4DL: a four-valued dynamic logic and its proof theory

Dynamic epistemic logic (\(\mathbf {DEL}\)) is known as a large family of logics that extend standard epistemic logic with dynamic operators. Such dynamic operators can be regarded as epistemic actions over Kripke semantics (or its variant). Therefore, \(\mathbf {DEL}\) is often used to model changes of agents’ knowledge, belief or preference over Kripke semantics in terms of dynamic operators in many literatures. As a variant of \(\mathbf {DEL}\), (van Benthem and Liu, J Appl Non-Classical Logics 17(2):157–18 2007; Liu, Reasoning about preference dynamics, Springer Science & Business Media, Berlin 2011) proposed dynamic logic of relation changers (\(\mathbf {DLRC}\)). They provided a general framework to capture many dynamic operators in terms of relation changing operation written in programs of propositional dynamic logic, and they also provided a sound and complete Hilbert-style axiomatization for \(\mathbf {DLRC}\). While \(\mathbf {DLRC}\) can cover many dynamic operators in a uniform manner, proof theory for \(\mathbf {DLRC}\) is not well-studied except the Hilbert-style axiomatization. Therefore, we propose a cut-free labelled sequent calculus for \(\mathbf {DLRC}\). We show that our sequent calculus is equipollent with the Hilbert-style axiomatization.

During the recent years the smart cities knows a great extension as a modern shape of sustainable expansion. It's a urban area that utilize various devices connected with internet and integrates them with ICTs to promote goodness and execution of services for the best interaction among citizens and city's government. The basic for smart cities is distributed and independent information infrastructure. Using information effectively is going to be a main factor for success in the smart cities. The sources of information's (models, experts, and sensors) must be reason, perfect and complete. The generated information from independent and distributed sources can be imprecise, uncertain, and/or incomplete in real life. Any deficiency in gathered information will have a negative effect on the performance of services and decision making process within smart cities. So, we need a general framework to represent all types of imperfect and incomplete information. Since the classical methods fails to deal with vague, inconsistent and incomplete information, the fuzzy set was introduced to solve this drawback. The fuzzy set was not the perfect method for dealing with these drawbacks because it considers only truthiness and fails to deal with indeterminacy. The efficient mathematical tool for dealing with uncertain, vague and inconsistent objects is rough sets theory which introduced by Pawlak. The theory of neutrosophic rough sets is powerful for dealing with incompleteness and neutrosophic set deals with indeterminate and inconsistent data efficiently through considering truthiness, indeterminacy and falsity degrees. So, in this research we will propose a general framework for dealing with imperfect and incomplete information through using single valued neutrosophic and rough set theories. The combination of two sets will deal with all aspects of vagueness, inconsistency and incompleteness of data and information, and then will enhance the quality of introduced services and decisions from smart cities to their citizens. As experimentation, we applied the proposed framework for modeling imperfect and incomplete data in healthcare field.

Paraconsistent soft sets can address issues involving incomplete and inconsistent information simultaneously. In this paper, paraconsistent soft sets are furtherly extended to dynamic paraconsistent soft sets by introducing time factors. We define some basic operations, such as dynamic paraconsistent soft subsets, complement, “AND”, restricted intersection, relaxed intersection, restricted cross and relaxed cross. Subsequently, we also propose the definitions of dynamic paraconsistent soft decision system, compound time choice value, compound time decision value, compound time weighting vector and final decision value. Additionally, we construct a decision making algorithm for dynamic paraconsistent soft sets, which can address issues involving both dynamic incomplete and inconsistent information. Furthermore, we apply the proposed algorithm to a practical loan problem for small and micro enterprises. Finally, we perform a sensitivity analysis and a comparative analysis to prove the effectiveness and feasibility of the proposed algorithm.

We combine first-order dynamic logic for reasoning about possible behaviour of hybrid systems with temporal logic for reasoning about the temporal behaviour during their operation. Our logic supports verification of hybrid programs with first-order definable flows and provides a uniform treatment of discrete and continuous evolution. For our combined logic, we generalise the semantics of dynamic modalities to refer to hybrid traces instead of final states. Further, we prove that this gives a conservative extension of dynamic logic. On this basis, we provide a modular verification calculus that reduces correctness of temporal behaviour of hybrid systems to non-temporal reasoning. Using this calculus, we analyse safety invariants in a train control system and symbolically synthesise parametric safety constraints.Keywordsdynamic logictemporal logicsequent calculuslogic for hybrid systemsdeductive verification of embedded systems

Non-dual modal operators as a basis for 4-valued accessibility relations in Hybrid logic

The proof theory of quasi-classical logic (QC logic) allows the derivation of non-trivializable classical inferences from inconsistent information. A non-trivializable, or paraconsistent, logic is, by necessity, a compromise, or weakening, of classical logic. The compromises on QC logic seem to be more appropriate than other paraconsistent logics for applications in computing. In particular, the connectives behave in a classical manner. Here we motivate the need for QC logic, present a proof theory, and semantics for the logic, and compare it to other paraconsistent logics.

The purpose of this study is to establish levels from an inductive proof to an algebraic demonstration in lower secondary school mathematics. I propose that we can establish six levels of proof in lower secondary school mathematics as steps from an inductive proof to an algebraic demonstration on the basis of three axes (contents of proof, representation of proof, and students' thinking). To reach this conclusion, I firstly examine the meaning of ‘demonstration in lower secondary school mathematics’ and ‘proof in lower secondary school mathematics’, and show the relationships between them. Secondly, I set out four basic levels of proof, as seen from two aspects (contents and representation of proof). Thirdly, I subdivide them into six levels from the third aspect of students' thinking. Finally, I illustrate my discussion with a 7th grader's activities.

A new method is developed to represent the phase behavior of multicomponent systems. This method uses fewer pseudocomponents than true components, but unlike conventional methods in which pseudocomponents are often chosen arbitrarily, the pseudocomponents are often chosen arbitrarily, the method uses regression analysis to find a "best" set of pseudocomponents.The method is applied to two surfactant systems of the type used for tertiary oil recovery. One system contains a crude oil (from the North Burbank Unit, Osage County, OK) and the other contains a pure hydrocarbon, 1-phenyltetradecane. For both systems the representation in terms of the pseudocomponents chosen by the regression analysis is significantly more faithful than that obtained by conventional methods.Since the considerations discussed here are general, they should be applicable to a wide range of phase studies in multicomponent systems. For example, they should be illuminating when applied to oil recovery by gas injection (carbon dioxide, natural gas, etc.), and to extraction processes, as well as to surfactant systems. Introduction Systems containing surface active agents have attracted a great deal of attention in connection with tertiary oil recovery. In many of these systems optimum oil recovery has been found to be strongly correlated with the phase behavior of these systems. To understand completely the basis for these correlations, one must be able to represent the phase behavior of systems containing surfactants adequately.Surfactant systems for tertiary oil recovery usually contain at least five components: oil, water, surfactant. cosurfactant, and electrolyte. The isothermal, isobaric phase diagram of these systems can be represented in a phase diagram of these systems can be represented in a four-dimensional space. Because physical representation (in three dimensions) of such a diagram is impossible. various techniques have been developed to attempt to represent the phase behavior in lower dimensional spaces. All these techniques correspond to projections of the original diagram onto lower dimensional spaces. Although almost unlimited methods of projection exist, only a small fraction convey useful information.Two projection schemes having some similarities but different intents and consequences are straight mathematical projection and "pseudocomponent" projection. Straight mathematical projection refers to the process of directly projecting the four-dimensional data for the entire phase diagram along some specified direction onto a three-dimensional space. All information parallel to the direction of the projection is lost. For example, if the rays of projections are parallel to the oil/water edge of the phase diagram, the resulting representation contains no information about the relative amounts of oil and water in the phases. In principle, this problem can be circumvented by generating two representations corresponding to projections of the same data along two different directions. For example. a second representation could be produced corresponding to a projection parallel to the water/alcohol edge of the phase diagram. parallel to the water/alcohol edge of the phase diagram. Although neither representation contains complete information, the pair of representations does contain all information about the system. The problem with this method is that the information is not perceived easily and, since the intent of using phase diagrams is usually to make visible a summary of the phase trends, such mathematical representations are not very useful.The second and generally more useful projection scheme uses pseudocomponents. A pseudocomponent is some mixture of pure components treated as a single component. SPEJ P. 289

Logics for social networks have been studied in recent literature. This paper presents a framework based on dynamic term-modal logic (\(\mathsf {DTML}\)), a quantified variant of dynamic epistemic logic (\(\mathsf {DEL}\)). In contrast with \(\mathsf {DEL}\) where it is commonly known to whom agent names refer, \(\mathsf {DTML}\) can represent dynamics with uncertainty about agent identity. We exemplify dynamics where such uncertainty and de re/de dicto distinctions are key to social network epistemics. Technically, we show that \(\mathsf {DTML}\) semantics can represent a popular class of hybrid logic epistemic social network models. We also show that \(\mathsf {DTML}\) can encode previously discussed dynamics for which finding a complete logic was left open. As complete reduction axioms systems exist for \(\mathsf {DTML}\), this yields a complete system for the dynamics in question.

This study tested mood effects on product evaluation, varying information type and time pressure. In this experiment, happy or sad moods were induced in participants, then they were asked to evaluate products. The results showed that in the happy mood/non‐time‐pressure condition, a favorable evaluation was seen when complete information was provided compared to incomplete. However, when time pressure was induced, no such difference was observed. On the other hand, in the sad mood/non‐time‐pressure condition, there was no difference in evaluation between complete and incomplete information, but when time pressure was induced, a less favorable evaluation of products was seen under incomplete relative to complete. The results provide support for the process in which participants depend on their mood as a source of information when inferring about missing information under reduced cognitive capacity.

Accurate and up-to-date models describing the behavior of software systems are seldom available in practice. To address this issue, software engineers may use specification mining techniques, which can automatically derive models that capture the behavior of the system under analysis. So far, most specification mining techniques focused on the functional behavior of the systems, with specific emphasis on models that represent the ordering of operations, such as temporal rules and finite state models. Although useful, these models are inherently partial. For instance, they miss the timing behavior, which is extremely relevant for many classes of systems and components, such as shared libraries and user-driven applications. Mining specifications that include both the functional and the timing aspects can improve the applicability of many testing and analysis solutions. This paper addresses this challenge by presenting the Timed k-Tail (TkT) specification mining technique that can mine timed automata from program traces. Since timed automata can effectively represent the interplay between the functional and the timing behavior of a system, TkT could be exploited in those contexts where time-related information is relevant. Our empirical evaluation shows that TkT can efficiently and effectively mine accurate models. The mined models have been used to identify executions with anomalous timing. The evaluation shows that most of the anomalous executions have been correctly identified while producing few false positives.

Public media serves as a significant source of scientific information for non-scientists. However, the simplifications and omissions inherent in media reporting often alter the nature of scientific information, potentially influencing understanding and perceptions of science and the nature of science. This study investigates how mechanistic explanations about biological processes are represented in public media, focusing on two forms of incomplete mechanistic information: “gray boxes” and “black boxes”. Using COVID-19 as a case study, we analyzed 122 media reports of biological mechanisms to understand how incomplete parts are masked by more complete explanations and their implications. Our findings highlighted three main points. First, incomplete information often appears alongside complete information within other parts of the explanation. Second, some parts of similar mechanisms are presented differently, which can create a sense of conflicting information if incompleteness is not recognized. Third, multiple filler terms are used to mask black boxes within biological explanations (e.g., “cause”, “fight”, or “mutate”). While filler terms enhance narrative flow, they can obscure gaps in scientific knowledge and lead to anthropocentric or teleological explanations. We categorized these filler terms into three groups and discussed their relevance to teaching and learning. Implications for addressing partial information in the science classroom are discussed.

The neutrosophic set model is an important tool for dealing with real scientific and engineering applications because it can handle not only incomplete information but also the inconsistent information and indeterminate information which exist commonly in real situations. In this paper, we firstly propose two practical techniques converting an interval neutrosophic set into a fuzzy set and a single-valued neutrosophic set, respectively. Then, we define the interval neutrosophic cross-entropy in two different ways, which are based on extension of fuzzy cross-entropy and single-valued neutrosophic cross-entropy. Additionally, two multicriteria decision-making methods using the interval neutrosophic cross-entropy between an alternative and the ideal alternative are developed in order to determine the order of the alternatives and choose most preferred one(s). Finally, an illustrative example is presented to verify the proposed methods and to demonstrate their effectiveness and practicality.

Total correctness assertions (TCAs) have long been considered a natural formalization of successful program termination. However, research dating back to the 1980s suggests that validity of TCAs is a notion of limited interest; we corroborate this by proving compactness and Herbrand properties for the valid TCAs, defining in passing a new sound, complete, and syntax-directed deductive system for TCAs. It follows that proving TCAs whose truth depends on underlying inductive data-types is impossible in logics of programs that are sound for all structures, such as dynamic logic based on Segerberg's PDL, even when augmented with powerful first-order theories like Peano arithmetic. Harel's convergence rule bypasses this difficulty, but is methodologically and conceptually problematic, in addition to being unsound for general validity. We propose instead to bind variables to inductive data via DL's box operator, leading to an alternative formalization of termination assertions, which we dub inductive TCA (ITCA). We observe that a TCA is provable in Harel's DL exactly when the corresponding ITCA is provable in Segerberg's DL, thereby showing that the convergence rule is not foundationally or practically necessary. We also show that validity of ITCAs is directly reducible to validity of partial correctness assertions, confirming the foundational importance of the latter.

Matching logic is a logic for specifying and reasoning about structure by means of patterns and pattern matching. This paper makes two contributions. First, it proposes a sound and complete proof system for matching logic in its full generality. Previously, sound and complete deduction for matching logic was known only for particular theories providing equality and membership. Second, it proposes matching μ -Iogic, an extension of matching logic with a least fixpoint μ -binder, It is shown that matching μ -Iogic captures as special instances many important logics in mathematics and computer science, including first-order logic with least fixpoints, modal μ -Iogic as well as dynamic logic and various temporal logics such as infinite/finite-trace linear temporal logic and computation tree logic, and notably reachability logic, the underlying logic of the \mathbbk framework for programming language semantics and formal analysis. Matching μ -logic therefore serves as a unifying foundation for specifying and reasoning about fixpoints and induction, programming languages and program specification and verification.