A Modified Completeness Theorem of KAT and Decidability of Term Reducibility

We show that Kleene algebra with tests (KAT) subsumes propositional Hoare logic (PHL). Thus the specialized syntax and deductive apparatus of Hoare logic are inessential and can be replaced by simple equational reasoning. In addition, we show that all relationally valid inference rules are derivable in KAT and that deciding the relational validity of such rules is PSPACE -complete.

The modern software engineering process is evolutionary, with commits/patches begetting new versions of code, progressing steadily toward improved systems. In recent years, program analysis and verification tools have exploited version-based reasoning, where new code can be seen in terms of how it has changed from the previous version. When considering program versions, refinement seems a natural fit and, in recent decades, researchers have weakened classical notions of concrete refinement and program equivalence to capture similarities as well as differences between programs. For example, Benton, Yang and others have worked on state-based refinement relations . In this paper, we explore a form of weak refinement based on trace relations rather than state relations. The idea begins by partitioning traces of a program C 1 into trace classes, each identified via a restriction r 1 . For each class, we specify similar behavior in the other program C 2 via a separate restriction r 2 on C 2 . Still, these two trace classes may not yet be equivalent so we further permit a weakening via a binary relation A on traces, that allows one to, for instance disregard unimportant events, relate analogous atomic events, etc. We address several challenges that arise. First, we explore one way to specify trace refinement relations by instantiating the framework to Kleene Algebra with Tests (KAT) due to Kozen. We use KAT intersection for restriction, KAT hypotheses for A , KAT inclusion for refinement, and have proved compositionality. Next, we present an algorithm for automatically synthesizing refinement relations, based on a mixture of semantic program abstraction, KAT inclusion, a custom edit-distance algorithm on counterexamples, and case-analysis on nondeterministic branching. We have proved our algorithm to be sound. Finally, we implemented our algorithm as a tool called Knotical, on top of Interproc and Symkat. We demonstrate promising first steps in synthesizing trace refinement relations across a hand-crafted collection of 37 benchmarks that include changing fragments of array programs, models of systems code, and examples inspired by the thttpd and Merecat web servers.

We study the complexity of reasoning in Kleene algebra and *-continuous Kleene algebra in the presence of extra equational assumptions E; that is, the complexity of deciding the validity of universal Horn formulas E/spl rarr/s=t, where E is a finite set of equations. We obtain various levels of complexity based on the form of the assumptions E. Our main results are: for *-continuous Kleene algebra, if E contains only commutativity assumptions pq=qp, the problem is II/sub 1//sup 0/-complete; if E contains only monoid equations, the problem is II/sub 2//sup 0/-complete; for arbitrary equations E, the problem is II/sub 1//sup 1/-complete. The last problem is the universal Horn theory of the *-continuous Kleene algebras. This resolves an open question of Kozen (1994).

Equational formulae are first-order formulae over an alphabet F of function symbols whose only predicate symbol is syntactic equality. Unification problems are an important special case of equational formulae, where no universal quantifiers and no negation occur. By the negation elimination problem we mean the problem of deciding whether a given equational formula is semantically equivalent to a unification problem. This decision problem has many interesting applications in machine learning, logic programming, functional programming, constrained rewriting, etc. In this work we present a new algorithm for the negation elimination problem of equational formulae with purely existential quantifier prefix. Moreover, we prove the coNP-completeness for equational formulae in DNF and the Π2p-hardness in case of CNF.

We propose a quasi-equational sound axiomatization of regular tree languages, called monodic tree Kleene algebra. The algebra is weaker than Kleene algebra introduced by Kozen. We find a subclass of regular tree languages, for which monodic tree Kleene algebra is complete. While regular tree expressions may have two or more kinds of place holders, the subclass can be equipped with only one kind of them. Along the lines of the original proof by Kozen, we prove the completeness theorem based on determinization and minimization of tree automata represented by matrices on monodic tree Kleene algebra.

We study Sowa’s conceptual graphs (CGs) with both existential and universal quantifiers. We explore in detail the existential fragment. We extend and modify Sowa’s original graph derivation system with new rules and prove the soundness and completeness theorem with respect to Sowa’s standard interpretation of CGs into first order logic (FOL). The proof is obtained by reducing the graph derivation to a question-answering problem. The graph derivation can be equivalently obtained by querying a Definite Horn Clauses program by a conjunction of positive atoms. Moreover, the proof provides an algorithm for graph derivation in a pure proof-theoretic fashion, namely by means of a slight enhancement of the standard PROLOG interpreter. The graph derivation can be rebuilt step-by-step and constructively from the resolution-based proof. We provide a notion of CGs in normal form (the table of the conceptual graph) and show that the PROLOG interpreter also gives a projection algorithm between normal CGs. The normal forms are obtained by extending the FOL language by witnesses (new constants) and extending the graph derivation system. By applying iteratively a set of rules the reduction process terminates with the normal form of a conceptual graph. We also show that graph derivation can be reduced to a question-answering problem in propositional datalog for a subclass of simple CGs. The embedding into propositional datalog makes the complexity of the derivation polynomial.

We investigate the equational theory for Kleene algebra terms with variable complements and constant complements — (language) complement where it applies only to variables or constants — w.r.t. languages. While the equational theory w.r.t. languages coincides with the language equivalence (under the standard language valuation) for Kleene algebra terms, this coincidence is broken if we extend the terms with complements. In this paper, we prove the decidability of some fragments of the equational theory: the universality problem is coNP-complete, and the inequational theory [Formula: see text] is coNP-complete when [Formula: see text] does not contain Kleene-star. To this end, we introduce words-to-letters valuations; they are sufficient valuations for the equational theory and ease us in investigating the equational theory w.r.t. languages. Additionally, we show a completeness theorem of the equational theory for words with variable complements and the non-empty constant.

Horn formulae are widely used in different settings that include logic programming, answer set programming, description logics, deductive databases, and system verification, among many others. One concrete example is concept subsumption in lightweight description logics, which can be reduced to inference in propositional Horn formulae. Some problems require one to reason with inconsistent Horn formulae. This is the case when providing minimal explanations of inconsistency. This paper proposes efficient algorithms for a number of decision, function and enumeration problems related with inconsistent Horn formulae. Concretely, the paper develops efficient algorithms for finding and enumerating minimal unsatisfiable subsets (MUSes), minimal correction subsets (MCSes), but also for computing the lean kernel. The paper also shows the practical importance of some of the proposed algorithms.

In this paper we study query and update operations for databases consisting of propositional clauses in definite Horn form. We assume that the operations are independent of the given form of the formula, i.e. operations applied to equivalent formulas result equivalent formulas. In order to estimate the required space and time we prove complexity bounds for each of the introduced operations. Finally our considerations are applied to so called interactive formulas, which are often found in knowledge bases of Expert systems.

A single reference point (SRP) is typically employed in traditional studies on the multi-attribute decision making (MADM) method. However, SRP lacks the advantage of the multiplicity of reference points and is thus unable to adequately describe loss aversion actions. With the goal of determining the loss aversion characteristics of decision makers, this work proposes an MADM method that is based on double reference points (DRPs) to solve the MADM problem using crisp and interval fuzzy numbers. First, this work describes the universality of decision-making problems with DRPs and then evaluates the characteristics of DRPs and their effects on decision making. Second, attitude and utility functions are established on the basis of the requirements of loss aversion. Third, considering the “one-vote veto”characteristic of a “dissatisfaction”attitude, a binary utility value is created by integrating the occurrence probability p, and the utility value u. The fourth, the main characteristics of and the aggregation method for the binary utility value are analyzed and established. Finally, a new decision-making method is applied to resolve the location decision of an agricultural product logistics center. Results indicate that the relationship between attribute and reference point significantly influences decision behavior. Compared with the traditional decision-making method, the proposed decision-making method can effectively identify the feasibility levels of alternatives. The ranking of the proposed decision-making method for feasible alternatives is basically the same as that of the traditional decision-making method. These results can effectively help solve agricultural engineering problems and broaden the coverage of and provide references for MADM research.

Horn clause logic is a part of first-order logic. It was first isolated by J. C. C. McKinsey (1943), who was working on decision problems. Between 1956 and 1970, Anatoliī Mal’tsev published a series of papers in which he showed that Horn clauses form the right setting for a large part of universal algebra—including the theory of presentations and initial models. Algebraic specification lies within this part of universal algebra.

In the face of competition, business managers should focus on the quality of decisionmaking. This paper provides an assessment of decision-making methods by discussing the current trends of managers referring to decision-making methods. The paper is an analysis of the attitudes of managers towards decision-making methods. This paper is conducted through questionnaires. It also discusses the impact of decision-making methods on business financial performance. Eventually, some conclusions will be drawn.

Z-numbers are a new concept considering both the description of cognitive information and the reliability of information. Linguistic terms are useful tools to adequately and effectively model real-life cognitive information, as well as to characterize the randomness of events. However, a form of Z-numbers, in which their two components are in the form of linguistic terms, is rarely studied, although it is common in decision-making problems. In terms of Z-numbers and linguistic term sets, we provided the definition of linguistic Z-numbers as a form of Z-numbers or a subclass of Z-numbers. Then, we defined some operations of linguistic Z-numbers and proposed a comparison method based on the score and accuracy functions of linguistic Z-numbers. We also presented the distance measure of linguistic Z-numbers. Next, we developed an extended TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method based on the Choquet integral for multi-criteria decision-making (MCDM) problems with linguistic Z-numbers. Finally, we provided an example concerning the selection of medical inquiry applications to demonstrate the feasibility of our proposed approach. We then verified the applicability and superiority of our approach through comparative analyses with other existing methods. Illustrative and comparative analyses indicated that the proposed approach was valid and feasible for different decision-makers and cognitive environments. Furthermore, the final ranking results of the proposed approach were closer to real decision-making processes. Linguistic Z-numbers can flexibly characterize real cognitive information as well as describe the reliability of information. This method not only is a more comprehensive reflection of the decision-makers’ cognition but also is more in line with expression habits. The proposed method inherited the merits of the classical TODIM method and considers the interactivity of criteria; therefore, the proposed method was effective for dealing with real-life MCDM problems. Consideration about bounded rational and the interactivity of criteria made final outcomes convincing and consistent with real decision-making.

Introduce the basic ideas, principle and modeling approach of the analytic hierarchy process and take examples of post selection and decision of graduates, making specific demonstration about the utilization of AHP to prove the scientificity and practicability of AHP for multi-attribute decision-making problems. There may be several kinds of natural conditions facing the people, and meanwhile there are also several kinds of action plans alternative for people when they are dealing with problems. Decision maker is required to make the decision based on the known information in such case, i.e. select the optimal action plan, and such problems are classified into decision-making problems (1-3). The several natural conditions faced by people are also known as natural state or abbreviated to state that is existed objectively and uncontrollable factor. The alternative action plans are collectively referred to as decision that is controlled factor. The decision maker is responsible for selecting the plan. As a major constituent part of the decision-making analytical problems, the multi- attribute decision-making is widely existed in the real life. For example, buy a house. It is necessary to comprehensively consider the price, area, floor, position, surrounding environment and other various factors of the house. Although the multi- attribute decision-making problems lie in different backgrounds, they usually have some common features: ①It is always mutual conflicting between attributes (with different dimensions); ②There may be qualitative attribute and quantitative attribute in an attribute set at the same time; ③Attribute normally constitute a hierarchical structure; ④Sometimes the decision information is not complete, and the decision maker can only provide the incomplete information of decision parameters; Based on the link where the decision maker provides information and its sufficient degree, the multi-attribute decision-making can be classified into 3 categories. They are decision without preference information, decision with attribute preference information and decision with preference information among the alternative plans. In decisions with attribute preference information, such decisions with the known attribute weight base information are applied most widely to create linear distribution method, simple weighting method, AHP and other classical decision-making methods. Mechanisms of various multi-attribute decision-making methods are also varied and have different application scopes in practice. By contrast, AHP is more useful

Purpose. The scientific paper is aimed: 1) to analyze the existing decision-making methods; 2) to characterize each type of the methods that have practical application in the project management theory; 3) to divide the presented methods into the sense groups; 4) to determine which of these groups should be used for decision-making in project management in three cases: a) when the object is a project; b) when the object is a program; c) when the object is a project portfolio. Methodology. It was carried out a comparative analysis of the existing decision-making methods, as well as it was studied the possibility of their use for different controlled objects of project manager. Findings. As a result of the analysis all of the methods have been divided into two categories: formal and logical-intuitive (informal) ones. It was determined which methods are equally appropriate to apply for all controlled objects of project manager, and which methods will not be effective for a particular object. The analysis showed that some of the decision-making methods are not suitable for use in the management of the certain object. It is concluded that they should be improved and adapted to the specific object of management. Originality. Solution of the problem of decision-making in project management was further developed, as well as it was analyzed the application of different decision-making methods for three controlled objects: project, program and portfolio. Practical value. Decision-making in project management has a direct influence on the result, which is expected to receive. The quality of decisions, in turn, depends on the correct method chosen in the given circumstances. The recommended improvement of certain methods will allow the project manager to make better and more efficient solution.