on the computational structure of the connected components of a hard problem

Information Processing Letters 72 (1999) 83–90

On the computational structure of the connected componentsof a hard problem

Martin Matamalaa,∗, Klaus Meerb,1a Universidad de Chile, Dpto. de Ingeneria Matemática, Beaucheff 850, Casilla 170-3, Santiago, Chile

b Fakultät für Informatik, TU Chemnitz, D-09107 Chemnitz, Germany

Received 4 August 1998; received in revised form 18 October 1999Communicated by P.M.B. Vitányi

Abstract

The study of sparse sets has tremendous importance in Turing complexity theory. Thus it is a natural task to work out a relatednotion for real number models of computation. This has not fully been done so far. In the present paper we suggest such a notionbased on the computational structure of the connected components a set has. Even though our notion of well-structured sets isdifferent in spirit from the classical sparseness property we will show that it shares some important features with the latter. Weare going to analyze the (non-)existence of well-structured complete sets within the Blum–Shub–Smale model of computationover the reals with linear operations and equality respectively inequality. Relations to exponential time classes are also drawn. 1999 Published by Elsevier Science B.V. All rights reserved.

Keywords:Connected components; Sparseness; Computational complexity over the reals

1. Introduction

One of the main important theorems on the structureof NP-complete sets is due to Mahaney [14]. It statesthat no sparse NP-hard sets exist unless P= NP. Heresparseness denotes “small” sets in the following sense:let Σ be the finite alphabet{0,1}; a setS ⊂ Σ∗ issparse if and only if there exists a polynomialp whichbounds for alln ∈N the census function

c(n) := |S ∩Σn|from above.

∗ Corresponding author. Email: [email protected]. Par-tially supported by Fondecyt 1970398.

1 Email: [email protected]. Partially sup-ported by Fondap on Applied Mathematics.

Beside the classical approach to complexity pro-vided by Turing algorithms on finite alphabets there isan increasing interest in computability and complex-ity over the reals. This is due to the fact that for manycomputational problems real numbers come into playquite naturally. Just to mention a few examples con-sider some of the most important algorithms in “clas-sical” mathematics such as Newton’s method (see [3])or the investigation of neural networks which involvereal (for example, physical) constants in the neu-rons [13]. An appropriate computational model wasintroduced in 1989 by Blum et al. [5]. It investigatesa notion of uniform algorithms over the real numbers(as well as other structures). Real number complexityclasses PR and NPR of problems being deterministi-cally decidable respectively non-deterministically ver-ifiable in polynomial time by such an algorithm are

0020-0190/99/$ – see front matter 1999 Published by Elsevier Science B.V. All rights reserved.PII: S0020-0190(99)00130-1

84 M. Matamala, K. Meer / Information Processing Letters 72 (1999) 83–90

defined straightforwardly. The question of whether PRequals NPR turns out to be of major importance in thisframework as well. Moreover, also specific variantsof the Blum–Shub–Smale model have been studied.In particular, so called linear models provide (besidebeing interesting in their own) close relations to openproblems in the Turing setting (see [8]). For an intro-duction into the field confer [4]. A survey on recentresults is given in [18].

We suppose the reader to be familiar with the Blum–Shub–Smale (shortly: BSS) approach and adapt thecommonly used notations.

In discrete complexity theory, i.e., overΣ = {0,1},Mahaney’s theorem states that an NP-hard problemmost likely must have many elements.

Because of its importance in classical complexitytheory it is a natural task to look for a meaningfulnotion of sparseness in the BSS model as well. Therehas not been much work on this topic so far (accordingto our knowledge only two papers deal with such a realversion of sparseness, see [7,17]).

The general idea of sparseness is to reduce thevalue of a specific measure for sets by an exponentialfactor. Over finite alphabets this measure is taken tobe the number of elements. But what should be asignificant counterpart in a real setting? In [7] theauthors propose the topological dimension of a set.In general, ifS is a decision problem overR∞ :=⋃∞i=1Ri , i.e., S ⊂ R∞, the intersectionsS ∩ Rn can

have a topological dimension close ton. Thus in [7]a sparse set is defined to be a setS ⊂ R∞ for whichthese quantities are bounded by log(n) ∀n ∈ N. Forthis notion it is then shown the non-completenessof any sparse set in the additive BSS model withbranching on equality. This corresponds to Mahaney’stheorem since in the above setting P is known tobe different from NP (see [12,15]). Unfortunately,nothing is known about the computational behavior ofsets having a low dimension in more powerful variantsof the BSS model, i.e., if branching on inequalitiesis allowed. Here it is not clear whether this notionof sparse real sets can serve as an analogue to theclassical case.

In the present paper we propose to study anotherproperty of real number problems which is related tothe computational structure of the connected compo-nents a set has. This is different in spirit to classi-cal sparseness in that a computational condition enters

into consideration—the reason why we prefer not tospeak of sparseness but of well-structured sets. How-ever, as will be shown our approach shares many ofthe properties of sparse sets in the Turing frameworkeven for ordered BSS models. The latter includes Ma-haney like theorems and connections to exponentialtime classes for linear BSS machines as well as re-lations with polynomial generators in arbitrary BSSmodels. Moreover, reduced to the classical frameworkour notion of being well-structured captures the origi-nal sparseness notion.

Let us be a little more precise: Instead of thetopological dimension we start with another importanttopological measure; in general, any semi-algebraicset in Rn defined by a Boolean combination ofpolynomial equalities and inequalities such that allinvolved polynomials have a degree bounded by afixed number has a bounded number of connectedcomponents; the bound is simply exponential in thenumbern of variables (see, for example, [2,11] andthe literature cited there). Moreover, given two pointsx, y ∈ Rn together with such a polynomial system, itis decidable in simply exponential time inn whetherx and y belong to the same connected componentof the semi-algebraic set defined by the system [11].Since then-dimensional part of any decision problemin NP∆, where∆ stands for the full, ordered linearor unordered linear BSS model (see below), definesa semi-algebraic set we take the above observationas starting point and propose the following notionof well-structured problems overR: A set S ⊂ R∞is called well-structured iff there is a polynomialpwhich bounds the number of connected components ofS ∩ Rn ∀n ∈ N and the problem of checking whethertwo points belong to the same connected componentof S ∩Rn is semi-decidable in polynomial time (for aprecise definition see Section 2). This definition asksfor some comments and explanations which will begiven in Section 2.

The main result of this paper is an analogue ofMahaney’s theorem in the linear model with order.Some further aspects of our notion of well-structuredsets are discussed.

2. Basic notations; well-structured sets

Let R∞ be the disjoint union of all the spacesRn, n ∈N. For anyx ∈Rn let size(x) := n. A decision

M. Matamala, K. Meer / Information Processing Letters 72 (1999) 83–90 85

problem overR is a subsetT ⊂ R∞ and its n-dimensional partTn is given as

Tn := T ∩Rn.We consider linear restrictions of the BSS model,

i.e., we allow as operations addition, subtraction andscalar multiplication by a scalar belonging to a fixed,finite set of machine constants. Referring to orderfree linear machines (or linear machines branching onequality, respectively) we mean linear machines whichcan only perform equality branches; similarly the termordered linear machines (or linear machines branch-ing on inequality, respectively) refers to machines per-forming branches “x > 0?”. The corresponding ana-logues of complexity classes PR and NPR are denotedby P=lin,NP=lin,P

6lin , and NP6lin . For explicit definitions

and properties of all these classes see [6,9,12,15] andespecially [8] for relations to some of the major openproblems in classical structural complexity theory.

Given a Boolean combination of polynomial equali-ties and inequalities the corresponding semi-algebraicset has at most exponentially (inn) many connectedcomponents (see [2]); checking whether two givenpoints in the set belong to the same component canalso be done in simply exponential time with respectto the number of variables (see [11]). Note that for adecision problem in NP (in any of the above models)it is easy to reduceS to another problemS ∈NP suchthat S ∩ Rn only consists of a single connected com-ponent. Just “combine” all components ofS ∩ Rn byconsidering the problem inRn+1 via

(x1, . . . , xn, y) ∈ S :⇔ (x1, . . . , xn) ∈ S or y 6= 0.

However, this reduction may “destroy” the computa-tional structure of the connected components of theoriginal problem.

In order to obtain a meaningful notion we willcombine both the number of connected componentsand their structure to define well-structured problems.

Definition 1. Fix one of the above variants of the BSSmodel. A setS ⊂ R∞ is called well-structured (withrespect to the corresponding model) iff(a) there exists a polynomialp such that the number

of connected components ofSn := S ∩ Rn is atmostp(n) ∀n ∈N;

(b) the functionχ :R∞ × S ∪ S × R∞ → {0,1},defined by

χ(x, y) :=

1 x, y belong to the same

component ofS,

0 x, y belong to differentcomponents ofS,

0 x /∈ S, y ∈ S or vice versa

is computable in polynomial time (with respect tothe specific BSS model under consideration).

Our notion isdifferentfrom sparseness for discretesets. This is due to the presence of condition (b).Let us therefore comment on it. Firstly, note that wedo not ask for deciding whether an input belongsto R∞ × S ∪ S × R∞. If both arguments of aninput do not belong toS the corresponding algorithmfor χ can compute an arbitrary value—especially1. A slightly strange flavor of this condition mightbe the dependence on the machine model. We willdiscuss this issue at the end of the paper. However,condition (b) is natural with respect to the sparsenessnotion over finite alphabets. There it is hidden behindthe fact that equality is a polynomial time computablebinary relation. Indeed, a set being sparse in theclassical sense satisfies condition (b) by takingχ tobe the equality relation on(Σ∗)2 (the “components”are just single elements). One further substantiationto include condition (b) is given by the relation withnon-uniform computations its presence provides. Thefollowing result holds true as well in the full BSSmodel as in its linear variants. It shows that well-structured sets are exactly those which can be used asoracles to capture sets having polynomial generators(where the relation between polynomial generatorsand non-uniform, non-deterministic computations canbe established as usual, see [1]).

Proposition 1. Fix one of the above versions of theBSS model(i.e., full, linear order-free or linear or-dered). Let S denote the family of all well-structuredsets in the corresponding model. ThenNP /poly=⋃S∈SNPS , whereNP stands for NPR,NP=lin or

NP6lin, respectively and oracle computations as well asnon-uniform computations are defined as usual, see,for example,[19]. (In [19], non-uniformity is definedrelying on families of circuits which use a finite num-ber of constants only. In our setting, we need such cir-


cuits with a polynomially bounded number of real con-stants. However, the generalization of the former to thelatter is straightforward and therefore omitted here).

Proof. The proof is almost straightforward from thatof the corresponding result in the Turing setting [1].Given a non-uniform, non-deterministic computationin NP /poly by a family of algebraic circuits{Cn} ofpolynomial size (inn), these circuits can be describedby polynomially many real numbers. These reals canbe enlarged to vectors inRn constituting the setSn;each such vector represents a connected componentof a single point. Clearly,S =⋃n>1Sn satisfies re-quirement (a) of the definition of well-structuredness.It also satisfies condition (b) by takingχ as the equal-ity relation onR∞. TheNPS computation for a giveninputx guesses the constants of circuitCn (wheren=sizeR(x)), checks by consulting the oracleS whetherthe guessed numbers are the correct constants and thenperforms the computation ofCn.

Vice versa supposeS to be a well-structured setandM to be a non-deterministic polynomial time or-acle machine in the corresponding model. A polyno-mially sized circuit simulatingM for inputs fromRncan be obtained as follows: the advice the circuit uses(in form of real constants) will code a set of represen-tatives of each connected component ofSn (withoutloss of generality we assume the oracle only to askfor membership inSn, but the proof works as well formembership in anySi with i polynomially boundedin n). Whenever the machineM enters an oracle stateasking for somey to belong toS, in the circuit therewill be a subroutine included for checkingy to be-long to the same component as one of the representa-tives. This subroutine uses the polynomial time algo-rithm originating in condition (b) of the definition ofbeing well-structured. After a polynomial number ofsteps the circuit finds the correct answer and continuesin simulating machineM. 2Remark 1. In the Turing theory also

P/poly=⋃S∈SPS

holds, whereS denotes the set of all sparse sets. Thecorresponding relation⊃ is valid in our setting aswell. However, for showing the converse one usuallyhas to compute the advice function of a problem in

P /poly by means of a sparse oracle. Over the realsthe latter task can never be accomplishedno matterwhich notion of sparsenessone defines. This is dueto a simple transcendency argument when the familyof constants in the circuits{Cn} does not belong tothe field extension ofQ by finitely many reals (theconstants of the P-machine inPS).

3. Mahaney-like results for linear BSS machines

Throughout this section let us consider the linearBSS model. If no order is available then there existproblems in NP=lin not being reducible to a well-structured set. More explicitly, one can show

Theorem 1. The real Knapsack decision problem isnot polynomial time reducible to a well-structured setwithin the linear order-free BSS model.

This result is analogous to Mahaney’s original onesince in the linear order-free framework the separationP=lin 6= NP=lin is known to hold true and the Knapsackproblem is one example witnessing this fact (see [12,15]). We omit the proof here because lack of space.The main idea is to show that any reduction of theKnapsack problem to another decision problem has tobe an injective map on large parts of the input. Thiscontradicts the condition of being well-structured. Theproof is similar to that showing the main result in [7].

Now we consider the linear BSS model branchingon 6. Here it is not known whether P6lin 6= NP6lin .Thus we cannot expect to show non-existence of well-structured sets, since this would prove the formerseparation (obviously, there are well-structured sets inP6lin). Nevertheless, Mahaney’s theorem holds true.

Theorem 2. There exists anNP6lin -complete well-struc-

tured set if and only ifP6lin =NP6lin.

Proof. Assume a decision problemT ⊂ R∞ belong-ing to NP6lin is given; denote byM a correspondingnon-deterministic linear machine. LetS be a well-structured set such thatT is polynomial time reducibleby a reductionρ to S (in the linear model with or-der). Without loss of generality we assume for nota-tional simplicity thatsize(ρ(x)) = size(x) ∀x ∈ R∞.


The number of connected components ofS ∩ Rn isbounded from above by a polynomialp(n).

Suppose for the moment given an input dimensionn we would already know a set{x1, . . . , xk} ⊂ Rn ofk 6 p(n) many representatives for each component ofS ∩ Rn. Then for anyx ∈ Rn membership inT canbe decided by the application of a P6lin-algorithm to in-puts of dimensionn as follows: For eachxi, 16 i 6 k,compute the value ofχ(ρ(x), xi) whereχ as usual de-notes the function given by Definition 1 forS. Acceptx if and only if at least one of the outcomes equals 1.Due to the fact thatk is polynomially bounded insize(x) andχ is polynomially time computable the as-sertion follows.

Thus it remains to be shown how to obtain{x1, . . . ,

xk} in polynomial time. This in fact is the heart of theproof.

To do so for every input dimensionn a vectorI (n) ∈ {0,1}∗ is constructed. This “oracle” vectorcodes the information we are looking for. It consistsof three different parts and can be used to compute arepresentative set within a polynomial amount of time.Most important, the length ofI (n) is polynomiallybounded. Thus, using an idea given in [5], the entireset of information vectors{I (n), n ∈N} can be codedin a single “magic” real numbers. Given n and sthe informationI (n) can be obtained froms by apolynomial (inn) decoding process, thus providing auniform way to access the needed information.

Before going into details let us make two moreremarks. Firstly, we assumeρ to mapT onto S. Ingeneral, there could exist connected components ofS not containing any image pointρ(x). However,in such a situation one can replace without loss ofgeneralityS by ρ(T ) =: S ⊆ S. The setS is well-structured because it is a subset of a well structuredset. Secondly, it is known from [12] that in the linearmodel (general) non-determinism is already capturedby digital non-determinism, i.e., guessing bits insteadof real numbers is sufficient to obtain the accordingclass NP6lin . This will be useful in Step 2 below.

Step1. Suppose the input dimension to ben, i.e.,consider the reduction forx ∈ Rn. Let ρ(T ∩ Rn) =Sn ⊂ S and assumeSn to split into k connected com-ponentsC1, . . . ,Ck . Due to S being well-structuredwe knowk to be bounded by the value of some polyno-mial p(n). However, using the above mentioned ideaof coding a discrete set by a real number, in the linear

BSS models the census function ofS is computablein polynomial time. The numberk thus will representthe first part of information coded inI (n). Note thatk 6 p(n) and therefore can be expressed by polyno-mially (in fact logarithmically) many bits.

Step2. Next supposey1, . . . , yk to be elements ofC1, . . . ,Ck , respectively. To eachyi there correspondsa pre-image underρ. Our aim now is to include inI (n)all necessary information in order to compute pointsxisuch thatρ(xi) ∈Ci .

Let a point xi in T ∩ Rn being given such thatρ(xi) ∈ Ci . To xi there essentially correspond twocomputations; the non-deterministic machineM ac-ceptsxi for at least one specific binary guessv alongan accepting pathγ . Here, the above mentioned equal-ity of digital and general non-determinism is used.

For the subsequent arguments fix the guessv andnote that the setVγ of inputs branched alongγ isa convex, connected subset ofT ∩ Rn. Furthermorethere is the computation of the reduction valueρ(xi).Once again a computation pathδ as well as the setVδ of inputs branched alongδ can be attached toxi .Vδ is convex and connected as well, and because ofits polyhedral structure the same holds true for theintersectionVγ ∩ Vδ. For all connected componentsof Sn we collect the guessesv together with a digitalcoding ofγ andδ. This builds up the second part ofthe information vectorI (n).

Step 3. The proof is finished if we succeed tocompute a pointxi within the intersectionP :=Vγ ∩ Vδ . In fact, convexity ofP together with affinelinearity (and hence: continuity) ofρ on Vδ imply allpoints of P to be mapped into the same connectedcomponent ofS. Sincexi ∈ P this component isCi .

The computation of such a pointxi—provided somediscrete information is given—can be achieved usinga technique from [16]. We therefore refer to this paperand just recall the basic ideas briefly. The setP is apolyhedron, given by polynomially many inequalitiesand strict inequalities:

P = {x ∈Rn |A · x 6 b, C · x < d}.Note that all these inequalities can be explicitly ob-tained from the machineM, the polynomial reductionmachine and the information vectorI (n) (its secondpart). Note furthermore thatP is non-empty.

If P would be closed (i.e.,P = {x | A · x 6 b}),we could continue exactly as in [16]. There has to


Fig. 1. (a) Number of lowest-dimensional faces is at least three. (b) Number of lowest-dimensional faces is two. (c) Number oflowest-dimensional faces is one.

be a point inP on a face of lowest dimension, andall such points are solutions of a specific subsystemA · x = b of linear equations; these equations arise byreplacing the “6” in some of the original inequalitiesby an “=”. The information about which inequalitiesare active in order to obtainA · x = b can be codedby a vector of polynomially many bits. However, thepresence of strict inequalities enforces a little moresubtle reasoning.

We first proceed as above, this time for the closureP of P which is given by

P = {x ∈Rn |A · x 6 b, C · x 6 d}.Now the foregoing depends on the number of lowest-dimensional facesP has.

In case this number is at least three compute in asimilar way as described above three pointsz1, z2, andz3 on three different such faces. Now

z= 14 · (z1+ z2)+ 1

2 · z3

belongs toP .If there are only two faces of lowest dimension first

compute one point on each of them, sayz1 and z2.Sincez1 is laying on a face of lowest dimension ithas to belong to the intersection of at least two facesof higher dimension (if the two faces are parallel apoint in P is given by 1

2 · (z1 + z2)). At least one ofthese faces does not containz2. Compute any arbitrarypoint z3 on that face (possibly outsideP ). Afterwardscompute

z4 := 2 · z1− z3

which is also lying on that face. We obtain a point inP among one of the points

A= 14 · (z1+ z2)+ 1

2 · z3 and

B = 14 · (z1+ z2)+ 1

2 · z4.

Finally, if there is only one face of lowest dimensionandz1 denotes a point on it, the face must lay in theintersection of two other faces of higher dimension.On each of them we compute an arbitrary pointz1 andz2 together with its reflectionz′1 andz′2 with respect toz1. One of the lines combining the four correspondingpoints has its middle-point inP (see Fig. 1).

All the information which allows to obtain thecorrect equations of the important faces as well asa point on it can be coded in a polynomially sizedvector of bits. This vector (again collected for allcomponents) constitutes the third part ofI (n).

The proof is finished. 2

4. Concluding remarks; open problems

Beside Mahaney’s theorem there are numerous fur-ther results concerning the properties of sparse sets.In particular there is a strong relationship betweensparseness and the collapse of higher time complexityclasses in the Turing model. More explicitly, Hartma-nis [10] has shown that NEXP6= EXP if and only ifthere exists a sparse set in NP\ P.

In the linear ordered BSS model one can show

Theorem 3. If NEXP6lin = EXP6lin every well-struc-

tured set inNP6lin is in P6lin .

The proof once more mainly relies on the techniqueof coding a point in a polyhedron by a vector of digitsused in Theorem 2. The reverse implication remainsopen.

One important feature of Theorem 2 is its differ-ent proof in comparison with the one of Mahaney’soriginal theorem. Mahaney’s proof relies on the factthat there exists an NP-complete problem being self-reducible; it thus only works for self-reducible prob-


lems. Contrary, the proof of Theorem 2 shows thatany problem being reducible to a well-structured setin the ordered linear BSS model is polynomial timedecidable. It does not depend on a specific structure ofcomplete problems in NP6lin (the existence of completeproblems was established in [9]).

Even in the full BSS model Mahaney’s proof, ap-plied to our notion and to a self-reducible problem willwork. However, it is not at all known whether there ex-ist such problems being NPR-complete. We thus con-sider it to be important presenting a proof without aself-reducibility assumptions on the structure of theproblems.

As is already obvious from the above proof the cod-ing power of real numbers turns out to be extremelyhelpful. With respect to well-structured sets it providesanother effect which is unknown in the discrete set-ting. There it is just known that a sparse set in NP be-longs to the second levelL2 of Schöning’s low hier-archy (see [20,1]). It is not known whether such a setalready belongs to the first levelL1=NP∩ co-NP. Inreal models the situation is different. Since any func-tion from N to N is BSS computable this especiallyholds true for the census function of the number ofconnected components. Thus if any setS ⊂ R∞ iswell-structured and belongs to the complexity classNPR (or to the NP class in any other of the modelsconsidered above), its complementR∞ \ S belongs toNPR as well. This can be seen as follows: For an in-put x ∈ Rn first compute the censusp(n) of S ∩ Rn.Due to the well-structuredness ofS the censusp isa polynomial and thus the entire sequence of integersp(1),p(2)p, . . . can be coded in a single real numbers such that each valuep(n) can be decoded fromsin polynomial time with respect ton (see [5]). Next,guessp(n) many realsx1, . . . , xp(n) in S ∈NP6lin suchthat no two of them belong to the same component.The latter can be done in polynomial time according toS being well structured. Finally, acceptx if and onlyif χ(x, xi)= 0 ∀i ∈ {1, . . . , p(n)}. Thenx is acceptediff x ∈R∞ \ S.

Therefore, the assumption of the existence of aNPR-complete well-structured set will already implyNPR = co-NPR. However, there are of course unde-cidable well-structured sets and Proposition 1 provesour notion not being too strong.

Finally let us consider again condition (b) in Defin-ition 1.

A purely topological notion of sparseness over thereals was introduced in [7]. There, a Mahaney-likeresult was shown for the order-free linear case. Itseems to be not at all clear how to use such a purelytopological property for obtaining Mahaney’s theoremin models other than order-free linear machines, ifthere is not any additional “computable” informationavailable about the “sparse” object. As we have seenin the classical setting the latter intrinsically is givenin a hidden way. For the real models we believe thatincluding such an information could help to providemeaningful concepts of sparseness. In the case ofmeasuring the dimension of a set [7] such an additionalknowledge could address the structure of the semi-algebraic maps which relate a given semi-algebraicset of dimensiond to Rd . In [17] a specialization ofthe sparseness notion used in [7] and including suchan additional computability property is investigated torelate exponential time classes in the BSS model withthe complexity of the Boolean Satisfiability problemwithin real models.

Referring to the approach towards a sparseness-likenotion followed in the present paper condition (b)could play the role of such an information.

It remains an open question to study well-structuredsets in the full BSS model. This holds true as well withrespect to Mahaney’s theorem as to the relations withexponential time classes. It remains open as well tostudy whether there are closer relationships betweenour notion and the classical one beside the fact thatthey imply similar properties for the correspondingcomputational models.

Acknowledgment

This work was done while the second author wasvisiting the Universidad de Chile. The stay was madepossible as part of the FONDAP program in AppliedMathematics. We gratefully acknowledge this support.

References

[1] J.L. Balcázar, J. Diaz, J. Gabarró, Structural Complexity I,II, EATCS Monographs of Theoretical Computer Science,Vol. 11, Vol. 12, Springer, Berlin, 1988, 1990.

[2] R. Benedetti, J.J. Risler, Real Algebraic and Semi-AlgebraicSets, Hermann, Paris, 1990.


[3] L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and realcomputation: A manifesto, Internat. J. Bifurcation Chaos 6(1996) 3–26.

[4] L. Blum, F. Cucker, M. Shub, S. Smale, Complexity and RealComputation, Springer, Berlin, 1997.

[5] L. Blum, M. Shub, S. Smale, On a theory of computation andcomplexity over the real numbers: NP-completeness, recursivefunctions and universal machines, Bull. Amer. Math. Soc. 21(1989) 1–46.

[6] F. Cucker, P. Koiran, Computing over the reals with additionand order. Higher complexity classes, J. Complexity 11 (3)(1995) 358–376.

[7] F. Cucker, P. Koiran, M. Matamala, Complexity and Dimen-sion, Inform. Process. Lett. 62 (1997) 209–212.

[8] H. Fournier, P. Koiran, Are lower bounds easier over the reals?,in: Proc. Symposium on Theory of Computing, 1998.

[9] C. Gassner, On NP-completeness for linear machines, J. Com-plexity 13 (2) (1997) 259–271.

[10] J. Hartmanis, On sparse sets in NP-P, Inform. Process. Lett. 16(1983) 55–60.

[11] J. Heintz, M.F. Roy, P. Solernò, Single exponential path findingin semi-algebraic sets II: The general case, in: C.L. Bajaj (Ed.),Algebraic Geometry and its Applications, Springer, Berlin,1993, pp. 467–481.

[12] P. Koiran, Computing over the reals with addition and order,Theoret. Comput. Sci. 133 (1994) 35–47.

[13] W. Maass, Bound for the computational power and learningcomplexity of analog neural nets, in: Proc. 25th Symposiumon the Theory of Computing, 1995, pp. 335–344.

[14] S.R. Mahaney, Sparse complete sets for NP: Solution of aconjecture by Berman and Hartmanis, J. Computer SystemSci. 25 (1982) 130–143.

[15] K. Meer, A note on a P6=NP-result for a restricted class of realmachines, J. Complexity 8 (1991) 451–453.

[16] K. Meer, On the complexity of quadratic programming in realnumber models of computations, Theoret. Comput. Sci. 133(1994) 85–94.

[17] K. Meer, On the relations between discrete and continuouscomplexity theory, Math. Logic Quarterly 41 (1995) 281–286.

[18] K. Meer, C. Michaux, A survey on real structural complexitytheory, Bull. Belgian Math. Soc. Simon Stevin 4 (1997) 113–148.

[19] B. Poizat, Les Petits Cailloux, Aléas, Lyon, 1995.[20] U. Schöning, A low and an high hierarchy within NP, J. Com-

put. System Sci. 27 (1983) 14–28.

on the computational structure of the connected components of a hard problem

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