Pure future local temporal logics are expressively complete for Mazurkiewicz traces

The paper settles a long standing problem for Mazurkiewicz traces: the pure future local temporal logic defined with the basic modalities exists-next and until is expressively complete. This means every first-order definable language of Mazurkiewicz traces can be defined in a pure future local temporal logic. The analogous result with a global interpretation has been known, but the treatment of a local interpretation turned out to be much more involved. Local logics are interesting because both the satisfiability problem and the model checking problem are solvable in Pspace for these logics whereas they are non-elementary for global logics. Both, the (previously known) global and the (new) local results generalize Kamp’s Theorem for words, because for sequences local and global viewpoints coincide.     

A comparison of two modelling paradigms in the Semantic Web

Classical logics and Datalog-related logics have both been proposed as underlying formalisms for conceptual modelling in the context of the Semantic Web. Although these two different formalism groups have some commonalities, and look similar in the context of expressively impoverished languages like RDF, their differences become apparent at more expressive language levels. After considering some of these differences, we argue that, although some of the characteristics of Datalog have their utility, the open environment of the Semantic Web is better served by standard logics.     

Partial Order Trace Analyzer (POTA) for Distributed Programs

Checking the correctness of software is a growing challenge. In this paper, we present a prototype implementation of Partial Order Trace Analyzer (POTA), a tool for checking execution traces of both message passing and shared memory programs using temporal logic. So far runtime verification tools have used the total order model of an execution trace, whereas POTA uses a partial order model. The partial order model enables us to capture possibly exponential number of interleavings and, in turn, this allows us to find bugs that are not found using a total order model. However, verification in partial order model suffers from the state explosion problem – the number of possible global states in a program increases exponentially with the number of processes.POTA employs an effective abstraction technique called computation slicing. A slice of a computation (execution trace) with respect to a predicate is the computation with the least number of global states that contains all global states of the original computation for which the predicate evaluates to true. The advantage of this technique is that, it mitigates the state explosion problem by reasoning only on the part of the global state space that is of interest. In POTA, we implement computing slicing algorithms for temporal logic predicates from a subset of CTL. The overall complexity of evaluating a predicate in this logic upon using computation slicing becomes polynomial in the number of processes compared to exponential without slicing.We illustrate the effectiveness of our techniques in POTA on several test cases such as the General Inter-ORB Protocol (GIOP)[18] and the primary secondary protocol[32]. POTA also contains a module that translates execution traces to Promela[16] (input language SPIN). This module enables us to compare our results on execution traces with SPIN. In some cases, we were able to verify traces with 250 processes compared to only 10 processes using SPIN.     

A concurrency-preserving translation from time Petri nets to networks of timed automata

Several formalisms to model distributed real-time systems coexist in the literature. This naturally induces a need to compare their expressiveness and to translate models from one formalism to another when possible. The first formal comparisons of the expressiveness of these models focused on the preservation of the sequential behavior of the models, using notions like timed language equivalence or timed bisimilarity. They do not consider preservation of concurrency. In this paper we define timed traces as a partial order representation of executions of our models for real-time distributed systems. Timed traces provide an alternative to timed words, and take the distribution of actions into account. We propose a translation between two popular formalisms that describe timed concurrent systems: 1-bounded time Petri nets (TPN) and networks of timed automata (NTA). Our translation preserves the distribution of actions, that is we require that if the TPN represents the product of several components (called processes), then each process should have its counterpart as one timed automaton in the resulting NTA.     

Definable transductions and weighted logics for texts

A text is a word together with an additional linear order on it. We study quantitative models for texts, i.e. text series which assign to texts elements of a semiring. We introduce an algebraic notion of recognizability following Reutenauer and Bozapalidis as well as weighted automata for texts combining an automaton model of Lodaya and Weil with a model of Ésik and Németh. After that we show that both formalisms describe the text series definable in a certain fragment of weighted logics as introduced by Droste and Gastin. In order to do so, we study certain definable transductions and show that they are compatible with weighted logics.     

An Expressively Complete Linear Time Temporal Logic for Mazurkiewicz Traces

A basic result concerning LTL, the propositional temporal logic of linear time, is that it is expressively complete; it is equal in expressive power to the first order theory of sequences. We present here a smooth extension of this result to the class of partial orders known as Mazurkiewicz traces. These partial orders arise in a variety of contexts in concurrency theory and they provide the conceptual basis for many of the partial order reduction methods that have been developed in connection with LTL-specifications.We show that LTrL, our linear time temporal logic, is equal in expressive power to the first order theory of traces when interpreted over (finite and) infinite traces. This result fills a prominent gap in the existing logical theory of infinite traces. LTrL also constitutes a characterisation of the so-called trace consistent (robust) LTL-specifications. These are specifications expressed as LTL formulas that do not distinguish between different linearisations of the same trace and hence are amenable to partial order reduction methods.     

Future temporal logic needs infinitely many modalities

Kamp’s theorem states that there is a temporal logic with two modalities (“until” and “since”) which is expressively complete for the first-order monadic logic of order over the real line. In this paper we show that there is no temporal logic with finitely many modalities which is expressively complete for the future fragment of first-order monadic logic of order over the real line (a future formula over the real time line is a formula whose truth value at a point is independent of what happened in the past).     

A study on multi-dimensional products of graphs and hybrid logics

In this work, we address some issues related to products of graphs and products of modal logics. Our main contribution is the presentation of a necessary and sufficient condition for a countable and connected graph to be a product, using a property called intransitivity. We then proceed to describe this property in a logical language. First, we show that intransitivity is not modally definable and also that no necessary and sufficient condition for a graph to be a product can be modally definable. Then, we exhibit a formula in a hybrid language that describes intransitivity. With this, we get a logical characterization of products of graphs of arbitrary dimensions. We then use this characterization to obtain two other interesting results. First, we determine that it is possible to test in polynomial time, using a model-checking algorithm, whether a finite connected graph is a product. This test has cubic complexity in the size of the graph and quadratic complexity in its number of dimensions. Finally, we use this characterization of countable connected products to provide sound and complete axiomatic systems for a large class of products of modal logics. This class contains the logics defined by product frames obtained from Kripke frames that satisfy connectivity, transitivity and symmetry plus any additional property that can be defined by a pure hybrid formula. Most sound and complete axiomatic systems presented in the literature are for products of a pair of modal logics, while we are able, using hybrid logics, to provide sound and complete axiomatizations for many products of arbitrary dimensions.     

Maximal traces and path-based coalgebraic temporal logics

This paper gives a general coalgebraic account of temporal logics whose semantics involves a notion of computation path. Examples of such logics include the logic CTL* for transition systems and the logic PCTL for probabilistic transition systems. Our path-based temporal logics are interpreted over coalgebras of endofunctors obtained as the composition of a computation type (e.g. non-deterministic or stochastic) with a general transition type. The semantics of such logics relies on the existence of execution maps similar to the trace maps introduced by Jacobs and co-authors as part of the coalgebraic theory of finite traces (Hasuo et al., 2007 [1]). We consider finite execution maps derived from the theory of finite traces, and a new notion of maximal execution map that accounts for maximal, possibly infinite executions. The latter is needed to recover the logics CTL* and PCTL as specific path-based logics.     

Rasiowa–Sikorski deduction systems in computer science applications

A Rasiowa-Sikorski system is a sequence-type formalization of logics. The system uses invertible decomposition rules which decompose a formula into sequences of simpler formulae whose validity is equivalent to validity of the original formula. There may also be expansion rules which close indecomposable sequences under certain properties of relations appearing in the formulae, like symmetry or transitivity. Proofs are finite decomposition trees with leaves having “fundamental”, valid labels. The author describes a general method of applying the R-S formalism to develop complete deduction systems for various brands of C.S and A.I. logic, including a logic for reasoning about relative similarity, a three-valued software specification logic with McCarthy's connectives and Kleene quantifiers, a logic for nondeterministic specifications, many-sorted FOL with possibly empty carriers of some sorts, and a three-valued logic for reasoning about concurrency.     

Preservation of Interpolation Features by Fibring

Fibring is a metalogical constructor that permits to combinedifferent logics by operating on their deductive systems undercertain natural restrictions, as for example that the two givenlogics are presented by deductive systems of the same type.Under such circumstances, fibring will produce a new deductivesystem by means of the free use of inference rules from bothdeductive systems, provided the rules are schematic, in thesense of using variables that are open for application to formulaswith new linguistic symbols (from the point of view of eachlogic component). Fibring is a generalization of fusion, a lessgeneral but wider developed mechanism which permits resultsof the following kind: if each logic component is decidable(or sound, or complete with respect to a certain semantics)then the resulting logic heirs such a property. The interestfor such preservation results for combining logics is evident,and they have been achieved in the more general setting of fibringin several cases. The Craig interpolation property and the Maeharainterpolation have a special significance when combining logics,being related to certain problems of complexity theory, someproperties of model theory and to the usual (global) metatheoremof deduction. When the peculiarities of the distinction betweenlocal and global deduction interfere, justifying what we callcareful reasoning, the question of preservation of interpolationbecomes more subtle and other forms of interpolation can bedistinguished. These questions are investigated and several(global and local) preservation results for interpolation areobtained for fibring logics that fulfill mild requirements.     

Expressive Completeness of Duration Calculus

This paper compares the expressive power of first-order monadic logic of order, a fundamental formalism in mathematical logic and the theory of computation, with that of the propositional version of duration calculus (PDC), a formalism for the specification of real-time systems. Our results show that the propositional duration calculus is expressively complete for first-order monadic logic of order. Our semantics for PDC conservatively extends the standard semantics to all positive (including infinite) length intervals. Hence, in view of the expressive completeness, liveness properties can be specified in PDC. This observation refutes a widely believed misconception that the duration calculus cannot specify liveness properties.     

Model Checking Games for the Quantitative μ-Calculus

We investigate quantitative extensions of modal logic and the modal μ-calculus, and study the question whether the tight connection between logic and games can be lifted from the qualitative logics to their quantitative counterparts. It turns out that, if the quantitative μ-calculus is defined in an appropriate way respecting the duality properties between the logical operators, then its model checking problem can indeed be characterised by a quantitative variant of parity games. However, these quantitative games have quite different properties than their classical counterparts, in particular they are, in general, not positionally determined. The correspondence between the logic and the games goes both ways: the value of a formula on a quantitative transition system coincides with the value of the associated quantitative game, and conversely, the values of quantitative parity games are definable in the quantitative μ-calculus.     

Characterization of Interval Fuzzy Logic Systems of Connectives by Group Transformations

The global structure of various systems of logic connectives is investigated by looking at abstract group properties of the group of transformations of these. Such characterizations of fuzzy interval logics are examined in Sections 4–9. The paper starts by introducing readers to the Checklist Paradigm semantics of fuzzy interval logics (Sections 2 and 3). In the Appendix we present some basic notions of fuzzy logics, sets and many-valued logics in order to make the paper accessible to readers not familiar with fuzzy sets.     

G-逻辑及其归结推理

该文提出了一种粒-逻辑，简记为G-逻辑，并构造了这种逻辑的近似推理系统，定义了G-公式、G-子句和G-文字，提出了这种逻辑的G-归结方法．G-归结的完备性定理也被证明了．这种逻辑公式的结构是有序二元对，第一元是断言；第二元是对应于这个断言的可定义集或不可定义域集的近似集．这种逻辑是定义在信息系统IS=(U，A)上，所以其公式中的个体变量被赋予U上的实体．公式中的命题或谓词被解释为属性集A上的属性，因此命题或谓词的意义集是U上的一个子集、属性及其意义集一起构成的二元对，被称做一个基本粒(granule)．而这种基本粒被当做这种逻辑中的一个G-原子，用G逻辑联结词组合这些G-原子便得到这种逻辑中的G-公式．公式的可满足性是其相应断言的意义集不空．当这种公式的定义域集不可定义时，则可将它移到其定义域集的Rough下和上近似集上去讨论．G-逻辑的提出为经典逻辑的应用开辟了新途径，也为处理非规范知识提供了较好的理论工具．G-逻辑的运算涉及整体到局部的分解和局部到整体的合并，以此提供了AI中问题求解的新思路．G-逻辑也是Rough逻辑的新扩充，其真值概念及其运算都不同于经典逻辑，也不同于其它非标准逻辑．这种逻辑中的演算既是逻辑的，又是集合论的．于是当处理真值及其运算时适合使用逻辑方法；而处理归结中的文字合一时可用集合论方法，这样可避免复杂的文字合一计算．最后，用实例说明了这种逻辑的G-归结方法的可行性和有效性，并给出了G-逻辑中机器定理证明的相关定理，讨论了G-归结反演的完备性和完全性．     

Difficult Configurations—On the Complexity of <Emphasis Type="Italic">LTrL</Emphasis>

The complexity of LTrL, a global linear time temporal logic over traces is investigated. The logic is global because the truth of a formula is evaluated in a global state, also called configuration. The logic is shown to be non-elementary with the main reason for this complexity being the nesting of until operators in formulas. The fragment of the logic without the until operator is shown to be EXPSPACE-hard.Supported by Polish KBN grant No. 8 T11C 002 11.     

Annotated Logic Applications for Imperfect Information

Imperfect information is a very general term that comprises different types of information, such as uncertain, vague, fuzzy, inconsistent, possibilistic, probabilistic, partially or totally incomplete information [2]. In the literature of knowledge representation we find a different formal model for each one of these distinct types. For example, annotated logic is a formal model to represent inconsistent information.Annotated logics are non-classical logics introduced in [20] as a logic programming theory. They were proved to be paraconsistent. Based on [5], we present in this work the annotated logic programming theory and some of its applications in Artificial Intelligence (AI). We present it as a formalism to reason with inconsistent information and investigate its possibility to represent other types of imperfect information, such as possibilistic and non-monotonic reasoning. Our main goal is to verify and confirm the importance of annotated logics as a tool for developing knowledge-based and automated reasoning systems in AI.