Computational Limitations of Stochastic Turing Machines and Arthur-Merlin Games with Small Space Bounds

A Stochastic Turing machine (STM) is a Turing machine that can perform nondeterministic and probabilistic moves and alternate between both types. Such devices are also called games against nature, Arthur-Merlin games, or interactive proof systems with public coins. We give an overview on complexity classes de ned by STMs with space resources between constant and logarithmic size and constant or sublinear bounds on the number of alternations. New lower space bounds are shown for a speci c family of languages by exploiting combinatorial properties. These results imply an in nite hierarchy with respect to the number of alternations of STMs, and nonclosure properties of certain classes. 1 The Computational Model A stochastic Turing machine (STM) is a nondeterministic machine with the additional ability to perform random moves, also called games against nature by Papadimitriou [Pa85]. Alternative characterizations can be given by so called Arthur-Merlin-games [Ba85, Co89] and interactive proof systems with public coins [GoSi86]. A stochastic Turing machine M models a 2-person game where one of the players determines his moves randomly. In a probabilistic move M chooses among the successor con gurations with equal probability. A computation of M on an input X can be described by a computation tree. To de ne acceptance of X , for each nondeterministic { also called existential { con guration one chooses a successor that maximizes the probability of reaching an accepting leaf. The acceptance probability of X is then given by the acceptance probability of the starting con guration in this truncated tree. In this paper, we will concentrate on machines with bounded error probability. M accepts a language L in space S if for every X it never uses more than S(jX j) space and strings in L are accepted with probability more than 3=4 , while for strings not in L this probability is less than 1=4 . Later, we will also consider machines with space bounds below the twice iterated logarithm. In this case the space bounds can only be maintained with high probability since those functions are no longer space-constructible. De nition. Let MA k Space(S) (resp. AM k Space(S) ) denote the set of languages that can be accepted by an S -space-bounded STM that makes at most k 1 alternations between nondeterministic and probabilistic con gurations and starts with a nondeterministic (resp. probabilistic) move. For such a machine we also say that it works in space S and k rounds. The corresponding time classes will be denoted by MA k T ime(T ) and AM k T ime(T ) . If there is no bound on the number of alternations we simply write AMSpace(S) and AMTime(T ) . The power of polynomial time bounded stochastic machines has been investigated quite extensively using the characterization by Arthur-Merlin-games and interactive proof systems. The complexity classes de ned by stochastic machines can equivalently be described with the help of existential and probabilistic quanti ers. For these classes strong connections to standard classes have been obtained. Machines that arbitrarily often may alternate between existential and probabilistic con gurations have exactly the same power as alternating Turing machines. Alternating machines work in existential and universal con gurations, and one can directly interpret them as games, in which both players try to optimize their moves (existential and universal quanti ers). The corresponding complexity classes are denoted by ATime(T ) . Thus it holds AM Time(POL) = A Time(POL) = PSPACE ; where POL denotes the set of polynomial time bounds. One could therefore say that given polynomial time and an unbounded number of quanti ers, universal quanti ers can be \interchanged" with probabilistic ones. If one however bounds the number of quanti ers the situation is di erent. For the combination of existential/universal quanti ers it is still unknown whether an interchange of quanti ers or the addition of a quanti er changes the complexity classes. Since it is generally believed that the Polynomial Hierarchy, which can be obtained this way, is strict it has been surprising that on the contrary for existential/probabilistic quanti er expressions any number of quanti ers can be replaced by just two [Ba85], more precisely using CON for the union of all constant bounds MA 2 Time(POL) AM 2 T ime(POL) = AM CON T ime(POL) : 2 Sublogarithmic Space Classes: What is Known? Switching from time bounds to space bounds changes the situation completely. Now, due to Immerman's and Szel epcsenyi's simulation result alternating machines with a constant number of alternations have no more power than nondeterministic machines { at least for space bounds above the logarithm. In other words, existential and universal quanti ers can be interchanged and then merged to a single one: k Space(S) = k Space(S) = 1 Space(S) = NSpace(S) for every k 1 and S log , where k Space(S) (resp. k Space(S) ) denotes the alternating S -space bounded complexity class, where the machine starts in an existential (resp. universal) state and alternates k 1 . However, for space bounds below the logarithm this is no longer true. De nition. Let log (i) denote the i -times iterated logarithmic function n 7! log 2 : : : log 2 n . For the twice iterated logarithm instead of log (2) we will also use the shorter notation llog . SUBLOG := (llog ) \ o(log ) denotes the set of all nontrivial (with resp. to DTM and NTM) sublogarithmic space bounds. Several authors, Ge ert, von Braunm uhl/Gengler/Rettinger, and we have shown that the space hierarchy de ned by such alternating machines is in nite [BGR93, Ge94, LiRe93, LiRe96a]: 1 Space(SUBLOG) 2 Space(SUBLOG) 3 Space(SUBLOG) : : : Furthermore the situation becomes trivial for space bounds below the twice iterated logarithm, since then even alternating machines accept only regular languages [Iw93]. On the other hand, it is easy to see that all these classes lie inside P since, by de nition, they are contained in AL := ASpace(log) , which has shown to be equal to P [CKS81]. For a recent overview on alternating sublogarithmic space classes see [LiRe97]. Space-bounded stochastic machines seem to be quite di erent. This even holds for the subclass of probabilistic machines without any nondeterminism. Freivalds has shown that probabilistic machines with constant space (probabilistic nite automata) are quite powerful already. They can accept nonregular languages [Fr81]. An example is the language COUNT := f1 n 01 m jn = mg which can be accepted by a probabilistic TM in constant space with an arbitrarily small error probability, that means COUNT 2 AM 1 Space(CON) = BPSpace(CON) ; where BP stands for bounded error probabilistic machines (Monte Carlo). Furthermore, probabilistic space classes cannot be separated by standard diagonalization techniques. Only recently, the rst nontrivial separation results for such classes have been established, and these apply only to sublogarithmic space classes [DwSt92], [FrKa94]. Dwork and Stockmeyer have investigated stochastic Turing machines and interactive proof systems with small space bounds rst. Extending an impossibility result for 2-way probabilistic nite automata they have shown that the language CENTER := fw0x j w; x 2 f0; 1g and jwj = jxjg cannot be recognized by a probabilistic TM even when sublogarithmic space is available [DwSt92]. However, a stochastic automata alternating between probabilistic and nondeterministic states can recognize this language easily, that is CENTER 2 AMSpace(CON) n AM 1 Space(SUBLOG) : It has been conjectured to be crucial that constant space-bounded STMs use (expected) exponential time for nonregular languages. This has been known for probabilistic machines, and has been shown for stochastic machines with a 1-way input restriction in [CHPW94] only recently, but in general is still open. For sublogarithmic space bounds it makes a dramatic di erence whether the random moves are known, as it is the case for stochastic machines and Arthur-Merlin games, or whether they are hidden as in standard interactive proof systems. Let use denote the corresponding complexity classes for the secret randomness by IPSpace(S) . The language PALINDROME of all strings that are palindromes, is an example that can easily be recognized by an interactive proof system with a constant space-bounded veri er and hidden random moves, but requires logarithmic space in case of public coins [DwSt92]: PALINDROME 2 IPSpace(CON) n AMSpace(SUBLOG) : Note that for the corresponding time classes public or private coins do not make an essential di erence [GoSi86]. By results of Condon/Lipton [CoLi89] and [DwSt92] (for an overview see [Co93]) we know that interactive proof systems even with only constant space { that means the veri er is a probabilistic automata { are already extremely powerful. A precise characterization by standard complexity classes, however, is still open. Let EXL , resp. EEXL denote the class of linear exponential ( 2 O(n) ), resp. linear double exponential bounds. The best known upper and lower bounds seem to be DTime(EXL) IPSpace(CON) ATime(EEXL) : Fortunately, in case of stochastic machines with public random moves the gap is not that huge. Condon has shown that such machines with logarithmic space still stay inside P , more precisely [Co89] AM Space(log ) = P = A Space(log ) : This means that for logarithmic space { as we have seen above for polynomial time { unbounded sequences of alternating quanti ers, either existential/probabilistic or existential/universal, de ne the same complexity class. For quanti er sequences of xed length the situation is not clear. We will consider this question for sublogarithmic space bounds, where nothing has been known so far. Motivated by the separation results for alternating Turing machines, we will investigate stochastic space complexity classes without any restrictions on the time complexity, but with bounds on the number of alternations, or equivalently interactions between the prover and the veri er. A sublogarithmic STM may run for exponential expected time. It has been shown in [DwSt92] that some languages cannot be recognized faster { CENTER is one of such example. On the other hand, the power of sublogarithmic spacebounded STMs restricted to polynomial expected time is still an open problem. Our approach will be to de ne a sequence of simple languages parameterized by an index k , and to show that in order for a stochastic machine to accept them a certain number of alternations are necessary and su cient, where the bound depends on k . As a consequence, almost tight separations for the corresponding complexity classes and an in nite hierarchy are obtained. These results and proof methods will be discussed in the following chapters. 3 Lower Bounds for STMs The results for the languages CENTER and PALINDROME cited above, for any space bound S 2 o(log) , yield the separations BPSpace(S) = AM 1 Space(S) AMSpace(S) AMSpace(log) = P : By limiting the number of alternations this chain will be re ned. De nition. Let be an alphabet, and bin(i) denote the binary representation of integer i of length maxf1; dlog 2 (i + 1)eg and BIN(m) := bin(0)#bin(1)#bin(2)# : : :#bin(m) : For a natural number k and strings U;u 2 + we say that u occurs k times in U if there exist W 0 ;W 1 ; : : : ;W k 2 such that U =W 0 uW 1 uW 2 : : :W k 1 uW k , and de ne the following languages PATTERN k := f U#u#BIN(2 d ) j U; u 2 f0; 1; 2g + ; d 2 IN; juj = d; u occurs k times in U g: As a rst step towards separating the round hierarchy for space-bounded stochastic machines we have shown in [LiRe96b] Theorem1. For S 2 o(log) , no S {space-bounded STM starting with a probabilistic move can recognize PATTERN 1 in 2 rounds , that is PATTERN 1 62 AM 2 Space(o(log)) : This result has been extended in [Li96] by showing that for every k > 1 the language PATTERN k k 1 requires more than k rounds on sublogarithmic STMs. Developing a simpler and more general proof technique we are now able to improve this separation substantially [LiRe96c]. Theorem2. For S 2 o(log ) and for any integer k 2 , no S {space-bounded STM can recognize PATTERN k in d2k=3e rounds, that is PATTERN k 62 AM d2k=3e Space(o(log)) [ MA d2k=3e Space(o(log)) : These impossibility results for space bounds in SUBLOG can be translated to arbitrary small bounds (remember functions in SUBLOG grow at least as the twice iterated logarithm). Such a technique will not work for deterministic or nondeterministic machines. Similarly to [FrKa94] for Monte Carlo machines, this can be done by a translation technique based on a suitable encoding. De nition. For a string W = w 0 w 1 : : : w n over and a symbol a 62 de ne E(W;a) := w 0 a 2 0 w 1 a 2 1 w 2 a 2 2 : : : w i a 2 i : : : w n a 2 n : Let # 1 ;# 2 ; : : : be symbols not in . Then E 0 (W ) := W ; and for i 1 E i (W ) := E(E i 1 (W );# i ) : For integers k and i a padded version of PATTERN is constructed by PATTERN k (i) := f U#u#E i (BIN(2 d )) j U; u 2 f0; 1; 2g + ; d 2 IN; juj = d; u occurs at least k times in U g: Note that PATTERN k (0) is just the language PATTERN k . Combining the lower bound argument and this translation technique one can show Theorem3. For k 1 and i 0 , it holds PATTERN 2k (i) 62 AM k Space(o(log (i+1) )) [ MA k Space(o(log (i+1) )) : 4 An In nite Hierarchy and Nonclosure under