scholarly journals Partial Correctness of a Power Algorithm

2019 ◽  
Vol 27 (2) ◽  
pp. 189-195
Author(s):  
Adrian Jaszczak
Keyword(s):  
Formal Verification ◽  
Complex Number ◽  
Hoare Logic ◽  
Inference System ◽  
Partial Correctness ◽  
Complex Valued

Summary This work continues a formal verification of algorithms written in terms of simple-named complex-valued nominative data [6],[8],[15],[11],[12],[13]. In this paper we present a formalization in the Mizar system [3],[1] of the partial correctness of the algorithm: i := val.1 j := val.2 b := val.3 n := val.4 s := val.5 while (i <> n) i := i + j s := s * b return s computing the natural n power of given complex number b, where variables i, b, n, s are located as values of a V-valued Function, loc, as: loc/.1 = i, loc/.3 = b, loc/.4 = n and loc/.5 = s, and the constant 1 is located in the location loc/.2 = j (set V represents simple names of considered nominative data [17]). The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2],[4] with partial pre- and post-conditions [14],[16],[7],[5].

scholarly journals Partial Correctness of a Factorial Algorithm

2019 ◽  
Vol 27 (2) ◽  
pp. 181-187
Author(s):  
Adrian Jaszczak ◽  
Artur Korniłowicz
Keyword(s):  
Formal Verification ◽  
Natural Number ◽  
Hoare Logic ◽  
Inference System ◽  
Partial Correctness ◽  
Complex Valued

Summary In this paper we present a formalization in the Mizar system [3],[1] of the partial correctness of the algorithm: i := val.1 j := val.2 n := val.3 s := val.4 while (i <> n) i := i + j s := s * i return s computing the factorial of given natural number n, where variables i, n, s are located as values of a V-valued Function, loc, as: loc/.1 = i, loc/.3 = n and loc/.4 = s, and the constant 1 is located in the location loc/.2 = j (set V represents simple names of considered nominative data [16]). This work continues a formal verification of algorithms written in terms of simple-named complex-valued nominative data [6],[8],[14],[10],[11],[12]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2],[4] with partial pre- and post-conditions [13],[15],[7],[5].

scholarly journals Partial Correctness of a Fibonacci Algorithm

2020 ◽  
Vol 28 (2) ◽  
pp. 187-196
Author(s):  
Artur Korniłowicz
Keyword(s):  
Fibonacci Number ◽  
Iterative Algorithms ◽  
Hoare Logic ◽  
Inference System ◽  
Partial Correctness ◽  
Complex Valued

Summary In this paper we introduce some notions to facilitate formulating and proving properties of iterative algorithms encoded in nominative data language [19] in the Mizar system [3], [1]. It is tested on verification of the partial correctness of an algorithm computing n-th Fibonacci number: i := 0 s := 0 b := 1 c := 0 while (i <> n)   c := s   s := b   b := c + s   i := i + 1 return s This paper continues verification of algorithms [10], [13], [12] written in terms of simple-named complex-valued nominative data [6], [8], [17], [11], [14], [15]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and post-conditions [16], [18], [7], [5].

scholarly journals Partial Correctness of GCD Algorithm

2018 ◽  
Vol 26 (2) ◽  
pp. 165-173
Author(s):  
Ievgen Ivanov ◽  
Artur Korniłowicz ◽  
Mykola Nikitchenko
Keyword(s):  
Greatest Common Divisor ◽  
Hoare Logic ◽  
Natural Numbers ◽  
Inference System ◽  
Partial Correctness ◽  
Euclid’S Algorithm ◽  
System 2 ◽  
Euclid's Algorithm ◽  
Complex Valued

Summary In this paper we present a formalization in the Mizar system [2, 1] of the correctness of the subtraction-based version of Euclid’s algorithm computing the greatest common divisor of natural numbers. The algorithm is written in terms of simple-named complex-valued nominative data [11, 4]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [7]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic with partial pre- and post-conditions [8, 10, 5, 3].

scholarly journals Reasoning about Partial Correctness Assertions in Isabelle/HOL

2020 ◽  
Vol 27 (3) ◽  
pp. 84-101
Author(s):  
Alfio Ricardo Martini
Keyword(s):  
Formal Verification ◽  
Broad Class ◽  
Object Oriented ◽  
Concurrent Programs ◽  
Proof Assistant ◽  
Hoare Logic ◽  
Partial Correctness ◽  
Human Understanding ◽  
Isabelle Proof Assistant

Hoare Logic has a long tradition in formal verification and has been continuously developed and used to verify a broad class of programs, including sequential, object-oriented and concurrent programs. The purpose of this work is to provide a detailed and accessible exposition of the several ways the user can conduct, explore and write proofs of correctness of sequential imperative programs with Hoare logic and the ISABELLE proof assistant. With the proof language Isar, it is possible to write structured, readable proofs that are suitable for human understanding and communication.

scholarly journals General Theory and Tools for Proving Algorithms in Nominative Data Systems

2020 ◽  
Vol 28 (4) ◽  
pp. 269-278
Author(s):  
Adrian Jaszczak
Keyword(s):  
General Theory ◽  
Iterative Algorithms ◽  
Data Systems ◽  
Hoare Logic ◽  
Inference System ◽  
The Future ◽  
Complex Valued

Summary In this paper we introduce some new definitions for sequences of operations and extract general theorems about properties of iterative algorithms encoded in nominative data language [20] in the Mizar system [3], [1] in order to simplify the process of proving algorithms in the future. This paper continues verification of algorithms [10], [13], [12], [14] written in terms of simple-named complex-valued nominative data [6], [8], [18], [11], [15], [16]. The validity of the algorithm is presented in terms of semantic Floyd-Hoare triples over such data [9]. Proofs of the correctness are based on an inference system for an extended Floyd-Hoare logic [2], [4] with partial pre- and postconditions [17], [19], [7], [5].

scholarly journals Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 206
Author(s):  
Ji-Eun Kim
Keyword(s):  
Series Expansion ◽  
Taylor Series ◽  
Complex Number ◽  
Complex Function ◽  
Taylor Series Expansion ◽  
Complex Numbers ◽  
Definition Of ◽  
Generalized Quaternion ◽  
Complex Valued ◽  
Complex Basis

The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.

scholarly journals Functional Equations of Dirichlet Series Derived from Non-Analytic Automorphic Forms of a Certain Type

1975 ◽  
Vol 18 (1) ◽  
pp. 87-94 ◽  
Author(s):  
V. Venugopal Rao
Keyword(s):  
Real Number ◽  
Dirichlet Series ◽  
Complex Number ◽  
Functional Equations ◽  
Fourier Expansion ◽  
Positive Real Number ◽  
Automorphic Forms ◽  
Positive Real ◽  
Real Value ◽  
Complex Valued

Let f(τ) be a complex valued function, defined and analytic in the upper half of the complex τ plane (τ = x+iy, y > 0), such that f(τ + λ)= f(τ) where λ is a positive real number and f(—1/τ) = γ(—iτ)kf(τ), k being a complex number. The function (—iτ)k is defined as exp(k log(—iτ) where log(—iτ) has the real value when —iτ is positive. Every such function is said to have signature (λ, k, γ) in the sense of E. Hecke [1] and has a Fourier expansion of the type f(τ) = a0 + σ an exp(2πin/λ), (n = 1,2,…), if we further assume that f(τ) = O(|y|-c) as y tends to zero uniformly for all x, c being a positive number.

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