Closed classes of three-valued logic generated by periodic symmetric funtions that equal 1 in tuples from {1, 2} and equal 0 on the rest tuples are considered. Criteria for bases existence and finite bases existence for these classes is obtained. The problem of the bases existence for some families of closed classes of the three-valued logic functions is considered in the paper. E. Post [1] (se...

Partial functions are abundant in mathematics and program specifications. Unfortunately, their importance has been mostly ignored in automated theorem proving. In this paper, we develop a theorem proving strategy for Partial Classical Logic (PCL). Proof search takes place in Kleene Logic. We show that PCL theories can be translated into equivalent sets of formulas in Kleene logic. For proof sea...

Łukasiewicz three-valued logic Ł3 is often understood as the set of 3-valued valid formulas according to Łukasiewicz’s 3-valued matrices. Following Wojcicki, in addition, we shall consider two alternative interpretations of Ł3: “well-determined” Ł3a and “truth-preserving” Ł3b defined by two different consequence relations on the 3-valued matrices. The aim of this paper is to provide (by using D...

in this paper, the notions of $(t,s)$-composition matrix and$(t,s)$-interval-valued intuitionistic fuzzy equivalence matrix areintroduced where $(t,s)$ is a dual pair of triangular module. theyare the generalization of composition matrix and interval-valuedintuitionistic fuzzy equivalence matrix. furthermore, theirproperties and characterizations are presented. then a new methodbased on $tilde{...

We study L-categories of lattice-valued convergence spaces. Suchcategories are obtained by fuzzifying" the axioms of a lattice-valued convergencespace. We give a natural example, study initial constructions andfunction spaces. Further we look into some L-subcategories. Finally we usethis approach to quantify how close certain lattice-valued convergence spacesare to being lattice-valued topologi...

This paper is devoted to consider the notions of complementary problem (CP) and set-valued complementary problem (SVCP). The set-valued complementary problem is compared with the classical single-valued complementary problem. Also, the solution set of the set-valued complementary problem is characterized. Our results illustrated by some examples. This paper is devoted to co...

Let p be a prime, and let f(x) be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the p-adic order of the sum ∑ k≡r (mod pβ) (n k ) (−1)f (⌊ k − r pα ⌋) , where α > β > 0, n > pα−1 and r ∈ Z. This polynomial extension of Fleck’s congruence has various backgrounds and several consequences such as ∑ k≡r (mod pα) (n k ) a ≡ 0 ( mod p ⌊ n−pα−1 φ(pα) ...

Let p be a prime, and let f (x) be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the p-adic order of the sum k≡r (mod p β) n k (−1) k f k − r p α , where α β 0, n p α−1 and r ∈ Z. This polynomial extension of Fleck's congruence has various backgrounds and several consequences such as k≡r (mod p α) n k a k ≡ 0 mod p n−p α−1 ϕ(p α) provided that ...

We give a classification of dual functions, they are m -al functions. We call a function m-al with respect to an operator if the operator lives any function unchanged after m times of using the operator. And 2 ≤ m ≤ k. Functions with different m have very different properties. We give theoretical results for clones of self-dual (m = 2) and selfk -al (m = k) functions in k -valued logic at k ≤ 3...

Computing certain answers is the preferred way of answering queries in scenarios involving incomplete data. This, however, is computationally expensive, so practical systems use efficient techniques based on a particular three-valued logic, although this often leads to