and preferences

Abstract: One of the major problems of intrusion detection concerns the large amount of alerts that intrusion detection systems (IDS) produce. Security operator who analyzes alerts and takes decisions, is often submerged by the high number of alerts to analyze. In this paper, we present a new alert correlation approach based on knowledge and preferences of security operators. This approach, which is complementary to existing ones, allows to rank-order produced alerts on the basis of a security operator knowledge about the system, used IDS and his preferences about alerts that he wants to analyze or to ignore. Our approach is based on the development of a new non-classical logic for representing preferences, called FO-MQCL (First Order - Minimal Qualitative Choice Logic). Our logic extends a fragment of the first order logic by adding a new logical connective. The general idea is to present only alerts that fully fit security operator's preferences and knowledge. And if needed, less preferred alerts can also be pres...

On Hamblin's 15 Tense Theorem

Abstract: It is demonstrated that Hamblin's 15 tense theorem does not only hold for temporal logic with linear time but also for branching time. Furthermore three other theorems with finitely many tenses are shown.

A duration calculus with neighborhood modalities

Abstract: To reason about continuous processes in some areas of artificial intelligence and embedded systems one has to express real-time properties. For such purpose a real-time logic has to be considered. ...