in [8]. A class F is closed under f if the composition of f with any function in F is again a function in F. Of highest importance are (non-constructi

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1 A Note on Unambiguous Function Classes Sven Kosub Theoretische Informatik, Julius-Maximilians-Universitat Wurzburg, Am Hubland, D Wurzburg, Germany Key words: Computational complexity, unambiguous computation, function classes 1 Introduction Unambiguous computation according to UP has become a classical notion in computational complexity theory. Unambiguity is also used in a theorem of Wagner [14]. A set L is in P NP i there are a set A 2 NP and a polynomial p such that for all x and y with jyj p(jxj), if (x; y) 2 A then (x; y? 1) 2 A, and x 2 L i the maximal y with (x; y) 2 A is odd. Here, essentially for each x all positive y's must be neighbored, they form a cluster, where 0 is always in the cluster if one exists. Unambiguity is reected in the existence of at most one such cluster. This construction is crucially relevant for parallel access to NP (see [14]). The same idea occurs in the denition of the complexity-theoretic operator F [13]: FK is the class of all functions f for which there exist an A 2 K and polynomial p as above with f(x) = supfyjjyj p(jxj) ^ (x; y) 2 Ag. This operator plays a key role as part of an operator pair to compare function classes and set classes. Note that maximization can be replaced with cardinality whenever K is closed under P m-reductions. Above, all clusters contain 0 as minimal element. In this note, we drop this requirement. Let us consider (polynomially time-bounded) nondeterministic machines where the set of accepting paths is always a cluster. As we will see, such cluster machines can be easily transformed into UP-machines. The point why it is nevertheless sensible to study these machines is that cluster machines capture UP-machines that actually can count. This feature is expressed in the unambiguous counting class c#p{the #P-version for cluster machines. The class c#p shows astonishing behavior with respect to closure properties. A theory of closure properties for complexity classes of functions was established Preprint submitted to Elsevier Preprint 22 November 1999

2 in [8]. A class F is closed under f if the composition of f with any function in F is again a function in F. Of highest importance are (non-constructively) hard closure properties{a function f is hard for F if the case that F is closed under f is equivalent to F being closed under every function from F. Using the structural assumption PP = UP, it was proved that, e.g., division is a hard closure property for #P [8]. First, we prove that c#p also possesses hard closure properties. This is surprising insofar as c#p is a promise class, i.e., a class without an obvious enumeration of its underlying machines. Even so, we identify operations (expressed by c#p-functions) being able to simulate any other operation (expressible as a c#p-function) over cluster machines. A second remarkable property is the following: It was observed that #P has intermediate closure properties (e.g., minimization [8]), i.e., functions that are neither hard nor is #P closed under them. Provably, c#p does not have (unary) intermediate closure properties. We give a criterion for deciding if a function is c#p-hard or c#p is closed under it. From [8,2,3], hard closure properties are expected to reduce the number of witnesses, e.g., division for #P. Here, however, the situation is completely dierent. We show that incrementation is c#p-hard, while c#p is closed under division. Hence, this is an instructive example where the term \witness reduction" is not to be taken literally, but rather in the sense of simplifying the witness set (see the discussion in [2, pp ]). Unambiguous clustering is furthermore studied in a more general setting for counting and optimization. Remember that for \xed" clusters (as in the denition of the operator F), maximization and counting coincide. For \free" clusters, this is not to be expected. We consider three unambiguous operators corresponding to those in [11,4]: counting (c#), maximization (cmax), and minimization (cmin) within clusters. These operators are compared to each other{giving a taxonomy of classes similar to [4]{and to the related ambiguous operators in the context of classes of the polynomial hierarchy. Mostly, proofs are straightforward, but they show in an intuitive way how strong clustering and unambiguity are connected. 2 Preliminaries We use the nite alphabet = f0; 1g. The cardinality of a set L is denoted by #L. Let c L denote the characteristic function of L. Usually we interpret a word x as the binary encoding of a natural number. For closure properties, we need a bijection between and N. We suppose the standard 2

3 lexicographical ordering of. For a function f :!, the notation x:f(x) stands for the mapping x 7! f(x). The function sgn : N! N is dened as sgn x = 1, if x > 0, and sgn x = 0, if x = 0. Our computational model is the standard nondeterministic Turing machine [9]. M(x) denotes the computation tree of a machine M on input x. In case of polynomial time resources we suppose complete binary computation trees. For computation trees M(x), we dene acc M (x) as the set of all accepting paths of M(x), and out + M(x) (out? M(x)) as the set of all outputs on accepting (rejecting, resp.) paths of M(x). We use some shorthands for machine types: NTM stands for nondeterministic Turing machine, and NPTM for polynomial-time NTM. We deal with the standard complexity classes P; UP; coup; NP; conp; PP; C = P; PH; CH, and their relativizations [5,9]. To describe function classes, we use the modern operator notation provided in [11,13,4]. A function f is in #K if there exist a set A 2 K and a polynomial p such that f(x) = #fyjjyj = p(jxj) and (x; y) 2 Ag for every x [11]. We abbreviate such witness sets as A p x. A function f is in max K if there is an A 2 K and a polynomial p such that f(x) = max A p for every x [4]. Replacing max by min gives the class x min K. To consider total functions we dene maximum and minimum of the empty set to be 0, what is in the second case a little dierent from the original denition [4] but is more compatible with the partial case. FP = F P [13] denotes the class of functions computable in deterministic polynomial time. 3 Clusters, Unambiguous Counting, and Closure Properties We formalize the concept of clusters by introducing certain equivalence relations. Let M be an NTM, and let y and z be paths of M(x) for input x 2. We write y M(x) z i jy? zj = 1 (i.e., y and z are neighboring paths) and M(x) accepts on path y if and only if M(x) accepts on path z. The relation M(x) (the reexive and transitive closure of M(x)) is an equivalence relation partitioning M(x) in equivalence classes. A cluster is an equivalence class whose representatives are accepting paths of M(x). Denition 1 An NTM M is a cluster machine if and only if for every x 2, there is a path y of M(x) such that fzjz M(x) y ^ y 2 acc M(x)g = acc M (x). An NTM M computes a function f uniquely i out + M(x) = ff(x)g and #acc M (x) = 1 for every x, and M computes f almost-uniquely i f(x) > 0 implies out + M(x) = ff(x)g ^ #acc M (x) = 1 and f(x) = 0 implies out + M(x) =?, both 3

4 for every x. Theorem 2 (1) If an NPTM M computes f uniquely, then there is a polynomialtime cluster machine (PCM) N such that #acc N (x) = f(x) for all x 2. (2) For every PCM N fullling #acc N (x) > 0 for every x 2, there is an NPTM M that computes the function #acc N uniquely. (3) There is an NPTM M that computes the function f almost-uniquely if and only if there exists a PCM N fullling #acc N (x) = f(x) for every x 2. PROOF. (1) is obvious. For (2), let N be a PCM such that smallest and largest computation paths of N are rejecting, and N is time-bounded by polynomial p. Dene K to be the machine that, on input x, (a) guesses paths y 1 and y 2 with y 1 < y 2 ; jy 1 j = jy 2 j = p(jxj), (b) simulates successively N(x) on y 1 (result 1 ) and on y (result 1 ), (c) simulates successively N(x) on y 2 (result 2 ) and on y mod 2 p(jxj) (result 2 ), and (d) outputs y 2? y 1 and accepts i ( 1 1 ) ^ ( 2 2 ) is true. Clearly, K is an NPTM, and since #acc N (x) > 0 we get #acc K (x) = 1 and out + K(x) = ff(x)g. (3) combines (1) and (2). 2 Corollary 3 For every set A 2 NP there is a PCM accepting A in the sense of NP if and only if NP = UP. The following results concern the class of counting functions computed by PCM's. Denition 4 c#p = f#acc M jm is a PCMg: The class UPSV [1] consists of all partial functions f such that there is an NPTM M with at most one accepting path, #acc M (x) = 1 i f(x) is dened, and #acc M (x) = 1 implies out + M(x) = ff(x)g. UPSV t is the class of all total functions in UPSV. Theorem 5 (1) UPSV t c#p. (2) UPSV t = c#p if and only if UP = coup. (3) A function f is in UPSV t if and only if its incremented version f 0, dened as f 0 (x) = f(x) + 1, is in c#p. PROOF. (1) and (3) are immediate from Theorem 2. For (2), rst, suppose UPSV t = c#p. Let A 2 UP, then c A 2 c#p UPSV t (the latter via NPTM M). Modifying M so that on unique accepting paths it is accepted i output is 0, yields A 2 coup. Conversely, suppose UP = coup. Let f 2 c#p. Dene A = fxjf(x) 1g. Then A; A 2 UP. Consider an NPTM M that, on input x, 4

5 on paths beginning with 0 decides x 2 A, on paths beginning with 1 decides x 2 A, if the accepting path gives x 2 A then outputs f(x), and if the accepting path gives x 2 A then outputs 0. Thus, f 2 UPSV t. 2 Closure properties for complexity-theoretic function classes were rst studied in [8]: A closure property is a function f : N n! N for n 2 N, an f 2 F is called an F-closure property. A class F is closed under f i for all g 1 ; : : : ; g n 2 F the function h is in F where h(x) = f(g 1 (x); : : : ; g n (x)). F is called G-closed i F is closed under all G-closure properties. A function f 2 F is F-hard whenever F is closed under f i F is F-closed. Theorem 6 The following statements are equivalent. (1) UP = coup. (2) c#p is closed under incrementation. (3) c#p is FP-closed. (4) c#p is c#p-closed. PROOF. Use Theorem 5 and note that UPSV t is UPSV t -closed. 2 It is well known that strong collapses of complexity classes such as PP = UP, PP = NP, or P PP = NP, can be characterized by closure properties of some function classes (see [8,3]). The preceding theorem shows that, surprisingly, the very weak collapse UP = coup has such a characterization, too. Corollary 7 There is relativized world in which c#p is not c#p-closed. PROOF. Observe that relativizably, UP = coup implies NP UP = NP. However in [10], an oracle was given which contradicts this equality. Note that all our proofs relativize. 2 A function f : N n! N is said to be 0-reproducing in its k-th component i m k = 0 implies f(m 1 ; : : : ; m n ) = 0 for all m 1 ; : : : ; m k?1 ; m k+1 ; : : : ; m n 2 N. Moreover, f is 0-reproducing i f is 0-reproducing in every component. Theorem 8 Let f be an n-ary c#p-closure property. (1) If f is constant or 0-reproducing, then c#p is closed under f. (2) Let h 1 ; : : : ; h k?1 ; h k+1 ; : : : ; h n 2 FP. If f is 0-reproducing in its k-th component, then c#p is closed under x:f(h 1 (x); : : : ; h k?1 (x); x; h k+1 (x); : : : ; h n (x)): 5

6 PROOF. (1): The case if f is constant is trivial. So, let f be 0-reproducing, let g 1 ; : : : ; g n 2 c#p be almost-uniquely computed by NPTM's M 1 ; : : : ; M n, and let D be a PCM with f = #acc D. Dene h(x) = f(hg 1 (x); : : : ; g n (x)i), and consider a machine N that, on x, (a) simulates M i (x) successively for all i in increasing order, (b) computes hz 1 ; : : : ; z n i if all simulations in (a) accept, where z i is the output of M i (x) on the accepting path, and (c) simulates D on hz 1 ; : : : ; z n i. Since there is always at most one path where all simulations of M i accept, N is a PCM with h(x) = #acc N (x). Thus h 2 c#p. (2) is similar to (1). 2 As an example, Theorem 8 immediately gives that c#p is closed under division. This should be contrasted to the results for #P in [8]. Theorem 9 Let f be an n-ary c#p-closure property. If there are k 2 N with 1 k n, and x 1 ; : : : ; x k?1 ; x k+1 ; : : : ; x n 2 N such that x:f(x 1 ; : : : ; x k?1 ; x; x k+1 ; : : : ; x n ) is neither 0-reproducing nor constant, then f is c#p-hard. PROOF. Suppose k; x 1 ; : : : ; x k?1 ; x k+1 ; : : : ; x n make the premise true. To prove that f is c#p-hard, assume c#p is closed under f. We have to show that c#p is c#p-closed. So, let '(x) = f(x 1 ; : : : ; x k?1 ; x; x k+1 ; : : : ; x n ). Thus, c#p is closed under '. Since ' is not 0-reproducing, 0 < '(0) = c 1. Since ' is not constant, there is a minimal z 0 so that c 1 6= '(z 0 ) = c 2. Dening (x) = '(min(z 0 ; z 0 x)), we have x = 0 ) (x) = c 1 and x > 0 ) (x) = c 2. Since multiplication and minimization are 0-reproducing and since c#p is closed under ', c#p is closed under. Now, there are two cases depending on c 2. If c 2 = 0, then dene t = sgn, i.e., x = 0 ) t(x) = 1 and x > 0 ) t(x) = 0. Observe that sgn is 0-reproducing. If c 2 > 0, then (x) > 0 for all x, thus is almost-uniquely computable, and there is t 2 c#p with x = 0 ) t(x) = 1 and x > 0 ) t(x) = 0. In both cases, c#p is closed under t, and t 2 c#p. Now, let A 2 UP. Then, c A (x) = 1, t(c A (x)) = 0 and c A (x) = 0, t(c A (x)) = 1. Hence, A 2 coup. Thus, UP = coup and using Theorem 6 we get that c#p is c#p-closed. 2 Though we do not know whether c#p is c#p-closed, Theorem 8 and Theorem 9 yield a criterion to decide whether a unary function f 2 c#p is hard for c#p or not: If f is neither 0-reproducing nor constant, then it is hard, otherwise c#p is closed under f. Corollary 10 For every unary c#p-closure property f, it holds that c#p is closed under f in all relativizations if and only if f is 0-reproducing or constant. 6

7 P = UP PH = UP NP = UP NP P NP = UP NP P P = UP c# P NP cmax P NP = cmin P NP FP NP cmin conp cmax conp P NP = UP NP P NP = UP NP P NP = UP NP P NP = UP NP c# conp cmax NP = c# NP cmin NP PH = UP PH = UP c# P cmax P = cmin P FP Key: a bold line means the inclusion of the lower in the upper class F F F G G G means F G =) means F G () means F G () and F G () Fig. 1. A taxonomy of clustering. 4 A Taxonomy of Clustering in Counting and Optimization In this section we generalize our denitions to arbitrary sets. For that, let A, and let A x = fyjhx; yi 2 Ag. For x; y; z 2, we write y x z i jy? zj = 1 and hx; yi 2 A, hx; zi 2 A. The relation is dened as the x equivalence relation generated by x. A is said to be a cluster set i for every 7

8 x there is an y such that A x = fzjz x y ^ hx; yi 2 Ag. The unambiguous cluster versions of the ordinary operators are obtained if we allow the operators to vary only over cluster sets in a class K. A function f is in c# K i there are a cluster set A 2 K and a polynomial p so that f(x) = #A p for every x. Evidently, c# P = c#p: The denitions of cmax x and cmin are similar. Note that, though the value f(x) above depends on A p x, we require that A is a cluster set, not only the set A p dened as fhx; yijjyj p(jxj) ^ hx; yi 2 Ag. Generally, to require that only A p is a cluster set leads to larger classes. However, for classes K closed under P m-reductions or closed under intersection with P-sets, both approaches are equivalent. Theorem 11 All inclusions, equivalences, and implications presented in Fig. 3 hold. PROOF. The techniques are straightforward. Thus we only prove one result for each type of statement as examples. The remaining claims are left to the reader. Observe from [4] that for complexity classes C; K closed under P m- reductions, it holds cmin C cmax K, cmax C cmin K. (I) Inclusions. c#np cmaxnp: Let f 2 c#np by A 2 NP and polynomial p. Consider B = fhx; yijjyj = p(jxj) ^ (9z 1 ; z 2 ; jz 1 j = jz 2 j = p(jxj))(hx; z 1 i 2 A ^ hx; z 2 i 2 A ^ y = jz 1? z 2 j + 1)g. Then B 2 NP and B = fhx; yijjyj = p(jxj) ^ y f(x)g. Thus, B is a cluster set and f(x) = max B p x. Hence, f 2 cmax NP. (II) Equivalences. PH = UP, cmaxnp = c#p: For [)] suppose PH = UP. Then conclude: c# P cmax NP cmax P NP = cmin P NP (cmax UP \ cmin UP). An easy analysis shows that the latter class is equal to c# P. For [(] suppose cmax NP = c# P. We show conp UP. Let A 2 NP accepted by NPTM M. Dene N to be the machine that, on input x, simulating M(x) outputs 1 on rejecting paths, outputs 2 on accepting paths, and always accepts. Note that out + N(x) f1; 2g and fhx; yijy 2 out + N(x)g 2 NP. Set f(x) = max out + N(x). Then, f 2 cmax NP and x 2 A ) f(x) = 1 and x =2 A ) f(x) = 2. Since cmax NP = c# P, there is an NPTM K that computes f uniquely. Thus K can be modied to accept A in the sense of UP. (III) Implications. Use the monotonic operator technique (see [12,4]). For a function class F, say A 2 U F i there is an f 2 F so that for every x, x 2 A, f(x) = 1 and x =2 A, f(x) = 0. Say A 2 9 F i an f 2 F exists satisfying x 2 A, f(x) 1. With #P NP = #conp [6], it is easily seen that UP NP = U#P NP = U#coNP (9cmaxcoNP\9cmincoNP\9c#coNP). Further, 9 FP NP = 9 cmaxp NP = P NP. 2 8

9 Main open problems concern the position of c# conp in the taxonomy of the gure. Especially, it is not known whether c# P NP = c# conp. As already mentioned this is true for the usual counting operator [6]. A partial answer gives the next theorem. Theorem 12 The following statements are equivalent. (1) c# P NP = c# conp. (2) cmax P NP c# conp. (3) FP NP c# conp. PROOF. [(1) ) (2)] and [(2) ) (3)] are trivial. For [(3) ) (1)] suppose FP NP c# conp. Let f 2 c# P NP. From relativized Theorem 2 we obtain an NPTM M having oracle access and A 2 NP, such that #acc M (A)(x) 1 and #acc M (A)(x) = 1 ) f(x) = out + (x). Dene g(x; y) = out + (x) if M (A) M (A) y 2 acc M (A)(x) and g(x; y) = 0 otherwise. Then g 2 FP NP c# conp. Consequently, there are a cluster set C 2 conp and polynomials p and r with g(x; y) = #fzjjzj = r(jxj) ^ jyj = p(jxj) ^ hx; y; zi 2 Cg. Note that for at most one y it holds g(x; y) > 0, when x is given. Dene C 0 = fhx; yzijjyzj = r(jxj)+p(jxj)^hx; y; zi 2 Cg. Then, C 0 2 conp, and C 0 is a cluster set. Dene g 0 (x) = #fwjjwj = r(jxj) + p(jxj) ^ hx; wi 2 C 0 g. Then, g 0 2 c# conp and g 0 (x) = g(x; y) = f(x) if such y exists, otherwise g 0 (x) = f(x) = 0. Hence, f 2 c# conp. 2 Next we compare cluster classes and their corresponding ambiguous classes. Trivially, the unambiguous classes are subclasses of the ambiguous one's. Theorem 13 (1) c# P = # P () PP = UP. (2) c# NP = # NP () PP = NP. (3) If c# conp = # conp then PP UP NP. PROOF. (1): For [)] suppose c# P = # P. Let A 2 PP via f 2 # P and g 2 FP with x 2 A, f(x) g(x) and f(x) > 0 for every x. Then, f 2 c# P, i.e., there is an NPTM M that computes f uniquely. Modify M such that M accepts on the only accepting path if f(x) g(x). Then A 2 UP. For [(] suppose PP = UP, equivalently C = P = UP [8]. Let f 2 # P. Dene C = fhx; yijjyj = p(jxj) ^ f(x) = yg. Then C 2 C = P UP. Dene N to be a machine that, on input x, checks hx; yi 2 C for a guessed word y; jyj = p(jxj), outputs y, and accepts i checking is successful. N computes f uniquely. Thus, f 2 c# P. (2): For [)] suppose c# NP = # NP. Let A 2 PP via f 2 # P # NP and g 2 FP with x 2 A, f(x) g(x) and f(x) > 0 for every x. Then, 9

10 f 2 c# NP = cmax NP, i.e., there exist a cluster set B 2 NP, accepted by NPTM M, and polynomial p with f(x) = maxfyjjyj = p(jxj) ^ hx; yi 2 Bg. Dene N to be a machine that, on input x, simulates M on input hx; yi with y being a guessed word, jyj = p(jxj), computes deterministically g(x), and accepts i M(hx; yi) is accepting on the simulation path and it holds y g(x). Thus, A 2 NP. For [(] suppose PP = NP and conclude c# NP # NP c# P #NP c# CH = c# NP since PP = NP, CH = NP (corollary of Toda's Theorem [11]). (3) is due to # P NP = # conp and to relativization of (1). 2 Theorem 14 (1) cmaxp = maxp () cminp = minp () NP = P. (2) cmax NP = max NP and cmin NP = min NP. (3) If cmax conp = max conp or cmin conp = min conp, then PH = NP NP. PROOF. Due to duality, we only prove the statements for the maximization parts. (1): For [)] suppose cmaxp = maxp. Let A 2 NP, accepted by NPTM M in runtime p. Dene B = fhx; 1yijjyj = p(jxj) ^ y 2 acc M (x)g [ fhx; 0 p(jxj) 1ijx 2 g. Thus, B 2 P. Dene f(x) = maxfzjjzj = p(jxj) + 1 ^ hx; zi 2 Bg. Then, f 2 max P cmax P (the latter via cluster set C 2 P and polynomial q with q(n) > p(n) for all n 2 N). Consider D = fxjhx; 0 q(jxj)?1 1i 2 C ^ hx; 0 q(jxj)?2 10i =2 Cg. Then, D 2 P and x 2 A, f(x) = 1, x 2 D. Hence, A 2 P. For [(] suppose P = NP. Let f 2 max P via A 2 P and polynomial p. Dene B = fhx; yijjyj = p(jxj) ^ (9z; jzj = p(jxj))(y z ^ hx; zi 2 A)g. Then, B 2 NP P. Binary search using B shows f 2 FP cmax P. (2) is the well-known construction for max NP # NP [6,4]. (3) follows from cmax conp cmax P NP max P NP \ min P NP and from max conp min P NP implies PH = NP NP [4]. 2 Acknowledgements I am grateful to Prof. Gerd Wechsung for supervising my diploma thesis [7] wherefrom material is taken here, and to Sanjay Gupta, Heribert Vollmer, and an anonymous referee for valuable hints. 10

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