TMPR: A tree-structured modified problem reduction proof procedure and its extension to three-valued logic

In this scientific paper, a software library for Ternary logic gates will be built based on VHDL language to be used in the implementation of combinational Ternary logic circuits. The VHDL language had been designed to be used to automate and design Binary electronic logic systems. Therefore, several problems have been arised in adapting the VHDL language to be used with Ternary electronic logic systems. However, these problems have been solved and the result were encouragement to continue the study with the other Ternary components, so that a complete software library based on VHDL language for different Ternary components will be used as a tool in the design and simulation of Ternary electronic logic circuits and systems. 
 Keywords: Ternary logic, Ternary logic gates, VHDL language.

The article proposes a method for synthesizing ternary binary logic functions using a multi-threshold element, which significantly extends the capabilities of ternary logic compared to traditional binary logic. The aim of the work is to develop an algorithm for synthesizing ternary logic functions, based on the use of a multi-threshold element, which simplifies the design of logical circuits and reduces energy consumption while maintaining high data processing speed. The study includes a comparative analysis of ternary and binary logic functions, allowing an assessment of the advantages of ternary logic, particularly its ability to process more values than binary logic, which opens up new possibilities for creating more efficient computing devices. Specifically, ternary logic can reduce the complexity of circuits and hardware resource usage, while also enhancing performance in tasks where energy efficiency is critical. An essential part of the research is the development of an algorithm for synthesizing ternary binary logic functions based on a four-threshold multi-valued element. Furthermore, a software module was implemented to automate the synthesis of ternary functions. The program not only generates a graphical representation of the logical circuit but also formulates the analytical expression corresponding to the given logical function. This significantly simplifies the design and analysis of logical systems and enhances the efficiency of computations. Considering the high potential of ternary logic, the results of this work are significant not only for theoretical research but also for practical applications in the development of new computing devices and technologies. The further development of these ideas could contribute to the progress in computational systems working on new principles and open up new possibilities for optimizing technological processes in various fields of science and engineering.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

In this paper we extend the models for the “logic of categories” to a wider class of categories than is usually considered. We consider two kinds of logic, a restricted first-order logic and the full higher-order logic of elementary topoi.The restricted first-order logic has as its only logical symbols ∧, ∃, Τ, and =. We interpret this logic in a category with finite limits equipped with a factorization system (in the sense of [4]). We require to satisfy two additional conditions: ⊆ Monos, and any pullback of an arrow in is again in . A category with a factorization system satisfying these conditions will be called an EM-category.The interpretation of the restricted logic in EM-categories is given in §1. In §2 we give an axiomatization for the logic, and in §§3 and 5 we give two completeness proofs for this axiomatization. The first completeness proof constructs an EM-category out of the logic, in the spirit of Makkai and Reyes [8], though the construction used here differs from theirs. The second uses Boolean-valued models and shows that the restricted logic is exactly the ∧, ∃-fragment of classical first-order logic (adapted to categories). Some examples of EM-categories are given in §4.The restricted logic is powerful enough to handle relations, and in §6 we assign to each EM-category a bicategory of relations Rel() and a category of “functional relations” fr. fr is shown to be a regular category, and it turns out that Rel( and Rel(fr) are biequivalent bicategories. In §7 we study complete objects in an EM-category where an object of is called complete if every functional relation into is yielded by a unique morphism into . We write c for the full subcategory of consisting of the complete objects. Complete objects have some, but not all, of the properties that sheaves have in a category of presheaves.

We introduce here the study of generalnonmonotonic rule systems. These deal with situations where a conclusion is drawn from a system of beliefsS (and seen to be inS), basedboth on some being inS and on some restraints not being inS. In the monotone systems of traditional logic there are no restraints, conclusions are drawn solely based on premises being inS. Nonmonotonic rule systems capture the essential syntactic, semantic, and algorithmic features of many nonmonotone systems such as default logic, negation as failure, truth maintenance, autoepistemic logic, and also important combinatorial questions from mathematics such as the marriage problem. This reveals semantics and syntax and proof procedures and algorithms for computing belief sets in many cases where none were previously available and entirely uniformly. In particular, we introduce and study deductively closed sets, extensions and weak extensions. Semantics of nonmonotonic rule systems is studied in part II of this paper and extensions to predicate classical, intuitionistic, and modal logics are left to a later paper.

Even though the strong relationship between proof-theoretic and model-theoretic notions in one’s logical theory can be shown by soundness and completeness proofs, whether we can define the model-theoretic notions by means of the inferences in a proof system is not at all trivial. For instance, provable inferences in a proof system of classical logic in the logical framework SET-FMLA do not determine its intended models as shown by Carnap (Formalization of logic, Harvard University Press, Cambridge, 1943), i.e., there are non-Boolean models that satisfy its provable inferences. In the literature, this is known as the Categoricity problem or Carnap’s problem. In this paper, we will discuss the Categoricity problem (or Carnap’s problem) for three-valued logics K3 and LP. We will provide three different restrictions on admissible models that will deliver us categoricity results, some of which draw from the solutions provided for the Categoricity problem for classical logic in Belnap and Massey (Stud Log 49(1):67–82, 1990) and Bonnay and Westerståhl (Erkenntis 81(4):721–739, 2016). We will then argue that two of those solutions are philosophically well-motivated: (1) restricting the admissible models where negation is interpreted as a Strong Kleene truth-function, and (2) restricting the admissible models where a complex formula is assigned the third value when its immediate subformulas are assigned the third value.

For a very long time truth-value gaps were under a cloud of suspicion because they were considered logically unmanageable. So Frege declared that:“as regards concepts we have a requirement of sharp delimitation; if this were not satisfied it would be impossible to set forth logical laws about them”.Three-valued logic promised to dispel the cloud but in the eyes of many it had promised more than it could deliver. So in response to Reichenbach's plea for a three-valued quantum logic Russell replied:“This is one of a number of questions as to which I am prevented from accepting a certain view by difficulties in carrying it out, but am prepared to alter my opinion if technical skill supplies an answer to my difficulties”.‘By now technical skill has supplied many alternative systems of three-valued logic, but few answers and numerous additional difficulties.’

A ternary logic system to realize the simplest multivalued logic architecture can enhance energy efficiency compared to a binary logic system by reducing the number of transistors and interconnections. For the ternary logic system, a ternary logic device to harness three stable states is needed. In this study, a vertically integrated complementary metal-oxide-semiconductor ternary logic device is demonstrated by monolithically integrating a thin-film transistor (TFT) over a transistor-based threshold switch (TTS). Because the TFT and the TTS have their own source (S), drain (D), and gate (G), there are physically six electrodes. But the hybrid ternary logic device of the TFT over the TTS has only four electrodes: S, D, GTFT, and GTTS like a single MOSFET. It is because the D of the underlying TTS is electrically tied with the S of the superjacent TFT. By combining an on- and off-state of the TFT and the TTS, ternary logic values of low current ("0"-state), middle current ("1"-state), and high current ("2"-state) are realized. Particularly, static power consumption at the "1"-state is decreased by employing the TTS with low off-state leakage current compared to previously reported other ternary logic devices. In addition, a footprint of the ternary logic device with the vertically overlaying structure that has a framework of "one over the other" can be lowered by roughly twice compared to that with the laterally deployed structure that has an organization of "one alongside the other".

SQL uses three-valued logic for evaluating queries on databases with nulls. The standard theoretical approach to evaluating queries on incomplete databases is to compute certain answers. While these two cannot coincide, due to a significant complexity mismatch, we can still ask whether the two schemes are related in any way. For instance, does SQL always produce answers we can be certain about? This is not so: SQL's and certain answers semantics could be totally unrelated. We show, however, that a slight modification of the three-valued semantics for relational calculus queries can provide the required certainty guarantees. The key point of the new scheme is to fully utilize the three-valued semantics, and classify answers not into certain or non-certain, as was done before, but rather into certainly true, certainly false, or unknown. This yields relatively small changes to the evaluation procedure, which we consider at the level of both declarative (relational calculus) and procedural (relational algebra) queries. We also introduce a new notion of certain answers with nulls, which properly accounts for queries returning tuples containing null values.

Full classical S5 in natural deduction with weak normalization

In this article, we propose to adopt the SCIFF abductive logic language to specify business contracts, and show how its proof procedures are useful to verify contract execution and fulfilment. SCIFF is a declarative language based on abductive logic programming, which accommodates forward rules, predicate definitions, and constraints over finite domain variables. Its declarative semantics is abductive, and can be related to that of deontic operators; its operational specification is the sound and complete SCIFF proof procedure, defined as a set of transition rules, which has been implemented and integrated into a reasoning and verification tool. A variation of the SCIFF proof-procedure (g-SCIFF) can be used for static verification of contract properties. We demonstrate the use of the SCIFF language for business contract specification and verification, in a concrete scenario. In order to accommodate integration of SCIFF with architectures for business contract, we also propose an encoding of SCIFF contract rules in RuleML.

In resolution proof procedures refinements based on A-orderings of literals have a long tradition and are well investigated. In tableau proof procedures such refinements were only recently introduced by the authors of the present paper. In this paper we prove the following results: we give a completeness proof of A-ordered ground clause tableaux which is a lot easier to follow than the one published previously. The technique used in the proof is extended to the non-clausal case as well as to the non-ground case and we introduce an ordered version of Hintikka sets that shares the model existence property of standard Hintikka sets. We show that A-ordered tableaux are a proof confluent refinement of tableaux and that A-ordered tableaux together with well-known connection refinements yield an incomplete proof procedure. We introduce A-ordered first-order NNF tableaux, prove their completeness, and we briefly discuss implementation issues.

Bitwise operation is a basic operation in C program. According to the statistics, the bitwise operation accounts for 12% in all numerical operations. There are two problems for the bitwise operation when search technology is used to generate test input: one is the low efficiency for it has no corresponding interval arithmetic rule to reduce the search space; the other is the absence of a unified approach when a variable is simultaneously involved in the bitwise and non-bitwise operations. For problem one, classical two-valued logic is extended to four-valued logic to express the two states of logic variable’s value, ‘uncertain’ and ‘neither true nor false’. According to the operation rules of the four-valued logic, the search space can be reduced. For problem two, the conversion between interval representation and four-valued logic representation is given, and then provided with a unified approach based on the search algorithm. Experiments show that this method can improve the efficiency of test input generation of bitwise operation.

Classical first-order logic is in many ways central to work in mathematics, linguistics, computer science and artificial intelligence, so it is worthwhile to define it in full detail. We present soundness and completeness proofs of a sequent calculus for first-order logic, formalized in the interactive proof assistant Isabelle/HOL. Our formalization is based on work by Stefan Berghofer, which we have since updated to use Isabelle's declarative proof style Isar (Archive of Formal Proofs, Entry FOL-Fitting, August 2007 / July 2018). We represent variables with de Bruijn indices; this makes substitution under quantifiers less intuitive for a human reader. However, the nature of natural numbers yields an elegant solution when compared to implementations of substitution using variables represented by strings. The sequent calculus considered has the special property of an always empty antecedent and a list of formulas in the succedent. We obtain the proofs of soundness and completeness for the sequent calculus as a derived result of the inverse duality of its tableau counterpart. We strive to not only present the results of the proofs of soundness and completeness, but also to provide a deep dive into a programming-like approach to the formalization of first-order logic syntax, semantics and the sequent calculus. We use the formalization in a bachelor course on logic for computer science and discuss our experiences.

First-order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like Frege's morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its formal semantics, and a proof procedure using tableaus. Naturally the tableau rules are not complete, but they are with respect to a Henkinization of the true semantics. We demonstrate the use of the tableau rules by proving one of the theorems involved in Godel's ontological argument, one of the rare instances in the literature where higher-order modal constructs have appeared. A fuller treatment of the material presented here is in preparation