On the power of real Turing machines over binary inputs Felipe Cucker, Dima Grigoriev To cite this version: Felipe Cucker, Dima Grigoriev. On the power of real Turing machines over binary inputs. SIAM Journal on Computing, Society for Industrial and Applied Mathematics, 1997. hal-03049470 HAL Id: hal-03049470 https://hal.archives-ouvertes.fr/hal-03049470 Submitted on 9 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the p ower of real Turing machines over binary inputs Felipe Cucker Universitat Pompeu Fabra Balmes Barcelona SPAIN email cuckerupfes y Dima Grigoriev Depts of Comp Science and Mathematics Penn State University University Park PA USA email dimacsepsuedu In recent years the study of the complexity of computational problems involving real numbers has b een an increasing research area A founda tional pap er has b een where a computational mo del the real Turing machine for dealing with the ab ove problems was developed One research direction that has b een studied intensively during the last two years is the computational p ower of real Turing machines over binary inputs The general problem can b e roughly stated in the following way Let us consider a class C of real Turing machines that work under some resource b ound for instance p olynomial time branching only on equality etc If we restrict these machines to work on binary inputs ie nite words over f g they dene a class of binary languages D The question is what can we say ab out D dep ending on C More formally let us denote by IR the direct sum of countably many copies of IR and let P IR b e the set of its subsets Also let us denote by the subset f g of IR and as usual by the subset of IR consisting Partially supp orted by DGICyT PB the ESPRIT BRA Program of the EC under contracts no and pro jects ALCOM I I and NeuroCOLT y Partially supp orted by VolkswagenStiftung of those vectors whose comp onents are in Given any complexity class C P IR we dene its Bo olean part to b e the class of binary languages BPC fX X C g Our problem now can b e stated as given a complexity class of real sets C characterize BPC A p ossible origin of the problem is the recent interest in the computa tional p ower of neural networks The rst results characterized the p ower of nets with rational weights working within p olynomial time by showing that they compute exactly the sets in P cf The same problem was then considered for neural networks with real weights and it was shown that the p ower of these nets working within p olynomial time is exactly Ppoly cf and This latter problem considers in a natural way a setting in which an algebraic mo del having real constants op erates over binary inputs A next step was then taken by P Koiran who passed from a structured mo del the neural net to a general one the real Turing machine However he did not deal with the real Turing machine as it was introduced in but with a restricted version of it that can do only a mo derate use of mul tiplication namely all rational functions intermediately computed in the input variables as well as in the machines constants must have degree and co ecient size b ounded by the running time For this weak mo del he con sidered the class P of sets accepted in p olynomial time and he proved that W BPP Ppoly see W Subsequently several pap ers exhibited new results on Bo olean parts In it was shown that BPPAR PSPACEpoly where PAR is the class W W of subsets of IR decided in weak parallel p olynomial time Also for additive machines ie real Turing machines that do not p erform multiplications at all it was shown in that BPP Ppoly and that BPNP add add NPpoly Here P and NP denote the obvious classes but we recall that add add the nondeterministic guesses in this mo del are real numbers Moreover if the machines are orderfree ie they are required to branch only on equality NP P and that BPNP tests we now have that BPP add add These results were subsequently generalized in to all the levels of the p olynomial hierarchy constructed up on NP or NP add add None of the mentioned results was done for the unrestricted real Turing machine In fact for this case it was even asked whether it existed a subset of not b elonging to the BPP cf First steps in this direction IR were done in where it is shown that if we consider orderfree machines then we have the inclusion BPP BPP the class of sets decided by IR randomized machines in p olynomial time with b ounded probability error see ch as well as a p ositive answer to the question ab ove In fact if PH is the p olynomial hierarchy constructed up on NP the existence of IR IR binary languages not b elonging to BPPH and a fortiori nor to BPP IR IR was also proved in The aim of this pap er is to prove that BP PAR PSPACEpoly IR where PAR is the class of sets computed in parallel p olynomial time by IR ordinary real Turing machines As a consequence we obtain the existence of binary sets that do not b elong to the Bo olean part of PAR an extension IR of the result in since PH PAR and a separation of complexity IR IR classes in the real setting Some geometrical background In the rest of the pap er IN ZZ Q IR and C denote the sets of natural integer rational real and complex numbers resp ectively By IR we denote the alg real closure of Q ie the eld of all real algebraic numbers Also for any p olynomial f with integer co ecients we shall denote by jco ef j the maximal absolute value of its co ecients The aim of this section is to show how to nd real algebraic p oints in the connected comp onents of nonempty op en sets We closelly follow Thus let g g ZZX X and let N k k V fx IR g x g x g N b e an op en nonempty semialgebraic set For the rest of this section we consider d a b ound on the degree of each g and L b e a b ound for all i jco eg j i Q N g Let also d be the degree of Lemma Lemma Let g i i g and L jco eg j Then there exists a positive integer such that any connected component of V has a nonempty intersection with the ball B R where k 1 d 1 R L k Let us recall now see section that a p oint a IR is a critical k p oint for a function f IR IR when it satises f f a a X X k In this case the value b f a is said to b e a critical value of f In the case when f is a p olynomial function Sards lemma theoreme or implies that there are only a nite number of critical values of f This last fact was used in and in several subsequent pap ers to re duce the dimension of nonempty semialgebraic sets to zero avoiding thus cascading of pro jections in the algorithm for deciding emptiness of semial gebraic sets Let us now consider the p olynomials k X g R X i i and N Y G g g i i O (k ) d k O N 2 We have that deg G d N d and L jco eGj L Ld due to lemma The following result gives a b ound on the small critical values of G Lemma There exists a positive integer such that for every nonzero critical value a of G we have jaj C where k 2 d 2 C L Proof Let us consider the system of equations in the variables X X Z k G G G Z X X k k as well as its set of solutions S IR On any connected comp onent of S the co ordinate Z b eing the critical value of G is constant since G is continuous and due to Sards lemma Now since the degrees and the co ecients of the p olynomials app earing in this system are b ounded by d and O L d resp ectively if we apply the quantier elimination algorithm given in or along X X onto Z we get a nite set of p oints in k IR just the critical values such that each non zero one has absolute value greater than k 2 d 2 L 2 Remark In the preceding pro of the use of quantier elimination is not strictly necessary One can use instead the b ounds for the representative p oints from the connected comp onents of S given in the main theorem of Because of the preceding lemma we have that the algebraic set k W fx IR Gx C g is a nonsingular closed hypersurface with the prop erty that each con nected comp onent of V B R contains at least one b ounded connected comp onent of W cf Note that W do not intersect the b oundary of B R Now lemma of asserts the existence of integers v v n k d such that the system G v v G k G C k k X d k X d k k P k G k where has a nite number of solutions in IR Moreover i X i each of these solutions is an absolutely irreducible dimensional comp onent k of the variety in C given by this system of equations Due to Bezouts k inequality the number of real solutions is b ounded by d Besides cf lemma in each b ounded connected comp onent of W contains a p oint satisfying the system We can summarize the preceding results in the following theorem that will b e our main technical to ol in the next section Theorem Let g g ZZX X satisfy for every i N the N k bounds degg d and jco eg j L Then with the notations introduced i i k k above there are integers v v d such that the set W IR k dened by