Functional completeness and canonical forms in many-valued logics1

Chapter 20 - Automated Deduction for Many-Valued Logics

Many-valued logic based interval reasoning plays an increasingly important role in fuzzy and other many-valued extensions of two-valued (crisp) logic. It is often overlooked that many-valued logic (MVL) reasoning has a richer inference rule base than the crisp (i.e. classical two-valued) logic. Indeed some rules that are not possible in the crisp logic (which is point-based) do come to existence when we accept intervals as the basic elements of the semantic space ([0, 1] or a more general lattice) into which the logic expressions are valuated [11, 8, 37].

Resolution-based Theorem Proving for Many-valued Logics

In this paper, we study alethic modal operators in Łukasiewicz’ many-valued logics where truth values are certain rational numbers from the closed unit interval [0,1]. The alethic modal operators necessary and possible are added to Ł3 using Tarski’s idea. The result is a modal logic denoted by Ł\(_{3}^{\mathrm{mod}}\). A formula-equivalency between Ł n and Ł\(_{n}^{\mathrm{mod}}\) is stated for n ≥ 3. Truth value assignments, or valuations, in Łukasiewicz’ n-valued modal logic are considered. The laws of excluded middle and contradiction are considered and found that they are possible in Ł\(_{n}^{\mathrm{mod}}\). Normal modal many-valued systems based on axiom schemes (K), (T), (S4), (B) and (S5) are considered.

Combining knowledge with many-valued logics

To enhance the expressive power and the declarative ability of a deductive database, various CWA (Closed World Assumption) formalizations including the naive CWA, the generalized CWA and the careful CWA are extended to multi-valued logics. The basic idea is to embed logic formulas into some polynomial ring. The extensions can be applied in a uniform manner to any finitely multi-valued logics. Therefore they are also of computational significance.

Multiple-valued logic (MVL) is a logic in which there are more than two truth values. MVL is used due to difficulty in interconnection problems in binary system. Carbon nanotube field-effect transistors (CNTFET) is used to design the Quaternary Half Subtractor (QHS) circuit. Sub blocks of Module (0210), Module (1021), Module (2102), Module (3210) using three supply voltage devices and Quaternary Multiplexer 2:1 using pass transistor have designed in this paper. Therefore, compared to existing designs, less no. of transistors are needed. Comparing proposed Half subtractor using 4:1 MUX modules to current one’s reveals that the 2:1 multiplexer-based method leads in lower power usage. These proposed designs have proven to work satisfactorily under a variety of operating situations, including power, delay, and power delay product (PDP).

Linguistic Geometry for autonomous navigation

Analogic: Inference beyond logic

(1978). Orthomodular Lattices and Quantum Physics. Mathematics Magazine: Vol. 51, No. 5, pp. 299-303.

Babbitt's Canonical Form Revisited:Codes and Metaphors for Epistemology Joshua Banks Mailman (bio) Even among Milton Babbitt's composition titles (such as It takes Twelve to Tango, Whirled Series, and the Joy of More Sextets) his title Canonical Form is unusually mysterious, not for being abstruse or impenetrable, but rather for being so multiply penetrable; that is, for being so easily unwrapped to reveal several possible meanings. Until recently, the titles of Babbitt's compositions have not been subject to much scrutiny, usually being treated merely as casual diversions, prompting mere casual speculation. Alison Maggart's (2017; 2021) recent work, however, delves into aspects of Babbitt's biography, identity, and personality to develop an unprecedented hermeneutical account of his music, foregrounding Babbitt's predilection for wordplay in the form of puns, considering the possible power dynamics of humor, Babbitt's Jewish identity, and his nostalgia for baseball. The present essay pursues such a hermeneutical account, focusing on Babbitt's solo piano work Canonical Form (1983). The third of Joseph Dubiel's "Three Essays on Milton Babbitt" (1990; 1991; 1992) addresses the overall grandeur of Babbitt's Canonical Form, suggesting some possible meanings of the title, which [End Page 215] provide part of the foundation that my remarks build on. Yet since the time of Dubiel's writing, not only have Babbitt's titles begun to be scrutinized more closely (Maggart 2017; 2021), but also a number of analytical technologies and attitudes have arisen which enable, or underwrite, an even broader spectrum of interpretations than was perhaps possible, or tenable. For instance, Gilles Fauconnier and Mark Turner's (1998; 2002) conceptual blending theory, including the analytical technology of conceptual integration networks (CINs), is enlisted to model the multifaceted, multivalent nature of Babbitt's music and titles.1 It provides a framework for interpreting various analytical results, as I've demonstrated before. For instance, as I showed previously (Mailman 2020) and show below, Schenkerian analysis can be deployed to reveal tonality in Babbitt's music, such that it can be read as a conceptual blend of serial and tonal syntaxes. Later in the analyses below I also apply, to pitch registers, both oscillator modeling and David Lewin's (1995) technique of binary state operators, which he had applied specifically to rhythms and instrumentation in two other Babbitt works. Finally I present a technique of cumulative pattern matching (across domains of pitch class and pitch register) based on Boolean functions (Boolean algebra) which reveals a hidden meaning in Babbitt's title. Yet another arises from an important matrix pattern in linear algebra. Ultimately, I suggest, as one aspect of Babbitt's compositional poetics, a certain pragmatism regarding ontologies in music theoretical discourse. That is, this piano work and its title, taken together, endorse a certain pragmatism as instrumental to the progress of musical inquiry. Canonicity One of the possible meanings of the title "canonical form" that is at least worth considering is the work's engagement with canonical repertoire, not in terms of anything as overt as quoting, but rather through subtler means. In this work, and others from the 1980s, Babbitt's pitch configurations dance with the tonal languages of canonic repertoire, by eluding to brief triadic chord progressions and even longer voice-leading motions. Canonical Form is based on the same array as Whirled Series, Quartet No. 5, and Around the Horn (Mead 1994; Dubiel 1997), an array whose row features three consonant triads as consecutive segments. Furthermore these works are based on superarrays, which enable a surprisingly vast flexibility in aligning pitches to create various configurations (Mailman 2019). This is because these superarrays feature three or more [End Page 216] simultaneous arrays whose ordering relative to each other is undetermined. Thus Babbitt can choose to align pitches from different array lynes, not only to forge consonant triads but also to suggest cadential tonal gestures and, in some cases, longer chromatic tonal progressions. A passage near the beginning of Canonical Form exemplifies this strikingly. Shown in Example 1, a perfect authentic (I-V-I) cadence in F# minor launches m. 21, prepared by a melodic lead-in (A and G# from m. 20). In m. 22, six pitches, drawn...

This is a contribution to the discussion on the role of truth degrees in manyvalued logics from the perspective of abstract algebraic logic. It starts with some thoughts on the so-called Suszko’s Thesis (that every logic is two-valued) and on the conception of semantics that underlies it, which includes the truth-preserving notion of consequence. The alternative usage of truth values in order to define logics that preserve degrees of truth is presented and discussed. Some recent works studying these in the particular cases of Łukasiewicz’s many-valued logics and of logics associated with varieties of residuated lattices are also presented. Finally the extension of this paradigm to other, more general situations is discussed, highlighting the need for philosophical or applied motivations in the selection of the truth degrees, due both to the interpretation of the idea of truth degree and to some mathematical difficulties.

As we sketched in [Godo and Hajek, 1996], most of fuzzy inference in Zadeh’s style is well formalized in the many-sorted many-valued Pavelka-style fuzzy predicate logic — and this can be done in several ways. The most important fact is that patterns of fuzzy inference can be presented as sound deduction rules, with all necessary assumptions expressed by some formulas of predicate logic (with the usual quantifiers ∀, ∃) in the premisse part of the rule. Here we complement the presentation of [Godo and Hajek, 1996] by a somewhat unexpected formalization of Zadeh’s Generalized Modus Ponens (GMP, cf. e.g. Zadeh [1988; 1990; 1994]). We shall be sketchy; a full detailed presentation is to be found in [Godo and Hajek, to appear]. For the reader’s convenience we repeat the definition of the many-sorted RQL (rational quantification logic, cf. [Hajek, 1995; Hajek, 1997]).

not when n = 2 or 3; and he gave one solution for n = 4. Squares of such a type have uses in experimental designs. We explain below a simple construction for squares where n has the form 22m1. Each square constructed in this way is represented in a canonical form by applying a well-known theorem of J. Singer [2]. In this form as soon as the top row of entries in a square is known, all the other entries may be written down immediately by means of a straight-forward cyclic process. Thus an index of first rows is all that is necessary to catalogue squares in their canonical forms. It may be permissible to give here a slight modification of the proof of Singer's theorem in order to show a natural application of the regular representation of linear algebras.

It was investigated the Boolean satisfiability (SAT) problem defined as follows: given a Boolean formula, check whether an assignment of Boolean values to the propositional variables in the formula exists, such that the formula evaluates to true. If such an assignment exists, the formula is said to be satisfiable; otherwise, it is unsatisfiable. With using of analytical expressions of multi-valued logic 2SAT boolean satisfiability was formulated as linear programming optimization problem. The same linear programming formulation was extended to find 3SAT and kSAT boolean satisfiability for k greater than 3. So, using new analytic multi-valued logic expressions and linear programming formulation of boolean satisfiability proposition that kSAT is in P and could be solved in linear time was proved.KeywordsBoolean satisfiabilityConjunctive normal formConvex functionLinear programmingMulti-valued logicNP problemTime complexity.