Modal Logics Definable by Universal Three-Variable Formulas

In this paper, we present a Satisfiability Modulo Theory based (SMT-based) bounded model checking (BMC) method for Timed Real-Weighted Interpreted Systems and for the existential fragment of the Weighted Epistemic Computation Tree Logic. SMT-based bounded model checking consists in translating the existential model checking problem for a modal logic and for a model to the satisfiability problem of a quantifier-free first-order formula. We have implemented the SMT-BMC method and performed the BMC algorithm on Timed Weighted Generic Pipeline Paradigm benchmark. The preliminary experimental results demonstrate the feasibility of the method. To perform the experiments, we used the state of the art SMT-solver Z3.

The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

The modal satisfiability problem is solved either by using a specifically designed algorithm, or by translating the modal logic formula into an instance of a different class of problem, such as a first-order logic problem, a propositional satisfiability problem, or, more recently, a constraint satisfaction problem. In the latter approach, the modal formula is translated into layered propositional formulae. Each layer is translated into a constraint satisfaction problem which is solved using a constraint solver. We extend this approach to the modal logics KT and S4 and introduce a range of optimizations of the basic prototype. The results compare favorably with those of other solvers, and support the adoption of constraint programming as implementation platform for modal and other related satisfiability solvers.

Modal dependence logic was introduced very recently by Väänänen. It enhances the basic modal language by an operator dep. For propositional variables p 1,...,p n , dep(p 1,...,p n − 1)p n intuitively states that the value of p n only depends on those of p 1,...,p n − 1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time.In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfibility for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using ∧, □, \(\Diamond\) (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend Väänänen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy.In this way we completely classifiy the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster.Keywordsdependence logicsatisfiability problemcomputational complexitypoor man’s logic

While a number of weak consistency mechanisms have been developed in recent years to improve performance and ensure availability in distributed, replicated systems, ensuring the correctness of transactional applications running on top of such systems remains a difficult and important problem. Serializability is a well-understood correctness criterion for transactional programs; understanding whether applications are serializable when executed in a weakly-consistent environment, however remains a challenging exercise. In this work, we combine a dependency graph-based characterization of serializability and leverage the framework of abstract executions to develop a fully-automated approach for statically finding bounded serializability violations under any weak consistency model. We reduce the problem of serializability to satisfiability of a formula in First-Order Logic (FOL), which allows us to harness the power of existing SMT solvers. We provide rules to automatically construct the FOL encoding from programs written in SQL (allowing loops and conditionals) and express consistency specifications as FOL formula. In addition to detecting bounded serializability violations, we also provide two orthogonal schemes to reason about unbounded executions by providing sufficient conditions (again, in the form of FOL formulae) whose satisfiability implies the absence of anomalies in any arbitrary execution. We have applied the proposed technique on TPC-C, a real-world database program with complex application logic, and were able to discover anomalies under Parallel Snapshot Isolation (PSI), and verify serializability for unbounded executions under Snapshot Isolation (SI), two consistency mechanisms substantially weaker than serializability.

One-dimensional fragment of first-order logic is obtained by restricting quantification to blocks of existential quantifiers that leave at most one variable free. We investigate one-dimensional fragment over words and over omega-words. We show that it is expressively equivalent to the two-variable fragment of first-order logic. We also show that its satisfiability problem is NExpTime-complete. Further, we show undecidability of some extensions, whose two-variable counterparts remain decidable.

In a very short time Saul Kripke provided a suitable and rigorous semantics for different axiomatic modal systems and established a series of related results. Many key ideas were already in the air in the late Fifties, but it was Kripkean articles’ merit to system atically introduce comprehensive devices and solutions. Later on, the spreading of possible-worlds semantics massively changed the approach to modal logic, which enormously increased in popularity after that. Since Kripke’s work in modal logic is central to the development of the discipline, the aim of this essay is to present the fundamental results published between 1959 and 1965. Indeed, it was in such a brief and early phase of his career that Kripke was able to conceive the main novelties that would become central to the subsequent academic debates about modality. Here, their presentation will follow the original historical progressive introduction. Particular attention will be given to the interconnection between articles, their similarities in structure and the unified analysis produced by means of them. It actually appears quite impressive that, already in 1959, Kripke seemed to have planned all the developments he would present, one after the other, in the following years. First, an overview of the background where Kripke’s ideas start to rise is given. Then, each text’s results are individually briefly analysed.

A dynamic model of socio-technical change : institutions, actors and technologies in interaction

One of the most important open problems in finite model theory is the question whether there is a logic characterising efficient computation. While this question usually concerns Ptime, it can also be applied to other complexity classes, and in particular to Logspace which can be seen as a formalisation of efficient computation for big data. One of the strongest candidates for a logic capturing Ptime is Choiceless Polynomial Time (CPT). It is based on the idea of choiceless algorithms, a general model of symmetric computation over abstract structures (rather than their encodings by finite strings). However, there is currently neither a comparably strong candidate for a logic for Logspace, nor a logic transferring the idea of choiceless computation to Logspace. We propose here a notion of Choiceless Logarithmic Space which overcomes some of the obstacles posed by Logspace as a less robust complexity class. The resulting logic is contained in both Logspace and CPT, and is strictly more expressive than all logics for Logspace that have been known so far. Further, we address the question whether this logic can define all Logspace-queries, and prove that this is not the case.

The problem of finding small unsatisfiable cores for SAT formulas has recently received a lot of interest, mostly for its applications in formal verification. However, propositional logic is often ...

The exponential-time hypothesis (ETH) states that 3-SAT is not solvable in subexponential time, i.e. not solvable in O(c^n) time for arbitrary c > 1, where n denotes the number of variables. Problems like k-SAT can be viewed as special cases of the constraint satisfaction problem (CSP), which is the problem of determining whether a set of constraints is satisfiable. In this paper we study the worst-case time complexity of NP-complete CSPs. Our main interest is in the CSP problem parameterized by a constraint language Gamma (CSP(Gamma)), and how the choice of Gamma affects the time complexity. It is believed that CSP(Gamma) is either tractable or NP-complete, and the algebraic CSP dichotomy conjecture gives a sharp delineation of these two classes based on algebraic properties of constraint languages. Under this conjecture and the ETH, we first rule out the existence of subexponential algorithms for finite domain NP-complete CSP(Gamma) problems. This result also extends to certain infinite-domain CSPs and structurally restricted CSP(Gamma) problems. We then begin a study of the complexity of NP-complete CSPs where one is allowed to arbitrarily restrict the values of individual variables, which is a very well-studied subclass of CSPs. For such CSPs with finite domain D, we identify a relation SD such that (1) CSP({SD}) is NP-complete and (2) if CSP(Gamma) over D is NP-complete and solvable in O(c^n) time, then CSP({SD}) is solvable in O(c^n) time, too. Hence, the time complexity of CSP({SD}) is a lower bound for all CSPs of this particular kind. We also prove that the complexity of CSP({SD}) is decreasing when |D| increases, unless the ETH is false. This implies, for instance, that for every c>1 there exists a finite-domain Gamma such that CSP(Gamma) is NP complete and solvable in O(c^n) time.

Many writers have commented on the revival of philosophical theology that began in analytic circles in the 1960s. Those of us who received our training in the 50s were reacting to two things. First, our predecessors’ preoccupation with the question of religious discourse’s meaningfulness. While not necessarily disagreeing with Basil Mitchell’s, say, or John Hick’s responses to Antony Flew’s charge of ‘death by a thousand qualifications’, most of us lost interest in the debate. Second, a conviction that Hume’s and Kant’s vaunted critiques of natural theology didn’t withstand careful scrutiny. On the positive side, developments in modal logic, probability theory, and so on, offered tools for introducing a new clarity and rigour to traditional disputes. Alvin Plantinga’s work on the ontological argument, or Richard Swinburne’s use of Bayesian techniques to formulate his cumulative argument for God’s existence, are paradigmatic examples.

Previous intuitionistic justification logics included explicit justifications for all admissible rules of intuitionistic logic in order to get completeness with respect to provability semantics. We present the justification logic iJT4, which does not have these additional justification terms. We establish that iJT4 is complete with respect to modular models and that there is a realization of intuitionistic S4 into iJT4. Hence iJT4 can be seen as an explicit version of intuitionistic S4.

In the propositional modal (and algebraic) treatment of two-variable first-order logic equality is modelled by a `diagonal' constant, interpreted in square products of universal frames as the identity (also known as the `diagonal') relation. Here we study the decision problem of products of two arbitrary modal logics equipped with such a diagonal. As the presence or absence of equality in two-variable first-order logic does not influence the complexity of its satisfiability problem, one might expect that adding a diagonal to product logics in general is similarly harmless. We show that this is far from being the case, and there can be quite a big jump in complexity, even from decidable to the highly undecidable. Our undecidable logics can also be viewed as new fragments of first- order logic where adding equality changes a decidable fragment to undecidable. We prove our results by a novel application of counter machine problems. While our formalism apparently cannot force reliable counter machine computations directly, the presence of a unique diagonal in the models makes it possible to encode both lossy and insertion-error computations, for the same sequence of instructions. We show that, given such a pair of faulty computations, it is then possible to reconstruct a reliable run from them.

The essence and significance of tasting events as an element of integration marketing communications are revealed. The approaches, statements and opinions of modern scientists are analyzed. The mechanism, components, algorithm of construction and the procedure of carrying out a complex of tasting events are investigated. The basic problems of management of marketing communication policy of business structures have been formed. The basics of the application of integrated marketing communications are defined. The problems of the correctness of the tasting events as the main element of marketing management on the way of the enterprise to conquering the market niche are considered. Management of marketing is represented by the main element of influence on the consumer decision, the formation of demand and sales promotion of their products by modern enterprises. The introduction of integrated marketing communications into the marketing management of enterprises provides opportunities to study how proper, expedient, expensive, and fast is the use of marketing tools. One such marketing tool is the implementation of presentation measures regarding the products. Effective management of marketing communications requires consideration of such functions as: developing a marketing strategy, adhering to the logical relationship between goals and business planning, taking into account previous experience, number of clients, message functions. It is necessary to create a slogan, theme, idea and a unique business proposal. All this will attract an additional consumer or business partner. Tasting a product can be seen as a mechanism for investing in innovation and image of the company (knowledge of its business, product, brand). The purpose of this mechanism is to create a brand image and gain popularity, which over time will provide a more reliable guarantee of interaction with suppliers, resellers and consumers. Integrated marketing communications include the development and use of a variety of communication measures (active use of ATL / IBTL technologies) to introduce and market various TM food manufacturers based on brand-oriented technologies. Tasting gives people the opportunity to taste and buy products for free. The best, effective and efficient measure to familiarize the consumer with the taste and quality of the products, appearance, quality is tasting events among all integrated marketing communications.