Abstract
We show that, for nondeterministic and alternating machines with weak space bounds, the minimal space that is required for accepting a nonregular language by real-time or one-way multicounter automata is \((\log n)^{\varepsilon }\!\). The same space is required for two-way multicounter automata, independent of whether they are deterministic, nondeterministic, or alternating, and of whether they work with strong or weak space bounds. On the other hand, for deterministic, nondeterministic, and alternating machines with strong space bounds, and also for deterministic machines with weak space bounds, we show that the minimal space required for accepting a nonregular language by real-time or one-way multicounter automata is \(n^{\varepsilon }\!\). All these bounds hold both for unary and general nonregular languages. Here \(\varepsilon \) represents an arbitrarily small—but fixed—real positive constant; the “space” refers to the values stored in the counters, rather than to the lengths of their binary representation.
Supported under contracts VEGA 1/0056/18 and APVV-15-0091.
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Notes
- 1.
Throughout the paper, \(\log x\) denotes the binary logarithm of x.
- 2.
Equivalently, the counter may be viewed as a special case of the pushdown store, the contents of which are always in the form \(\vdash \!X^h\!\), where \(\vdash \) denotes the bottom-of-pushdown-store-endmarker.
- 3.
An important detail is that leaving this loop is disabled, whenever \(\mathcal {C}_3=0\) or \(\mathcal {C}_4=0\). This ensures that we cannot choose k be equal to a power of two—to be described later.
- 4.
Throughout the entire computation, \(s=\mathsf {even}\) if and only if \(\mathcal {C}_1 + \mathcal {C}_2\) is even.
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Geffert, V., Bednárová, Z. (2018). Minimal Useful Size of Counters for (Real-Time) Multicounter Automata. In: Durand-Lose, J., Verlan, S. (eds) Machines, Computations, and Universality. MCU 2018. Lecture Notes in Computer Science(), vol 10881. Springer, Cham. https://doi.org/10.1007/978-3-319-92402-1_6
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