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Turing's Algorithmic Lens ARBOR Ciencia, Pensamiento y Cultura Vol. 189-764, noviembre-diciembre 2013, a080 | ISSN-L: 0210-1963 doi: http://dx.doi.org/10.3989/arbor.2013.764n6003 EL LEGADO DE ALAN TURING / THE LEGACY OF ALAN TURING TuRing’s AlgoriThmiC lens: LA LENTE ALGORÍTMICA DE From comPutabiliTy to TURING: DE LA COMPUTABILIDAD comPlexiTy Theory A LA TEORÍA DE LA COMPLEJIDAD Josep Díaz* and Carme Torras** *Llenguatges i Sistemes Informàtics, UPC. [email protected] **Institut de Robòtica i Informàtica Industrial, CSIC-UPC. [email protected]. edu Citation/Cómo citar este artículo:Díaz, J. and Torras, C. (2013). Copyright: © 2013 CSIC. This is an open-access article distributed “Turing’s algorithmic lens: From computability to complexity under the terms of the Creative Commons Attribution-Non theory”. Arbor, 189 (764): a080. doi: http://dx.doi.org/10.3989/ Commercial (by-nc) Spain 3.0 License. arbor.2013.764n6003 Received: 10 July 2013. Accepted: 15 September 2013. ABsTRACT: The decidability question, i.e., whether any RESUMEN: La cuestión de la decidibilidad, es decir, si es posible de- mathematical statement could be computationally proven mostrar computacionalmente que una expresión matemática es true or false, was raised by Hilbert and remained open until verdadera o falsa, fue planteada por Hilbert y permaneció abierta Turing answered it in the negative. Then, most efforts in hasta que Turing la respondió de forma negativa. Establecida la theoretical computer science turned to complexity theory no-decidibilidad de las matemáticas, los esfuerzos en informática and the need to classify decidable problems according to teórica se centraron en el estudio de la complejidad computacio- their difficulty. Among others, the classes P (problems nal de los problemas decidibles. En este artículo presentamos una solvable in polynomial time) and nP (problems solvable in breve introducción a las clases P (problemas resolubles en tiempo non-deterministic polynomial time) were defined, and one polinómico) y nP (problemas resolubles de manera no determinis- of the most challenging scientific quests of our days arose: ta en tiempo polinómico), al tiempo que exponemos la dificultad whether P = nP. This still open question has implications de establecer si P = nP y las consecuencias que se derivarían de not only in computer science, mathematics and physics, but que ambas clases de problemas fueran iguales. Esta cuestión tiene also in biology, sociology and economics, and it can be seen implicaciones no solo en los campos de la informática, las mate- as a direct consequence of Turing’s way of looking through máticas y la física, sino también para la biología, la sociología y la the algorithmic lens at different disciplines to discover how economía. La idea seminal del estudio de la complejidad computa- pervasive computation is. cional es consecuencia directa del modo en que Turing abordaba problemas en diferentes ámbitos mediante lo que hoy se denomi- na la lupa algorítmica. El artículo finaliza con una breve exposición de algunos de los temas de investigación más actuales: transición de fase en problemas nP, y demostraciones holográficas, donde se trata de convencer a un adversario de que una demostración es correcta sin revelar ninguna idea de la demostración. Keywords: Computational complexity; the permanent PAlabrAs ClAve: Complejidad computacional; el problema problem; P and NP problems; Interactive proofs and holographic de calcular la permanente de una matriz; problemas P y NP; proofs. demostraciones interactivas y holográficas. 1. inTRODuCTiOn nary sets. A set is defined to be extraordinary if it is a member of itself, otherwise it is called ordinary. Is The English mathematician Alan Mathison Turing the set of all ordinary sets also ordinary? (see Doxi- (1912-1954) is well-known for his key role in the a080 adis, Papadimitriou, Papadatos and di Donna, 2009, development of computer science, but he also Turing’s algorithmic lens: From computability to complexity theory complexity to computability lens: From algorithmic Turing’s pp. 169 to 171 for a particular charming account of made important contributions to the foundations the effect of Russell's letter to Frege). of mathematics, numerical analysis, cryptography, quantum computing and biology. In the present Frege's work was the spark for 30 years of re- paper we focus exclusively on the influence of search on the foundations of mathematics, and Turing on computability and complexity theory, the end of the 19th c. and beginning of the 20th leaving aside other aspects of his work even within c. witnessed strong mathematical, philosophical computer science, most notably his contribution and personal battles between members of the in- to the birth of the digital computer and artificial tuitionist and formalist European schools (Davis, intelligence. 2000; Doxiadis, Papadimitriou, Papadatos and di Donna, 2009). Turing was an important player in the settle- ment of the decidability issue, which led naturally To better frame further developments, let us re- to the study of the complexity of decidable prob- call some notions. A formal system is a language, lems, i.e., once it was proved that there are prob- a finite set of axioms, and a set of inference rules lems unsolvable by a computer, research turned used to derive expressions (or statements) from the to the question of “how long would it take to solve set of axioms; an example is mathematics with the a decidable problem?” This is an exciting research first-order logic developed by Frege. A system is said field with lots of open questions. In this paper, we to be complete if every statement can be proved or try to give a gentle introduction to the historical disproved; otherwise the system is said to be incom- perspective of the search for efficient computing plete. A system is said to be consistent if there is not and its limitations. a step-by-step proof within the system that yields a false statement. 2. BeFORe TuRing: The DReAm OF meChAniCAl A system is said to be decidable if, for every state- ReAsOning AnD DeCiDABiliTy ment, there exists an algorithm to determine whether Gottfried W. Leibniz (1646-1716) was an impor- the statement is true. tant mathematician, philosopher, jurist and inven- In 1928, at the International Congress of Math- tor, whose dream was to devise an automatic rea- ematicians in Bologna, David Hilbert (1862-1943), soning machine, based on an alphabet of unam- a leading figure in the formalist school, presented biguous symbols manipulated by mechanical rules, to his fellow mathematicians three problems, the which could formalize any consistent linguistic or second of which would have a big impact on the mathematical system. He produced a calculus ra- development of computer science: the Entschei- tiocinator, an algebra to specify the rules for ma- dungsproblem. In English, it is named the decision nipulating logical concepts. More than a century problem, and it can be formulated as follows: “pro- later, George Boole (1815-1864) and Augustus De vide a method that, given a first-order logic state- Morgan (1806-1871) converted the loose ideas ment, would determine in a finite number of steps of Leibniz into a formal system, Boolean algebra, whether the statement is true". In other words, the from which Gottlob Frege (1848-1925) finally pro- Entscheidungsproblem asks for the existence of a duced the fully developed system of axiomatic finite procedure that could determine whether a predicate logic. statement in mathematics is true of false. Hilbert Frege went on to prove that all the laws of arith- was convinced that the answer would be yes, as metic could be logically deduced from a set of axi- mathematics should be complete, consistent and oms. He wrote a two-volume book with the formal decidable. development of the foundations of arithmetic, and when the second volume was already in print, Frege 3. TuRing esTABlishes The limiTs OF COmPuTABiliTy received the celebrated letter from Bertrand Russell In 1936 Turing completed the draft of his paper (1872-1970) showing his theory was inconsistent. "On Computable Numbers, with an Application to the The counterexample was the paradox of extraordi- Entscheidungsproblem", where he proposed a formal 2 ARBOR Vol. 189-764, noviembre-diciembre 2013, a080. ISSN-L: 0210-1963 doi: http://dx.doi.org/10.3989/arbor.2013.764n6003 model of computation, the a-machine, today known ity, as it could be used to compare the relative dif- as the Turing machine (TM), and he proved that any ficulty of problems; thus he defined the concept of function that can be effectively computed (in Hilbert’s Turing-reducibility (T-reducibility). Given problems P1 a080 T sense) by a human being could also be mechanically and P2, P1 is said to be Turing-reducible to P2 (P1 ≤ P2) if calculated in a finite number of steps by a mechanical P2 can be used to construct in a finite number of steps, Josep T procedure, namely the TM. Turing used his formalism a program to solve P1. Note that if P1 ≤ P2 and P2 is de- Díaz & T to define the set of computable numbers, those reals cidable then P1 is also decidable, and if P1 ≤ P2 and P1 for which their ith decimal can be computed in a finite is undecidable, then P2 is also undecidable. Therefore, Carme Torras number of steps. The number of computable numbers T-reducibility can be used to enlarge the class of know is countable but the number of reals is uncountable, undecidable problems. so most of the reals are not computable. Some well- known computable numbers are 1/7, π and e. 4. AFTeR TuRing: PROBlem COmPlexiTy ClASSES In his model, Turing introduced two essential as- We have seen that the work of Turing and others let sumptions, the discretization of time and the discre- us determine, for any given problem, whether there tization of the state of mind of a human calculator.
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