Tabular Logics Research Articles

Computer-Aided Searching for a Tabular Many-Valued Discussive Logic—Matrices

Abstract In the paper, we tackle the matter of non-classical logics, in particular, paraconsistent ones, for which not every formula follows in general from inconsistent premisses. Our benchmark is Jaśkowski’s logic, modeled with the help of discussion. The second key origin of this paper is the matter of being tabular, i.e. being adequately expressible by finitely many finite matrices. We analyse Jaśkowski’s non-tabular discussive (discursive) logic $ \textbf {D}_{2}$, one of the first paraconsistent logics, from the perspective of a trivalent tabular logic. We are motivated to step down from the ongoing modal $ \textbf {S5}$-perspective of developing $ \textbf {D}_{2}$ both by certain mysteries that have been surrounding it and by gaps in Jaśkowski’s arguments contra the multivalent tabular perspective. Although Jaśkowski’s idea to use $ \textbf {S5}$ in order to define $ \textbf {D}_{2}$ is very attractive since it allows one to benefit from the tools and results of modal logic, it also gives a ‘non-direct’ formulation and, as it appeared later, is superfluous with respect to what is meant to be achieved since one can define the very same logic but using modal logics weaker than S5. It is also known, due to Kotas, that discussive logic is not finite-valued. So, in light of Kotas’s result that $ \textbf {D}_{2}$ is non-tabular, we propose to associate it with a few dozen discussive formulae that Jaśkowski unequivocally and illustratively suggests to be its theorems or non-theorems rather than with axioms of its modern axiomatizations (one of which Kotas employs) in order to be capable of performing a computer-aided brute-force search for suitable trivalent matrices in the cases of one and two designated values. As a result, we find trivalent matrices with two designated values that might be dubbed ‘discussive’ because they meet Jaśkowski’s suggestion to validate and invalidate the litmus theorems and non-theorems, respectively, despite the fact that none of them validates all the negation axioms in the modern axiomatizations of $ \textbf {D}_{2}$. The matrices found are then analysed along with highlighting the ones that were previously mentioned in the literature. We conclude the paper with a comparative analysis of Omori’s results and a test of Karpenko’s hypothesis.

Optimization of a Mealy Automaton Circuit in a Mixed Element Basis

Introduction. The control device is one of the most important blocks of any digital system. The main function of the control device is to coordinate the interaction of the remaining units of the system. Therefore, the characteristics of the control device circuit have a significant impact on the quality of the overall system. To represent the law of functioning of the control device, the models of the microprogrammed automaton (MPA) by Moore and Mealy are used. When synthesizing MPA circuits, it is necessary to solve a number of optimization problems: reducing hardware costs, increasing performance, minimizing power consumption, and jointly optimizing hardware-time characteristics. Methods for solving these problems largely depend on the elemental basis used. Currently, one of the main bases in which modern digital systems are implemented is the FPGA. The main blocks in the FPGA are configurable logic blocks, a programmable interconnect matrix, a timing tree, and programmable inputs and outputs. To implement MPA schemes, two types of configurable logic blocks can be used: tabular logic elements (TLE) and built-in memory blocks (VBP), which have the property of reconfiguration. However, VBPs are widely used to implement various operating blocks of digital systems. Therefore, the controller circuit designer can use a limited number of such memory blocks. Purpose of the article. The article deals with the issues of MPA synthesis when there are a limited number of "free" blocks of EBP. In this case, the microprogram automaton circuit is represented by a network consisting of VBP and TLE blocks. A method for the synthesis of a microprogram automaton with optimization of the number of TLEs is proposed when only one VBP can be used in the microprogram automaton circuit. The proposed method is based on the use of a built-in memory block that performs the replacement of input variables and the coding of the automaton outputs. Results. Studies of the effectiveness of the proposed method were carried out on standard machines. FPGAs of the Virtex-7 family from Xilinx were used as the elemental basis. To implement the proposed MPA, the Vivado package was used. The results of the research showed that the use of the VBP block made it possible to reduce the number of SLE blocks by an average of 14 % – 18 % compared to schemes consisting only of SLE. For the Virtex-7 family FPGA, the number of TLE inputs Io= 6 was sufficient for a single-level implementation of the output system. Conclusions. The effectiveness of the proposed method makes it possible to recommend it for use in the synthesis of microprogram automata under conditions of an extremely limited number of BVPs. Keywords: Mealy automaton, synthesis, coding of inputs, coding of sets of outputs.

Explicit basis for admissible rules in K-saturated tabular logics

Abstract We construct an explicit finite basis for admissible rules in K-saturated tabular logics that extend the logic Grz.

QUANTUM COMPRESSION OF BINARY MESSAGES

In the proposed work, the problem of compressing binary messages in one cycle is solved. A method for solving it in relation to Bernoulli sources of information is proposed, using tabular logic, according to which the original message is converted by the decoder of the compression system into a pre-recorded number in one of its memory cells together with a number representing the number of units in the original message. Restoration of compressed information also occurs according to tabular logic, in which the number of the compressed message excites the original message recorded in the corresponding memory cell, and it is transmitted to the system output. A significant advantage of the proposed method, in addition to speed, is that the compression effect approaches the entropy of the original messages and makes it possible to build quantum processors both on ordinary elements and on elements that use quantum effects.

TABULARITY AND POST-COMPLETENESS IN TENSE LOGIC

AbstractA new characterization of tabularity in tense logic is established, namely, a tense logic L is tabular if and only if $\mathsf {tab}_n^T\in L$ for some $n\geq 1$ . Two characterization theorems for the Post-completeness in tabular tense logics are given. Furthermore, a characterization of the Post-completeness in the lattice of all tense logics is established. Post numbers of some tense logics are shown.

Diagram and Discourse in the Anatomy of Melancholy

Abstract This article examines the methodological, epistemological, aesthetic, and affective tensions between the promise of diagrammatic representation and the practice of discursive expression in Robert Burton’s The Anatomy of Melancholy (1621, first edition). It closely interprets the Anatomy’s tabular logic and analytic claims per se and in conjunction with Burton’s inductive, imaginative prose. While the discursive gathering of copious particulars aims to cure and to console, by ‘rectification’ and ‘recreation’ respectively, the synoptic tables introducing the book’s three partitions represent the ambiguous promise of human scientia, thus becoming yet another cause of melancholy. Compared with other early modern instances of tabular and encyclopedic reason, and interpreted in light of recent scholarship on the diagram, Burton’s tables play a critical, subtle role not only in the Anatomy’s invention and arrangement of topics, but also on the local level where the struggle for meaning and the experience of affect occurs.

Глобально допустимые правила вывода

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).

Explicit basis for admissible rules in $K$-saturated tabular logics

В статье получен явный конечный базис для допустимых правил $K$-насыщенных табличных логик, расширяющих логику $Grz$.

ON EQUATIONAL COMPLETENESS THEOREMS

AbstractA logic is said to admit an equational completeness theorem when it can be interpreted into the equational consequence relative to some class of algebras. We characterize logics admitting an equational completeness theorem that are either locally tabular or have some tautology. In particular, it is shown that a protoalgebraic logic admits an equational completeness theorem precisely when it has two distinct logically equivalent formulas. While the problem of determining whether a logic admits an equational completeness theorem is shown to be decidable both for logics presented by a finite set of finite matrices and for locally tabular logics presented by a finite Hilbert calculus, it becomes undecidable for arbitrary logics presented by finite Hilbert calculi.

Classical Logic with n Truth Values as a Symmetric Many-Valued Logic

We introduce Boolean-like algebras of dimension n (n{mathrm {BA}}s) having n constants {{{mathsf {e}}}}_1,ldots ,{{{mathsf {e}}}}_n, and an (n+1)-ary operation q (a “generalised if-then-else”) that induces a decomposition of the algebra into n factors through the so-called n-central elements. Varieties of n{mathrm {BA}}s share many remarkable properties with the variety of Boolean algebras and with primal varieties. The n{mathrm {BA}}s provide the algebraic framework for generalising the classical propositional calculus to the case of n–perfectly symmetric–truth-values. Every finite-valued tabular logic can be embedded into such a n-valued propositional logic, n{mathrm {CL}}, and this embedding preserves validity. We define a confluent and terminating first-order rewriting system for deciding validity in n{mathrm {CL}}, and, via the embeddings, in all the finite tabular logics.

Bisimulation games and locally tabular logics

Bisimulation games and locally tabular logics

Evaluation of knowledge resources for public health reporting logic: Implications for knowledge authoring and management

To control disease, laboratories and providers are required to report conditions to public health authorities. Reporting logic is defined in a variety of resources, but there is no single resource available for reporters to access the list of reportable events and computable reporting logic for any jurisdiction. In order to develop evidence-based requirements for authoring such knowledge, we evaluated reporting logic in the Council of State and Territorial Epidemiologist (CSTE) position statements to assess its readiness for automated systems and identify features that should be considered when designing an authoring interface; we evaluated codes in the Reportable Condition Mapping Tables (RCMT) relative to the nationally-defined reporting logic, and described the high level business processes and knowledge required to support laboratory-based public health reporting. We focused on logic for viral hepatitis. We found that CSTE tabular logic was unnecessarily complex (sufficient conditions superseded necessary and optional conditions) and was sometimes true for more than one reportable event: we uncovered major overlap in the logic between acute and chronic hepatitis B (52%), acute and Past and Present hepatitis C (90%). We found that the RCMT includes codes for all hepatitis criteria, but includes addition codes for tests not included in the criteria. The proportion of hepatitis variant-related codes included in RCMT that correspond to a criterion in the hepatitis-related position statements varied between hepatitis A (36%), acute hepatitis B (16%), chronic hepatitis B (64%), acute hepatitis C (96%), and past and present hepatitis C (96%). Public health epidemiologists have the need to communicate parameters other than just the name of a disease or organism that should be reported, such as the status and specimen sources. Existing knowledge resources should be integrated, harmonized and made computable. Our findings identified functionality that should be provided by future knowledge management systems to support epidemiologists as they communicate reporting rules for their jurisdiction.

Decidability of the interpolation problem and of related properties in tabular logics

Propositional modal and positive logics are considered as well as extensions of Johansson’s minimal logic. It is proved that basic versions of the interpolation property and of the Beth definability property, and also the Hallden property, are decidable on the class of tabular logics, i.e., logics given by finitely many finite algebras. Algorithms are described for constructing counterexamples to each of the properties mentioned in handling cases where the logic under consideration does not possess the required property.

Substitution Frege and extended Frege proof systems in non-classical logics

We investigate the substitution Frege ( SF ) proof system and its relationship to extended Frege ( EF ) in the context of modal and superintuitionistic (si) propositional logics. We show that EF is p-equivalent to tree-like SF , and we develop a “normal form” for SF -proofs. We establish connections between SF for a logic L , and EF for certain bimodal expansions of L . We then turn attention to specific families of modal and si logics. We prove p-equivalence of EF and SF for all extensions of KB , all tabular logics, all logics of finite depth and width, and typical examples of logics of finite width and infinite depth. In most cases, we actually show an equivalence with the usual EF system for classical logic with respect to a naturally defined translation. On the other hand, we establish exponential speed-up of SF over EF for all modal and si logics of infinite branching, extending recent lower bounds by P. Hrubeš. We develop a model-theoretical characterization of maximal logics of infinite branching to prove this result.

Semi-structured interview protocol for constructing logic models

This paper details a semi-structured interview protocol that evaluators can use to develop a logic model of a program's services and outcomes. The protocol presents a series of questions, which evaluators can ask of specific program informants, that are designed to: (1) identify key informants basic background and contextual information, (2) generate logic model elements, (3) model program inputs, activities, outputs, and outcomes, (4) build a rational theory, (5) develop a program theory, (6) prioritize logic model elements, and (7) build a graphical or tabular logic model. The paper will also provide an example of how this approach was used to develop a logic model for a youth mentoring program. It is our hope and belief that with this interview protocol, novice evaluators will be able to generate comprehensive logic models like seasoned professional evaluators.

Transfer theorems for extensions of the paraconsistent Nelson logic

Descriptions of tabular logics, pretabular logics, and logics satisfying Craig’s interpolation property are transferred from the class of superintuitionistic logics to the class of extensions of the paraconsistent Nelson logic.

Tabular Logics with no Finite Bases for Inference Rules

Tabular Logics with no Finite Bases for Inference Rules

Euclidean Hierarchy in Modal Logic

For a Euclidean space $$\mathbb{R}^n $$ , let L n denote the modal logic of chequered subsets of $$\mathbb{R}^n $$ . For every n ≥ 1, we characterize L n using the more familiar Kripke semantics, thus implying that each L n is a tabular logic over the well-known modal system Grz of Grzegorczyk. We show that the logics L n form a decreasing chain converging to the logic L ∞ of chequered subsets of $$\mathbb{R}^\infty $$ . As a result, we obtain that L ∞ is also a logic over Grz, and that L ∞ has the finite model property. We conclude the paper by extending our results to the modal language enriched with the universal modality.

Heterotopology and Geography

Topology is the search for the places in history. It is the possibility to suspend time. Here, starting from Edward Soja and Michel Foucault, it legitimates the returning to the very beginning of the modern age-there, where we can perceive that original and premodern heterotopia concretely ordered by a mirror and a panel. This heterotopia reflects and inverts a space ordered by analogies, a space that is not still fixed by the mortifying tabular logic proper of the modern age. Or else a space whose main working might be similar to the postmodern wrapping. This heterotopia—the specular projection by Filippo Brunelleschi—reflects the subjection in which the space submits whoever looks towards it. The aim is to understand the premodern origins of that mechanism through which postmodern geographers could redraw their maps.

An Inheritance Criterion for the Admissible Inference Rules of K4

We give a criterion for inheritance of the admissible inference rules of the modal logic K4 by the finitely approximable modal logics extending K4. We give a negative solution to the question of inheritance of the admissible rules of K4 by tabular logics. We exhibit a series of examples of modal logics which inherit or fail to inherit the admissible inference rules of K4.