P versus NP: el problema estrella de la matemàtica de la computació

El problema «P versus NP» és un dels set Problemes del Mil.lenni de l?Institut Clay de Matemàtiques, la solució del qual estaria premiada amb un milió de dòlars. En aquest article presentem de manera divulgativa el problema i els seus orígens, donant pel camí exemples de problemes computacionals de diferents nivells de dificultat, alguns algoritmes no trivials, la definició de màquina de Turing ?el model matemàtic d?ordinador? i el concepte de reducció polinòmica entre problemes. La part més avançada de l?article presenta una demostració del teorema de Razborov sobre circuits monòtons de l?any 1985 que resol un cas especial de la conjectura. També donem una traducció al català d?una carta de Gödel a Von Neumann de l?any 1956 que es va descobrir l?any 1988 i que es pot considerar com la primera formulació per escrit del problema «P versus NP».The problem ?P versus NP? is one of the seven Millennium Prize Problems of the Clay Mathematics Institute, whose solution would be awarded one million dollars. In this article we present in an informal manner the problem and its origin, giving along the way examples of computational problems of diferent levels of hardness, some non-trivial algorithms, the definition of Turing machine ? the mathematical model of a computer ? and the concept of polynomial reduction between problems. The most advanced part of the article presents a proof of Razborov?s 1985 theorem for monotone circuits, which solves a special case of the conjecture. We also give a translation into Catalan of a letter from Gödel to Von Neumann from 1956 which was discovered in 1988 and can be considered the first written formulation of the problem ?P versus NP?

A Computable Economist’s Perspective on Computational Complexity

A computable economist's view of the world of computational complexity theory is described. This means the model of computation underpinning theories of computational complexity plays a central role. The emergence of computational complexity theories from diverse traditions is emphasised. The unifications that emerged in the modern era was codified by means of the notions of efficiency of computations, non-deterministic computations, completeness, reducibility and verifiability - all three of the latter concepts had their origins on what may be called 'Post's Program of Research for Higher Recursion Theory'. Approximations, computations and constructions are also emphasised. The recent real model of computation as a basis for studying computational complexity in the domain of the reals is also presented and discussed, albeit critically. A brief sceptical section on algorithmic complexity theory is included in an appendix

Topics in Programming Languages, a Philosophical Analysis through the case of Prolog

[EN]Programming languages seldom find proper anchorage in philosophy of logic, language and science. is more, philosophy of language seems to be restricted to natural languages and linguistics, and even philosophy of logic is rarely framed into programming languages topics. The logic programming paradigm and Prolog are, thus, the most adequate paradigm and programming language to work on this subject, combining natural language processing and linguistics, logic programming and constriction methodology on both algorithms and procedures, on an overall philosophizing declarative status. Not only this, but the dimension of the Fifth Generation Computer system related to strong Al wherein Prolog took a major role. and its historical frame in the very crucial dialectic between procedural and declarative paradigms, structuralist and empiricist biases, serves, in exemplar form, to treat straight ahead philosophy of logic, language and science in the contemporaneous age as well. In recounting Prolog's philosophical, mechanical and algorithmic harbingers, the opportunity is open to various routes. We herein shall exemplify some: - the mechanical-computational background explored by Pascal, Leibniz, Boole, Jacquard, Babbage, Konrad Zuse, until reaching to the ACE (Alan Turing) and EDVAC (von Neumann), offering the backbone in computer architecture, and the work of Turing, Church, Gödel, Kleene, von Neumann, Shannon, and others on computability, in parallel lines, throughly studied in detail, permit us to interpret ahead the evolving realm of programming languages. The proper line from lambda-calculus, to the Algol-family, the declarative and procedural split with the C language and Prolog, and the ensuing branching and programming languages explosion and further delimitation, are thereupon inspected as to relate them with the proper syntax, semantics and philosophical élan of logic programming and Prolog

Decidability vs. undecidability. Logico-philosophico-historical remarks

The aim of the paper is to present the decidability problems from a philosophical and historical perspective as well as to indicate basic mathematical and logical results concerning (un)decidability of particular theories and problems

Décidabilité et Complexité

International audienceL'informatique fondamentale est un vaste sujet, comme en témoignent les 2 283 et 3 176 pages des "Handbooks" (228; 1). Couvrir en quelques dizaines de pages, l'ensemble de l'in- formatique nous a semblé une entreprise hors de notre portée. De ce fait, nous nous sommes concentrés sur la notion de calcul, sujet qui reflète le goût et la passion des auteurs de ce chapitre. La notion de calcul est omniprésente et aussi ancienne que les mathématiques

Quantum Limits, Computational Complexity and Philosophy – A Review: Shamaila Shafiq

Quantum computing physics uses quantum qubits (or bits), for computer’s memory or processor. They can perform certain calculations much faster than a normal computer. The quantum computers have some limitations due to which the problems belonging to NP- Complete are not solved efficiently. This paper covers effective quantum algorithm for solving NP-Complete problems through some features of complexity theory, that we can simplify some of the philosophical interest problems

Ernst Zermelo's Project of Infinitary Logic

This paper is a summary of a more comprehensive work Infinitarna Logika Ernsta Zermela (The Infinitary Logic of Ernst Zermelo) being currently under preparation for the research grant KBN 2H01A 00725 Metody nieskończonościowe w teorii definicji (Infinitary methods in the theory of definitions) headed by Professor JANUSZ CZELAKOWSKI at the Institute of Mathematics and Information Science of the University of Opole, Poland. The presentation of Zermelo's ideas is accompanied with some remarks concerning the development of infinitary logic.This paper is a summary of a more comprehensive work Infinitarna Logika Ernsta Zermela (The Infinitary Logic of Ernst Zermelo) being currently under preparation for the research grant KBN 2H01A 00725 Metody nieskończonościowe w teorii definicji (Infinitary methods in the theory of definitions) headed by Professor JANUSZ CZELAKOWSKI at the Institute of Mathematics and Information Science of the University of Opole, Poland. The presentation of Zermelo's ideas is accompanied with some remarks concerning the development of infinitary logic.