Разрешимость глобальной допустимости правил вывода в логике 𝑆4

Setting the basic rules of inference is fundamental to logic. The most general variant of possible inference rules are admissible inference rules: in logic 𝐿, a rule of inference is admissible if the set of theorems 𝐿 is closed with respect to this rule. The study of admissible inference rules was stimulated by Friedman’s problem: Is there an algorithm for recognizing the admissibility of an inference rule in intuitionistic logic? For a wide class of non-classical logics the problem of recognizing with respect to the admissibility of inference rules was solved in 1980s. Another way of describing all admissible rules in logic goes back to the problem of A. Kuznetsov (1975): specifying a certain (finite) set of admissible rules, from which all other admissible rules in logic will be derived as consequences, i.e. setting a (finite) basis. It turned out that most basic non-classical logics do not have a finite basis for admissible inference rules. In the early 2000s, for most basic non-classical logics and some tabular logics, the Fridman-Kuznetsov problem was solved by describing an explicit basis for admissible rules. The next stage in the study of admissible inference rules for non-classical logics can be considered the concept of a globally admissible inference rule. Globally admissible rules in the logic 𝐿 are those inference rules that are admissible simultaneously in all (with finite model property) extensions of the given logic. Such rules develop and generalize the concept of an admissible inference rule. The present work is devoted to the study of globally admissible rules of logic 𝑆4. Conditions for global admissibility in the logic 𝑆4 were obtained, a characteristic (universal) model was constructed (checking global admissibility is reduced to checking the truth of a rule on its submodels), a basis was described (all globally admissible rules are derived from it) and an anti-basis (from which all rules not available globally in 𝑆4).

Setting the basic rules of inference is fundamental to logic. The most general variant of possible inference rules are admissible inference rules:in logic L, a rule of inference is admissible if the set of theorems L is closed with respect to this rule. The study of admissible inference rules was stimulated by the formulation of problems about decidability by admissibility (Friedman) and the presence of a finite basis of admissible rules (Kuznetsov) in Int logic. In the early 2000s, for most basic non-classical logics and some tabular logics, the Fridman-Kuznetsov problem was solved by describing an explicit basis for admissible rules. The next stage in the study of admissible inference rules for non-classical logics can be considered the concept of a globally admissible inference rule. Globally admissible rules in the logic L are those inference rules that are admissible simultaneously in all (with finite model property) extensions of the given logic. Such rules develop and generalize the concept of an admissible inference rule. The presented work is devoted to the study of bases for globally admissible rules of logic S4. An algorithm for constructing a set of inference rules in a reduced form was described, forming the basis for globally admissible inference rules in S4 logic.

The paper discusses the admissibility and controller design of nonlinear systems based on the Takagi-Sugeno descriptor model, which is a well-known system that can represent a wide class of nonlinear systems. Due to the importance of the model, many results on the admissibility problem have been in the literature and have shown the improved results. However, there is still room to reduce the conservativeness of the existing admissibility conditions. In this paper, new admissibility conditions and control design methods are proposed. To this end, new Lyapunov functions, which have an integral structure of the membership functions, are introduced. These classes of Lyapunov functions eventually reduces the conservatism in admissibility and control design conditions. Furthermore, a relaxation lemma is used to further reduce the conservatism. Finally, illustrative examples are given to show the effectiveness of our control design methods.

We consider HNN extensions where U 1 and U 2 are inverse monoids of an inverse semigroup S such that, for any and with in S, there exists with in S, for we say that U 1 and U 2 are lower bounded in S. We construct and describe the Schützenberger automata of and give conditions for to have decidable word problem. Homomorphisms of the Schützenberger graphs of are studied and conditions are given for to be completely semisimple. When S has decidable word problem and U 1 and U 2 are finite, we show that has decidable word problem. The class of HNN extensions considered here is surprisingly useful and generalizes the class introduced by Jajcayová. A future paper intends to show that any HNN extension of an inverse semigroup can be embedded into an HNN extension where the subsemigroups are lower bounded.

We study intransitive temporal logic implementing multi-agent’s approach and formalizing knowledge and uncertainty. An innovative point here is usage of non-transitive linear time and multi-valued models - the ones using separate valuations \(V_j\) for agent’s knowledge of facts and summarized (agreed) valuation together with rules for computation truth values for compound formulas. The basic mathematical problems we study here are - decidability and decidability w.r.t. admissible rules. First, we study general case - the logic with non-uniform intransitivity and solve its decidability problem. Also we consider a modification of this logic - temporal logic with uniform non-transitivity and solve problem of recognizing admissibility in this logic.

The paper studies temporal logic implementing multi-agent’s approach and formalizing knowledge and uncertainty. We consider non-transitive linear time and multi-valued models—the ones using separate valuations \(V_j\) for the agent’s knowledge of facts and summarized (agreed) valuation together with rules for computation truth values for compound formulas. The basic mathematical problems we study here are decidability and decidability w.r.t. admissible rules. First, we study the general case—the logic with non-uniform intransitivity—and solve its decidability problem. Also, we consider a modification of this logic—temporal logic with uniform non-transitivity—and solve the problem of recognizing admissibility in this logic. The conclusion contains a discussion and a list of open problems.

Unification in Description Logics has been introduced as a means to detect redundancies in ontologies. We try to extend the known decidability results for unification in the Description Logic EL to disunification since negative constraints on unifiers can be used to avoid unwanted unifiers. While decidability of the solvability of general EL-disunification problems remains an open problem, we obtain NP-completeness results for two interesting special cases: dismatching problems, where one side of each negative constraint must be ground, and local solvability of disunification problems, where we restrict the attention to solutions that are built from so-called atoms occurring in the input problem. More precisely, we first show that dismatching can be reduced to local disunification, and then provide two complementary NP-algorithms for finding local solutions of (general) disunification problems.

We describe an explicit basis for the admissible inference rules in the Godel-Lob logic. The basis consists of a sequence of inference rules in infinitely many variables. Inference rules in the reduced form play an important role in this study. Alongside a basis for the admissible rules we obtain a basis for the quasi-identities of the countable rank free algebra in the Godel-Lob logic.

We study bases for the admissible inference rules in a broad class of modal logics. We construct an explicit basis for all admissible rules in the logics S4.1, Grz, and their extensions whose number is at least countable. The resulting basis consists of an infinite sequence of rules in a concise and simple form. In the case of a logic of finite width a basis for all admissible rules consists of a finite sequence of rules.

Previous intuitionistic justification logics included explicit justifications for all admissible rules of intuitionistic logic in order to get completeness with respect to provability semantics. We present the justification logic iJT4, which does not have these additional justification terms. We establish that iJT4 is complete with respect to modular models and that there is a realization of intuitionistic S4 into iJT4. Hence iJT4 can be seen as an explicit version of intuitionistic S4.

We study an extension of temporal logic, a multi-agent logic on models with nontransitive linear time (which is, in a sense, also an extension of interval logic). The proposed relational models admit lacunas in admissibility relations among agents: information accessible for one agent may be inaccessible for others. A logical language uses temporary operators ‘until’ and ‘next’ (for each of the agents), via which we can introduce modal operations ‘possible’ and ‘necessary.’ The main problem under study for the logic introduced is the recognition problem for admissibility of inference rules. Previously, this problem was dealt with for a logic in which transitivity intervals have a fixed uniform length. Here the uniformity of length is not assumed, and the logic is extended by individual temporal operators for different agents. An algorithm is found which decides the admissibility problem in a given logic, i.e., it recognizes admissible inference rules.

The legacy of logical revisabilism is a hot issue in the philosophy of logic in China. Logical revisabilism holds that Quine is the source of this theory, and that non-classical logic is an instance of logical revision. Here, the reason for logical revisability is due to false descriptive elements in logic. Quine may not agree with logical revisabilism because he thinks that only first-order logic is the orthodox logic, there being no instance of logical revision. Logical revisabilists do not discuss the problem of logical revision on the same level. What’s more, there is an unsolved problem with logical revisabilism, which is explaining “the false descriptive elements in logic.”

We show that the unification problem “is there a substitution instance of a given formula that is provable in a given logic?” is undecidable for basic modal logics K and K4 extended with the universal modality. It follows that the admissibility problem for inference rules is undecidable for these logics as well. These are the first examples of standard decidable modal logics for which the unification and admissibility problems are undecidable. We also prove undecidability of the unification and admissibility problems for K and K4 with at least two modal operators and nominals (instead of the universal modality), thereby showing that these problems are undecidable for basic hybrid logics. Recently, unification has been introduced as an important reasoning service for description logics. The undecidability proof for K with nominals can be used to show the undecidability of unification for Boolean description logics with nominals (such as ALCO and SHIQO). The undecidability proof for K with the universal modality can be used to show that the unification problem relative to role boxes is undecidable for Boolean description logics with transitive roles, inverse roles, and role hierarchies (such as SHI and SHIQ).

We construct an explicit finite basis for admissible inference rules in an arbitrary modal logic of width 2 extending the logic Grz.

We have designed and implemented a set of software tools for the composition and evaluation of hypotheses about gene regulation in biological systems. Our software uses a unified formal grammar for the representation of both diagram-based and text-based hypotheses. The objective of this paper is to show how to use this grammar as the basis for an effective logic for specifying hypotheses about biological systems in precise model-theoretic terms. To accomplish this, we take inspiration from inflationary extensions to fixed point logics and define a new type of logic: a deflationary logic for describing the effects of experiments upon models of biological systems. We present results that characterize decidability, satisfiability, and inflationary/deflationary properties of this logic. We formally define what it means for a set of assertions to be discoverable under this new logic, and show that our software generates discoverable queries. Thus, we lay the groundwork for a formal treatment of machine-aided experimental design under the conceptual framework we have developed for our hypothesis evaluation software.