Отношения следования и интенсиональная силлогистика

The paper presents an original supraclassical nontrivial plausible entailment relation $\vapprox$ that employs Kolmogorov's probability theory. Its crucial feature is the primitiveness of a conditional probability, which one calculates with the help of the method of truth tables for classical propositional logic. I study the properties of the entailment relation in question. In particular, I show that while being supraclassical, i.e., all classical entailments and valid formulas are $\vapprox$-valid, but not vice versa, it is not trivial and enjoys the same form of inconsistency as classical entailment ⊧ does. I specify the place of the proposed probability entailment relation in certain classifications of nonclassical entailment relations. In particular, I use Douven's analysis of some probabilistic entailment relations that contains dozens of properties that are crucial for any probabilistic entailment relation, as well as Hlobil's choosing your nonmonotonic logic: shopper’s guide, due to the fact that $\vapprox$ is not monotonic, and Cobreros, Egré, Ripley, van Rooij's entailment relations for tolerant reasoning. At last, I perform a comparative analysis of classical, the proposed, and some other entailment relations closely related to the latter: those introduced by Bocharov, Markin, Voishvillo, Degtyarev, Ivlev, where the last two entailment relations are based on the so-called principle of reverse deduction, which is an intuitively acceptable way to connect classical and probabilistic entailment relations.

PurposeThe purpose of this paper is to describe an approach towards code validation of RISC microcontrollers, which helps to automate software debugging. A static machine code analysis which checks the appropriateness of instructions in a sequence to identify any logical mistakes and also to identify redundant codes appearing in a program for the target processor is presented.Design/methodology/approachValidation is done with the help of rules of inferences formulated for the target processor. The rules govern the occurrence of illegitimate/out of place instructions and code sequences for executing the computational and integrated peripheral functions. The stipulated rules are encoded in propositional logic formulae and their compliance is tested in all possible execution paths of the application programs. An incorrect sequence of machine code pattern is identified using slicing techniques on the control flow graph generated from machine code.FindingsThe results explain that the technique is independent of compiler/assembler and contributes to early detection of software bugs that are otherwise hard to detect. Program states are identified mainly with machine code pattern, which drastically reduces the state space creation contributing to an improved state‐of‐the‐art model checking.Research limitations/implicationsThough the technique described is general, the implementation is highly architecture oriented, and hence the feasibility study is conducted only on PIC16F87X microcontrollers.Practical implicationsThis validation tool can be integrated to the system development environment resulting in improved software quality and reduced debugging time.Originality/valueIt is a novel and original approach at machine code level applicable to a wide range of processors once appropriate rules are available.

In this paper we describe a teaching tool for proving equivalences between propositional logic formulae, using rewrite rules such as De Morgan’s laws and double negation. This tool is based on an earlier tool for rewriting logical formulae into disjunctive normal form (DNF). Both tools make use of a rewrite strategy, which specifies how an exercise can be solved stepwise. Different types of feedback can be calculated automatically from such a strategy specification. We describe a strategy for constructing expert-like equivalence proofs, and present two techniques for improving the proofs that are generated by the strategy.Keywordspropositional logicequivalencese-learningfeedback

A catalog of Boolean concepts

Space Bounds for Resolution

On the hydra number of disconnected graphs

AI systems often need to deal with inconsistent information. For this reason since the early 2000s some AI researchers have developed ways to measure the amount of inconsistency in a knowledge base. By now there is a substantial amount of research about various aspects of inconsistency measuring. The problem is that most of this work applies only to knowledge bases formulated as sets of formulas in propositional logic. Hence this work is not really applicable to the way that information is actually stored. The purpose of this paper is to extend inconsistency measuring to real world information. We first define the concept of general information space which encompasses various types of databases and scenarios in AI systems. Then, we show how to transform any general information space to an inconsistency equivalent propositional knowledge base, and finally apply propositional inconsistency measures to find the inconsistency of the general information space. Our method allows for the direct comparison of the inconsistency of different information spaces, even though the data is presented in different ways. We demonstrate the transformation on three general information spaces: a relational database, a graph database, and a Blocks world scenario, where we apply several inconsistency measures after performing the transformation.

Our terminology and notation is the same as that of [1], of which this note is a sequel.We wish to show that if we take the natural deduction system (N) described in [1], and delete the rules for the quantifiers, we obtain a complete system for propositional logic. [Of course we now construe “X”, “Y”, “Z” as syntactic variables ranging over formulas of propositional logic, rather than sentences of quantification theory.] Moreover the system serves as a neat decision procedure.

We describe an incremental algorithm for computing interpolants for a pair ϕ A , ϕ B of formulas in propositional logic. In contrast with the common approaches, our method does not require a proof of unsatisfiability of ϕ A ∧ ϕ B , and can be realized using any SAT solver as a black box. We achieve this by combining model enumeration with the ability to easily generate interpolants in the special case that one of the formulas is a cube.

Cut formulas in propositional logic

Reaction systems are a qualitative formalism for modeling systems of biochemical reactions characterized by the non-permanency of the elements: molecules disappear if not produced by any enabled reaction. Reaction systems execute in an environment that provides new molecules at each step. Brijder, Ehrenfeucht and Rozemberg introduced the idea of predictors. A predictor of a molecule s, for a given n, is the set of molecules to be observed in the environment to determine whether s is produced or not at step n by the system. We introduced the notion of formula based predictor, that is a propositional logic formula that precisely characterizes environments that lead to the production of s after n steps. In this paper we revise the notion of formula based predictor by defining a specialized version that assumes the environment to provide molecules according to what expressed by a temporal logic formula. As an application, we use specialized formula based predictors to give theoretical grounds to previously obtained results on a model of gene regulation.

Recognition and determination of fuzzy logical relationship in the system fault evolution process

Nontrivial reasoning under inconsistency, called paraconsistent reasoning, has been discussed mainly in two frameworks; one is the classical logic (consistency‐based) framework and the other is the three‐valued logic framework. In this paper, we propose a new entailment relation based on three‐valued logic for reasoning from an inconsistent knowledge base, and discuss the relation between the previous two frameworks. A U‐minimum model of the inconsistent knowledge base is defined in accordance with a principle of uncertainty minimization for three‐valued models. We show that every set of atomic formulas assigned the additional third value in the U‐minimum model is the minimum hitting set for the collection of sets each of which consists of atomic formulas occurring in the minimal inconsistent set of the knowledge base. Based on this result, we present an algorithm to compute the paraconsistent entailment relation by reducing the three‐valued entailment relation to a two‐valued classical entailment relation. © 2006 Wiley Periodicals, Inc. Syst Comp Jpn, 37(14): 44–51, 2006; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.20539

This paper introduces the use of propositional logic calculus in the elaboration of educational quizzes to assess the level understanding of students in a specific theme of their courses. The technique introduced in this paper goes beyond multiple-choice quizzes. The technique requires several steps like a) to give a phrase, b) to re-order words of the given phrase in order to form a propositional logic formula, c) to make use of background knowledge for performing substitutions, d) to answer questions from one of the person in the team, e) to change synonyms/antonyms (if this is feasible), f) to perform actions in order to give value to both or at least one operand of the logic formula and g) to conclude the final answer of the logic formula (true or false) depending of the logic values of the operands in the logic formula. As a working example, the author shows a quiz for universal history, however, the same technique could be used to assess students in different courses.

In the last decade, advances in satisfiability-modulo-theories (SMT) solvers have powered a new generation of software tools for verification and testing. These tools transform various program analysis problems into the problem of satisfiability of formulas in propositional or first-order logic, where they are discharged by SMT solvers, such as Z3 from Microsoft Research. This paper briefly summarizes four initiatives from Microsoft Research that build upon Z3 and move beyond first-order satisfaction: Fixed points—μZ is a scalable, efficient engine for discharging fixed point queries over recursive predicates with logical constraints, integrated in Z3; Interpolants—Interpolating Z3 uses Z3’s proof generation capability to generate Craig interpolants in the first-order theory of uninterpreted functions, arrays and linear arithmetic; Automata—The symbolic automata toolkit lifts classical automata analyses to work modulo symbolic constraints on alphabets; Polynomials—a new decision procedure for the existential theory of the reals allows efficient solving of systems of non-linear arithmetic constraints.KeywordsPoint QueryAbstract DomainSymbolic Model CheckLinear ArithmeticUninterpreted FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.