Balcazar J., Daz J. and Gabarro J.: Structural complexity II. EATCS Monographs on Theoretical Computer Science 22, Springer-Verlag (1990).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with 8. Similarly, SAT k denotes the set of all QBF formulas with at most k 1 quantifier alternations starting with 9. As usual, in the case of k = 1 we omit the index and simply write SAT for SAT 1 and TAUT for TAUT 1 . We assume some familiarity with complexity theory and refer the reader to [2, 3] for standard notions and for the definition of complexity classes. A language A many one reduces to a language B (in symbols: A m B) if there is a polynomial time computable function f (in symbols: f 2 FP) such that for all strings x, x 2 A if and only if f(x) 2 B. We use Turing machines as ....

J. L. Balc azar, J. D az, and J. Gabarr o. Structural Complexity II. EATCS Monographs on Theoretical Computer Science #22. Springer-Verlag, 1990.

....with 8. Similarly, SAT k denotes the set of all QBF formulas with at most k 1 quantifier alternations starting with 9. As usual, in the case of k = 1 we omit the index and simply write SAT for SAT 1 and TAUT for TAUT 1 . We assume some familiarity with complexity theory and refer the reader to [2, 3] for standard notions and for the definition of complexity classes. A language A many one reduces to a language B (in symbols: A m B) if there is a polynomial time computable function f (in symbols: f 2 FP) such that for all strings x, x 2 A if and only if f(x) 2 B. We use Turing machines as ....

J. L. Balc azar, J. D az, and J. Gabarr o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science #11. Springer-Verlag, 1988.

....circuit family by an in nite symbol sequence. When the circuits are of exponential size (with respect to input word length and where circuit size is the number gates in a circuit) for each L a circuit family to decide L. For a more thorough introduction to circuits we refer the reader to [9]. ARNNs are nite size feedback rst order neural networks with real weights [6,7] The state of each neuron at time t 1 is given by an update equation of the form a ij x j (t) b ij u j (t) c i A ; i = 1; N (9) where N is the number of neurons, M is the number of inputs, ....

J. L. Balcazar, J. Daz, J. Gabarro, Structural Complexity, Vol. 1 of EATCS Monographs on Theoretical Computer Science, Springer-Verlag, Berlin, 1988.

.... result we will derive in this paper is already subsumed, we think that our proof technique is interesting for its own, and hence, we encourage the reader to continue reading 2 Preliminaries We follow the standard definitions and notations in computational complexity theory (see, e.g. BDG88, BDG91] We fix an alphabet Sigma = f0; 1g. For any set X Sigma , we denote the complement of X as X = Sigma Gamma X. Natural numbers are encoded in Sigma by using their binary representation. For any string x, let jxj denote the length of x, and for any set X, let jj X jj denote the ....

J. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity II. EATCS Monographs on Theoretical Computer Science, SpringerVerlag (1991).

.... the main result we will derive in this paper is already subsumed, we think that our proof technique is interesting for its own, and hence, we encourage the reader to continue reading 2 Preliminaries We follow the standard definitions and notations in computational complexity theory (see, e.g. BDG88, BDG91] We fix an alphabet Sigma = f0; 1g. For any set X Sigma , we denote the complement of X as X = Sigma Gamma X. Natural numbers are encoded in Sigma by using their binary representation. For any string x, let jxj denote the length of x, and for any set X, let jj X jj denote the ....

J. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science, SpringerVerlag (1988).

....extend the techniques from Section 3 to study the classes NP= Mod NP[k] m . We also consider P evaluators and show equation (1) 2 Preliminaries We follow standard definitions and notations in computational complexity theory. Readers are referred to a standard reference (see, e.g. HU79] or [BDG88] for the definitions of common notations and concepts such as alphabets, strings, languages, Turing machines, polynomial time bounded computation, and nondeterminism. Throughout this paper, we use the alphabet Sigma = f0; 1g. If A is a set, we use A( Delta) to denote the characteristic function ....

J. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I. EATCS Monographs in Theoretical Computer Science. SpringerVerlag, 1988.

....reduction doesn t respect the structure of the solution spaces of the instances that are mapped to each other. It just guarantees the existence nonexistence of solutions. However, looking at the many one reductions of the NP completeness proofs in the standard textbooks (see for example in [BDG88, HU79]) we know that all the known NP complete sets in fact share the above described property with SAT. Let R be an NP relation such that DR is NP complete. We call R witness preserving complete, if, for any NP relation R 0 , there exist functions g, h 2 FP such that h is a many one reduction from ....

J. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I. EATCS Monographs in Theoretical Computer Science. SpringerVerlag, 1988.

....(in some cases) for analyzing relationships among counting classes. In Section 4, we provide such results with respect to closure properties median, plurality, and maximum. 2. Preliminaries In this paper, we follow standard definitions and notations in computational complexity theory (see, e.g. [BDG88, BDG91]) Throughout this paper, we fix our alphabet to Sigma = f0; 1g; by a string we mean an element of Sigma , and by a language we mean a subset of Sigma . Natural numbers are encoded in Sigma in an ordinary way, and let N denote the set of (encoded) natural numbers. For any string x, ....

J. Balc'azar, J. D'iaz, and J. Gabarr'o, Structural Complexity II, EATCS Monographs on Theoretical Computer Science, Springer-Verlag (1991).

....(in some cases) for analyzing relationships among counting classes. In Section 4, we provide such results with respect to closure properties median, plurality, and maximum. 2. Preliminaries In this paper, we follow standard definitions and notations in computational complexity theory (see, e.g. [BDG88, BDG91]) Throughout this paper, we fix our alphabet to Sigma = f0; 1g; by a string we mean an element of Sigma , and by a language we mean a subset of Sigma . Natural numbers are encoded in Sigma in an ordinary way, and let N denote the set of (encoded) natural numbers. For any string x, ....

J. Balc'azar, J. D'iaz, and J. Gabarr'o, Structural Complexity I, EATCS Monographs on Theoretical Computer Science, Springer-Verlag (1988).

....locating such, somehow involved classes, in the, much simpler defined, Boolean Hierarchy are interesting enough in its own and might find further applications in other settings. 2 Preliminaries We follow standard definitions and notations in computational complexity theory (see, e.g. HU79] or [BDG88] Throughout this paper, we use the alphabet Sigma = f0; 1g. For a predicate P , let [P ] denote 1 if P is true, and 0 if P is false. P (NP) denote the classes of languages that can be recognized by a polynomial time deterministic (nondeterministic) Turing machine. FP is the class of ....

J. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I. EATCS Monographs in Theoretical Computer Science. Springer-Verlag, 1988.

....with the help of a technical result established in section 4. Corollary 1 If PW = NPW , the polynomial hierarchy collapses at the second level. Proof. According to Theorem 4, PW = NPW = NP ae P=poly since NP ae BP(NPW ) If NP ae P=poly, the polynomial hierarchy collapses at the second level [2]. As was mentioned in the introduction, it was recently shown in [12] that PW 6=NPW holds without any assumption. The decoding argument of Theorem 4 also shows that all boolean languages can be recognized in weak exponential time. Theorem 5 BP(EXPW ) 2 f0;1g . Proof Sketch. The truth table ....

J.L. Balc'azar, J. Di'az, and J. Gabarr'o. Structural Complexity II. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1990.

.... states that the class of boolean languages recognizable in polynomial time in the weak BSS model is P=poly (P=poly is the class of sets recognized in polynomial time by Turing machines using advice functions of polynomial length or, equivalently, by non uniform boolean circuits of polynomial size [1]) The main tool for this proof is a new result on semi algebraic sets of independent interest, stating that a semi algebraic set of non empty interior defined by small equations must contain a small rational point. It follows from this characterization that P6=NP in the weak model under a ....

....= W ) P. In the full BSS model, we are only able to prove the following weaker result. Theorem 9 BP(P = R ) ae BPP: Recall that BPP is the class of (boolean) languages that can be recognized in polynomial time by a probabilistic Turing machine with a bounded probability of error (see, e.g. [1] for more details) Before moving to the proof of Theorem 9, we present two lemmas which are often used in the analysis of randomized algorithms (see [15] for a proof of the first one) Order free models 17 Lemma 7 (Schwartz) Let P (x 1 ; x p ) be a polynomial of degree d. Choose the x i ....

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J.L. Balc'azar, J. Di'az, and J. Gabarr'o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

....denote sets. A set S is called sparse if jS n j is bounded above by a polynomial. To encode pairs (or tuples) of strings we use a standard polynomial time computable pairing function denoted by h Delta; Deltai whose inverses are also computable in polynomial time. Following standard notation [BDG], we denote the complexity classes DTIME(2 n O(1) NTIME(2 n O(1) of languages accepted by (non)deterministic Turing machines in time 2 poly by EXP (resp. NEXP) The corresponding 2 linear time bounded classes are denoted by E and NE. Further, we denote the class of functions ....

....properties hold: ffl for all inputs x 2 L, Prob[M A (x) 1] 3=4, one sided helping) ffl for all inputs x 62 L and all oracles B, Prob[M B (x) 1] 1=4. robustness) For other standard definitions used in the paper we refer the reader to a standard book on structural complexity theory [BDG]. The next theorem states that BPP 1 Gammahelp exactly characterizes MIP, where the complexity of the helping oracle corresponds to the prover complexity in the MIP model. Theorem 2.6 [FRS88] For all sets A: BPP 1 Gammahelp (A) MIP[A] The following proposition shows that, interestingly, for ....

J.L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I,II. EATCS Monographs on Theoretical Computer Science, Springer Verlag, 1988.

....2 Preliminaries Let = f0; 1g. We denote the cardinality of a set X by kXk and the length of a string x 2 by jxj. The characteristic function of a language L is denoted by L . The de nitions of standard complexity classes like P, NP, E, EXP etc. can be found in standard books [7, 22]. A relativized complexity class C with oracle A is denoted by either C A or C(A) Likewise, we denote an oracle Turing machine M with oracle A by M A or M(A) For a class C of sets and a class F of functions from 1 to , let C=F [15] be the class of sets A such that there is a set B ....

.... Likewise, N D is overproductive if for each x we have set N D (x) 6= and N is said to be robustly overproductive if for each oracle D and input x we have set N D (x) 6= With standard arguments we can convert a sparse set into a polynomialsize advice string and vice versa (see, e.g. [7]) It follows that A 2 NP=poly co NP=poly if and only if there is a sparse set S and a nondeterministic machine N such that N S is both overproductive and underproductive and A = L(N S ) Similarly, A 2 (NP co NP) poly if and only if there is a sparse set S and a nondeterministic machine N ....

J. L. Balcazar, J. Daz, and J. Gabarro. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, second edition, 1995.

....constant of f . Hence p = k n 1 is sufficient. The converse is obvious. In order to deal with arbitrary real numbers, we need to consider the class C=poly for each class C defined above. The functions in these classes are computed by Turing machines using polynomial advice funtions [1]. Contrary to oracles, the value of the advice function does not depend fully on the input of the Turing machine, but only on its length. For instance, f 2 PE d =poly 4 if there is a function A : N f0; 1g (the advice function) and a Turing machine T such that on the input (p; x; A(p ....

....class P=poly. It will be shown that this is exactly the class of sets that can be recognized in polynomial time by iterated functions. A useful characterization of P=poly is the following. f : f0; 1g f0; 1g is in P=poly if and only if it can be computed by boolean circuits of polynomial size [1], i.e. if there is a family of circuits (C n ) n1 such that the size of C n is polynomial in n, and C n computes f jf0;1g n. It can be easily seen that there are non computable functions is P=poly, since the circuit family (C n ) n2N is not supposed to be recursive. Conversely, some computable ....

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J.L. Balc'azar, J. Di'az, and J. Gabarr'o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

....stack automaton, and since discrete two stack automata can simulate Turing machines [10] analog automata are able to simulate Turing machines. The exact computational power of analog automata is given by the following theorem (for the definition of the complexity classes P=poly and NP=poly, see [5]) Theorem 2.1 ffl Every language L ae f0; 1g can be recognized by a deterministic analog two stack automaton in exponential time. ffl The languages L ae f0; 1g accepted by deterministic (respectively: non deterministic) analog two stack automata in polynomial time are exactly the ....

....kp(n) the p(n) first letters of each of the k advices of M , and then simulates M . Hence the computational power of analog automata is bounded by P=poly. 4 Let L be a language in P=poly. By definition, L is recognized by a Turing machine M 0 with an advice function f : N f0; 1g (see [5]) We can construct a word fl of infinite length as the concatenation, with delimiters, of f(1) f(2) etc: In order to recognize L, an analog automaton M , on input w 2 f0; 1g , first makes advice fl appear. Then M seeks in fl the value of f(jwj) This operation can be done in polynomial ....

J. L. Balc'azar, J. Di'az, and J. Gabarr'o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

....n O(1) 2. Ba92] Every word decreasing self reducible set is in DTIME(2 O(n) 3. BBS86] Every polynomially related self reducible set is in PSPACE. For other standard definitions used in the paper we refer the reader to standard books on structural complexity theory (for example [BDG, Sch86] 3 Simple descriptions for deterministic reduction classes In this section we will investigate simplicity properties of the classes IC[log; poly] R p hd (R p c (SPARSE) and R p d (SPARSE) The inclusion relations between these classes are represented in Figure 1 (see [Ko89, ....

J.L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science, SpringerVerlag, 1988.

....Let Sigma = f0; 1g. We denote the cardinality of a set X by kXk and the length of a string x 2 Sigma by jxj. The characteristic function of a language L Sigma is denoted by L . The definitions of standard complexity classes like P, NP, E, EXP etc. can be found in standard books [7, 22]. A relativized complexity class C with oracle A is denoted by either C A or C(A) Likewise, we denote an oracle Turing machine M with oracle A by M A or M(A) For a class C of sets and a class F of functions from 1 to Sigma , let C=F [13] be the class of sets A such that there is a set B ....

.... Likewise, N D is overproductive if for each x we have set N D (x) 6= and N is said to be robustly overproductive if for each oracle D and input x we have set N D (x) 6= With standard arguments we can convert a sparse set into a polynomial size advice string and vice versa (see, e.g. [7]) It follows that A 2 NP=poly co NP=poly if and only if there is a sparse set S and a nondeterministic machine N such that N S is both overproductive and underproductive and A = L(N S ) Similarly, A 2 (NP co NP) poly if and only if there is a sparse set S and a nondeterministic machine N ....

J. L. Balc' azar, J. D' iaz, and J. Gabarr' o, Structural Complexity I, EATCS Monographs on Theoretical Computer Science. Springer-Verlag, second edition, 1995.

....flavour. However, in saturated structures non uniformity can be dropped. Thus, in domains like Z or C the results following from Ladner and our first approach can be recaptured. We assume the reader to be aware of the BSS model (see [5] 4] 14] as well as classical complexity theory (see [7] [2]) This paper summarizes in a shortened way ideas presented in [13] and [3] Interested readers are refered to these works for more explicit presentations 2 Diagonal problems for NP C n P C In this section we will show the following analogue of Ladner s result over the complex numbers. Theorem ....

J.L. Balc'azar, J. Diaz, J. Gabarr'o : Structural Complexity I. EATCS Monographs on Theoretical Computer Science 11, Springer, Berlin, 1988

....form g(Z 1 ; Z k ) 0 we use the same algorithm of step (s5) for determining the sign of g(Z 1 ; Z k ) Y on the point coded in (s5) The above considerations show that the described algorithm runs in parallel polynomial time. Since this is equivalent to polynomial space ( 6] [2] ch.4) we have shown that the set decided by the algorithm above belongs to PSPACE poly. 2 3 Binary inputs for parallel real Turing machines Our next goal is to extend our previous result to the class PAR IR of sets decided in parallel polynomial time. We recall from [9] the definition of a ....

J.L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity II. EATCS Monographs on Theoretical Computer Science, 22. SpringerVerlag, 1990.

.... to the BP(P IR ) cf. 13] First steps in this direction 2 were done in [20] where it is shown that if we consider order free machines then we have the inclusion BP(P = IR ) BPP (the class of sets decided by randomized machines in polynomial time with bounded probability error, see [1] ch. 6) as well as a positive answer to the question above. In fact if PH IR is the polynomial hierarchy constructed upon NP IR , the existence of binary languages not belonging to BP(PH IR ) and a fortiori nor to BP(P IR ) was also proved in [20] The aim of this paper is to prove that BP(PAR ....

....case F = poly, the class of functions f such that for some polynomial p we have jf(n)j p(n) for each n 2 IN. For the boolean case, one can find the main properties and characterizations of classes like P poly or PSPACE poly (as well as of some other non uniform complexity classes) in chapter 5 of [1]. Theorem 2 The inclusion BP(P IR ) PSPACE poly holds. 6 Proof. Let M be a RTM working in polynomial time, say n q , and let ff 1 ; ff k be its real constants. For any n 2 IN the machine M has an associated algebraic computation tree TM;n having depth n q and size bounded by 2 ....

J.L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science, 11. Springer-Verlag, 1988. 14

....(in some cases) for analyzing relationships among counting classes. In Section 4, we provide such results with respect to closure properties median, plurality, and maximum. 2. Preliminaries In this paper, we follow standard de nitions and notations in computational complexity theory (see, e.g. [BDG88, BDG91]) Throughout this paper, we x our alphabet to = f0; 1g; by a string we mean an element of , and by a language we mean a subset of . Natural numbers are encoded in in an ordinary way, and let N denote the set of (encoded) natural numbers. For any string x, let jxj denote the ....

J. Balcazar, J. Daz, and J. Gabarro, Structural Complexity II, EATCS Monographs on Theoretical Computer Science, Springer-Verlag (1991).

....(in some cases) for analyzing relationships among counting classes. In Section 4, we provide such results with respect to closure properties median, plurality, and maximum. 2. Preliminaries In this paper, we follow standard de nitions and notations in computational complexity theory (see, e.g. [BDG88, BDG91]) Throughout this paper, we x our alphabet to = f0; 1g; by a string we mean an element of , and by a language we mean a subset of . Natural numbers are encoded in in an ordinary way, and let N denote the set of (encoded) natural numbers. For any string x, let jxj denote the ....

J. Balcazar, J. Daz, and J. Gabarro, Structural Complexity I, EATCS Monographs on Theoretical Computer Science, Springer-Verlag (1988).

....(shown in [Ha87] that an oracle can be constructed relative to which all the levels of the polynomial time hierarchy are distinct, finally achieving the original goal of the Furst Saxe Sipser work. For more on oracles and relativized complexity, see the book by Balcazar, Diaz and Gabarro [BDG88], and its forthcoming sequel) The method of restrictions does not appear to extend to circuits with more general kinds of gates. For example, if a restriction is applied to an exclusive OR (i.e. a MOD 2 gate) a smaller gate of the same type results. A gate is not killed as in the AND or OR ....

J. L. Balc'azar, J. Diaz, and J. Gabarr'o, Structural Complexity I, EATCS Monographs on Theoretical Computer Science 11 (Berlin: Springer-Verlag, 1988).

....structures one can give a partial converse to this transfer theorem. In particular, it is known that the boolean part of P C is included in BPP (the boolean part of PM for the structure M = R; is included in BPP by Theorem 9 of [12] and the proof for C is identical) Since BPP P=poly [1] the collapse B 2 C = B 2 C would imply 2 =poly = 2 =poly. We conclude that the separation B 2 C 6= B 2 C is most likely true, but extremely hard to prove. The collapse B 2 R = B 2 R also seems highly unlikely, but we cannot point to such a dramatic consequence as 2 ....

J.L. Balcazar, J. Daz, and J. Gabarro. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

.... R p T (SPARSE) 19,16, 10,18] and in terms of the robustness of R p T (SPARSE) R p T (SPARSE) is indeed quite robust; in addition to its characterization in terms of small circuits, R p T (SPARSE) is easily noted equivalent to R p T (TALLY) R p tt (TALLY) and R p tt (SPARSE) see [6]) Nonetheless, Book and Ko showed that there were limits to the robustness of R p T (SPARSE) they initiated the study of the classes of languages reducible to sparse (and tally) sets under various weak notions of polynomial time reducibility, and proved that such classes di ered both from R p ....

J. Balc azar, J. D az, and J. Gabarr o, Structural Complexity I, EATCS Monographs in Theoretical Computer Science, Springer-Verlag, 1988.

....satis able propositional formulas suitably encoded as strings in . Similarly, let TAUT denote the language of propositional tautologies. For the de nitions of standard complexity classes like P, NP, coNP, NEXP, the polynomial hierarchy PH etc. we refer the reader to a standard textbook, e.g. [1]. The class p 2 (cf. 6] also described as P NP [O(log n) consists of all sets decidable by a polynomial time oracle machine that can make O(log n) queries to an NP oracle. We start with a de nition of proof systems that is equivalent to the original Cook Reckhow formulation [4] De ....

J. L. Balc azar, J. D az, and J. Gabarr o, Structural Complexity I, EATCS Monographs on Theoretical Computer Science. Springer-Verlag, second edition, 1995.

....considered as an n ary boolean function that we denote by L =n . Conversely, each n ary boolean function g defines a finite language fx 2 Sigma n j g(x) 1g that we denote by L g . The definitions of complexity classes we consider like P, NP, AM, E, EXP etc. can be found in standard books [BDG95, BDG90, Pap94]. A set A f0g is called tally (A 2 Tally for short) By log we denote the function log x = maxf1; dlog 2 xeg and h Delta; Deltai denotes a standard pairing function. For a string x 2 Sigma , num(x) denotes the natural number whose binary representation is given by 1x, For a class C of ....

J. L. Balc' azar, J. D' iaz, and J. Gabarr' o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, second edition, 1995.

....considered as an n ary boolean function that we denote by L =n . Conversely, each n ary boolean function g defines a finite language fx 2 Sigma n j g(x) 1g that we denote by L g . The definitions of complexity classes we consider like P, NP, AM, E, EXP etc. can be found in standard books [BDG95, BDG90, Pap94]. A set A f0g is called tally (A 2 Tally for short) By log we denote the function log x = maxf1; dlog 2 xeg and h Delta; Deltai denotes a standard pairing function. For a string x 2 Sigma , num(x) denotes the natural number whose binary representation is given by 1x, For a class C of ....

J. L. Balc' azar, J. D' iaz, and J. Gabarr' o. Structural Complexity II. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1990.

....classes, their containments in the Extended Low Hierarchy (defined by [BBS86] and the lowness results in case of self reducibility as shown in this paper. 2 Preliminaries and notation In general the notations and definitions from the standard books on structural complexity theory (for example, BDG, Sch86] are used. Let A be a set. A =n (A n ) denotes the set of all strings in A of length n (up to length n, respectively) The cardinality of A is denoted by jAj. A set S is called sparse if for some polynomial p and every n jA n j p(n) We use SPARSE and co SPARSE to represent the ....

....and invertible pairing function. L(M ) is the set accepted by the Turing machine M , L(M;A) is the set accepted accepted by the oracle Turing machine M using the set A as oracle. The reducibilities discussed in this paper are standard polynomial time bounded reducibilities defined (see [BDG] and the Hausdorff reducibility p hd by Wagner [Wag87] At first the definitions of some of the non standard reducibilities are repeated. They are generalizations of the many one reducibility p m and restrictions of the truth table reducibility p tt . Definition 2.1 A set A is ....

J.L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science, Springer Verlag, 1988.

....formulas suitably encoded as strings in Sigma . Similarly, let TAUT denote the language of propositional tautologies. For the definitions of standard complexity classes like P, NP, coNP, Theta p 2 , NEXP, the polynomial hierarchy PH etc. we refer the reader to a standard textbook, e.g. [1]. We start with a definition of proof systems that is equivalent to the original CookReckhow formulation [4] Definition 1 [2] A (sound and complete) propositional proof system S is defined to be a polynomial time predicate S such that for all F , F 2 TAUT , 9p : S(F; p) 2 In other words, a ....

J. L. Balc' azar, J. D' iaz, and J. Gabarr' o, Structural Complexity I, EATCS Monographs on Theoretical Computer Science. Springer-Verlag, second edition, 1995.

....of a finite set X by jjX jj. Let [n] denote the initial segment f1; 2; Delta Delta Delta ; ng of the set of natural numbers N. We use basic complexity theoretic concepts like many one reducibility, truth table reducibility, and interactive proof systems defined in standard textbooks like [3, 18]. A useful reducibility that is not standard is the NC truth table reducibility: 2 Definition 3 For two sets, A; B Sigma , we say that A is NC truth table reducible to B, A NC tt B) if A can be computed by a uniform family of NC circuits with query gates for B, with the additional ....

J. L. Balc' azar, J. D' iaz, J. Gabarr' o, Structural Complexity I & II, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1989.

..... If S q 1 i=1 (E t i ) K q then one of the translates of E must be dense in K q . This implies that E is dense, too. The three theorems of section 4. 2 are adaptations to the BSS model of classical theorems of complexity theory (BPP P=poly, RP P=poly and BPP Sigma 2 ) See e.g. [2] for the classical theory and [7, 11] for adaptations of these results to the BSS model of computation over the reals. 11 4.3 Construction in Characteristic 0 In this subsection we assume that the algebraically closed field K is of characteristic 0. We will see in Theorem 4.11 that it is ....

J.L. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

....more complicated and can be found in [BHT89] Proof. We first show this result for the class NEXP and then use a simple padding argument to obtain it for NE. Let K be a polynomial time paddable p m complete set for NE. It is easy to see (for example see Balc azar, D iaz, and Gabarr o [BDG88]) that K is p m complete for NEXP as well. The set B will be constructed so that its only elements are of the form (e; x; l; i) i = 0 or i = 1. B will be complete via the p 2 Gammatt reduction: e; x; l) 2 K [ e; x; l; 0) 2 B (e; x; l; 1) 2 B] To ensure that B is not p m complete ....

J. L. Balc'azar, J. D'iaz and J. Gabarr'o. Structural Complexity I. W. Brauer, G. Rozenberg and A. Salomaa (eds.) EATCS Monographs on Theoretical Computer Science 11, Springer Verlag, 1988.

....machines has been investigated in [10, 11] It has been proved that the discrete languages recognized in polynomial time are precisely the languages belonging to the complexity class P=poly, and that any discrete language L ae N can be recognized in exponential time. For a definition of P=poly see [4]. Note that linear machines with rational constants are equivalent to Turing machines. 2.5 Arithmetical hierarchy We recall: Definition 2.7 (Arithmetical hierarchy [17] The classes Sigma k ; Pi k ; Delta k , for k 2 N, are defined inductively by: ffl Sigma 0 is the class of the languages ....

J. L. Balc'azar, J. Di'az, and J. Gabarr'o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science, 1988.

....locating such, somehow involved classes, in the, much simpler defined, Boolean Hierarchy are interesting enough in its own and might find further applications in other settings. 2 Preliminaries We follow standard definitions and notations in computational complexity theory (see, e.g. HU79] or [BDG88] Throughout this paper, we use the alphabet Sigma = f0; 1g. P (NP) denote the classes of languages that can be recognized by a polynomial time deterministic (nondeterministic) Turing machine. FP is the class of polynomial time computable total functions. The Boolean hierarchy is the closure ....

J. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I. EATCS Monographs in Theoretical Computer Science. Springer-Verlag, 1988.

....central facts about these hierarchies to non uniform computations, reductions to sets of subexponential density and in particular to small circuits. We now briefly review the definitions of these non uniform complexity measures. For more detailed definitions see Balc azar, D iaz and Gabarr o [4]. Definition 2.2 An advice function is a function f : IN Sigma . Let C be a complexity class and F be a class of advice functions. The class C=F is the collection of all sets A such that for some B 2 C and some f 2 F , A = fxj x; f(jxj) 2 Bg. In this paper we will mainly consider three ....

....(x) is one where the state, the tape contents and the place of the head change accordingly M e s transition relation. Note that in order to check one bit of c j one only needs to consider a constant number of bits of c j and c j Gamma1 . For a more detailed description of this see, for example, [4]. As in the proof of Theorem 4 we need to define a complete set for EXP NP . We use almost the same set U as in theorem 4. Define U 0 to be the following set: U 0 = f e; x; t; j; k; b jthe k th bit of the j th configuration of M e (x) s leftmost accepting path of length t equals bg A ....

Balc'azar J.L., J. D'iaz & J. Gabarr'o. Structural Complexity I . W. Brauer, G. Rozenberg & A. Salomaa (eds.) EATCS Monographs on Theoretical Computer Science 11 (1988) Springer Verlag.

....machine which on input n constructs Cn in time polynomial in n. This de nition makes sense since a parameter free circuit is a purely boolean object. To be completely precise one should specify how circuits are encoded in binary words [35] there is no signi cant di erence with the classical case [1]. This complexity class can also be de ned with Turing machines over M instead of circuits [16, 35] In the case where M is a ring, these Turing machines are equivalent to Blum Shub Smale machines [5] The class NPM of non deterministic polynomial time problems is obtained from PM in the same way ....

....positive answer for many structures of interest. For instance, as pointed out before, in the standard structure M = f0; 1g any boolean function of n variables can be computed by a tree of depth n, but a simple counting argument shows that most boolean functions have exponential circuit complexity [1]. Bounds on the number of consistent sign conditions ( a la Thom Milnor, see end of section 5.1) show that most boolean functions also have exponential circuit complexity over R and C , even if arbitrary real or complex parameters are allowed [28] It is nevertheless possible to construct a ....

J.L. Balcazar, J. Daz, and J. Gabarro. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1988.

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Balcazar J., Daz J. and Gabarro J.: Structural complexity II. EATCS Monographs on Theoretical Computer Science 22, Springer-Verlag (1990).

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J.L. Balcazar, J. Daz, and J. Gabarro. Structural Complexity I/II. EATCS Monographs on Theoretical Computer Science. Springer Verlag, 1988.

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Balc azar, J. L., D az, J., and Gabarr o, J. Structural complexity II. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1990.

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Balc azar, J. L., D az, J., and Gabarr o, J. Structural complexity I, second revised ed. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1995. 108

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J. L. Balcazar, J. Daz, and J. Gabarro. Structural Complexity II. EATCS Monographs on Theoretical Computer Science #22. Springer-Verlag, 1990.

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J. L. Balcazar, J. Daz, and J. Gabarro. Structural Complexity I. EATCS Monographs on Theoretical Computer Science #11. Springer-Verlag, 1988.

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J. Balcazar, J. Daz, and J. Gabarro. Structural Complexity I. EATCS Monographs in Theoretical Computer Science. Springer-Verlag, 1988. 2nd edition 1995.

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J. L. Balc azar, J. D az, J. Gabarr o, Structural Complexity I, EATCS Monographs on Theoretical Computer Science, Springer-Verlag, 1989.

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J. L. Balc azar, J. D az, and J. Gabarr o. Structural Complexity I. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, second edition, 1995.

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J. L. Balc azar, J. D az, and J. Gabarr o. Structural Complexity II. EATCS Monographs on Theoretical Computer Science. Springer-Verlag, 1990.

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J. L. Balcazar, J. Daz, and J. Gabarro, \Structural Complexity I", EATCS Monographs on Theoretical Computer Science. Springer-Verlag, second edition, 1995.

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J. Balc'azar, J. D'iaz, and J. Gabarr'o. Structural Complexity I & II. EATCS Monographs on Theoretical Computer Science, Springer-Verlag (

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