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Symmetric implicational restriction method of fuzzy inference

Yiming Tang, Wenbin Wu, Youcheng Zhang, Witold Pedrycz, Fuji Ren, Jun Liu (2021)

Kybernetika

The symmetric implicational method is revealed from a different perspective based upon the restriction theory, which results in a novel fuzzy inference scheme called the symmetric implicational restriction method. Initially, the SIR-principles are put forward, which constitute optimized versions of the triple I restriction inference mechanism. Next, the existential requirements of basic solutions are given. The supremum (or infimum) of its basic solutions is achieved from some properties of fuzzy...

(T, ⊥, N) fuzzy logic.

Y. Xu, J. Lin, Da Ruan (2001)

Mathware and Soft Computing

To investigate more reasonable fuzzy reasoning model in expert systems as well as more effective logical circuit in fuzzy control, a (T, ⊥, N) fuzzy logic is proposed in this paper by using T-norm, ⊥-norm and pseudo-complement N as the logical connectives. Two aspects are discussed: (1) some concepts of (T, ⊥, N) fuzzy logic are introduced and some properties of (T, ⊥, N) fuzzy logical formulae are discussed. (2) G-fuzzy truth (falsity) of (T, ⊥, N) fuzzy logical formulae are investigated and also...

Taclets: a new paradigm for constructing interactive theorem provers.

Bernhard Beckert, Martin Giese, Elmar Habermalz, Reiner Hähnle, Andreas Roth, Philipp Rümmer, Steffen Schlager (2004)

RACSAM

Frameworks for interactive theorem proving give the user explicit control over the construction of proofs based on meta languages that contain dedicated control structures for describing proof construction. Such languages are not easy to master and thus contribute to the already long list of skills required by prospective users of interactive theorem provers. Most users, however, only need a convenient formalism that allows to introduce new rules with minimal overhead. On the the other hand, rules...

Tarski Geometry Axioms – Part II

Roland Coghetto, Adam Grabowski (2016)

Formalized Mathematics

In our earlier article [12], the first part of axioms of geometry proposed by Alfred Tarski [14] was formally introduced by means of Mizar proof assistant [9]. We defined a structure TarskiPlane with the following predicates: of betweenness between (a ternary relation), of congruence of segments equiv (quarternary relation), which satisfy the following properties: congruence symmetry (A1), congruence equivalence relation (A2), congruence identity (A3), segment construction (A4), SAS (A5), betweenness...

The Axiomatization of Propositional Logic

Mariusz Giero (2016)

Formalized Mathematics

This article introduces propositional logic as a formal system ([14], [10], [11]). The formulae of the language are as follows φ ::= ⊥ | p | φ → φ. Other connectives are introduced as abbrevations. The notions of model and satisfaction in model are defined. The axioms are all the formulae of the following schemes α ⇒ (β ⇒ α), (α ⇒ (β ⇒ γ)) ⇒ ((α ⇒ β) ⇒ (α ⇒ γ)), (¬β ⇒ ¬α) ⇒ ((¬β ⇒ α) ⇒ β). Modus ponens is the only derivation rule. The soundness theorem and the strong completeness theorem are proved....

The Basic Existence Theorem of Riemann-Stieltjes Integral

Kazuhisa Nakasho, Keiko Narita, Yasunari Shidama (2016)

Formalized Mathematics

In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15],...

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