Abstract
We consider one-way logspace counting classes which are defined via Turing machines that scan their input only in one direction. The one-way logspace counting classes #1L and span-1L are strict subclasses of the corresponding (two-way) logspace classes #L and span-L, resp. We separate the one-way classes 1UL and 1NL which correspond to the classes UL and NL. It follows that F1L ⊂1L ⊂span-lL ⊂#P.
We generalize first-order counting classes to use <, SUCC, and + as linear orderings. It turns out that with respect to certain natural encodings for op ε { <, SUCC, +} the classes #gSo[op] and #gS1[op] are subclasses of #1L and span-1L. It holds that #Π2[<] = #Π1[SUCC] = #Π1[+], and that this class characterizes #P. From that, we obtain a characterization of #P via universally branching alternating logtime Turing machines.
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Burtschick, HJ. (1995). Comparing counting classes for logspace, one-way logspace, and first-order. In: Wiedermann, J., Hájek, P. (eds) Mathematical Foundations of Computer Science 1995. MFCS 1995. Lecture Notes in Computer Science, vol 969. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-60246-1_120
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DOI: https://doi.org/10.1007/3-540-60246-1_120
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