ON DEFINITIONAL TRANSFORMATIONS TO NORMAL FORM FOR INTUITIONISTIC LOGIC

1997 ◽  
Vol 29 (1,2) ◽  
pp. 165-201 ◽  
Author(s):  
Uwe Egly
Keyword(s):  
Normal Form ◽  
Intuitionistic Logic

scholarly journals A Prolog-based Proof Tool for Type-Theory TA-lambda and Implicational Intuitionistic-Logic

10.29007/8ttk ◽  
2018 ◽  
Author(s):  
L. Yohanes Stefanus ◽  
Ario Santoso
Keyword(s):  
Normal Form ◽  
Computer Science ◽  
Intuitionistic Logic ◽  
Type Theory ◽  
General Type ◽  
Search Algorithm ◽  
Principal Type ◽  
Web Based ◽  
Type Algorithm ◽  
Important Variant

Studies on type theory have brought numerous important contributions to computer science. In this paper we present a GUI-based proof tool that provides assistance in constructing deductions in type theory and validating implicational intuitionistic logic formulas. As such, this proof tool is a testbed for learning type theory and implicational intuitionistic logic. This proof tool focuses on an important variant of type theory named TA<sub>λ</sub>, especially on its two core algorithms: the principal-type algorithm and the type inhabitant search algorithm. The former algorithm finds a most general type assignable to a given λ-term, while the latter finds inhabitants (closed λ-terms in β-normal form) to which a given type can be assigned. By the Curry-Howard correspondence, the latter algorithm provides provability for implicational formulas in intuitionistic logic. We elaborate on how to implement those two algorithms declaratively in Prolog and the overall GUI-based program architecture. In our implementation, we make some modification to improve the performance of the principal-type algorithm. We have also built a web-based version of the proof tool called λ-Guru.

Proofs, Snakes and Ladders

Dialogue ◽  
1974 ◽  
Vol 13 (4) ◽  
pp. 723-731 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Normal Form ◽  
Intuitionistic Logic ◽  
Natural Deduction ◽  
Tree Structure ◽  
Deduction System ◽  
Syntactical Structure ◽  
Form Theorem ◽  
Structural Identity ◽  
Natural Deduction System ◽  
Normal Form Theorem

Anyone who has worked at proving theorems of intuitionistic logic in a natural deduction system must have been struck by the way in which many logical theorems “prove themselves.” That is, proofs of many formulas can be read off from the syntactical structure of the formulas themselves. This observation suggests that perhaps a strong structural identity may underly this relation between formulas and their proofs. A formula can be considered as a tree structure composed of its subformulas (Frege 1879) and by the normal form theorem (Gentzen 1934) every formula has a normalized proof consisting of its subformulas. Might we not identify an intuitionistic theorem with (one of) its proof(s) in normal form?

scholarly journals Normalisation and subformula property for a system of intuitionistic logic with general introduction and elimination rules

Synthese ◽  
2021 ◽  
Author(s):  
Nils Kürbis
Keyword(s):  
Normal Form ◽  
Intuitionistic Logic ◽  
General Introduction ◽  
Main Method ◽  
Philosophical Importance ◽  
Subformula Property

AbstractThis paper studies a formalisation of intuitionistic logic by Negri and von Plato which has general introduction and elimination rules. The philosophical importance of the system is expounded. Definitions of ‘maximal formula’, ‘segment’ and ‘maximal segment’ suitable to the system are formulated and corresponding reduction procedures for maximal formulas and permutative reduction procedures for maximal segments given. Alternatives to the main method used are also considered. It is shown that deductions in the system convert into normal form and that deductions in normal form have the subformula property.

Closing the GAP—A 5Å Scanning Microscope

1969 ◽  
Vol 27 ◽  
pp. 6-7
Author(s):  
A. V. Crewe
Keyword(s):  
Normal Form ◽  
The Other ◽  
Resolving Power ◽  
Scanning Microscope ◽  
Conventional Microscope ◽  
Other Hand ◽  
Closing The Gap ◽  
Thermal Sources ◽  
Better Than ◽  
Transmission Microscope

We have become accustomed to differentiating between the scanning microscope and the conventional transmission microscope according to the resolving power which the two instruments offer. The conventional microscope is capable of a point resolution of a few angstroms and line resolutions of periodic objects of about 1Å. On the other hand, the scanning microscope, in its normal form, is not ordinarily capable of a point resolution better than 100Å. Upon examining reasons for the 100Å limitation, it becomes clear that this is based more on tradition than reason, and in particular, it is a condition imposed upon the microscope by adherence to thermal sources of electrons.

METHODS OF REDUCING OF LARGE BOOLEAN FORMULAS REPRESENTED IN A CONJUNCTIVE NORMAL FORM FOR DETERMINING THEIR SATISFIABILITY

2020 ◽  
Author(s):  
N.I. Gdansky ◽  
◽  
A.A. Denisov ◽  
Keyword(s):  
Normal Form ◽  
Normal Forms ◽  
Conjunctive Normal Form ◽  
Boolean Formulas

The article explores the satisfiability of conjunctive normal forms used in modeling systems.The problems of CNF preprocessing are considered.The analysis of particular methods for reducing this formulas, which have polynomial input complexity is given.

Analysis of a 2DOF Nonlinear Mechanical System Using the Normal Form Method

2019 ◽  
Author(s):  
David Julian Gonzalez Maldonado ◽  
Peter Hagedorn ◽  
Thiago Ritto ◽  
Rubens Sampaio ◽  
Artem Karev
Keyword(s):  
Normal Form ◽  
Mechanical System ◽  
Nonlinear Mechanical ◽  
Nonlinear Mechanical System ◽  
Normal Form Method

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

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