Logical Properties Research Articles

A method of constructing fuzzy implications from the FIφ-construction

In the recent years, several new construction methods of fuzzy implications have been proposed. However, these construction methods actually care about that the new implication could preserve more properties. In this paper, we introduce a new method for constructing fuzzy implications based on an aggregation function with F (1, 0) =1, a fuzzy implication I and a non-decreasing function φ, called FIφ-construction. Specifically, some logical properties of fuzzy implications preserved by this construction are studied. Moreover, it is studied how to use the FIφ-construction to produce a new implication satisfying a specific property. Furthermore, we produce two new subclasses of fuzzy implications such as UIφ-implications and GpIφ-implications by this method and discuss some additional properties. Finally, we provide a way to generate fuzzy subsethood measures by means of FIφ-implications.

Formal logic and natural ways of reasoning

In the paper I ask the question about the relation between formal logic and the natural logic of human mind. By a natural logic I mean the ways of thinking of a person that is intelligent but untrained in formal logic. As it turns out that the laws, rules or properties of formal logic in some cases diverge from the natural ways of reasoning, I explain the causes of this divergence. Since the majority of research in this area has been carried out from the standpoint of psychology, as a logician I suggest a slight change of the angle from which we look at the problem. I argue that certain narrowing of an interdisciplinary research would be helpful in getting a better picture of natural logic, and might provide a new stimulus for formal investigations.

From LTL to rLTL monitoring: improved monitorability through robust semantics

Runtime monitoring is commonly used to detect the violation of desired properties in safety critical cyber-physical systems by observing its executions. Bauer et al. introduced an influential framework for monitoring Linear Temporal Logic (LTL) properties based on a three-valued semantics for a finite execution: the formula is already satisfied by the given execution, it is already violated, or it is still undetermined, i.e., it can still be satisfied and violated by appropriate extensions of the given execution. However, a wide range of formulas are not monitorable under this approach, meaning that there are executions for which satisfaction and violation will always remain undetermined no matter how it is extended. In particular, Bauer et al. report that 44% of the formulas they consider in their experiments fall into this category. Recently, a robust semantics for LTL was introduced to capture different degrees by which a property can be violated. In this paper we introduce a robust semantics for finite strings and show its potential in monitoring: every formula considered by Bauer et al. is monitorable under our approach. Furthermore, we discuss which properties that come naturally in LTL monitoring—such as the realizability of all truth values—can be transferred to the robust setting. We show that LTL formulas with robust semantics can be monitored by deterministic automata, and provide tight bounds on the size of the constructed automaton. Lastly, we report on a prototype implementation and compare it to the LTL monitor of Bauer et al. on a sample of examples.

Choice-Driven Counterfactuals

In this paper, we investigate the semantics and logic of choice-driven counterfactuals, that is, of counterfactuals whose evaluation relies on auxiliary premises about how agents are expected to act, i.e., about their default choice behavior. To do this, we merge one of the most prominent logics of agency in the philosophical literature, namely stit logic (Belnap et al. 2001; Horty 2001), with the well-known logic of counterfactuals due to Stalnaker (1968) and Lewis (1973). A key component of our semantics for counterfactuals is to distinguish between deviant and non-deviant actions at a moment, where an action available to an agent at a moment is deviant when its performance does not agree with the agent’s default choice behavior at that moment. After developing and axiomatizing a stit logic with action types, instants, and deviant actions, we study the philosophical implications and logical properties of two candidate semantics for choice-driven counterfactuals, one called rewind models inspired by Lewis (Nous13(4), 455–476 1979) and the other called independence models motivated by well-known counterexamples to Lewis’s proposal Slote (Philos. Rev.87(1), 3–27 1978). In the last part of the paper we consider how to evaluate choice-driven counterfactuals at moments arrived at by some agents performing a deviant action.

Automaton-ABC: A statistical method to estimate the probability of spatio-temporal properties for parametric Markov population models

We present an adaptation of the Approximate Bayesian Computation method to estimate the satisfaction probability function of a temporal logic property for Markov Population Models.In this paper, we tackle the problem of estimating the satisfaction probability function of a temporal logic property w.r.t. a parametric Markovian model of Chemical Reaction Network. We want to assess the probability with which the trajectories generated by a parametric Markov Population Model (MPM) satisfy a logical formula over the whole parameter space. In the first step of the work, we formally define a distance between a trajectory of an MPM and a logical property. If the distance is 0, the trajectory satisfies the property. The larger the distance is, further the trajectory is from satisfying the property. In the second step, we adapt the Approximate Bayesian Computation method using the distance defined in the first step. This adaptation yields a new algorithm, called automaton-ABC, whose output is a density function that directly leads to the estimation of the desired satisfaction probability function. We apply our methodology to several examples and models, and we compare it to state-of-the-art techniques. We show that the sequential version of our algorithm relying on ABC-SMC leads to an efficient exploration of the parameter space with respect to the formula and gives good approximations of the satisfaction probability function at a reduced computational cost.

Automatic Generation of Route Control Chart From Validated Signal Interlocking Plan

Railway signalling is a complex and safety critical problem that has been extensively studied and standardised over a long period of time. The signalling equipment is typically procured from standard vendors and configured with yard specific application logic for which the route control chart (RCC) is a key input. As the yard size increases, the number of routes also increases and accordingly the difficulty of RCC preparation increases rapidly. RCC preparation for big yards may take months and affects the project deadlines adversely. In this work we report computational procedures to address: <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$a$ </tex-math></inline-formula> ) capturing of signal interlocking plan (SIP) given on paper and storing it using suitable data structures, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$b$ </tex-math></inline-formula> ) generating the RCC automatically from the captured SIP supported by procedures based on graph theoretic formulation, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$c$ </tex-math></inline-formula> ) storing the SIP graphically in memory, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$d$ </tex-math></inline-formula> ) application of formal methods towards validation of yard structure and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$e$ </tex-math></inline-formula> ) generation of temporal logic properties for formal verification of electronic interlocking (EI) logic. Several important steps towards RCC generation, such as conflict identification and isolation determination and also validation and verification (V&V) covering yard layout and safety property generation based on graph theoretic modelling are the most interesting aspect of this work. The described techniques have been tested successfully on many actual yards.

LOGICALITY AND MODEL CLASSES

AbstractWe ask, when is a property of a model a logical property? According to the so-called Tarski–Sher criterion this is the case when the property is preserved by isomorphisms. We relate this to model-theoretic characteristics of abstract logics in which the model class is definable. This results in a graded concept of logicality in the terminology of Sagi [46]. We investigate which characteristics of logics, such as variants of the Löwenheim–Skolem theorem, Completeness theorem, and absoluteness, are relevant from the logicality point of view, continuing earlier work by Bonnay, Feferman, and Sagi. We suggest that a logic is the more logical the closer it is to first order logic. We also offer a refinement of the result of McGee that logical properties of models can be expressed in $L_{\infty \infty }$ if the expression is allowed to depend on the cardinality of the model, based on replacing $L_{\infty \infty }$ by a “tamer” logic.

Inconsistent Models for Relevant Arithmetics

&#x0D; &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; This paper develops in certain directions the work of Meyer in [3], [4], [5] and [6] (see also Routley [10] and Asenjo [11]). In those works, Peano’s axioms for arithmetic were formulated with a logical base of the relevant logic R, and it was proved finitistically that the resulting arithmetic, called R♯, was absolutely consistent. It was pointed out that such a result escapes incau- tious formulations of Goedel’s second incompleteness theorem, and provides a basis for a revived Hilbert programme. The absolute consistency result used as a model arithmetic modulo two. Modulo arithmetics are not or- dinarily thought of as an extension of Peano arithmetic, since some of the propositions of the latter, such as that zero is the successor of no number, fail in the former. Consequently a logical base which, unlike classical logic, tolerates contradictory theories was used for the model. The logical base for the model was the three-valued logic RM3 (see e.g. [1] or [8]), which has the advantage that while it is an extension of R, it is finite valued and so easier to handle. &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; The resulting model-theoretic structure (called in this paper RM32) is interesting in its own right in that the set of sentences true therein consti- tutes a negation inconsistent but absolutely consistent arithmetic which is an extension of R♯. In fact, in the light of the result of [6], it is an extension of Peano arithmetic with a base of a classical logic, P♯. A generalisation of the structure is to modulo arithmetics with the same logical base RM3, but with varying moduli (called RM3i here). We first study the properties of these arithmetics in this paper. The study is then generalised by vary- ing the logical base, to give the arithmetics RMni, of logical base RMn and modulus i. Not all of these exist, however, as arithmetical properties and logical properties interact, as we will show. The arithmetics RMni give rise, on intersection, to an inconsistent arithmetic RMω which is not of modulo i for any i. We also study its properties, and, among other results, we show by finitistic means that the more natural relevant arithmetics R♯ and R♯♯ are incomplete (whether or not consistent and recursively enumerable). In the rest of the paper we apply these techniques to several topics, particularly relevant quantum arithmetic in which we are able to show (unlike classical quantum arithmetic) that the law of distribution remains unprovable. Aside from its intrinsic interest, we regard the present exercise as a demonstration that inconsistent theories and models are of mathematical worth and interest. &#x0D; &#x0D; &#x0D; &#x0D; &#x0D; &#x0D;

Compound nouns do not always fulfill the conjunction thesis: An analysis using the mental model theory

This paper is intended to show that the properties classical logic attributes to conjunction are not always found in real compound nouns in natural language. This fact is described by means of several examples taken from different natural languages. The main conclusion is that perhaps a better option to understand human language and communicative phenomena is to assume a basically semantic approach with the mental model theory proposed as a more suitable alternative. This is because the fact that the logical properties that are not always present in real compound nouns is not considered a problem under the framework of this theory.

Collapsible Pushdown Parity Games

This article studies a large class of two-player perfect-information turn-based parity games on infinite graphs, namely, those generated by collapsible pushdown automata. The main motivation for studying these games comes from the connections from collapsible pushdown automata and higher-order recursion schemes, both models being equi-expressive for generating infinite trees. Our main result is to establish the decidability of such games and to provide an effective representation of the winning region as well as of a winning strategy. Thus, the results obtained here provide all necessary tools for an in-depth study of logical properties of trees generated by collapsible pushdown automata/recursion schemes.

Certifying proofs for SAT-based model checking

In the context of formal verification, certifying proofs are evidences of the correctness of a model in a deduction system produced automatically as outcome of the verification. They are quite appealing for high-assurance systems because they can be verified independently by proof checkers, which are usually simpler to certify than the proof-generating tools. Model checking is one of the most prominent approaches to formal verification of temporal properties and is based on an algorithmic search of the system state space. Although modern algorithms integrate deductive methods, the generation of proofs is typically restricted to invariant properties only. Moreover, it assumes that the verification produces an inductive invariant of the original system, while model checkers usually involve a variety of complex pre-processing simplifications. In this paper we show how, exploiting the k-liveness algorithm, to extend proof generation capabilities for invariant checking to cover full linear-time temporal logic (LTL) properties, in a simple and efficient manner, with essentially no overhead for the model checker. Besides the basic k-liveness algorithm, we integrate in the proof generation a variety of widely used pre-processing techniques such as temporal decomposition, model simplification via computation of equivalences with ternary simulation, and the use of stabilizing constraints. These techniques are essential in many cases to prove that a property holds, both for invariant and for LTL model checking, and thus need to be considered within the proof. We implemented the proof generation techniques on top of IC3 engines, and show the feasibility of the approach on a variety of benchmarks taken from the literature and from the Hardware Model Checking Competition. Our results confirm that proof generation results in negligible overhead for the model checker.

From Generative Linguistics to Categorial Grammars: Overt Subjects in Control Infinitives

AbstractThe present thesis lies at the interface of logic and linguistics; its object of study are control sentences with overt pronouns in Romance languages (European and Brazilian Portuguese, Italian and Spanish). This is a topic that has received considerably more attention on the part of linguists, especially in recent years, than from logicians. Perhaps for this reason, much remains to be understood about these linguistic structures and their underlying logical properties. This thesis seeks to fill the lacunas in the literature or at least take steps in this direction by way of addressing a number of issues that have so far been under-explored. To this end, we put forward two key questions, one linguistic and the other logical. These are, respectively, (1) What is the syntactic status of the surface pronoun? and (2) What are the available mechanisms to reuse semantic resources in a contraction-free logical grammar? Accordingly, the thesis is divided into two parts: generative linguistics and categorial grammar. Part I starts by reviewing the recent discussion within the generative literature on infinitive clauses with overt subjects, paying detailed attention to the main accounts in the field. Part II does the same on the logical grammar front, addressing in particular the issues of control and of anaphoric pronouns. Ultimately, the leading accounts from both camps will be found wanting. The closing chapter of each of Part I and Part II will thus put forward alternative candidates, that we contend are more successful than their predecessors. More specifically, in Part I, we offer a linguistic account along the lines of Landau’s T/Agr theory of control. In Part II, we present two alternative categorial accounts: one based on Combinatory Categorial Grammar, the other on Type-Logical Grammar. Each of these accounts offers an improved, more fine-grained perspective on control infinitives featuring overt pronominal subjects. Finally, we include an Appendix in which our type-logical proposal is implemented in a categorial parser/theorem-prover.Abstract prepared by María Inés Corbalán.E-mail: inescorbalan@yahoo.com.arURL: http://repositorio.unicamp.br/jspui/handle/REPOSIP/331697

Strategic coalitions in stochastic games

AbstractThe article compares two different approaches of incorporating probability into coalition logics. One is based on the semantics of games with stochastic transitions and the other on games with the stochastic failures. The work gives an example of a non-trivial property of coalition power for the first approach and a complete axiomatization for the second approach. It turns out that the logical properties of the coalition power modality under the second approach depend on whether the modal language allows the empty coalition. The main technical results for the games with stochastic failures are a strong completeness theorem for the logical system without the empty coalition and an incompleteness theorem which shows that there is no strongly complete logical system in the language with the empty coalition.

Algorithmic properties of first-order modal logics of linear Kripke frames in restricted languages

Abstract We study the algorithmic properties of first-order monomodal logics of frames $\langle {\textrm{I}\!\textrm{N}}, \leqslant \rangle $, $\langle {\textrm{I}\!\textrm{N}}, &amp;lt; \rangle $, $\langle \mathbb {Q}, \leqslant \rangle $, $\langle \mathbb {Q}, &amp;lt; \rangle $, $\langle {\textrm{I}\!\textrm{R}}, \leqslant \rangle $, $\langle {\textrm{I}\!\textrm{R}}, &amp;lt; \rangle $, as well as some related logics, in languages with restrictions on the number of individual variables as well as the number and arity of predicate letters. We show that the logics of frames based on $ {\textrm{I}\!\textrm{N}}$ are $\varPi ^1_1$-hard—thus, not recursively enumerable—in languages with two individual variables, one monadic predicate letter and one proposition letter. We also show that the logics of frames based on $\mathbb {Q}$ and ${\textrm{I}\!\textrm{R}}$ are $\varSigma ^0_1$-hard in languages with the same restrictions. Similar results are obtained for a number of related logics.

Edge-Based Runtime Verification for the Internet of Things

Complex distributed systems such as the ones induced by Internet of Things (IoT) deployments, are expected to operate in compliance to their requirements. This can be checked by inspecting events flowing throughout the system, typically originating from end-devices and reflecting arbitrary actions, changes in state or sensing. Such events typically reflect the behavior of the overall IoT system – they may indicate executions which satisfy or violate its requirements. This article presents a service-based software architecture and technical framework supporting runtime verification for widely deployed, volatile IoT systems. At the lowest level, systems we consider are comprised of resource-constrained devices connected over wide area networks generating events. In our approach, monitors are deployed on edge components, receiving events originating from end-devices or other edge nodes. Temporal logic properties expressing desired requirements are then evaluated on each edge monitor in a runtime fashion. The system exhibits decentralization since evaluation occurs locally on edge nodes, and verdicts possibly affecting satisfaction of properties on other edge nodes are propagated accordingly. This reduces dependence on cloud infrastructures for IoT data collection and centralized processing. We illustrate how specification and runtime verification can be achieved in practice on a characteristic case study of smart parking. Finally, we demonstrate the feasibility of our design over a testbed instantiation, whereupon we evaluate performance and capacity limits of different hardware classes under monitoring workloads of varying intensity using state-of-the-art LPWAN technology.

PHRASEOLOGICAL UNITS OF EXPRESSION IN GERMAN AND UZBEK LANGUAGES

Phraseology is the most lively, vivid and original part of the vocabulary of a language. Phrase logical sky units, with their appropriate use, enliven and decorate speech, are an important building material of any language. The emergence of such units in the language is associated with the desire to more clearly to express what has already been expressed by other means. As I. I. Chernysheva emphasizes, “The development of phrase logical units goes along the line of creating stable phrases expressing emotionally rich characteristic of the qualities, properties, actions of the subject by phrase logical means properties of the language, i.e. based on the rethinking of variable phrases with the most apt for a given concepts in the manner”.

Temporal Logic Inference for Fault Detection of Switched Systems With Gaussian Process Dynamics

In this article, we present a method for constructing the fault detector in the form of signal temporal logic (STL) formulas, which can be understood by human users and formally proven to detect faults with probabilistic satisfaction guarantees, for a class of switched nonlinear systems with partially unknown dynamics. First, the partially unknown internal dynamics are approximated by the Gaussian process with stability guarantees. Second, a novel temporal logic inference algorithm is proposed to find the fault detector, which takes advantage of the internal properties of temporal logic and searches for the optimal formula along a partially ordered direction. Moreover, the algorithm is not allowed for missing faults but allowed for false alarms during the temporal logic inference process. In addition, we simulate finitely many trajectories with Chua’s circuit and infer the temporal logic formulas with the Gaussian optimization. The results show that the proposed method can find a temporal logic formula to detect the faulty trajectory with a probability guarantee. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The method proposed in this article can be used to detect faults for switched systems with partially unknown dynamics. STL is used to describe the behaviors of the system, which acts as a classifier and detector, such that all normal behaviors of the system will satisfy the description, while the faulty behaviors will violate the description. Moreover, STL formulas can be understood by human operators, which is important for the timely response to faulty events. For example, the normal behavior of a smart grid can be described as follows: “if the smart grid is safe, it should reach 9 kV within 15 min when the voltage to region A is above 12 kV,” which can be expressed with STL. Due to the unknown dynamics, the Gaussian process regression is applied to estimate the model and the region that is robust to noises.

On some logical relation properties for rough neutrosophic set

In this paper, we mainly investigate the relation theory and topological structures of rough neutrosophic sets which constructed by combining rough sets and neutrosophic sets. Firstly, we introduce the concept of algebra which is operations and identity relation. Then, discuss the properties of inverse, reflexive, symmetric as well as transitive relation of rough neutrosophic set. Further, regular sets form a Boolean algebra is introduced from the interior and closure of interior in rough neutrosophic sets. The results establish can help to gain more insight in rough neutrosophic sets.

Higher-order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set of elements for which a formula holds), and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems, we provide an effective solution. This is obtained, thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

Deriving invariant checkers for critical infrastructure using axiomatic design principles

Cyber-physical systems (CPSs) in critical infrastructure face serious threats of attack, motivating research into a wide variety of defence mechanisms such as those that monitor for violations of invariants, i.e. logical properties over sensor and actuator states that should always be true. Many approaches for identifying invariants attempt to do so automatically, typically using data logs, but these can miss valid system properties if relevant behaviours are not well-represented in the data. Furthermore, as the CPS is already built, resolving any design flaws or weak points identified through this process is costly. In this paper, we propose a systematic method for deriving invariants from an analysis of a CPS design, based on principles of the axiomatic design methodology from design science. Our method iteratively decomposes a high-level CPS design to identify sets of dependent design parameters (i.e. sensors and actuators), allowing for invariants and invariant checkers to be derived in parallel to the implementation of the system. We apply our method to the designs of two CPS testbeds, SWaT and WADI, deriving a suite of invariant checkers that are able to detect a variety of single- and multi-stage attacks without any false positives. Finally, we reflect on the strengths and weaknesses of our approach, how it can be complemented by other defence mechanisms, and how it could help engineers to identify and resolve weak points in a design before the controllers of a CPS are implemented.