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Two-Way Automata Versus Logarithmic Space

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Abstract

We strengthen a previously known connection between the size complexity of two-way finite automata () and the space complexity of Turing machines (tms). Specifically, we prove that

  • every s-state has a poly(s)-state that agrees with it on all inputs of length ≤s if and only if NL⊆L/poly, and

  • every s-state has a poly(s)-state that agrees with it on all inputs of length ≤2s if and only if NLL⊆LL/polylog.

Here, and are the deterministic and nondeterministic , NL and L/poly are the standard classes of languages recognizable in logarithmic space by nondeterministic tms and by deterministic tms with access to polynomially long advice, and NLL and LL/polylog are the corresponding complexity classes for space O(loglogn) and advice length poly(logn). Our arguments strengthen and extend an old theorem by Berman and Lingas and can be used to obtain variants of the above statements for other modes of computation or other combinations of bounds for the input length, the space usage, and the length of advice.

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Notes

  1. Note the unusual forms ‘A⊇B’ and ‘A⊉B’. In cases where A⊆B (e.g., when A=2D & B=2N), these are of course equivalent to the more usual ‘A=B’ and ‘A⊆̷B’. However, we will encounter cases where A⊆B is not true (e.g., when A=2D & B=2N/poly or when A=L/poly & B=NL—see below) and hence ‘A⊇B’ and ‘A⊉B’ are appropriate. We thus use these forms throughout the article, so that all statements are easy to compare. We read and think of these forms as ‘A covers/does not cover B’.

  2. Note that a set of designated final states F is not really necessary in this definition. Since the accepts by falling off ⊣, designating all states as accepting or just one state as accepting (e.g., q 0) would have had the same effect. With F, our definition stays comparable to other standard finite automata definitions.

  3. Note that, in contrast, for an with n>m we have no guarantee that A m  agrees with M under . It may be that the computation tree τ n of M on input x and advice y n =ε uses f(n) work tape cells on every accepting branch. Hence, if f(n)>f(m), our A m will be missing some of the states necessary to simulate these accepting branches in its computation tree \(\tau _{m}'\) on x. Therefore, τ n  will be accepting but \(\tau _{m}'\) will be rejecting—despite the fact that y n =y m =ε.

  4. These trailing 0s are, in fact, redundant. They are included in the definition just for symmetry with the definitions of the variants in the next two paragraphs.

  5. More tightly, the expansion of the first reduction is O(s 2 σ)+mlgσ. But the looser O(s 2 σm) is simpler and does not harm our conclusions, since it is later (Theorem 3.0, forward direction) fed to an unspecified polynomial. Similar looseness is adopted elsewhere, too.

  6. The actual statement of [3, Theorem 6] is very close to the usual citation (8); however, it also includes a pointer to a Remark a few pages later (p. 17), which explains that the promised can be constructed in logarithmic space and that with this observation the theorem becomes an equivalence. We also note that the actual statement of [3, Theorem 6] uses s⋅m as the length bound, for s the number of states in the , whereas the full statement (\(8^{\scriptscriptstyle +}\)) uses just m; the two statements are equivalent, but (\(8^{\scriptscriptstyle +}\)) is simpler and facilitates comparison with subsequent statements.

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Authors and Affiliations

  1. LIAFA, Université Paris Diderot—Paris VII, Case 7014, 75205, Paris Cedex 13, France

    Christos A. Kapoutsis

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Correspondence to Christos A. Kapoutsis.

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Research funded 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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Kapoutsis, C.A. Two-Way Automata Versus Logarithmic Space. Theory Comput Syst 55, 421–447 (2014). https://doi.org/10.1007/s00224-013-9465-0

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