Are Cook and Karp Ever the Same?

Home , Reduction (complexity), Turing reduction

Richard Beigel∗ Lance Fortnow† 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 equivalent∈ ⇔ to B”)∈ 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 ⊕ { | ∈ } ∪ Selman [Sel79] shows that no tally set collapses. Un- 1x x B . The set A B is many-one reducible to { | ∈ } ⊕ 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 = †Address: 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 . { } 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 ): {h i | ··· of length n in A x 1 B = x Σ∗ x A()A(1)A(11) A(1| |− ) } { ∈ | ≤ ··· } C = n, w, i, j There exist x1 < x2 < < xw {h i | ··· where denotes lexicographic ordering. of length n in A with the jth bit of xi is 1 } 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 and∈ f(x) could≤ be in B, it∈ 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 . {h i | ∃ | | } 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 ≤ | bounded.| We have f(Σn) f(|A Σn|) + f(A Σn) . Each of those terms is | | ≤ | ∩ | | ∩ | 4 Separations polynomially bounded because A is sparse and f(A) A. By Lemma 4.3, B is P-printable. Consider C B⊆ defined as ⊆ In this section we show that under a hard-to-disprove assumption, 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. {h i | h i ∈ ∈ } 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 ∈ A 1| |, f(y) / C. ⇐⇒ h i ∈ 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. Define{ the set D as } 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. { | ∈ } 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 6∈ already currently set.∈ pB pB C A C A. The first set of conditions we use will require that yi ≡T ⇐⇒ ≡m ∈ A yi + 1 A for each i. ⇔ 6∈ Proof: Let M1,M2,... be an enumeration of Eventually we will connect all the members of U. For all B polynomial-time oracle Turing machines. Note that C p pairs u and v of elements in U we will encode at B( u, v ) ≡T h i A is the same as saying there exist machines Mi and Mj whether u A v A or u A v A. B A B C ∈ ⇔ ∈ ∈ ⇔ 6∈ such that C = L(Mi ⊕ ) and A = L(Mj ⊕ ). Let We will now work in substages j = 1, . . . , s. In each R1,R2,... and S1,S2,... be enumerations of polynomial- substage we will handle requirement Fj. time machines such that for every i and j there is a k such In substage j and later we guarantee not to add any 10 that Rk = Mi and Sk = Mj. We assume without loss of strings to B of length less than nj . Thus DTIME(nj) i generality that Mi, Ri and Si run in time at most n for n machines or even a composition of five of them acting on the length of the input. strings of length n will not be affected by strings we add to We need to fulfill the following requirements: B.

B We first describe how to construct the many-one reduc- 1. Ei: There is an x such that x A Mi (x) rejects. B A ∈ ⇔ tion from C to A. Fix an input w and simulate Sj ⊕ (w), B A B C pB the potential Turing reduction from C to A. If Sj makes a 2. Fi: If C = L(Ri ⊕ ) and A = L(Si ⊕ ) then A m B ≤ query to B we just query it. Now consider all of the queries C and C p A. ≤m made to A. Since we have encoded the relationship between B A B all pairs u and v in U, we can reduce S ⊕ (w) to a single The Ei requirements guarantee that A is not in P while j query yi in A. If Sj accepts or rejects regardless of whether the Fi requirements guarantee that A is collapsing relative to B. yi is in A then just accept or reject respectively. If Sj accepts iff yi is in A then we have our many-one reduction. When we fulfill our Fi requirements we will allow our many-one reductions to occasionally just accept and reject. Otherwise reduce to yi + 1. We now describe how to construct the many-one reduc- Since our Ei requirements will guarantee that A is not in PB we can just map to fixed strings known to be in or out tion from A to C. Fix a u of length n. Check with B to see of A and C. if u is in U. If not then we just reject. B C We will define our oracle B as a function B : 0, 1 n Simulate Rj ⊕ (u), the potential reduction from A to C. nk { } → Consider the strings it queries in C in order. Let w be the 0, 1 for some fixed k. One can use standard coding B A { } next string queried and consider the reduction Sj ⊕ (w). ideas to create a subset of 0, 1 ∗ equivalent to B. { } B A We build the oracle in stages. In each stage s we will Note that Sj ⊕ cannot query any string in U not connected ns 1 to u or we would violate the randomness of the string used pick ns = 2 − . log ns 1 to generate the yi’s. ns 1 − We will set B(1 − ) to be a list of the elements of ns 1 By the above argument we have only three possibilities. A Σ − . This guarantees that any reduction that looks at If S accepts or rejects regardless of A then we answer R previous∩ strings in A can just find them easily in B. j j accordingly and go to R ’s next query. If SB A(w) accepts Fix s and let n = n . In stage s we will fulfill require- j j ⊕ s iff u A then we just output w as our many-one reduction ment E for some string of length n. We will also guarantee s ∈ u U log ns 1 and we then go on to the next in . that F1,...,Fs hold for strings of length between ns 1 − The problem occurs if Sj accepts iff u A. Set w = wu and nlog ns . At the end of the construction, each require-− s and we’ll handle this case later. Go on and6∈ handle the next ment F will have been fulfilled for all but a finite number i u in U. of strings. Suppose for some u, RB A(u) runs out of queries and Choose a Kolmogorov random string of length n2 and j ⊕ just accepts or rejects. We then set u in A to diagonalize break it into n pieces of length n each and call these strings against R and go on to the next substage. y , . . . y . Let U = y , y + 1, . . . , y , y + 1 where j 1 n 1 1 n n Now consider the w ’s. If we have a w and a w (with y + 1 is the lexicographically{ next string after y .} We will u u v i i u connected to v) such that u A v A, then guarantee that A Σn is a subset of U. we can just use the many-one reduction∈ ⇐⇒ of v to6∈w and u We set B(u) =∩ 1 for each u in U. u to w . Otherwise pick a w and a w not connected, and We will put conditions on A of the form u A v v u v ∈ ⇔ 6∈ add the condition u A v A. We encode this A. The construction will not build a contradictory set of 10 ∈ ⇐⇒ 6∈ nj conditions and will allow some freedom so at the end we connection in the oracle by setting B( 1 , u ) = v and 10 h i nj can choose A to fulfill Es. B( 1 , v ) = u. We then have the many-one reduction h i map all of the r with r A u A to wv and all of [Kar72] R. Karp. Reducibility among combinatorial ∈ ⇐⇒ ∈ the s with s A v A to wu. We then remove from problems. In R. Miller and J. Thatcher, editors, ∈ ⇐⇒ ∈ consideration all of these wr’s and ws’s and repeat. Complexity of Computer Computations, pages If we have one wu remaining without a wv to connect to 85–103. Plenum Press, 1972. it then just set u A and encode all of the r connected to j1∈0 [Sel79] A. Selman. P-selective sets, tally languages, u B B( 1n , u ) = in as TRUE or FALSE as appropriate and the behavior of polynomial time reducibil- h i u and have the many-one reduction on just accept. ities on NP. Mathematical Systems Theory, s n At the beginning of stage we have connected compo- 13:55–65, 1979. nents of elements of U. In each stage we shrink the number of components in at most half and possibly determine the elements of one component. Since s log n, we still have components remaining after substages. Pick some u in one of these components and set u in A to diagonalize against the Ms(u) to fulfill Es.

6 Further Research

Of course, the question whether there exists a collapsing set still remains open. In particular can one modify the oracle construction in Section 5 to simply create a (nonsparse) collapsing set without needing a relativized world? Another related question is whether all Turing complete sets for NP are many-one complete. This question is in- comparable to the existence of a collapsing set since in this case we restrict ourselves to sets in the class NP, which is not necessarily closed under Turing reductions. Neverthe- less, perhaps our techniques could also shed light on that problem.

Acknowledgments

We thank Bill Gasarch, Aduri Pavan, Marcus Schaefer, Alan Selman, and Frank Stephan for helpful discussions. Bill Gasarch also provided a careful proofreading of this exposition.

References

[ABF+00] K. Ambos-Spies, L. Bentzien, P. Fejer, W. Merkle, and F. Stephan. Collapsing polynomial-time degrees. In Logic Colloquium ’98, volume 13 of Lecture Notes in Logic, pages 1–24. Association of Symbolic Logic, 2000. [Coo71] S. Cook. The complexity of theorem-proving procedures. In Proceedings of the 3rd ACM Symposium on the Theory of Computing, pages 151–158. ACM, New York, 1971. [HIS85] J. Hartmanis, N. Immerman, and V. Sewelson. Sparse sets in NP P: EXPTIME versus NEX- PTIME. Information− and Control, 65:158–181, 1985.