M. Sipser, "Halting space-bounded computations," Theoretical Computer Science, 10: 335-338, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a linear ordering on the transitions of M (the order in which the transitions appear in the string representing M ) This extends to a linear ordering on the incoming edges for each node in the configuration graph of M . The simulation is just a depth first search as found in LMT[5] and Sipser[7]. DFS(M,C,S, q) 0) If C is in state q, then return C. If C uses more than space S, return undefined. 1) If M entered C via the last incoming edge to C (with respect to the ordering) a) If M is in state q, then return C. b) Let (C, D) be the outgoing edge of C. If D uses space S 1, ....

M. Sipser, Halting space-bounded computations, Theor. Comput. Sci., 10 (1980) 335-338. 8

....S, then the space use is S 2 . This method can be modelled by a reversible pebble game. Reference [12] demonstrated that Bennett s method is optimal for reversible pebble games and that simulation space can be traded off against limited erasing. In [9] it was shown that using a method by Sipser [16] one can reversibly simulate using only O(S) extra space but at the cost of using exponential time. Results: Previous results seem to suggest that a reversible simulation is stuck with either quadratic space use or exponential time use. This impression turns out to be false: Here we prove a ....

M. Sipser, Halting space-bounded computation, Theoret. Comp. Sci., 10(1990), 335--338. 11

....alternately in the directions (2; 1) and ( 1; 2) to verify that it is in a square of side 2 n . These kinds of arithmetic properties would require a context sensitive grammar to recognize in one dimension. DFAs can get stuck in loops and run forever. However, we can use an argument of Sipser [55] to convert any DFA into one which always arrives in the lower right corner (in an accepting or non accepting state) and never gets stuck in a loop. This works by starting in the lower right corner in an accepting state, doing a depth rst backwards search of the tree of all possible trajectories ....

....The class of DFA languages is closed under intersection, union and complement. Proof. For intersection, run the rst DFA and then the second, returning to the upper left corner after the rst one accepts. For complement, use the backwards search from the accepting state described above from [55]. Then the union can be written using De Morgan s law, L 1 [ L 2 = L 1 L 2 . 20 Proposition. The class of NFA languages is closed under union and intersection. Proof. For intersection, run one DFA after the other. For union, nondeterministically choose at the outset which one to run. We use ....

M. Sipser, \Halting space-bounded computations." Theoretical Computer Science 10 (1980) 335-338.

....appearing in the upper left corner of the picture. A fair amount is known about the closure properties of these classes. The DFAs, NFAs, AFAs and REC are all closed under intersection and union using straightforward constructions. DFAs are also closed under complement by an argument of Sipser [Sip80] which allows us to remove the danger that a DFA might loop forever and never halt. We construct a new DFA that starts in the nal halt state, which we can assume without loss of generality is in the lower right hand corner. Then this DFA does a depth rst search backwards, attempting to reach the ....

....than the number of states in B plus two. This contradicts the corollary to Lemma 4, so we have proved Theorem 4. The NFA recognizable picture languages are not closed under complementation, even for a one symbol alphabet. Since the DFA recognizable languages are closed under complementation [Sip80] it follows that S is not in DFA. We can show this directly, without the help of closure properties, using the deterministic variants of Lemma 3 and Lemma 4. Furthermore, it is easy to see that if a language is recognized by a loop free NFA, then its complement is accepted by an AFA whose states ....

M. Sipser, \Halting space-bounded computations." Theoretical Computer Science 10 (1980) 335-338.

....is known about the closure properties of these classes as well. The DFA, NFA, and h(LLL) languages are all closed under intersection and union using straightforward constructions. The situation for complement is somewhat more complicated. DFAs are closed under complement by an argument of Sipser [12] which allows us to remove the danger that a DFA might loop forever and never halt. We construct a new DFA that starts in the final halt state, which we can assume without loss of generality is in the lower right hand corner. Then this DFA does a depth first search backwards, attempting to reach ....

....than the number of states in B plus two. This contradicts the corollary to Lemma 2.2, so we have proved Theorem 2.1. The family of rectangular sets recognizable by NFAs is not closed under complementation. Because the family of rectangular sets recognizable by DFA is closed under complementation [12] we have Corollary 2.2. There are rectangular sets recognizable by NFAs that are not recognized by any DFA. An example of such a set is our set S above. We can show it directly, without the help of closure properties, using the deterministic variants of Lemmmas 2.1 and 2.2. 3. RECOGNIZING ....

M. Sipser (1980) Halting space-bounded computations. Theoretical Computer Science 10 335338.

....at a time. Now we turn to the presentation of the main results of this section. 5 Theorem 3 S = rat ae 6 Gamma Dspace(log n) S 6= rat ae 6 Gamma Dspace(log n) and both inclusions are optimal. Proof. We prove the first relation. Because Dspace(log n) is closed under complementation [Sip 1980] and S 6= rat is the complement of S = rat , it follows the second relation. Let L belong to S = rat and let A be a PFA that recognizes L and computes the probability p. Let w be an input word of length n. w 2 L ( 1=2 = p(w) 2p 0 (w) b n : We check the last equality (whose ....

Sipser, M. Halting space-bounded Computation. Theoretical Computer Science 10, 1980, pp. 335-338.

....reversibly cycle through the configuration tree of the machine. For our purposes, it will suffice to consider the configuration tree as an undirected tree, in which each edge is duplicated. We will then in effect perform an Euler tour of the resulting tree. A similar technique was used by Sipser [Si80] to simulate an S(n) space bounded Turing machine by another S(n) space bounded machine which never loops on a finite amount of tape. Let G1 (M) be the infinite configuration graph of a single worktape linear space deterministic Turing machine M . Write C 0 (w) Gamma B q 0 Delta y w 1 w ....

....of w, and hence, of the true initial configuration from which it started. We can also adapt the above simulation to the case of general (non linear and not necessarily constructible) space bounds. We make use of the standard technique for avoiding constructibility, as applied for example by Sipser [Si80]: Theorem 3.3 Any function f computable irreversibly in space S(n) can be computed by a reversible Turing machine in the same space. Proof. Recall the proof of Theorem 3.1. To meaningfully discuss non constructible space bounds, we must drop the assumption that the space allotted to the ....

Michael Sipser, Halting Space-Bounded Computations, Theoretical Computer Science 10 (1990),pp. 335--338.

....standard techniques used for machine simulations do not work in this case, and for many standard properties it was not clear whether they also hold for sublogarithmic space bounds. For example, a novel technique was necessary to prove for small bounds S that DSpace(S) is closed under complement [Si80], where DSpace(S) denotes the set of all languages that can be recognized by S space bounded deterministic Turing machines (DTM) Savitch s technique to simulate nondeterministic machines by deterministic machines requires space (maxflog ; Sg) 2 because of the recursion depth and the necessity ....

....is closed under complement. This method cannot be applied in the sublogarithmic space world. Sipser, however, found another way to check whether the starting configuration can reach an accepting configuration without the danger of being trapped in a loop. He simulates backwards . Theorem 5 [Si80] DSpace(S) is closed under complement for arbitrary S . As the next result will show this property absolutely fails for weakly space bounded complexity classes. A special situation holds for bounded languages containing only strings of a certain block structure. Definition 3 Let Z : IN IN be a ....

M. Sipser, Halting space-bounded computations, Theoret. Comput. Sci. 10, 1980, 335-338.

....the standard techniques used for machine simulations did not work for sublogarithmic space, and it was not clear whether evident properties also hold for sublogarithmic space bounds. For example, a novel technique was necessary to prove for small bounds S that DSpace(S) is closed under complement [Si80], where DSpace(S) denotes the set of all languages that can be recognized by 2 Maciej Li skiewicz , Rudiger Reischuk S space bounded deterministic Turing machines (DTM) And Savitch s technique to simulate nondeterministic machines (NTM) by deterministic machines requires space (maxflog ; Sg) ....

....This method cannot be applied in the sublogarithmic space world. Sipser, however, found another way to check whether the starting configuration can reach an accepting configuration without the danger of being trapped in a loop. He simulates backwards . As a consequence we obtain Theorem 5 [Si80] For arbitrary S it holds: DSpace(S) is closed under complement . A special situation holds for bounded languages containing only strings of a certain block structure. Definition 3 Let Z : IN IN be a function. A language L f0; 1g is Z bounded if each X 2 L contains at most Z(jXj) zeros. ....

M. Sipser, Halting space-bounded computations, Theor. Comput. Sci. 10, 1980, 335-338.

....in the directions (2; Gamma1) and ( Gamma1; 2) to verify that it is in a square of side 2 n . These kinds of arithmetic properties would require a context sensitive grammar to recognize in one dimension. DFAs can get stuck in loops and run forever. However, we can use an argument of Sipser [50] to convert any DFA into one which always arrives in the lower right corner (in an accepting or non accepting state) and never gets stuck in a loop. This works by starting in the lower right corner in an accepting state, doing a depth first backwards search of the tree of all possible ....

....The class of DFA languages is closed under intersection, union and complement. Proof. For intersection, run the first DFA and then the second, returning to the upper left corner after the first one accepts. For complement, use the backwards search from the accepting state described above from [50]. Then the union can be written using De Morgan s law, L 1 [ L 2 = L 1 L 2 . Proposition. The class of NFA languages is closed under union and intersection. Proof. For intersection, run one DFA after the other. For union, nondeterministically choose at the outset which one to run. We use the ....

M. Sipser, "Halting space-bounded computations." Theoretical Computer Science 10 (1980) 335-338.

....reversibly cycle through the configuration tree of the machine. For our purposes, it will suffice to consider the configuration tree as an undirected tree, in which each edge is duplicated. We will then in effect perform an Euler tour of the resulting tree. A similar technique was used by Sipser [Si80] to simulate an S(n) space bounded Turing machine accepting a language Y (i.e. with no bounds on space when the input w = 2 Y ) by a Turing machine deciding Y in space S(n) Let G1 (M) be the infinite configuration graph of a single worktape linear space deterministic Turing machine M . Write C ....

MICHAEL SIPSER, Halting Space-Bounded Computations, Theoretical Computer Science 10 (1990),pp. 335--338.

.... It is an interesting open problem whether Pi k Space(S) co Sigma k Space(S) for k = 1; 2; see the discussion in [14] Here, we obtain the following partial solution generalizing Sipser s result on halting spacebound computation for sublogarithimic space bounded deterministic TMs [19]: For bounded languages it can be shown that there exist equivalent ATMs that always halt. This implies Theorem 1.8. Let S 2 SUBLOG be a space bound and Z be a function computable in space S with Z exp S. Then for all k 1 and for every Z bounded language L f0; 1g holds: L 2 Sigma k ....

M. Sipser, Halting space-bounded computations, Theoret. Comput. Sci., 10 (1980), pp. 335--338.

....simulations of an irreversible computation can essentially be represented as the pebble game defined below, and that consequently the lower bound of Corollary 2 applies to all reversible simulations of irreversible computations. This conjecture was refuted in [11] using a technique due to [18] to show that there exists a general reversible simulation of an irreversible computation using only order S space at the cost of using a thoroughly unrealistic simulation time exponential in S. In retrospect the conjecture is phrased too general: it should be restricted to useful ....

M. Sipser, Halting space-bounded computations, Theoretical Computer Science, 10(1980), 335-338.

....to enable consideration of smaller space bounds on the storage tape [MW79] Neither of these reports distinguished between space bounds that were within a constant factor of each other, and neither handled space bounds smaller than logarithmic in the input length. Subsequently, Sipser s discovery [Si80] of a space efficient way for a Turing machine to detect its own infinite loops made it possible to overcome both these limitations [Le79] Our exposition includes these refinements and also careful extension of the results to bounds and problems that are not total. The latter formulations follow ....

....0 (x) then [ 0 is computable by a third STM M 00 in space S 00 (x) min(S(x) S 0 (x) The idea is for M 00 alternately to attempt simulation of M and M 0 from scratch within successively larger space preallocations, until one concludes. By employing Sipser s loop testing method [Si80], M 00 can insure that no simulation attempt runs forever. A minor subtlety in the simulations of M and M 0 above is recognition of the right ends of these machines storage tapes, since the simulator s actual storage tape is usually already longer. To mark the right end of its own tape, each ....

[Article contains additional citation context not shown here]

M. Sipser, Halting space-bounded computations, Theoretical Computer Science 10, 3 (March 1980), 335--338.

....standard techniques used for machine simulations do not work in this case, and for many standard properties it was not clear whether they also hold for sublogarithmic space bounds. For example, a novel technique was necessary to prove for small bounds S that DSpace(S) is closed under complement [Si80], where DSpace(S) denotes the set of all languages that can be recognized by S space bounded deterministic Turing machines (DTM) Savitch s technique to simulate nondeterministic machines by deterministic machines requires space (maxflog ; Sg) 2 because of the recursion depth and the necessity ....

....is closed under complement. This method cannot be applied in the sublogarithmic space world. Sipser, however, found another way to check whether the starting configuration can reach an accepting configuration without the danger of being trapped in a loop. He simulates backwards . Theorem 5 [Si80] DSpace(S) is closed under complement for arbitrary S . As the next result will show this property absolutely fails for weakly space bounded complexity classes. A special situation holds for bounded languages containing only strings of a certain block structure. Definition 3 Let Z : IN IN be a ....

M. Sipser, Halting space-bounded computations, Theoret. Comput. Sci. 10, 1980, 335-338.

....be in n different positions. However, with less than lg n bits, the machine cannot count up to n. So, the standard configuration counting argument does not work. Nevertheless, Sipser showed by an elegant argument that all deterministic space bounded classes are indeed closed under complementation [14]. We state a special case of Sipser s theorem. Theorem 8 SPACE [lg lg n] co SPACE [lg lg n] Proof: Sketch) For a detailed proof see [14] The proof of this theorem is based on the observation that the SPACE [lg lg n] machine accepts if and only if there is a backwards path from the ....

....does not work. Nevertheless, Sipser showed by an elegant argument that all deterministic space bounded classes are indeed closed under complementation [14] We state a special case of Sipser s theorem. Theorem 8 SPACE [lg lg n] co SPACE [lg lg n] Proof: Sketch) For a detailed proof see [14] . The proof of this theorem is based on the observation that the SPACE [lg lg n] machine accepts if and only if there is a backwards path from the unique accepting configuration to the unique initial configuration. By backwards , we mean that the path begins with the accepting configuration ....

M. Sipser. Halting space-bounded computations. Theoretical Computer Science, 10:335--338, 1980.

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M. Sipser, "Halting space-bounded computations," Theoretical Computer Science, 10: 335-338, 1980.

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M. Sipser, Halting space bounded computations, Theoretical Computer Science, 10 (1980), 335 -- 338.

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