La lente algoritmica de Touring

7/27/2019 La lente algoritmica de Touring

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EL LEGADO DE ALAN TURING / THE LEGACY OF ALAN TURING

ARBOR Cca, Pato Ctra

Vol. 189-764, noviembre-diciembre 2013, a080 | ISSN-L: 0210-1963

doi: hp://dx.doi.org/10.3989/arbor.2013.764n6003

TuRings AlgORiThmiC lens:FROm COmPuTABiliTy TO

COmPlexiTy TheORy

LA LENTE ALGORTMICA DE

TURING: DE LA COMPUTABILIDAD

A LA TEORA DE LA COMPLEJIDAD

Jop Daz* and Car Torra***Llenguatges i Sistemes Informcs, UPC.

[email protected]**Instut de Robca i Informca Industrial, CSIC-UPC.

[email protected]. edu

Coprt: 2013 CSIC. This is an open-access arcle distributedunder the terms of the Creave Commons Aribuon-NonCommercial (by-nc) Spain 3.0 License.

Ctao/Co ctar t arco: Daz, J. and Torras, C. (2013).Turings algorithmic lens: From computability to complexitytheory. Arbor, 189 (764): a080. doi: hp://dx.doi.org/10.3989/arbor.2013.764n6003

Received: 10 July 2013. Accepted: 15 September 2013.

Resumen: La cuesn de la decidibilidad, es decir, si es posible de-mostrar computacionalmente que una expresin matemca esverdadera o falsa, fue planteada por Hilbert y permaneci abiertahasta que Turing la respondi de forma negava. Establecida lano-decidibilidad de las matemcas, los esfuerzos en informcaterica se centraron en el estudio de la complejidad computacio-nal de los problemas decidibles. En este arculo presentamos unabreve introduccin a las clases P(problemas resolubles en empopolinmico) y nP(problemas resolubles de manera no determinis-ta en empo polinmico), al empo que exponemos la dicultadde establecer si P= nPy las consecuencias que se derivaran deque ambas clases de problemas fueran iguales. Esta cuesn eneimplicaciones no solo en los campos de la informca, las mate-

mcas y la sica, sino tambin para la biologa, la sociologa y laeconoma. La idea seminal del estudio de la complejidad computa-cional es consecuencia directa del modo en que Turing abordabaproblemas en diferentes mbitos mediante lo que hoy se denomi-na la lupa algortmica. El arculo naliza con una breve exposicinde algunos de los temas de invesgacin ms actuales: transicinde fase en problemas nP, y demostraciones hologrcas, dondese trata de convencer a un adversario de que una demostracin escorrecta sin revelar ninguna idea de la demostracin.

PAlABRAs ClAve: Complejidad computacional; el problemade calcular la permanente de una matriz; problemas P y NP;demostraciones interacvas y hologrcas.

ABsTRACT: The decidability queson, i.e., whether anymathemacal statement could be computaonally proventrue or false, was raised by Hilbert and remained open unlTuring answered it in the negave. Then, most eorts intheorecal computer science turned to complexity theoryand the need to classify decidable problems according totheir diculty. Among others, the classes P (problemssolvable in polynomial me) and nP (problems solvable innon-determinisc polynomial me) were dened, and oneof the most challenging scienc quests of our days arose:whether P = nP. This sll open queson has implicaonsnot only in computer science, mathemacs and physics, butalso in biology, sociology and economics, and it canbe seen

as a direct consequence of Turings way of looking throughthe algorithmic lens at dierent disciplines to discover howpervasive computaon is.

KeywORDs: Computaonal complexity; the permanentproblem; P and NP problems; Interacve proofs and holographicproofs.
http://dx.doi.org/10.3989/arbor.2013.764n6003http://1si.upc.edu/http://dx.doi.org/10.3989/arbor.2013.764n6003http://dx.doi.org/10.3989/arbor.2013.764n6003http://dx.doi.org/10.3989/arbor.2013.764n6003http://dx.doi.org/10.3989/arbor.2013.764n6003http://1si.upc.edu/http://dx.doi.org/10.3989/arbor.2013.764n6003


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ARBOR Vol. 189-764, noviembre-diciembre 2013, a080. ISSN-L: 0210-1963 doi: hp://dx.doi.org/10.3989/arbor.2013.764n6003

Turingsalgorithmiclens:Fromcomputabilityto

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1. inTRODuCTiOn

The English mathematician Alan Mathison Turing

(1912-1954) is well-known for his key role in the

development of computer science, but he alsomade important contributions to the foundations

of mathematics, numerical analysis, cryptography,

quantum computing and biology. In the present

paper we focus exclusively on the influence of

Turing on computability and complexity theory,

leaving aside other aspects of his work even within

computer science, most notably his contribution

to the birth of the digital computer and artificial

intelligence.

Turing was an important player in the settle-

ment of the decidability issue, which led naturally

to the study of the complexity of decidable prob-lems, i.e., once it was proved that there are prob-

lems unsolvable by a computer, research turned

to the question of how long would it take to solve

a decidable problem?This is an exciting research

field with lots of open questions. In this paper, we

try to give a gentle introduction to the historical

perspective of the search for efficient computing

and its limitations.

2. BeFORe TuRing: The DReAm OF meChAniCAlReAsOning AnD DeCiDABiliTy

Gottfried W. Leibniz (1646-1716) was an impor-tant mathematician, philosopher, jurist and inven-

tor, whose dream was to devise an automatic rea-

soning machine, based on an alphabet of unam-

biguous symbols manipulated by mechanical rules,

which could formalize any consistent linguistic or

mathematical system. He produced a calculus ra-

tiocinator, an algebra to specify the rules for ma-

nipulating logical concepts. More than a century

later, George Boole (1815-1864) and Augustus De

Morgan (1806-1871) converted the loose ideas

of Leibniz into a formal system, Boolean algebra,

from which Gottlob Frege (1848-1925) finally pro-

duced the fully developed system of axiomatic

predicate logic.

Frege went on to prove that all the laws of arith-

metic could be logically deduced from a set of axi-

oms. He wrote a two-volume book with the formal

development of the foundations of arithmetic, and

when the second volume was already in print, Frege

received the celebrated letter from Bertrand Russell

(1872-1970) showing his theory was inconsistent.

The counterexample was the paradox of extraordi-

nary sets. A set is defined to be extraordinary if it is

a member of itself, otherwise it is called ordinary. Is

the set of all ordinary sets also ordinary? (see Doxi-

adis, Papadimitriou, Papadatos and di Donna, 2009,

pp. 169 to 171 for a particular charming account ofthe effect of Russell's letter to Frege).

Frege's work was the spark for 30 years of re-

search on the foundations of mathematics, and

the end of the 19 th c. and beginning of the 20 th

c. witnessed strong mathematical, philosophical

and personal battles between members of the in-

tuitionist and formalist European schools (Davis,

2000; Doxiadis, Papadimitriou, Papadatos and di

Donna, 2009).

To beer frame further developments, let us re-

call some noons. A formal system is a language,a nite set of axioms, and a set of inference rules

used to derive expressions (or statements) from the

set of axioms; an example is mathemacs with the

rst-order logic developed by Frege. A system is said

to be complete if every statement can be proved or

disproved; otherwise the system is said to be incom-

plete. A system is said to be consistent if there is not

a step-by-step proof within the system that yields a

false statement.

A system is said to be decidable if, for every state-

ment, there exists an algorithm to determine whether

the statement is true.

In 1928, at the International Congress of Math-

ematicians in Bologna, David Hilbert (1862-1943),

a leading figure in the formalist school, presented

to his fellow mathematicians three problems, the

second of which would have a big impact on the

development of computer science: the Entschei-

dungsproblem. In English, it is named the decision

problem, and it can be formulated as follows: pro-

vide a method that, given a first-order logic state-

ment, would determine in a finite number of steps

whether the statement is true". In other words, the

Entscheidungsproblem asks for the existence of afinite procedure that could determine whether a

statement in mathematics is true of false. Hilbert

was convinced that the answer would be yes, as

mathematics should be complete, consistent and

decidable.

3. TuRing esTABlishes The limiTs OF COmPuTABiliTy

In 1936 Turing completed the dra of his paper

"On Computable Numbers, with an Applicaon to the

Entscheidungsproblem", where he proposed a formal
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model of computaon, the a-machine, today known

as the Turing machine (TM), and he proved that any

funcon that can be eecvely computed (in Hilberts

sense) by a human being could also be mechanically

calculated in a nite number of steps by a mechanicalprocedure, namely the TM. Turing used his formalism

to dene the set of computable numbers, those reals

for which their ith decimal can be computed in a nite

number of steps. The number of computable numbers

is countable but the number of reals is uncountable,

so most of the reals are not computable. Some well-

known computable numbers are 1/7, and e.

In his model, Turing introduced two essenal as-

sumpons, the discrezaon of me and the discre-

zaon of the state of mind of a human calculator. A

TM consisted of an innite tape divided into squares,

a nite input on a nite alphabet, and a write-read

head that could be in a nite number of states repre-

senng those of the human calculator's brain (Davis,

2000; Moore and Mertens, 2011). The TM was a theo-

recal model of a stored-in program machine. An out-

standing achievement of Turing was the concept of

Universal Turing Machine (UTM). Given the codied

descripon of a TM as input, the UTM is able to simu-

late the computaon of that machine and write the

result. Hence, the UTM can simulate any other Turing

machine, thus opening the way to digital computers.

Relying on his UTM, Turing made a decisive contri-

buon to computability, namely he produced a prob-

lem that was undecidable: the halng problem. It can

be stated as follows: Given a codied descripon of

a TM and an input to it, decide if the UTM will halt.

The proof was a mimic of Geodel's incompleteness

one, where a diagonalizaon argument produces a

paradox (see Davis, 2000, chapter 7 for a nice infor-

mal explanaon). Note that the halng problem is a

counterexample, a negave soluon to the Entschei-

dungsproblem1.

Turings PhD. thesis extended Gdel's incomplete-ness theorem by proving that when new axioms areadded to an incomplete formal system the systemremains incomplete (Turing, 1939). A very importantcontribuon of that work was the Oracle Turing-ma-chine model. In the words of Turing: Let us supposewe are supplied with some unspecied means ofsolving number-theorecal problems, a kind of ora-

cle as it were. We shall not go any further into thenature of this oracle apart from saying that it can'tbe a machine.

Emil Post (1897-1954) realized that Turing's oracle

machine had important implicaons for computabil-

ity, as it could be used to compare the relave dif-

culty of problems; thus he dened the concept of

Turing-reducibility (T-reducibility). Given problems P1

and P2, P

1is said to be Turing-reducible to P

2(P

1TP

2) if

P2can be used to construct in a nite number of steps,a program to solve P

1. Note that if P

1TP

2and P

2is de-

cidable then P1is also decidable, and if P

1TP

2and P

1

is undecidable, then P2is also undecidable. Therefore,

T-reducibility can be used to enlarge the class of know

undecidable problems.

4. AFTeR TuRing: PROBlem COmPlexiTy ClAsses

We have seen that the work of Turing and others let

us determine, for any given problem, whether there

is an algorithm to solve it (i.e., the problem is decid-

able) or not. Let us turn into the realm of solvable

problems, and ask the queson of how long it takes

to solve a given decidable problem. It may seem that

this is a pure technological issue, however we will see

that there are problems for which today's computers

would take more that the age of universe to nd a so-

luon, when the input to the problem is a bit large.

The me complexity of a problem with input xis a

measure of the number of steps that an algorithm will

take to solve it, as a funcon of the size of the input

|x| = n. Let us consider the worst-case complexity,

where among all the possible inputs of size n, we take

the one that behaves worst in terms of the number

of steps performed by the algorithm. As an example,

consider the school algorithms for mulplying two

integers aand b of sizes naand n

b, respecvely, size

measured as the number of digits in the binary ex-

pression (i.e., na= log

2a), and let n= max{n

a, n

b}, then

the worst-case complexity of the algorithm is T(n) =

O(n2).2

For decidable problems, the me complexity ex-

pression has a tremendous impact on the running

me of an algorithm. Figure 1 (taken from Moore and

Mertens, 2011) is an indicaon of the running me

of dierent worst-case complexity gures, assuminga processor can solve an instance of size n= 1 in 103

seconds, which is consistent with todays technology.

Note that an algorithm that solves a problem with

complexity O(2n), if given an input of size n= 90, may

yield the output aer the universe would be nished

(again, assuming today's technological level).

Let us try to get a feeling for the me-complexity

of some problems. Given an n x n matrix M = (mij), its

determinant is dened by the Leibniz formula


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where is one permutaon from the symmetric

group of permutaons Sn, and sgn () is the sign of

the permutaon. A similar associate value of a square

matrix is thepermanent, which is dened

For example if

then det(M) = m11

.m22 m

12 .m

21, while perm(M) =

m11

.m22

+ m12

.m21

.

A direct implementaon of the denion yields an

O(n!)algorithm for both problems. However, by using

the LU decomposion by Gauss and Turing, the deter-

minant of an nx nmatrix can be computed in O(n3)

steps, see for example in Cormen, Leiserson, Rivest

and Stein (2001), while the complexity of comput-

ing the permanent seems to be much more dicult

(Valiant, 1979). Recently D. Glynn has obtained a de-

terminisc bound of O(n2n) to the complexity of the

permanent (Glynn, 2010). Noce that if M1and M

2are

square matrices then det(M1M

2) = det(M

1) x det(M

2),

while perm(M1M

2) perm(M

1) x perm(M

2).

A problem is said to be feasible if there exists an

algorithm that solves it with polynomial me-com-

plexity. All other decidable problems are said to be

unfeasible, which does not imply that in the near or

distant future some problems may turn into feasible.

A parcularly interesng kind of problems are the

combinatorial opmizaon problems, where for anyinstance xand a soluon s

xfor that instance, there is a

cost funcon k(x,sx) +and the objecve is to nd

the sx*that opmizes (maximizes or minimizes) k(x,s

x)

over all possible soluons sx

3. Let Opt(x) denote the

cost of an opmal soluon.

For example, consider the chromac number prob-

lem of a graph: Given as input a graph G = (V, E), nd

the minimum number of dierent colors needed to

have a valid coloring of the verces in V, where a valid

coloring of G is an assignment of labels : V {1, 2,

, k}, each integer represenng a color, such that for

any (u, v) E, (u) (v). Noce that, for any input Gand soluon , Opt(G) is a valid coloring with minimum

number of colors.

4.1 T ca P, nP ad nP-copt

In the early 1950s, some mathemacians started to

realize that some decidable problems took too long to

be solvable for large inputs. In a series of leers to the

US Naonal Security Agency, John Nash proposed a

secure cryptographic encrypon system based on the

computaonal diculty of problems (Nissan, 2004).

Fr 1:Running mes as a funcon of input size n


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These leers foresaw the classicaon of decidable

problems according to their computaonal complex-

ity and could have given birth to the eld of complex-

ity theory if the leers had not remained a state secret

unl 2012.

The second important historical event for the studyof problem complexity was the leer Kurt Gdel sentto John von Neumann in March 1956. At the me, vonNeumann was dying and he did not read the leer,which was lost unl the 1980s. An English transla-on of the leer can be found in the appendix of Lip-tons book (2010), which is a compilaon of selectedposts of his useful blog. As it is beaufully explainedin Liptons poec version of the events, Kurt Gdel iswalking along through the falling snow, he is thinking.Gdel is the greatest logician of his me, perhaps of allmes, yet he is deeply troubled. He has found a prob-

lem he cannot solve. The problem concerns a simplebut fundamental queson. Suddenly he smiles, he hasan idea. If he cannot solve the problem, then he willwrite a leer explaining it to John von Neumann; per-haps John will be able to solve the problem. Aer allJohn von Neumann has one of the fastest minds in theworld. Gdel pulls his scarf ghter, gets his bearings inthe heavy falling snow, and heads to his oce to writehis leer.

In his leer, Gdel considered the truncated Entsc-heidungsproblem problem: Given any statement inrst-order logic (for example x, y, z, n - {0} : (n 3) (xn+ yn= zn))decide if there is a proof of thestatement with nite length of at most mlines, wherem could be any large (1010) constant. As any math-emacal proof could be represented using a smallconstant number c of symbols, the problem is decid-able; just construct all the exponenally many cmpos-sible proofs of length m, over the nite alphabet, andcheck if any of them works. Noce that most of theproofs will be gibberish. Moreover, for large m, thereis a chance that the process could take longer than theexistence of the earth! (see Fig. 1). Gdel in his leerasked if it was possible to do it in O(m2) steps, mak-ing explicit the possible paron between decidableproblems solved in polynomial me, and decidable

problems that cannot be solved in polynomial me.During the 1960s, the existence of easy and hard

problems started to be more obvious to researchers.

As computers were solving problems with increas-

ing input size, researchers begin to consider the ef-

fect of input size on the computaon me (number of

steps) and space (number of cells visited), using the

Turing machine as the computaonal model. An im-

portant contribuon appeared in 1965 by Hartmanis

and Stearns (1965), which dened the mul-tape Tur-

ing machine and laid the foundaons of complexity

theory. The same year, Edmonds (1965) gave an infor-

mal descripon of non-determinisc polynomial-me

problems, i.e., those that permit verifying in polyno-

mial me whether a guessed hypothecal soluon is

indeed a correct soluon to the problem. Nondeter-minism would play a key role in the development of

computaonal complexity theory. A nondeterminisc

algorithm could be seen as a game where a compu-

taonally omnipotent prover P provides a witnessto the soluon of the problem, and a verier has to

prove determiniscally that the witness is indeed a

correct soluon.

Consider the problem of the sasability of a con-

juncve normal form (SAT): Given a set of Boolean

variablesX and a Boolean formula on Xin conjunc-ve normal form, decide if there is an assignment A :

X{T, F} (truth, false) such that it sases (eval-uates to T). A conjuncve normal form formula con-sists of the conjuncon of clauses, where each clause

is a disjuncon of literals (Boolean variables or their

negaon). For example, if X = {x1, x

2,x3, x4}, consider

the Boolean formula

A backtracking determinisc algorithm would ndan assignment sasfying . In a nondeterminisc al-gorithm, Pwould supply as witness an assignment A(for example, A(x

1) = A(x

4)= T, A(x

2)= A(x

3)= F)

and V would have to verify that the substuon ofsuch an assignment makes sasable. Note that ifhas input size n(length of ),Vcould verify wheth-er the assignment sases in O(n)steps.

In 1971 Stephen Cook showed that the SAT problem

was a paradigmac unfeasible problem in the sense

that any problem that could be solved4in polynomial

me by a nondeterminisc Turing machine could be

reduced to the SAT problem, where the reducibility is

the same concept as Turing-reducibility but imposing

the condion that the construcon should be done in

polynomial me (Cook, 1971). The paper was present-

ed at the STOC-71 conference, and again using the

Lipton (2010) descripon: A tall gure walks slowly

to the front of the conference room. Steve Cook is a

young scienst, who is about to change the world. He

has independently discovered the problem that trou-

bled Gdel that snowy day. He gives his talk, and aer

there is a polite applause, as there is for every talk.


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One aendee who understood perfectly the mean-

ing of Cooks result was Richard Karp, who the follow-

ing year published a seminal paper formally dening

the classes P, nPand nP-complete, and provided the

proof that nine well-known problems were in the classnP-complete (Karp, 1972). His starng seed was that

SAT nP-complete, which he coined as Cooks Theo-

rem. At the same me, but independently from Cook

and Karp, a Russian mathemacian, Leonid Levin, was

basically dening the class nP-complete (Levin, 1973),

the arcle appeared translated into English the same

1973, but it took a while for the community working

in complexity theory to realize the meaning of Levins

result (see Trakhtenbrot, 1984 for a history of early

complexity theory developments in Russia). Today the

result that SAT is nP-complete is known as the Cook-

Levin Theorem.

We next give an intuive descripon of the com-

plexity classes. For a more technical descripon the

reader is referred to any of the complexity or algorith-

mic books listed in the bibliography of this paper.

P is the class of problems for which there is a de-

terminisc algorithm nding a soluon in polynomial

me, for any input.

nP is the class of problems such that if an omnis-

cient prover Pprovides a polynomial-length witness

to the soluon of the problem, a verier Vcan prove

in polynomial me whether the witness is indeed cor-rect. The term nPstands for non-determinisc poly-

nomial me.

From the previous remarks on the SAT problem, it

is clear that SAT nP: Psupplies a valid assignmentand Vchecks in linear me that the assignment sas-

es the formula. However, if we consider a combina-

torial opmizaon problem, as the chromac number

of a graph, it is not so easy. Given an input G = (V, E),

the chromac number problem consists of nding a

minimum valid coloring * for G, i.e., assigning the

minimum number of colors to V such that for every

edge (u,v) E, * (u) * (v). To prove this prob-lem is not in nP, assume that Pprovides a witness *to the colorability of V, then Vtests that indeed * is

a valid coloring by looking at every edge of G, which

has a cost of O(n2). Moreover, Vmust also verify that

there is no other valid coloring with less colors that

*, i.e., he must compare with all other possible valid

colorings for G, which could be exponenal. There-

fore, the problem is not in nP.

To circumvent this diculty, we consider the search

version of the combinatorial opmizaon problem,

where as a part of the input we also are given a value

b that the cost funcon can take. If the opmizaon

problem demands to maximize, the search version

would require that k(x) b, and if it is a minimizaon

problem the search version would require k(x) b.

Connuing with the example, the search version of

the chromac number problem can be stated as fol-

lows: Given as input G= (V, E) and a b > 0, nd a valid

coloring of Gsuch that |(V)| b.

The search versions of opmizaon problems are

obviously in nP. Furthermore, by using binary search

it is a standard exercise to prove that if the search

version of an opmizaon problem can be solved in

polynomial me then the opmizaon version can

also be solved in polynomial me (see Exercise 8.1 in

Dasgupta, Papadimitriou and Vazirani, 2008).Let us see some further examples of problems in P

and nP:

The primality problem: Given an integer a with

length n= log abits, decide whether ais prime. There

is a non-trivial proof that primality nP due to V.

Pra (1975), and for a long me it was open whether

the problem was in P. In 2002, Agrawal, Kayal and

Saxena provided a determinisc polynomial-me al-

gorithm to solve the primality problem, so the prob-

lem is in P (Agrawal, Kayal, and Saxena, 2002).

Thefactorizaon problem: Given an integer aof nbits, nd its prime decomposion. This is an impor-

tant problem as, among other things, it is the basis

of the RSA, one of the most used public-key cryp-

tographic systems (see Chapter 8 in Singh, 2000).

Factorizaon is in nP, since if P provides a witness

p1,,p

m, Vcan test that each p

iis prime (using the al-

gorithm in Agrawal, Kayal, and Saxena, 2002) and see

that the product is the given number. Nevertheless,

it is an important open queson whether there is a

polynomial-me algorithm to solve factorizaon5.

The problem of 3-sasability (3-SAT) is the parcu-

lar case of SAT where every clause in the input hasexactly 3 literals. The problem is also in nP. However,

an easy greedy algorithm puts 2-sasability in the

class P.

Finally, the truncated Entscheidungsproblem is also

in nP, since if we get a proof of less than 1010pages

as a cercate, any mathemacian can verify in a rea-

sonable me (polynomial in 1010) whether it is cor-

rect (independently that there exist other valid proofs

among the exponenal number of generated ones).


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Noce that P nP. IfVcan obtain a determinis-c soluon in polynomial me, he can also verify it in

polynomial me.

The problem to decide whether P=nP was consid-ered one of the nine millennium problems posed by

the Clay Mathemacs Instute in the year 2000. A 107

US$ prize is awarded for each soluon to one of the

problems.

The answer PnP would mean that, for many im-

portant search problems,nding is more dicult than

verifying. If, on the contrary, the answer is P=nP, then

there would be a feasible algorithm for the truncated

Entscheidungsproblem, i.e., proofs with a large but -

nite number of lines, which in pracce would suce

for most mathemacal theorems.

4.2 Karp rdcbt ad nP-copt

To further study the relaonship between search

problems and the P=nP queson, Karp dened the fol-

lowing simplied variaon of Turing-reducibility: Given

search problems A and B, with sets of instances IA

and IB

and sets of soluons SA

and SB, a Karp reducon fromA

to B (A PB) is a polynomial-me computable funcon

f: IA I

Bsuch that for any input x I

A, f(x)has a

soluon in SBi xhas a soluon in S

A.

In other words, ifA PBand we have a polynomial-

me algorithmAB

to solve B then we have a polyno-

mial-me algorithm to solveA, for any x IA

comput-

ed in polynomial meAB(f(x)). Noce, a polynomial

algorithm to solve Adoes not imply anything about

the existence of a polynomial algorithm to solve B.

A search problem B is nP-complete if BnPand for

any nP problemA, we haveA PB.

A useful property of reducons is that they are

transive. This property, together with the previous

remark about reducons as problems solvers, tells us

that the class nP-complete is the most dicult class

of nP problems, in the sense that if one nP-complete

problem is known to be in Pthen P=nP.

It is known that if PnP, there would be problems in

nP that are neither in Pnor nP-complete. These are in

the class nP-intermediate (see Figure 2).

It is beyond the scope of this paper to show de-tailed reducons between nP-complete problems,which can be found in any standard algorithmic book,see for example Garey and Johnson, 1979; Mooreand Mertens, 2011. But we would like to enumeratea few more examples of problems nP-complete and

Fr 2:Complexity classes inside nP

nP-intermediate. For a full taxonomy of nP-completeproblems see Crescenzi and Kann (2012).

As already menoned, SAT was the rst problem

known to be nP-complete; 3-SAT, the truncated Entsc-heidungsproblem, and chromac number of a graphare also nP-complete problems. Factoring an integeras a product of primes is in nP-intermediate. Anothersignicant problem in the class nP-intermediate isgraph isomorphism: Given two graphs G

1= (V

1, E

1) and

G2= (V

2, E

2), nd if there is a permutaon : V

1 V

2

such that (u, v) E1i ((u), (v)) E

2.

Consider the 3-coloring of a graph problem: Given

input graph G = (V, E), decide if there is a valid coloring

of V with exactly 3 colors (see Figure 3). The problem is

in nP: if Pprovides a 3-color witness , then Vcan ver-

ify in O(|V|

2

) whether is a valid coloring. It is knownthat this problem is nP-complete. Deciding whether a

graph is 2-colorable and, if so, nding the coloring is in

P(Dasgupta, Papadimitriou and Vazirani, 2008).

The following example taken from Moore and

Mertens (2011, p. 152) shows that nP-completeness

can appear in calculus problems. The cosine integra-

on problem: Given integers x1, ... , x

n, decide if

The class nP-complete contains many everyday

praccal problems. There is a series of problems deal-ing with minimizing delivery costs or me, which de-

rive from the Traveling Salesman Problem (TSP). The

search version of TSP has as input a complete graph G

= (V, E), with a weight w(vi, v

j) 0 on each edge (v

i, v

j),

together with a value c, the goal is to nd a tour visit-

ing all verces exactly once, such that


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This problem is nP-complete. Another importantThis problem is nP-complete. Another important set of

praccal problems derive from thejob scheduling prob-

lem. In its most general seng, the problem can be for-

mulated as follows: Given njobsJ1,...J

n, where eachJ

k

has a processing me tk, and given midencal machines

M1,..., M

m, then each jobJ

kmust run on a machine M

i

for tkconsecuve units of me, and during that me no

other job can run on the same machine. The problem

is to nd an assignment of jobs to machines such that it

minimizes the makespan of the schedule

where Tidenotes the me at which machine M

icom-

pletes its jobs. The search version of the problem is

nP-complete for m> 2. For more informaon on the

problem and its variaons, see Brucker (2006).

As a final example, we would like to present evi-

dence of the influence of the P versus nP prob-

lem in other disciplines, for example, economics.

The Efficient Market Hypothesis (EMH) states that

future prices cannot be predicted by analyzing

prices from the past (Fama, 1965). In an enjoyable

recent paper by the economist P. Maymin (2011),

he proves that the EMH is false iff P=nP.For peo-ple with a little background in complexity theory,

the paper is quite straightforward, still it presents

a clear example of the spread of complexity theory

to other fields.

4.3 Cop t nP-copt

What can be done when having to deal with an nP-

complete problem? For inputs of very small size n,

a complexity of O(2n) or O(n!) can be computed in

reasonable computer me, but as we showed in Fig-

ure 1, the problem becomes intractable as ngrows.Nevertheless, it turns out that for some nP-com-

plete problems there are only a few bad inputs.

The worst-case complexity assumes that there is a

devil adversary that chooses the worst possible in-

stance of the problem. In the early 1990s there was

empirical evidence that for some problems such as

graph coloring, sasability, integer paron, etc.,

there was a sharp jump from instances which were

easily solved to instances that were easily shown not

to have a soluon, and just a few instances were dif-

cult to solve. That phenomenon is denoted phase

transion as it is similar to the phenomena studiedby physics of sudden changes of states, between

solid, liquid and gas. For instance, consider the 3-SAT

problem. There is a well-know clever exhausve

search algorithm for SAT, the Davis-Putnam-Loge-

mann-Loveland (DPLL), which was shown to solve

many random instances of 3-SAT in polynomial me

(Mitchell, Selman, and Levesque, 1992). To produce

a random instance for 3-SAT with nboolean variables

and m = rnclauses, one has to choose uniformly at

random each clause, with probability 1/(n3) and then

go over the variables in each clause and negate each

with probability = 1/2. The density of one of these

formulae is dened as r = n/m.

For example, the random 3-SAT formula

has density r= 0.4.

Fr 3:3-colorable G with a valid coloring


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Mitchell, Selman, and Levesque (1992) rst experi-mentally showed that the number of random 3-SATformulae for which the DPLL algorithm took exponen-al me was small (see Figure 4a). Moreover, their ex-

periments also showed that, with high probability, fordensies r< 4.2 random 3-SAT formulae are sasableand for r> 4.2 random 3-SAT formulae are not sas-

able (see Figure 4b).

In 2002, using non-rigorous techniques from stas-cal physics (the replica method) on very large instanc-es of 3-SAT, Mzard, Parisi and Zecchina (2002) andMzard and Zecchina (2002) showed that the thresh-old for 3-SAT occurs at formulae with density rc= 4.27,i.e., random 3-SAT formulae with density < 4.27 aresasable with high probability, and random 3-SAT

formulae with density < 4.27 are sasable with high

probability, and radom 3-SAT formulae with density >4.27 are NOT sasable with high probability. Sincethen, there has been an eort to establish this sharpphase transion by rigorous analycal methods. Sofar for 3-SAT, the best lower bound is 3.52 (Hajiaghayiand Sorkin, 2003; Kaporis, Kirousis and Lalas, 2006)and the corresponding upper bound is 4.4907 (Daz,

Kirousis, Mitsche and Prez, 2009). Closing the gap re-mains an open problem. See Chapter 14 in Moore andMertens (2011).

The fact that some nP-complete problems have a nottoo large number of bad instances indicates that some-mes heuriscs can be used to achieve good results.Heuriscs are procedures with no guarantees either in

the running me or on the accuracy of the obtainedsoluon, but for many hard problems, like for example,the layout of VLSI circuits, heuriscs are the best prac-

cal soluon (Wolf, 2011). Some modern textbooks on al-gorithms include a chapter on heuristics, clever backtrackingtechniques, like the DPLL algorithm we menoned for solv-ing SAT, simulated annealing or dierent versions of localsearch. For a general textbook on heuriscs see Michale-wicz and Fogel (1998).

Another praccal alternave is using approximaon

algorithms. An algorithm is said to r-approximate an

opmizaon problem if, on every input, the algorithm

nds a soluon whose cost is 1/rif the problem asksfor the minimum value, or 1/rif the problem asks for

a maximum value.

Almost since the beginning of the development of

complexity theory, there was a parallel eort to develop

approximaon algorithms for nP-complete problems. Al-

though the rst such algorithm is due to Graham in 1966,

who developed it to approximate a version of scheduling,

the seminal paper for approximaon theory by Johnson

Fr 4: Experiments with random 3-SAT formulae, from Mitchell, Selman, and Levesque (1992). In (a) each dot

represents the me DPLL takes on a random instance of 3-SAT. Light dots represent sasable instances and darkdots represent non-sasable instances. (b) The phase transion sasable to non-sasable for a random 3-SATformula occurs at density 4.27.


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(1974) came very lile aer Cook, Levin and Karps papers.

Since then, the theory of approximaon and of inapprox-

imability has become a very fruiul eld in theorecal

computer science, with good books covering the topic,

for example Williamson and Smoys, 2010.

Parallelism is another powerful tool to speed up com-

putaon by a constant factor. Noce that anything that

can be done with 1010processors in T steps, can be done

with one processor in 1010Tsteps. But unless P=nP, the

dierence between Pand nP-complete problems is an

exponenal cost of computaon, therefore parallelism

would not be the tool to solve nP-complete problems in

polynomial me.

In the same way, a result by Impagliazzo and Widg-

erson (1997) implies that, unless P=nP, randomizaon

would not help to solve nP-complete problems. In fact,there is a stronger conjecture: randomizaon mainly

helps to improve the me complexity of problems in P.

At the present me, quantum computers are not an ex-

isng reality, although D-Wave Systems, Inc. has managed

to build a 128-qubit processor (not general purpose). At

the theory level, we know quantum computaon could

solve nP-intermediate problems, for example factorizaon

of an integer (Shor, 1997), but it is generally agreed that,

unless P=nP, quantum computaon would not help with

nP-complete problems (Bernstein and Vazirani, 1997).

See Aaronson (2008) for an extensive discussion on the

limitaons of quantum computers.

5. BeyOnD TuRing: inTeRACTive PROOFs AnDZeROKnOwleDge PROOFs

A fascinang line of research in theorecal com-

puter science started in the mid 1980s, which led

to fruiul results, namely Interacve Proofs. In this

secon, we aim to give a brief intuive introducon

to the topic, in parcular to Zero-Knowledge Proofs

(ZKP) and Probabiliscally Checkable Proofs (PCP). For

the interested reader, we recommend Chapter 11 in

Moore and Mertens (2011) and Chapters 8, 9 and 11

in Arora and Barak (2009). The topic has spanned 26years, with numerous papers and fruiul applicaons

in various elds, such as cryptography and approxima-

on algorithms, among others.

The basic idea is how one researcher (the prover P)

can convince another researcher (the verier V)6thathe has a correct proof to a dicult theorem by only

showing to Va few random bits of the proof, in such a

way that at the end Vis convinced that the proof is cor-

rect without having any insight into how it works.

To grasp the concept of zero-knowledge proof, let

us start with the very simple illustrave example from

Quisquater, Quisquater, Quisquater, Quisquater, Guil-

lou, Guillou, Guillou, Guillou, Guillou, Guillou, and

Berson (1989)7. There is a cave with two paths AandB, which are connected at their ends by a secret pas-

sage (see Figure 5). To cross that passage one must

know the magic words. In our case, Pwants to con-vince Vthat he knows the magic words without tell-

ing them to V. They agree on the following protocol:

Vwill remain outside of the cave, so he cannot see

which path P takes. When Parrives at the end of thecave, Vtells P which path to return along, and Ven-

ters the cave to make sure P is returning for the path

he indicated. The probability that P took the path V

asks him to return is 1/2, therefore if the experiment

is repeated a suciently large number of mes, andeach me P returns by the correct path, with prob-

ability 1 Pmust know the magic words to cross the

cave, and Vis convinced that Pknows them.

ZKP are a parcular case of the most general Inter-

acve Proofs Systems, introduced concurrently in Ba-

bai (1985) and Goldwasser, Micali and Racko (1989)8.

Technically the word proof refers to a randomized

interacve protocol between P and V, where P hasunlimited computaonal capabilies and tries to con-

vince V of the truth of a certain statement. Loosely

speaking, the two characteriscs that an interacve

protocol must have to be an interacve proof is that

an honest Vshould always be convinced by an honest

P, but a cheater P should have a very small probabil-

ity of convincing an honest Vthat a false statement is

true. The interacve proof system is zero-knowledge

if V is not going to learn anything from the interac-

on with P. In the previous example, V becomes

convinced that Pknows the magic words to cross be-

tween the two paths.

Fr 5:Ali Babas magic cave.


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Let us describe a more interesng example. Consid-

er a given graph G = (V, E), where V = n and E = m.As we saw before, to decide whether G has chromac

number 3 is an nP-complete problem, therefore in

general it is not a feasible problem to solve for largevalues of n. In this seng, Pwants to convince Vthathe has a valid 3-coloring of G, without revealing the

coloring. This example is from Goldreich, Micali and

Wigderson (1991). At each iteraon of the protocol, V

only has access to the colorings at the ends of a single

edge he chooses at the iteraon. The protocol is the

following: First Pselects a valid 3-coloring (for every

(, ) E,uand vmust have dierent colors) andgenerates all 3! = 6 permutaons of valid 3-colorings

for G; let Cbe the set of all 6 valid colorings. For n3

iteraons, at each iteraon i, P chooses with prob-

ability 1/6 a new coloring ci C, Vselects an edgeand veries that the edge is correctly colored. Assumethat G has a valid 3-coloring (see for example Figure

3). Noce that if V chooses the same edge (u, v) at

two dierent iteraons, the colors assigned to each

vertex may be dierent, as at each iteraon Pchooses

a new random coloring, therefore Vwill not be able

to learn a valid coloring for the whole G. On the other

hand, if G is not 3-colorable, at each round, at least

one edge will have the same color for both verces,

therefore the probability that Vwill discover a wrong

edge is at least 1/mper round. As m n2, aer n3itera-

ons with probability tending to 1, Vwill discover that

G is not 3-colorable.

Technically the way Pshows the colors to V is a bit

more complicated, using a one-way-funcon. One-way-

funcons are funcons that can be easily computed but

are hard (exponenal me) to invert. A trivial example

of one-way-funcon is integer mulplicaon: it is easy

to mulply m= alx a

2x ... x a

n, however as we already

menoned, there is no known polynomial-me algo-

rithm for factorizing m. Another more interesng ex-

ample of a one-way-funcon is the discrete logarithm:

Given n-bit integers x,y, z, nd whether there existsan integer wsuch that

y=

x

wmodz. Given

x,

wand

z,

it is easy to computey= f (x, w, z) = xwmod z, but it isconjectured that ndingf 1(y) takes exponenal me.

One-way-funcons are in standard use in cryptography.

By applying reducons to the 3-colorability prob-

lem, it was shown in Goldreich, Micali and Wigderson

(1991) that under the assumpon of the existence

of one-way-funcons, every problem in class nP has

a zero-knowledge proof. In fact, if we denote iP the

class of problems having an interacve protocol, it

is known that iP, as a complexity class, contains far

more dicult problems than nP (under the hypoth-

esis PnP)(Shamir, 1992). Compung the permanent

of a matrix is a problem in the class iP,which means

that even under the hypothesis P=nP,it would remain

a hard problem.

The culminaon of interacve proof systems re-

search was one of the most beauful and deep theo-

rems in computer science, the PCP-theorem. Although

the Gdel prize 2001 was shared by Arora and Safra

(1998), Arora, Lund, Motwani, Sudan and Szegedy

(1998) and Feigue, Goldwasser, Lovasz, Safra, and

Szegedy (1996) for their contribuon to Probabilis-

cally Checkable Proofs and the PCP-theorem, many

of the techniques and ideas are due to a much larg-

er number of researchers (see for example Johnson

(1992) for an extensive historical account, and Chap-

ter 16 in Williamson and Smoys (2010) for further re-

cent work using the PCP-theorem to obtain inapproxi-

mability results).

The rough idea of probabiliscally checkable proof

systems is: Given a convenonal mathemacal proof

in which a mistake could be hidden in any equaon,

transform the proof in such a way that the mistake

is spread almost everywhere. This kind of proof is

denoted a holographic proof. A PCP system for an nP

problem encodes the witness to the problem in a way

such that Vcan verify probabiliscally the witness by

looking only to a few of its bits, so that if it is true V

accepts with probability 1, and if it is false Vaccepts

with probability < 1/2.

The PCP-theorem states that holographic proofs ex-

ist for problems in nP, i.e., that any problem in nP has

a polynomial length probabiliscally checkable proof,

where Vips O(log n) random coins and need to lookonly at O(1) bits of the proof.

6. COnClusiOns

Contrary to Hilberts Entscheidungsproblem, it re-

mains an open problem to decide whether the trun-

cated Entscheidungsproblem is feasible, i.e., it re-mains open to decide whether P=nP.It follows from

the arguments in the present paper that a posive an-

swer to that queson may answer all seven remaining

open millennium problems.

At least there are 54 exisng bogus proofs of the

P=nP queson. Of them, 26 proving the equality,

24 proving the strict inclusion, and 3 proving that

the P=nP queson is itself undecidable. For further

details see Woeginger. Most sciensts working in

complexity theory believe that PnP, but there are


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some top sciensts in the eld that disagree with the

majority, see for instance Liptons blog.

The aim of this manuscript was not to review the

complexity eld or the status and future of the P=nPqueson. As we pointed out, there are some outstand-

ing textbooks dealing with all the past, present and fu-

ture aempts and results in complexity. Our only incur-

sion into the areas of modern complexity has been the

topic of interacve proofs, and this is because we think

that it is a natural connuaon to the truncated Entsc-

heidungsproblem, in the sense that PCP basically tells

us how to convince our colleagues that we have a cor-

rect proof without giving away any real details. Moreo-

ver, it turns out that PCP is a strong characterizaon of

nP problems. Could we one day even have a praccal

holographic way to check proofs?

Complexity theory has studied dierent models of

computaon (Turing machine with bounded number

of steps, Boolean circuits, quantum algorithms, rand-

omized algorithms), the complexity of measuring dif-

ferent parameters (space, number of iteraons, depth

of circuits), and several measures of complexity (worst

case, average, smoothed). All this work has created a

whole cosmos of about 500 complexity classes (Aar-

onson, Kuperberg and Granade). For most of them,

strict relaons are not known and, at the end, the

main issue boils down to the basic intuion by KurtGdel in 1954.

We tried to convey the idea that this is one of the

deepest quesons of todays science, aecng not

only computer science, mathemacs and physics, but

also biology, sociology, economics and most human

acvies. We see this broad coverage of this queson

as a direct consequence of Turings way of looking

through the algorithmic lens to problems in dierent

disciplines, from cryptography and physics to biology.

The spread of modern technologies is accelerang the

need for an algorithmic view of todays social, eco-

nomical, cultural, and polical interacons, leading

directly to the queson P=nP?For an excellent survey

of the role of the algorithmic view into the acvies

within the modern world, we recommend the book

by Easley and Kleinberg (Easley and Kleinberg, 2010).

nOTes

1 In parallel to Turing, A. Church also gave

a negave answer to the Entschei-

dungsproblem using a logic formalism,-calculus.

2 The complexity is expressed in asympto-

c notaon, i.e., for very large values of

the input. The notaon T(n) = O(f (n))

means that limn

T(n)/f(n) = c , where

c is a constant.

3 It may happen that there is no feasible

soluon for input x, then k(x, s) does

not exist.

4 The correct word is recognized, as the

problems are posed as recognion

problems, i.e. determining whether a

word belongs to a language over a nite

alphabet.

5 A posive answer will render all transac-

ons done using RSA insecure.

6Ina large part of the scienc bibliogra-

phy, the prover and the verier are res-

pecvely named Merlin and Arthur.

7 The authors frame their explanaon in

the arabic tale Ali Baba and the forty

thieves from the classic One thousand

and one nights.

8 The conference version of Goldwasser,

Micali and Racko (1989) appeared at

STOC-85.

ReFeRenCes

Aaronson, S. (2008). The limits of quan-

tum computers. Scienc American,

289, pp. 62-69.

Aaronson, S.; Kuperberg, G. and Granade,

C. Complexity Zoo. hp://qwiki.stan-

ford.edu/index.php/Complexity_Zoo.

Agrawal, M.; Kayal, N. and Saxena, N.

(2002). Primes is in P. Annals of Ma-

themacs, 2, pp. 781-793.

Arora, S. and Barak, B. (2009). Computa-

onal Complexity: A Modern Approach.

Cambridge University Press.

Arora, S.; Lund, C.; Motwani, R.; Sudan, M.

and Szegedy, M. (1998). Proof verica-

on and the hardness of approximaon

problems. Journal of the ACM, 45 (3),

pp. 501-555.

Arora, S. and Safra, S. (1998). Probabilisc

checking of proofs: A new characteriza-

on of NP.Journal of the ACM, 45 (1),

pp. 70-122.

Babai, L. (1985). Trading group theory for

randomness. In Proc. 17th. ACM Sym-

posium on the Theory of Compung,

pages 421-429.

Bernstein, E. and Vazirani, U. V. (1997).

Quantum complexity theory. SIAM Jo-

urnal Compung, 26 (5), pp. 1411-1473.

Brucker, P. (2006). Scheduling Algorithms.

Springer, h edion.

Cook, S. (1971). The complexity of theo-

rem-proving procedures. In 3rd. ACM

Symposium on the Theory of Compu-

ng, pages 151-158.

Cormen, T. H.;Leiserson, C.; Rivest, R. and

Stein, C. (2001). Introducon to Algori-

thms. The MIT Press, 3 edion.

Crescenzi, P. and Kann, V. (2012). A com-

pendium of NP opmizaon problems.

Dasgupta, S.; Papadimitriou, C. and Vazira-

ni, U. (2008).Algorithms. McGraw-Hill.
http://dx.doi.org/10.3989/arbor.2013.763n5011http://qwiki.stanford.edu/index.php/Complexity_Zoo.http://qwiki.stanford.edu/index.php/Complexity_Zoo.http://qwiki.stanford.edu/index.php/Complexity_Zoo.http://qwiki.stanford.edu/index.php/Complexity_Zoo.http://dx.doi.org/10.3989/arbor.2013.763n5011


13/13



13

a080

Davis, M. (2000). The universal computer:

the road from Leibniz to Turing. Norton.

Daz, J.; Kirousis, L.; Mitsche, D. and Prez,

X. (2009). On the sasability thres-

hold of formulae with three literals perclause. Theorecal Computer Science,

410, pp. 2920-2934.

Doxiadis, A.; Papadimitriou, C.; Papadatos,

A. and di Donna, A. (2009). LOGICOMIX:

an epic search for truth. Bloomsbury.

Easley, D. and Kleinberg, J. (2010). Net-

works, Crowds and Markets. Reasoning

about a highly connected world. Cam-

bridge University Press.

Edmonds, J. (1965). Paths, trees, and

owers. Canad. J. Math., 17, pp. 449-

467.

Fama, E. (1965). The behavior of stock-

market prices. The Journal of Business,

38 (1), pp. 34-105.

Feigue, U.; Goldwasser, S.; Lovasz, L.; Safra,

S. and Szegedy, M. (1996). Interacve

proofs and the hardness of approxima-

ng cliques.Journal of the ACM, 43 (2),

pp. 268-292.

Garey, M. R. and Johnson, D. S. (1979). Com-

puters and Intractability: A Guide to the

Theory of NP-Completeness. Freeman.

Glynn, D. G. (2010). The permanent of a

square matrix. European J. of Combina-

torics, 31 (7), pp. 1887-1891.

Goldreich, O.; Micali, S. and Wigderson, A.

(1991). Proofs that yield nothing but

their validity or all languages in NP have

Zero-Knowledge Proof Systems. Jour-

nal of the ACM, 38 (1), pp. 691-729.

Goldwasser, S.; Micali, S. and Racko, C.

(1989). The knowledge complexity of

interacve proof systems. SIAM J. Com-

pung, 18 (1), pp. 186-208.

Graham, R. (1966). Bounds for certain

mulprocessing anomalies. Bell System

Technology Journal, 45, pp. 1563-1581.

Hajiaghayi, M. T. and Sorkin, G. (2003). The

sasability threshold of random 3-SATis at least 3.52. Technical report, IBM

Research Report.

Hartmanis, J. and Steam, R. (1965). On

the computaonal complexity of algo-

rithms. Transacons of the American

Mathemacal Society, 117, pp. 285-306.

Impagliazzo, R. and Wigderson, A. (1997).P = BPP if E requires exponenal cir-

cuits: Derandomizing the XOR lemma.

In Proceedings of tTwenty-Ninth Annual

ACM Symposium on the Theory of Com-

pung, pages 220-229.

Johnson, D. J. (1974). Approximaon algo-

rithms for combinatorial problems. Jo-

urnal of Computer and System Sciences,

9, 256-278.

Johnson, D. S. (1992). The NP-comple-

teness column. The tale of the second

prover. Journal of Algorithms, 13 (3),

pp. 502-524.

Kaporis, A. C.; Kirousis, L. and Lalas, E. G.(2006). The probabilisc analysis of a

greedy sasability algorithm. Random

Struct. Algorithms, 28 (4), pp. 444-480.

Karp, R. M. (1972). Reducibility among

combinatorial problems. In R. E. Miller

and J. W. Thatcher (eds.), Complexity of

Computer Computaons, pp. 85-104.

NY: Plenum Press.

Levin, L. (1973). Universal sequenal

search problems. Probl. Peredachi Inf.,

9, pp. 115-116.

Lipton, R. Godels lost leer and P = NP.

hp://rjlipton.wordpress.com.

Lipton, R. (2010). The P=NP Queson and

Godels Lost Leer. Springer.

Maymin, P. (2011). Markets are ecient if

and only if P=NP. Algorithmic Finance,

1, pp. 1-11.

Mezard, M.; Parisi, G. and Zecchina, R.

(2002). Analyc and algorithmic solu-

on of random sasability problems.

Science, 297 (812).

Mezard, M. and Zecchina, R. (2002). The

random k-sasability problem: from

an analyc soluon to an ecient al-

gorithm. Physics Review, E 66-056126.

Michalewicz, Z. and Fogel, D. (1998). How

to solve it: Modern Heuriscs. Springer.

Mitchell, D.; Selman, B. and Levesque, H.

(1992). Hard and easy distribuons

of sat problems. In Proceedings of the

10th. Naonal Conference on Arcial

Intelligence (AAAI), pp. 459-465.

Moore, C. and Mertens, S. (2011). The Nature

of Computaon. Oxford University Press.

Nissan, N. (2004). John Nashs leer to

the NSA. hp: //agtb.wordpress.

com/2012/02/17/john-nashs-leer-to-

the-nsa/,February 17.

Pra, V. R. (1975). Every prime has a suc-

cinct cercate. SIAM J. Comput., 4 (3),

pp. 214-220.

Quisquater, J.-J.; Quisquater, M.; Quisqua-

ter, M.; Quisquater, M.; Guillou, L. C.;

Guillou, M. A.; Guillou, G.; Guillou, A.;

Guillou, G.; Guillou, S. and Berson, T. A.

(1989). How to explain zero-knowled-ge protocols to your children. In G.

Brassard, editor, CRYPTO-89, volume

435 of Lecture Notes in Computer Scien-

ce, pp. 628-631. Springer.

Shamir, A. (1992). IP = PSPACE.Journal of

the ACM, 39 (4), pp. 869-877.

Shor, P. W. (1997). Polynomial-me algori-

thms for prime factorizaon and discre-

te logarithms on a quantum computer.

SIAM J. Comput., 26 (5), pp. 1484-1509.

Singh, S. (2000). The Code Book. Anchor Books.

Trakhtenbrot, B. (1984). A survey of rus-

sian approaches to perebor (brute-forcesearches) algorithms.Annals of the His-

tory of Compung, 6 (4), pp. 384-400.

Turing, A. M. (1939). Systems of logic based

on ordinals. Proceedings of the London

Mathemacal Society-2, 45, pp. 161-228.

Valiant, L. G. (1979). The complexity of

enumeraon and reliability problems.

SIAM J on Compung, 8, pp. 410-421.

Williamson, D. and Smoys, D. (2010). The

Design of Approximaon Algorithms.

Cambridge University Press.

Woeginger, G. The P vs. NP page. hp://www.

win.tue.nl/ gwoegi/P-versus-NP.htm.Wolf, W. (2011). Modern VLSI Design. Pren-

ce-Hall, fourth edion.
http://dx.doi.org/10.3989/arbor.2013.763n5011http://rjlipton.wordpress.com/http://localhost/var/www/apps/conversion/tmp/scratch_4/.com/2012/02/17/john-nashs-letter-to-the-nsa/,http://localhost/var/www/apps/conversion/tmp/scratch_4/.com/2012/02/17/john-nashs-letter-to-the-nsa/,http://localhost/var/www/apps/conversion/tmp/scratch_4/.com/2012/02/17/john-nashs-letter-to-the-nsa/,http://localhost/var/www/apps/conversion/tmp/scratch_4/.com/2012/02/17/john-nashs-letter-to-the-nsa/,http://localhost/var/www/apps/conversion/tmp/scratch_4/.com/2012/02/17/john-nashs-letter-to-the-nsa/,http://localhost/var/www/apps/conversion/tmp/scratch_4/.com/2012/02/17/john-nashs-letter-to-the-nsa/,http://rjlipton.wordpress.com/http://dx.doi.org/10.3989/arbor.2013.763n5011

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