The Inverse Method for Many-Valued Logics

Multiple-valued logic can be viewed as an alternative approach to solving many problems in transmission, storage, and processing of large and even increasing amounts of information encoded in digital signals. Conversely, methods used in digital signal processing can be exploited to solve particular problems in the design of multiple-valued logic circuits and systems. This paper discusses applications of group-theoretic methods used in digital signal processing to derivation of compact representations for multiple-valued logic functions and the design of multiplevalued logic circuits with regular structures.

We describe an extension of Stalmarck’s method in First Order Logic. Stalmarck’s method is a tableaux-like theorem proving method for propositional logic, that uses a branch-and-merge rule known as the dilemma rule. This rule opens two branches and later merges them, by retaining their common consequences. The propositional version does this with normal set intersection, while the FOL version searches for pairwise unifiable formulae from the two branches. The proof procedure attempts to find proofs with as few simultaneously open branches as possible. We present the proof system and a proof procedure, and show soundness and completeness. We also present benchmarks for an implementation of the proof procedure.

The language of signed formulas offers a first-order classical logic framework for automated reasoning in multiple-valued logics. It is sufficiently general to include both annotated logics and fuzzy operator logics. Signed resolution unifies the two inference rules of annotated logics, thus enabling the development of an SLD-style proof procedure for annotated logic programs. Signed resolution also captures fuzzy resolution. The logic of signed formulas offers a means of adapting most classical inference techniques to multiple-valued logics.

A Multi-State System (MSS) is a mathematical model in reliability engineering that allows representing investigated system/object using several performance levels from perfect working to complete failure. A new approach for the analysis of a MSS based on mathematical methods of multiple-valued logic is proposed in this paper. In particular, the method for the calculation of critical system states is presented.

Tremendous amounts of data are being generated and should be processed during the massive employment of artificial intelligence and edge/cloud computing, necessitating computers with higher capabilities. However, as Moore's Law approaches its limit, enhancing computing capability through transistor miniaturization is becoming less feasible. New strategies for “more-than-Moore” are highly preferred to drive future technological advancements. Multiple-valued logic, especially ternary logic with its superior data density compared to traditional binary logic, offers a viable route to augment computing capability while reducing chip interconnections and dynamic power consumption. Despite the growing interest in ternary logic within academia and industry, its practical application remains limited due to the increased transistor count and connections required by earlier logic gate implementations. In this thesis, we present efficient implementation methods of basic ternary logic gates based on emerging nano-electronics including two-dimensional material-based transistors. The fabricated ternary devices show high performance in DC gain and low power consumption. Additionally, compact models of the devices have been established and calibrated, facilitating the circuit design and simulation in Electronic Design Automation (EDA) tools. At the circuit level, this thesis introduces novel ternary arithmetic circuits, including adders and multipliers, that are optimized for low power consumption and minimal delay. Finally, we showcase the practical benefits of ternary logic systems in data processing through the implementation of carry-free arithmetic circuits, underscoring their potential as a transformative computing paradigm.

This paper presents a method and circuit for correcting faults in logic systems. The method, called feedback (RFB), is similar in some respects to triple modular redundancy (TMR), but has an improved error probability with respect to transient errors. Simulation results indicate an improvement by about two orders of magnitude compared to traditional TMR. CMOS circuits are presented for implementing restorative feedback. For binary logic, a dynamic CMOS circuit is considered. For multiple-valued logic, a semi-floating gate implementation is presented.

Automated theorem provers for multiple-valued logics with satisfiability modulo theory solvers

Resolution-based Theorem Proving for Many-valued Logics

In this study, a fuzzy logic model was constituted by using the Fuzzy Logic (FL) method, which is one of the traditional artificial intelligence (AI) methods, in order to estimate the shear strength of reinforced concrete (RC) beams with limited ductility. In this model, beam width(bw), beam height(h), characteristic concrete compressive strength(fck), transverse reinforcement diameter(T), the number of arms bearing the shear force of the transverse reinforcement(n) and transverse reinforcement spacing(s) were taken into account as variable parameters. The model developed by using the problem data containing the solutions of shear force strength of 2640 beams with different cross-section properties were tested with 480 beam solutions different from these data. In the tests of the developed FL model, maximum percentage error, minimum percentage error, average percentage error and correlation coefficient values were obtained as 3.604, -0.091, 1.514 and R2=0.999678. By applying the fuzzy inverse logic method (FIL), which was recently developed by the author of this study, on the FL model, which is seen to have been developed quite sensitively from the test results, a total of 521 designs were obtained for 15 different RC beams with limited ductility subjected to shear. In order to check the accuracy of these designs, after shear strengths were obtained by conventional computations for these designs, % error and correlation coefficients were computed between the obtained strength values and the shear force values taken into account for the design. The promising results show that the FIL method can be used in the design of RC beams under shear force and even in other scientific studies such as design, optimization and control.

Context. Such perfect algebraic constructions of many-valued logic as ternary bent-functions and their truth tables which are called as 3-bent-sequences, are used very often in modern cryptographic algorithms, in particular, in pseudorandom sequence generators. However, today there are no methods for synthesizing the ternary bent-functions class for a number of variables greater than two, which significantly limits the ability to scale the number of protection levels of the of pseudorandom sequence generators based on the ternary bent-functions. This circumstance generates the task of developing methods for the synthesis of ternary bent-functions, which is solved in this paper for the case of ternary bent-functions of three variables. The object of this research is the process of efficiency increasing of the cryptographic algorithms based on the functions of many-valued logic.Objective. The purpose of the paper is to construct a method for the synthesis of the set of ternary bent-functions of three variables.Method. The mathematical apparatus of the Reed-Muller transform (algebraic normal form) was used as the basis of the proposed constructive method for the synthesis of ternary bent-functions of three variables. So, on the basis of the established properties of the algebraic normal form of ternary bent-functions and limited enumeration, the search for ternary bent-functions up to affine terms is performed, after which we apply the procedure of reproduction.Results. As a result of using of the proposed method for the synthesis of ternary bent-functions of three variables, 155844 3-bentsequences were found up to an affine term, while the cardinality of the full set of found 3-bent-sequences is 12623364. The research performed made it possible to determine that in this set there are 3-bent-sequences of six different weight structures, on the basis of which 12 different triple sets can be compiled for use in pseudorandom sequence generators. A scheme for a cryptographically stable pseudorandom sequence generator based on the found set of 3-bent-sequences of length N = 27 is proposed. It is shown that the protection levels number of such a generator of pseudorandom sequences is 41 7.041 10    which is comparable with the protection levels number of modern block symmetric cryptographic algorithms, for example, AES-128.Conclusions. The further development of modern cryptographic algorithms, in particular, cryptographically stable pseudorandom sequence generators, is largely based on the use of perfect algebraic constructions of many-valued logic. For the first time, a constructive method for the synthesis of ternary bent-functions of three variables is proposed. For the found set of ternary bent-functions, the distribution of weight structures is found, and the possible triple sets are established. Based on the constructed set of ternary bentfunctions, a pseudorandom sequence generator scheme is proposed that has a protection levels number that is comparable with modern block symmetric cryptographic algorithms. We note that the constructed class of ternary bent-functions can also be used for the synthesis of cryptographically strong S-boxes, codes of constant amplitude, as well as error correction codes. As an actual area of further research, we can note the development of methods for the synthesis of ternary bent-functions of a larger number of variables.

Data processing optimization is one of the main concerns for developing of multiple-valued logic. An advantage could be achieved by realization of new functions existing in non-binary logic. These new logic functions could be implemented using quaternary look-up tables. In this work, a quaternary multiplexer circuit is designed to implement any n-variable quaternary logic function based on its truth table. Voltage-mode CMOS with multi-threshold transistors and multi-Vdd quaternary design was suggested. The multiplexer circuit consists of quaternary down literal circuits, binary inverters and binary pass transistor gates. All circuits were simulated with the Spice tool using TSMC 0.18 mum technology and have shown improvements in performance and power consumption and using less transistors than their equivalent binary circuits.

It is demonstrated that the use of Fourier–Galois transforms significantly simplifies the conversion of various multi-valued logic operations into algebraic form. A method is proposed for constructing explicit algebraic expressions of this kind, including for logic functions defined by truth tables. The proposed approach is shown to notably reduce computational complexity compared to previously known methods. Furthermore, representing multi-valued logic operations in algebraic form has considerable practical value, particularly for enhancing control algorithms for groups of unmanned aerial vehicles (UAVs). This representation also improves the efficiency of onboard computing systems in UAV swarms managed by a single operator. The potential of the proposed method for mapping functions defined over Galois fields GF(pⁿ) is discussed.

Two open questions in ternary Boolean algebras (Not. Amer. Math. Soc., 16, No. 4, 654 (June 1969; Bull Amer, Math. Soc., 53, No. 6, 567-572 (June 1947); Proc. Eighth Int. Sypm. on Multiple-Valued Logic, pp. 251-256) were answered with the aid of an existing automated theorem-proving program without recourse to any additional programing. The new automated theorem-proving techniques developed in answering the open questions are presented in this paper; essentially the existing theorem prover is used in a nonstandard way to seek and verify small finite models and counterexamples for a first-order axiom system. Exhibiting a model of an axiom system proves it consistent; this facility complements traditional theorem-proving methods which can only prove inconsistency. 1 figure.

The past few decades have seen a resurgence of reasoning techniques in artificial intelligence involving both classical and non-classical logics. In his paper, ``Multi-valued Logics: A Uniform Approach to Reasoning in Artificial Intelligence'', Ginsberg has shown that through the use of bilattices, several reasoning techniques can be unified under a single framework. A bilattice is a structure that can be viewed as a class of truth values that can accommodate incomplete and inconsistent information and in certain cases default information. In bilattice theory, knowledge is ordered along two dimensions: truth/falsity and certainty/uncertainty. By defining the corresponding bilattices as truth spaces, Ginsberg has shown that the same theorem prover can be used to simulate reasoning in first order logic, default logic, prioritized default logic and assumption truth maintenance system. Although this is a significant contribution, Ginsberg's paper was lengthy and involved. This paper summarizes some of the essential concepts and foundations of bilattice theory. Furthermore, it discusses the connections of bilattice theory and several other existing multi-valued logics such as the various three-valued logics and Belnap's four-valued logic. It is noted that the set of four truth values in Belnap's logic form a lattice structure that is isomorphic to the simplest bilattice. Subsequently, Fitting proposed a conflation operation that can be used to select sub-sets of truth values from this and other bilattices. This method of selecting sub-sets of truth values provides a means for identifying sub-logic in a bilattice.

Beagle is an automated theorem prover for first-order logic modulo built-in theories. It implements a refined version of the hierarchic superposition calculus. This system description focuses on Beagle ’s proof procedure, background reasoning facilities, implementation, and experimental results.