Relationships Among PL, L, and the Determinant y z Eric Allender Mitsunori Ogihara Department of Computer Science Department of Computer Science Hill Center, Busch Campus, P.O. Box 1179 UniversityofRochester Piscataway, NJ 08855-117 9, USA Ro chester, NY 14627 [email protected] [email protected] Abstract Recent results byTo da, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the numb er of accepting compu- tations of a nondeterministic logspace-b ounded machine. This class of functions is known as L. By using that characterization and by establishing a few elementary closure prop erties, we givea very simple pro of of a theorem of Jung, showing that probabilistic logspace-b ounded PL machines lose none of their computational p ower if they are restricted to run in p olynomial time. We also present new results comparing and contrasting the classes of functions reducible to PL, L, and the determinant, using various notions of reducibility. 1 Intro duction One of the most imp ortant and in uential early results of complexity theory is the theorem of [47] showing that the complexity of computing the p ermanentofaninteger matrix is characterized by the complexity class P of functions that count the numb er of accepting computation paths of a nondeterministic p olynomial-time machine. It is p erhaps surprising that well over a decade passed b efore it was discovered that an equally-close connection exists b etween the complexityof computing the determinant of a matrix and the class L de ned in [5] of functions that count the numb er of accepting computation paths of a nondeterministic logspace-b ounded machine. In order to state this connection more precisely, let us de ne GapL to b e ff : f x=g x hx for some g and h in Lg. This de nition is by analogy to the class GapP, consisting of the di erence of P functions, which is de ned and studied in [17, 36, 22]. A preliminary version of this pap er app eared in Pro c. 9th IEEE Structure in Complexity Theory Conference, 1994 y Supp orted in part by NSF grants CCR-9204874 and CCR-9509603. Part of this work was done while on sabbatical at Princeton University. z Supp orted in part by the JSPS under grant NSF-INT-9116781/JSPS-ENG-207, and by the NSF under grant CCR-9002292. Work done in part while at University of Electro-Communications, Tokyo. 1 Although stated in di erentways, the results of [49, Theorem 6.5], [44, Theorem 2.1], [16], and [48, Theorem 2] show the following fact: Theorem 1 [44, 16, 49 , 48 ] A function f is in GapL i f is logspace many-one reducible to the determinant. A function f is logspace many-one reducible to the determinant if there is a function g computable in logspace such that f x viewed as a numb er written in binary is equal to the determinantof 1 matrix g x. The pro of given in [44] is particularly clear and direct: To da shows that the determinant of integer matrices is reducible to iterated matrix multiplication over the integers using [7], which in turn is reducible to iterated matrix multiplicationover f0; 1; 1g, which in turn is reducible to a canonical GapL-complete problem, which in turn is reducible to the determinant. We remark that the logspace many-one reductions presented in [44], can in fact b e computed by 0 uniform AC circuits; we will use this fact later. In this pap er we use Theorem 1 to giveavery simple pro of of a theorem of Jung [27], concerning the complexity class PL. PL is de ned to b e the class of languages A for which there exists a probabilistic Turing machine in the sense of [19]; that is, a Turing machine with access to a source of unbiased random bits, such that on input x the machine never uses more than log jxj space, and x 2 A if and only if the probability that the machine reaches an accepting con guration is greater than one half. Although the de nition of PL is straightforward, it turns out that the complexity of PL is rather dicult to analyze, in part b ecause probabilistic logspace machines can p erform useful work after exp onentially many computation steps. An easy way to see this is to note that the s-t connectivity problem a standard NL-complete set can b e accepted with zero error bya PL machine that follows a randomly-chosen path from vertex s, accepting if vertex t is reached, and otherwise trying another randomly-chosen path, and so on; if a path exists, it will eventually b e found with probability one. 6 Gill showed in [19] that PL is contained in PSPACE. This was improved to DSPACElog n in [41], but it was not until the app earance of [8] that it was even known that PL is contained 2 1 in P; [8] shows that PL is contained in NC , and this can b e improved slightly to TC the 1 This de nition of \functional many-one reducibili ty" is from [5]. It should b e noted that the notion of one function b eing \many-one reducible" to another is sometimes de ned in a much less restrictiveway.For example, it is shown in [53, 11 ] that the p ermanent of zero-one matrices is \many-one complete" for P, using a less restrictiveversion of \many-one reductions." On the other hand, it follows from [47] that the p ermanentofinteger matrices cannot b e complete for P or GapP using our de nition of \many-one reduction" unless P = P since the p ermanent and the determinant are equal mo d 2. That is, the relationship b etween the determinant and L is even closer than the relationship that is known to exist b etween the p ermanent and P. 2 2 class computed by logarithmic-depth threshold circuits. Similarly,PLwas not known to b e closed under complementationuntil a complicated pro of was given in [42];amuch simpler pro of subsequently app eared in [39]. The source of all this dicultywas the fact, already alluded to, that probabilistic logspace machines cannot obviously b e restricted to run in p olynomial time without loss of computational p ower. However, Jung showed in [27] that, at least in the unbounded error mo del which de nes the class PL, the p olynomial-time restriction causes no loss of p ower. The pro of presented in [27] is complicated; we present an easy pro of in Section 3. Note that Theorem 1, combined with our easy pro of of Jung's theorem, give an alternative pro of of the result of [8] concerning the complexity of PL, as well as a making closure of PL under complement completely obvious. Note that it would b e remarkable if a theorem analogous to Jung's theorem could also b e proved in the bounded error case. Wehave already seen that NL can b e accepted with b ounded in fact zero error; thus this would in some sense provide a non-uniform version of L=NL i.e., L/p oly = NL/p oly. Furthermore, Nisan [33] has shown that b ounded-error, p olynomial-time logspace can 2 b e simulated in p olynomial time and space log n; hence extending Jung's theorem to b ounded error would provide a small-space p olynomial-time algorithm for transitive closure. All of these consequences would b e surprising. We will not discuss b ounded-error probabilistic logspace in the remainder of the pap er; for more information on the b ounded-error classes, see [9]. Let DET denote the determinant function. Do not confuse this with the de nition of [15], 1 where DET is used to denote the class of functions NC -reducible to the determinant. Theorem DET L 0 0 1 1 allows one to show that L =L ,aswell as AC DET = AC L and NC DET = 1 PP P NC L, etc. However, no logspace-analog of the theorem P =P is known, and we susp ect that no such analog exists. For example, although we show that any of the high-order O log n PL bits of a L function can b e computed in L ,we know of no reason to susp ect that the middle or low-order bit of a L function can b e computed in this way. For most natural problems A, the class of things reducible to A remains the same regardless of the notion of reducibility that is used. For instance, NL is the class of languages reducible to s-t- connectivity under logspace many-one or Turing reductions, or under 1-L reductions or quanti er- 1 0 free pro jections or NC or AC reductions or under any of the many other low-level reductions that have b een studied in the literature. However, DET do es not seem to have this prop erty.In Section 6 we presentcharacterizations of the classes reducible to PL and L under various notions 2 The complete history of the complexity ofPLiseven more complicated. The interested reader may wish to consult [8, p. 115], [25 ], [43 , p. 111], and [20]. The related notion of probabili sti c nite automata has also b een studied in depth; for recent results and references, see [31, 32 ]. 3 :PL 0 PL: of reducibility.For example, AC PL is equal to the hierarchyPL .