Semantical proofs of correctness for programs performing non-deterministic tests on real numbers

2010 ◽  
Vol 20 (5) ◽  
pp. 723-751
Author(s):  
THOMAS ANBERRÉE
Keyword(s):  
Real Number ◽  
Denotational Semantics ◽  
Functional Language ◽  
Real Numbers ◽  
First Order

We consider a functional language that performs non-deterministic tests on real numbers and define a denotational semantics for that language based on Smyth powerdomains. The semantics is only an approximate one because the denotation of a program for a real number may not be precise enough to tell which real number the program computes. However, for many first-order total functions f : n → , there exists a program for f whose denotation is precise enough to show that the program indeed computes the function f. In practice, it is not difficult to find programs like this that possess a faithful denotation. We provide a few examples of such programs and the corresponding proofs of correctness.

scholarly journals A Denotational Investigation of Defunctionalization

2000 ◽  
Vol 7 (47) ◽  
Author(s):  
Lasse R. Nielsen
Keyword(s):  
Data Structures ◽  
Program Transformation ◽  
Denotational Semantics ◽  
Higher Order ◽  
Functional Language ◽  
First Order

<p>Defunctionalization was introduced by John Reynolds in his 1972<br />article Definitional Interpreters for Higher-Order Programming <br />Languages. Defunctionalization transforms a higher-order program into a first-order one, representing functional values as data structures. Since then it has been used quite widely, but we observe that it has never been proven correct. We formalize defunctionalization denotationally for a typed functional language, and we prove that it preserves the meaning of any terminating program. Our proof uses logical relations.</p><p>Keywords: defunctionalization, program transformation, denotational semantics, logical relations.</p>

scholarly journals Polyprograms and Polyprogram Bisimulation

2018 ◽  
Vol 25 (5) ◽  
pp. 534-548
Author(s):  
Sergei Grechanik
Keyword(s):  
Denotational Semantics ◽  
Weak Form ◽  
Functional Language ◽  
Transition Systems ◽  
Labelled Transition Systems ◽  
First Order ◽  
Single Function ◽  
The Difference ◽  
Two Phases ◽  
Transformation Systems

A polyprogram is a generalization of a program which admits multiple definitions of a single function. Such objects arise in different transformation systems, such as the Burstall-Darlington framework or equality saturation. In this paper, we introduce the notion of a polyprogram in a non-strict first-order functional language. We define denotational semantics for polyprograms and describe some possible transformations of polyprograms, namely we present several main transformations in two different styles: in the style of the Burstall-Darlington framework and in the style of equality saturation. Transformations in the style of equality saturation are performed on polyprograms in decomposed form, where the difference between functions and expressions is blurred, and so is the difference between substitution and unfolding. Decomposed polyprograms are well suited for implementation and reasoning, although they are not very human-readable. We also introduce the notion of polyprogram bisimulation which enables a powerful transformation called merging by bisimulation, corresponding to proving equivalence of functions by induction or coinduction. Polyprogram bisimulation is a concept inspired by bisimulation of labelled transition systems, but yet it is quite different, because polyprogram bisimulation treats every definition as self-sufficient, that is a function is considered to be defined by any of its definitions, whereas in an LTS the behaviour of a state is defined by all transitions from this state. We present an algorithm for enumerating polyprogram bisimulations of a certain form. The algorithm consists of two phases: enumerating prebisimulations and converting them to proper bisimulations. This separation is required because polyprogram bisimulations take into account the possibility of parameter permutation. We prove correctness of this algorithm and formulate a certain weak form of its completeness. The article is published in the author’s wording.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

scholarly journals Quasi-decidability of a Fragment of the First-Order Theory of Real Numbers

2015 ◽  
Vol 57 (2) ◽  
pp. 157-185 ◽  
Author(s):  
Peter Franek ◽  
Stefan Ratschan ◽  
Piotr Zgliczynski
Keyword(s):  
Order Theory ◽  
Real Numbers ◽  
First Order ◽  
First Order Theory

scholarly journals A fully abstract semantics for a first-order functional language with logic variables

1991 ◽  
Vol 13 (4) ◽  
pp. 577-625 ◽  
Author(s):  
Radha Jagadeesan ◽  
Keshav Pingali ◽  
Prakash Panangaden
Keyword(s):  
Functional Language ◽  
First Order ◽  
Abstract Semantics

scholarly journals A question of van den Dries and a theorem of Lipshitz and Robinson; Not everything is standard

2007 ◽  
Vol 72 (1) ◽  
pp. 119-122 ◽  
Author(s):  
Ehud Hrushovski ◽  
Ya'acov Peterzil
Keyword(s):  
Real Numbers ◽  
The Real ◽  
First Order ◽  
Real Closed Fields ◽  
Minimal Structure ◽  
Closed Fields ◽  
New Construction ◽  
The Relationship

AbstractWe use a new construction of an o-minimal structure, due to Lipshitz and Robinson, to answer a question of van den Dries regarding the relationship between arbitrary o-minimal expansions of real closed fields and structures over the real numbers. We write a first order sentence which is true in the Lipshitz-Robinson structure but fails in any possible interpretation over the field of real numbers.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

scholarly journals Oscillation Conditions for Difference Equations with a Monotone or Nonmonotone Argument

2018 ◽  
Vol 2018 ◽  
pp. 1-13 ◽  
Author(s):  
G. M. Moremedi ◽  
I. P. Stavroulakis
Keyword(s):  
Positive Integer ◽  
Difference Equation ◽  
Difference Equations ◽  
Delay Difference Equation ◽  
Real Numbers ◽  
First Order ◽  
All Solutions ◽  
Delay Difference ◽  
Constant Argument

Consider the first-order delay difference equation with a constant argument Δxn+pnxn-k=0,  n=0,1,2,…, and the delay difference equation with a variable argument Δxn+pnxτn=0,  n=0,1,2,…, where p(n) is a sequence of nonnegative real numbers, k is a positive integer, Δx(n)=x(n+1)-x(n), and τ(n) is a sequence of integers such that τ(n)≤n-1 for all n≥0 and limn→∞τ(n)=∞. A survey on the oscillation of all solutions to these equations is presented. Examples illustrating the results are given.

scholarly journals ON THE SPECTRA OF PISOT NUMBERS

2011 ◽  
Vol 54 (1) ◽  
pp. 127-132 ◽  
Author(s):  
TOUFIK ZAIMI
Keyword(s):  
Real Number ◽  
Fractional Part ◽  
Pisot Number ◽  
Real Numbers ◽  
Pisot Numbers

AbstractLet θ be a real number greater than 1, and let (()) be the fractional part function. Then, θ is said to be a Z-number if there is a non-zero real number λ such that ((λθn)) < for all n ∈ ℕ. Dubickas (A. Dubickas, Even and odd integral parts of powers of a real number, Glasg. Math. J., 48 (2006), 331–336) showed that strong Pisot numbers are Z-numbers. Here it is proved that θ is a strong Pisot number if and only if there exists λ ≠ 0 such that ((λα)) < for all$\alpha \in \{ \theta ^{n}\mid n\in \mathbb{N}\} \cup \{ \sum\nolimits_{n=0}^{N}\theta ^{n}\mid \mathit{\}N\in \mathbb{N}\}$. Also, the following characterisation of Pisot numbers among real numbers greater than 1 is shown: θ is a Pisot number ⇔ ∃ λ ≠ 0 such that$\Vert \lambda \alpha \Vert <\frac{1}{% 3}$for all$\alpha \in \{ \sum\nolimits_{n=0}^{N}a_{n}\theta ^{n}\mid$an ∈ {0,1}, N ∈ ℕ}, where ‖λα‖ = min{((λα)), 1 − ((λα))}.

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