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A graphical representation of relational formulae with complementation

Domenico Cantone, Andrea Formisano, Marianna Nicolosi Asmundo, Eugenio Giovanni Omodeo (2012)

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications

We study translations of dyadic first-order sentences into equalities between relational expressions. The proposed translation techniques (which work also in the converse direction) exploit a graphical representation of formulae in a hybrid of the two formalisms. A major enhancement relative to previous work is that we can cope with the relational complement construct and with the negation connective. Complementation is handled by adopting a Smullyan-like uniform notation to classify and decompose...

A graphical representation of relational formulae with complementation∗

Domenico Cantone, Andrea Formisano, Marianna Nicolosi Asmundo, Eugenio Giovanni Omodeo (2012)

RAIRO - Theoretical Informatics and Applications

We study translations of dyadic first-order sentences into equalities between relational expressions. The proposed translation techniques (which work also in the converse direction) exploit a graphical representation of formulae in a hybrid of the two formalisms. A major enhancement relative to previous work is that we can cope with the relational complement construct and with the negation connective. Complementation is handled by adopting a Smullyan-like...

A reduction-based theorem prover for 3-valued logic.

Gabriel Aguilera Venegas, Inmaculada Pérez de Guzmán, Manuel Ojeda Aciego (1997)

Mathware and Soft Computing

We present a new prover for propositional 3-valued logics, TAS-M3, which is an extension of the TAS-D prover for classical propositional logic. TAS-M3 uses the TAS methodology and, consequently, it is a reduction-based method. Thus, its power is based on the reductions of the size of the formula executed by the F transformation. This transformation dynamically filters the information contained in the syntactic structure of the formula to avoid as much distributions as possible, in order to improve...

Abstract Reduction Systems and Idea of Knuth-Bendix Completion Algorithm

Grzegorz Bancerek (2014)

Formalized Mathematics

Educational content for abstract reduction systems concerning reduction, convertibility, normal forms, divergence and convergence, Church- Rosser property, term rewriting systems, and the idea of the Knuth-Bendix Completion Algorithm. The theory is based on [1].

Algebra of Polynomially Bounded Sequences and Negligible Functions

Hiroyuki Okazaki (2015)

Formalized Mathematics

In this article we formalize negligible functions that play an essential role in cryptology [10], [2]. Generally, a cryptosystem is secure if the probability of succeeding any attacks against the cryptosystem is negligible. First, we formalize the algebra of polynomially bounded sequences [20]. Next, we formalize negligible functions and prove the set of negligible functions is a subset of the algebra of polynomially bounded sequences. Moreover, we then introduce equivalence relation between polynomially...

Algebraic Approach to Algorithmic Logic

Grzegorz Bancerek (2014)

Formalized Mathematics

We introduce algorithmic logic - an algebraic approach according to [25]. It is done in three stages: propositional calculus, quantifier calculus with equality, and finally proper algorithmic logic. For each stage appropriate signature and theory are defined. Propositional calculus and quantifier calculus with equality are explored according to [24]. A language is introduced with language signature including free variables, substitution, and equality. Algorithmic logic requires a bialgebra structure...

Algebraic Numbers

Yasushige Watase (2016)

Formalized Mathematics

This article provides definitions and examples upon an integral element of unital commutative rings. An algebraic number is also treated as consequence of a concept of “integral”. Definitions for an integral closure, an algebraic integer and a transcendental numbers [14], [1], [10] and [7] are included as well. As an application of an algebraic number, this article includes a formal proof of a ring extension of rational number field ℚ induced by substitution of an algebraic number to the polynomial...

All Liouville Numbers are Transcendental

Artur Korniłowicz, Adam Naumowicz, Adam Grabowski (2017)

Formalized Mathematics

In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real...

Altitude, Orthocenter of a Triangle and Triangulation

Roland Coghetto (2016)

Formalized Mathematics

We introduce the altitudes of a triangle (the cevians perpendicular to the opposite sides). Using the generalized Ceva’s Theorem, we prove the existence and uniqueness of the orthocenter of a triangle [7]. Finally, we formalize in Mizar [1] some formulas [2] to calculate distance using triangulation.

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