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Are Cook and Karp Ever the Same?

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Are Cook and Karp Ever the Same? Are Cook and Karp Ever the Same? Richard Beigel∗ Lance Fortnowy Temple University NEC Laboratories America Abstract relativized world where there is a sparse set that collapses. This is in fact the first relativized world in which any set has We consider the question whether there exists a set A been shown to collapse. such that every set polynomial-time Turing equivalent to A In Section 2 we give some background and formal defi- is also many-one equivalent to A. We show that if E = nitions of our problem. Section 4 shows that no sparse set NE then no sparse set has this property. We give the first collapses unless E = NE. Section 5 describes the rela- 6 relativized world where there exists a set with this property, tivized world giving a sparse collapsing set. and in this world the set A is sparse. 2 Preliminaries p 1 Introduction We say a set A m B (“A many-one reduces to B”) if there exists a polynomial-time≤ computable function f such that x A f(x) B and a set A p B (“A is many- The seminal papers of Cook [Coo71] and Karp [Kar72] m one equivalent2 , to B”)2 if A p B and≡B p A. Similarly define NP-completeness differently based on the kind of m m a set A p B (“A Turing≤ reduces to B”)≤ if there exists a reductions used. Cook allows Turing reductions, where one T polynomial-time≤ oracle Turing machine M such that A = can decide an instance of one problem by asking an arbi- L(M B), and a set A p B (“A is Turing equivalent to B”) trary collection of instances of a second problem. Karp al- T if A p B and B p≡A. lows only the more restrictive many-one reductions, which ≤T ≤T require mapping an instance of one problem into a single in- Hypothesis 2.1 (Collapse) There exists an A not in P such stance of the second. Computational complexity theory has p p that for all B, if A B then A m B. often looked at the relationship between these two notions ≡T ≡ of reducibility. We call a set A fulfilling Hypothesis 2.1 a collapsing set. In this paper, we will focus on a fundamental question The condition that A is not in P is not strictly necessary— along these lines: Does there exist a set A such that all sets for any A = in P, is Turing equivalent to A, but is 6 ; ; ; Turing equivalent to A are also many-one equivalent to A? not many-one equivalent to A. Nevertheless we will carry We say that such a set collapses or has the collapsing prop- along this condition for clarity. erty. While there has been some work on this problem, most Collapsing is invariant under Turing equivalence: If A is notably by Selman [Sel79] and by Ambos-Spies, Bentzien, collapsing and B is Turing equivalent to A then B is also Fejer, Merkle and Stephan [ABF+00], three decades after collapsing. the work of Cook and Karp this question remains open. The join of two sets A B is the set 0x x A ⊕ f j 2 g [ Selman [Sel79] shows that no tally set collapses. Un- 1x x B . The set A B is many-one reducible to f j 2 g ⊕ der the assumption E = NE we show that no sparse set any other set C such that A many-one reduces to C and collapses either. B many-one reduces to C. A similar statement holds for Some assumption appears to be necessary: We give a Turing reductions. A set A is sparse if for some polynomial p, there are at ∗Address: Dept. of Computer and Information Sciences, 1805 N. Broad most p(n) elements of A of length n, for all n. A set A is P- St. Floor 3, Philadelphia, PA 19122-6094, USA. Phone: (215)204- 8450, fax: (215)204-5082, email: [email protected], printable if there is a function h computable in polynomial- http://www.cis.temple.edu/ beigel. Supported in part by time such that h(1n) outputs a list of all strings in A having ∼ the National Science Foundation under grants CCR-9996021 and CCR- length n. Every P-printable set is sparse. 0049019. Work done while at the NEC Research Institute. The complexity class E = DTIME(2O(n)), and NE = yAddress: 4 Independence Way, Princeton, NJ O(n) 08540, USA. Email: [email protected]. NTIME(2 ). http://www.neci.nj.nec.com/homepages/fortnow. We use Σ to represent our alphabet 0; 1 . f g 3 Previous Work The set D is Turing equivalent to B so if B is collapsing then D is collapsing. But D is a tally set so D is not col- Selman [Sel79] looked at whether collapsing sets could lapsing by Theorem 3.1. be tally sets, i.e., subsets of 1∗. We also use the following lemma from Hartmanis, Im- merman and Sewelson [HIS85]: Theorem 3.1 (Selman) No tally set is collapsing. Lemma 4.3 (HIS) If E = NE then all sparse sets in NP Since we make use of Theorem 3.1 in this paper, we give a are P-printable. proof here for completeness. Proof: Let A be a sparse set in NP. Consider the sets B Proof of Theorem 3.1: Suppose A 1∗ is a collapsing and C defined as set. Consider the following set B (known⊆ as the left-cut of A B = n; w There exist x1 < x2 < < xw ): fh i j ··· of length n in A x 1 B = x Σ∗ x A()A(1)A(11) A(1j |− ) g f 2 j ≤ ··· g C = n; w; i; j There exist x1 < x2 < < xw fh i j ··· where denotes lexicographic ordering. of length n in A with the jth bit of xi is 1 g One≤ can easily Turing reduce B to A. A also Turing where the numbers n, w, i and j are written in binary. The reduces to B by using binary search to find A()A(1) . sets B and E are in NE = E. In particular, in polynomial The sets B and B are both Turing equivalent to ···A so in n time we can find the maximum w such that n; w is they are Turing equivalent to each other. By assumption in B. This tells us how many strings of length n areh iniA. this means there is a many-one reduction f reducing B to Note that there is only one possible choice for x ; : : : ; x . B. 1 w We now determine whether n; w; i; j in C for all of the Note that if x B and y x then y B. Since appropriate i and j to reconstructh the stringsi x ; : : : ; x in exactly one of x and2 f(x) could≤ be in B, it2 must be the 1 w A. lexicographically least one. This lets us test membership in B in polynomial-time. Since A is equivalent to B we also Proof of Theorem 4.1: Suppose E = NE and let A be have A in P. a sparse collapsing set. A is Turing equivalent to A so by Ambos-Spies, Bentzien, Fejer, Merkle and assumption there is a function f reducing A to A. Consider Stephan [ABF+00] proved many other results about the set collapsing sets including the following: B = 1n; x ( y)[ y = n and f(y) = x : fh i j 9 j j g Theorem 3.2 (ABFMS) If A is a collapsing set then The set B is in NP. 1. A is computable, and We also claim that B is sparse. The number of strings in m B of length n is bounded by m n f(Σ ) , so it suffices 2. A is not hard for EXP. to show that f(Σn) is polynomiallyP ≤ j bounded.j We have f(Σn) f(jA Σnj) + f(A Σn) . Each of those terms is j j ≤ j \ j j \ j 4 Separations polynomially bounded because A is sparse and f(A) A. By Lemma 4.3, B is P-printable. Consider C B⊆ de- fined as ⊆ In this section we show that under a hard-to-disprove as- sumption, no sparse set can be collapsing. Section 5 indi- C = 1n; x 1n; x B and x A : cates that some assumption appears to be needed. fh i j h i 2 2 g The set C is Turing equivalent to A: C is reducible to A Theorem 4.1 If E = NE then no sparse set is collapsing. because B is in P; for the other direction, we have y y 2 A 1j j; f(y) = C. () h i 2 To prove Theorem 4.1 we first need the following Since A is collapsing C is also collapsing. But C is lemma: a subset of a P-printable set so C does not collapse by Lemma 4.2. Lemma 4.2 No subset of a P-printable set is collapsing. Proof: Let B be a subset of a P-printable set C. Let 5 Collapse n h(1 ) = yn;1; : : : ; yn;k list the elements of C of length n. Definef the set D as g In this section we present the first relativized world where there exists a collapsing set, i.e., where Hypothe- n;i D = 1h i yn;i B : sis 2.1 holds. f j 2 g Theorem 5.1 There exists an oracle B and a sparse set We say u and v are connected if the value of A(u) is A PB such that for every set C forced by setting v A, given the other conditions we have 62 already currently set.2 pB pB C A C A: The first set of conditions we use will require that yi ≡T () ≡m 2 A yi + 1 A for each i.
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