Quantitative Model Checking of Linear-Time Properties Based on Generalized Possibility Measures

Model checking of linear-time properties based on possibility measures was studied in previous work (Y. Li and L. Li, Model checking of linear-time properties based on possibility measure, IEEE Transactions on Fuzzy Systems, 21(5)(2013), 842-854). However, the linear-time properties considered in the previous work was classical and qualitative, possibility information of the systems was not considered at all. We shall study quantitative model checking of fuzzy linear-time properties based on generalized possibility measures in the paper. Both the model of the system, as well as the properties the system needs to adhere to, are described using possibility information to identify the uncertainty in the model/properties. The systems are modeled by {\sl generalized possibilistic Kripke structures} (GPKS, in short), and the properties are described by fuzzy linear-time properties. Concretely, fuzzy linear-time properties about reachability, always reachability, constrain reachability, repeated reachability and persitence in GPKSs are introduced and studied. Fuzzy regular safety properties and fuzzy $\omega-$regular properties in GPKSs are introduced, the verification of fuzzy regular safety properties and fuzzy $\omega-$regular properties using fuzzy finite automata are thoroughly studied. It has been shown that the verification of fuzzy regular safety properties and fuzzy $\omega-$regular properties in a finite GPKS can be transformed into the verification of (always) reachability properties and repeated reachability (persistence) properties in the product GPKS introduced in this paper. Several examples are given to illustrate the methods presented in the paper.Comment: arXiv admin note: text overlap with arXiv:1409.646

Model-Checking of Linear-Time Properties Based on Possibility Measure

We study the LTL model-checking in possibilistic Kripke structure using possibility measure. First, the notion of possibilistic Kripke structure and the related possibility measure are introduced, then model-checking of reachability and repeated reachability linear-time properties in finite possibilistic Kripke structure are studied. Standard safety property and -regular property in possibilistic Kripke structure are introduced, the verification of regular safety property and -regular property using finite automata are thoroughly studied. It has been shown that the verification of regular safety property and -regular property in finite possibilistic Kripke structure can be transformed into the verification of reachability property and repeated reachability property in the product possibilistic Kripke structure introduced in this paper. Several examples are given to illustrate the methods presented in the paper.Comment: 22pages,5 figure

Semi-dynamic shortest-path tree algorithms for directed graphs with arbitrary weights

Given a directed graph $G$ with arbitrary real-valued weights, the single source shortest-path problem (SSSP) asks for, given a source $s$ in $G$, finding a shortest path from $s$ to each vertex $v$ in $G$. A classical SSSP algorithm detects a negative cycle of $G$ or constructs a shortest-path tree (SPT) rooted at $s$ in $O(mn)$ time, where $m,n$ are the numbers of edges and vertices in $G$ respectively. In many practical applications, new constraints come from time to time and we need to update the SPT frequently. Given an SPT $T$ of $G$, suppose the weight on a certain edge is modified. We show by rigorous proof that the well-known {\sf Ball-String} algorithm for positively weighted graphs can be adapted to solve the dynamic SPT problem for directed graphs with arbitrary weights. Let $n_0$ be the number of vertices that are affected (i.e., vertices that have different distances from $s$ or different parents in the input and output SPTs) and $m_0$ the number of edges incident to an affected vertex. The adapted algorithms terminate in $O(m_0+n_0 \log n_0)$ time, either detecting a negative cycle (only in the decremental case) or constructing a new SPT $T'$ for the updated graph. We show by an example that the output SPT $T'$ may have more than necessary edge changes to $T$. To remedy this, we give a general method for transforming $T'$ into an SPT with minimal edge changes in time $O(n_0)$ provided that $G$ has no cycles with zero length.Comment: 27 pages, 3 figure

Exogenous Quantum Operator Logic Based on Density Operators

Although quantum logic by using exogenous approach has been proposed for reasoning about closed quantum systems, an improvement would be worth to study quantum logic based on density operators instead of unit vectors in the state logic point of view. In order to achieve this, we build an exogenous quantum operator logic(EQOL) based on density operators for reasoning about open quantum systems. We show that this logic is sound and complete. Just as the exogenous quantum propositional logic(EQPL), by applying exogenous approach, EQOL is extended from the classical propositional logic, and is used to describe the state logic based on density operators. As its applications, we confirm the entanglement property about Bell states by reasoning and logical argument, also verify the existence of eavesdropping about the basic BB84 protocol. As a novel type of mathematical formalism for open quantum systems, we introduce an exogenous quantum Markov chain(EQMC) where its quantum states are labelled using EQOL formulae. Then, an example is given to illustrate the termination verification problem of a generalized quantum loop program described using EQMC.Comment: 26 pages, 1 figure

Monogamy of α\alphath Power Entanglement Measurement in Qubit Systems

In this paper, we study the $\alpha$th power monogamy properties related to the entanglement measure in bipartite states. The monogamy relations related to the $\alpha$th power of negativity and the Convex- Roof Extended Negativity are obtained for N-qubit states. We also give a tighter bound of hierarchical monogamy inequality for the entanglement of formation. We find that the GHZ state and W state can be used to distinguish the $\alpha$th power the concurrence for $0<\alpha<2$. Furthermore, we compare concurrence with negativity in terms of monogamy property and investigate the difference between them.Comment: 6 pages, 2 figure

Continuity in Information Algebras

In this paper, the continuity and strong continuity in domain-free information algebras and labeled information algebras are introduced respectively. A more general concept of continuous function which is defined between two domain-free continuous information algebras is presented. It is shown that, with the operations combination and focusing, the set of all continuous functions between two domain-free s-continuous information algebras forms a new s-continuous information algebra. By studying the relationship between domain-free information algebras and labeled information algebras, it is demonstrated that they do correspond to each other on s-compactness

Quantitative Computation Tree Logic Model Checking Based on Generalized Possibility Measures

We study generalized possibilistic computation tree logic model checking in this paper, which is an extension of possibilistic computation logic model checking introduced by Y.Li, Y.Li and Z.Ma (2014). The system is modeled by generalized possibilistic Kripke structures (GPKS, in short), and the verifying property is specified by a generalized possibilistic computation tree logic (GPoCTL, in short) formula. Based on generalized possibility measures and generalized necessity measures, the method of generalized possibilistic computation tree logic model checking is discussed, and the corresponding algorithm and its complexity are shown in detail. Furthermore, the comparison between PoCTL introduced in (2013) and GPoCTL is given. Finally, a thermostat example is given to illustrate the GPoCTL model-checking method

On Quotients of Formal Power Series

Quotient is a basic operation of formal languages, which plays a key role in the construction of minimal deterministic finite automata (DFA) and the universal automata. In this paper, we extend this operation to formal power series and systemically investigate its implications in the study of weighted automata. In particular, we define two quotient operations for formal power series that coincide when calculated by a word. We term the first operation as (left or right) \emph{quotient}, and the second as (left or right) \emph{residual}. To support the definitions of quotients and residuals, the underlying semiring is restricted to complete semirings or complete c-semirings. Algebraical properties that are similar to the classical case are obtained in the formal power series case. Moreover, we show closure properties, under quotients and residuals, of regular series and weighted context-free series are similar as in formal languages. Using these operations, we define for each formal power series $A$ two weighted automata ${\cal M}_A$ and ${\cal U}_A$. Both weighted automata accepts $A$, and ${\cal M}_A$ is the minimal deterministic weighted automaton of $A$. The universality of ${\cal U}_A$ is justified and, in particular, we show that ${\cal M}_A$ is a sub-automaton of ${\cal U}_A$. Last but not least, an effective method to construct the universal automaton is also presented in this paper.Comment: 48 pages, 3 figures, 30 conference

Relational reasoning in the region connection calculus

This paper is mainly concerned with the relation-algebraical aspects of the well-known Region Connection Calculus (RCC). We show that the contact relation algebra (CRA) of certain RCC model is not atomic complete and hence infinite. So in general an extensional composition table for the RCC cannot be obtained by simply refining the RCC8 relations. After having shown that each RCC model is a consistent model of the RCC11 CT, we give an exhaustive investigation about extensional interpretation of the RCC11 CT. More important, we show the complemented closed disk algebra is a representation for the relation algebra determined by the RCC11 table. The domain of this algebra contains two classes of regions, the closed disks and closures of their complements in the real plane.Comment: Latex2e, 35 pages, 2 figure

Coherent-induced state ordering with fixed mixedness

In this paper, we study coherence-induced state ordering with Tsallis relative entropy of coherence, relative entropy of coherence and $l_{1}$ norm of coherence. Firstly, we show that these measures give the same ordering for single-qubit states with a fixed mixedness or a fixed length along the direction $\sigma_{z}$. Secondly, we consider some special cases of high dimensional states, we show that these measures generate the same ordering for the set of high dimensional pure states if any two states of the set satisfy majorization relation. Moreover, these three measures generate the same ordering for all $X$ states with a fixed mixedness. Finally, we discuss dynamics of coherence-induced state ordering under Markovian channels. We find phase damping channel don't change the coherence-induced state ordering for some single-qubit states with fixed mixedness, instead amplitude damping channel change the coherence-induced ordering even though for single-qubit states with fixed mixedness