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Two-Way Automata Characterizations of L/poly Versus NL

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Abstract

Let L/p o l y and N L be the standard complexity classes, of languages recognizable in logarithmic space by Turing machines which are deterministic with polynomially-long advice and nondeterministic without advice, respectively. We recast the question whether \(\mathsf {L}/\mathsf {poly}~{\supseteq }~{\mathsf {NL}}\) in terms of deterministic and nondeterministic two-way finite automata (2dfas and 2nfas). We prove it equivalent to the question whether every s-state unary 2nfa has an equivalent poly(s)-state 2dfa, or whether a poly(h)-state 2dfa can check accessibility in h-vertex graphs (even under unary encoding) or check two-way liveness in h-tall, h-column graphs. This complements two recent improvements of an old theorem of Berman and Lingas. On the way, we introduce new types of reductions between regular languages (even unary ones), use them to prove the completeness of specific languages for two-way nondeterministic polynomial size, and propose a purely combinatorial conjecturethat implies \(\mathsf {L}/\mathsf {poly}~{\nsupseteq }~\mathsf {NL}\).

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Notes

  1. It is tempting to consider the following, simpler simulation. From (q,i) on \({\vdash }\), \(M^{\prime }\) sweeps the tape deterministically up to cell i (using states of the form (q,i,j) with j<i), where it records x i in its memory; it then completes the sweep in state (q,i,x i ), jumping back to \({\vdash }\) in that same state. From there, it transitions nondeterministically to states of the form (p,i±1) according to the choices of M from q on x i . This can be implemented with O(s l 2+s l σ) states, where σ=|Σ|. If σ is constant in s, then this is O(s l 2) states, better than in Lemma 1. If σ=poly(s), then it increases to poly(s)⋅l 2 states, still good for Corollary 1. But if σ is super-polynomial in s, then the size of \(M^{\prime }\) becomes super-polynomial as well, and Corollary 1 cannot follow.

  2. As suggested by a reviewer, if we require that \(M^{\prime }\) agrees with M only on inputs of length exactly l, then the simulation can be improved reducing to O(s 2 l) the number of states of \(M^{\prime }\). This can be done as follows. The event of M being in state q and on tape cell i is directly simulated by \(M^{\prime }\) in the same state q on the same tape cell i. From this configuration, using a counter bounded by l, \(M^{\prime }\) can ‘rotate’ along the input to reach again cell i. During this phase, when \(M^{\prime }\) reaches \({\vdash }\), it makes a nondeterministic ‘guess’ of a pair (p,d)∈S×{L,R}, and when it finally reaches cell i, \(M^{\prime }\) verifies that the guess corresponds to a valid transition of M. If this is the case, then \(M^{\prime }\) moves to cell i+d (in the case d=−1 this is done again using a rotation and a counter), otherwise it hangs.

  3. Actually the construction in [3] assumes acceptance on \({\vdash }\) in \(\tilde {q}_{f}\). Furthermore, \(\tilde {q}_{f}\) can be entered only from some states in \(\tilde {S}_{B}\) on \({\vdash }\), without moving the input head. To be consistent with the acceptance condition given in Section 2.1, we make the following minor changes to \(\tilde {M}\), which do not affect the set of states. Each transition entering \(\tilde {q}_{f}\) on \({\vdash }\) moves the head to the right. Furthermore, in the state \(\tilde {q}_{f}~\tilde {M}\) always moves to the right, without changing state. This is done even on ⊣. In this way, once \(\tilde {q}_{f}\) is entered, \(\tilde {M}\) completely scans the input by remaining in \(\tilde {q}_{f}\), in order to accept after violating ⊣.

  4. We remind the reader that, since \(\tilde {M}\) is a sofa, the direction of the head in its transitions is implicit, i.e, the set of transitions is a subset of \(\tilde {S}\times ({\varSigma }\cup \{{\vdash }{,}{\dashv }\})\times \tilde {S}\).

  5. Notice that \(\tilde {M}\) has the transition \((q_{\mathrm {f}},{\dashv },\tilde {q}_{\mathrm {f}})\) (cf. note 3). Hence, q≠q f here.

  6. The two-way liveness problem is classically defined using two-column graphs as alphabet symbols. As pointed out by a reviewer, we could use a modified version of this problem, where each alphabet symbol is defined by just one column with outgoing edges, to the left and to the right, that reach, when symbols are concatenated, vertices in the adjacent symbols. Let \(\text {TWL}^{\prime }_{h}\) be the problem so modified for graphs of height h. Then in the statement of Lemma 5 we can write \(\mathfrak {L}\leq _{\text {h}}\text {TWL}^{\prime }_{s}\).

  7. Notice that even in this case, the string produced by T in a printing step (which contains one or more infixes #z i ) does not need to be explicitly represented: it only depends on the simulated printing step of T.

  8. When a printing step of T produces an infix #z i #z i+1⋯#z k , M 1 can compute in a single step \(\bigl (\bigl (\cdots \bigl ((\rho \cdot n_{i} \bmod l_{j} ) \cdot n_{i+1} \bigr ) \bmod l_{j} \cdots \bigr ) \cdot n_{k} \bigr ) \bmod l_{j} \,. \) Since the infix depends only on the printing step of T, this can be embedded in the transition function of M 1, without explicitly representing the infix #z i #z i+1⋯#z k .

  9. Fix any bijection π from the binary relations on [h] to the numbers in \([2^{h^{2}}]\). Then the homomorphic image f(A,L) of a left symbol (A,L) is the partial function that only maps the number π(A) to itself. The image f(B,R) of a right symbol (B,R) is the partial function in which a number i can only map to itself, and this holds iff i=π(A) for some A that matches with B. Clearly then, A,B match iff f(A,L),f(B,R) match (by the one two-step cycle going from π(A) on the left column to π(A) on the right column and back).

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Acknowledgments

The first author has been supported by a Marie Curie Intra-European Fellowship (pief-ga-2009-253368) within the European Union Seventh Framework Programme (fp7/2007-2013).

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Authors and Affiliations

  1. Carnegie Mellon University in Qatar - Education City, P.O. Box 24866, Doha, Qatar

    Christos A. Kapoutsis

  2. DI, Università degli Studi di Milano, I-20135, Milano, Italy

    Giovanni Pighizzini

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  1. Christos A. Kapoutsis
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  2. Giovanni Pighizzini
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Correspondence to Giovanni Pighizzini.

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Preliminary version presented in the 7th International Computer Science Symposium in Russia, Nizhny Novgorod, July 3-7, 2012 [Lecture Notes in Computer Science vol. 7353, Springer-Verlag, pp. 217-228].

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Kapoutsis, C.A., Pighizzini, G. Two-Way Automata Characterizations of L/poly Versus NL. Theory Comput Syst 56, 662–685 (2015). https://doi.org/10.1007/s00224-014-9560-x

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