Abstract
We propose a logical characterization of problems solvable in deterministic polylogarithmic time (\(\mathrm {PolylogTime}\)). We introduce a novel two-sorted logic that separates the elements of the input domain from the bit positions needed to address these elements. In the course of proving that our logic indeed captures \(\mathrm {PolylogTime}\) on finite ordered structures, we introduce a variant of random-access Turing machines that can access the relations and functions of the structure directly. We investigate whether an explicit predicate for the ordering of the domain is needed in our logic. Finally, we present the open problem of finding an exact characterization of order-invariant queries in \(\mathrm {PolylogTime}\).
The research reported in this paper results from the project Higher-Order Logics and Structures supported by the Austrian Science Fund (FWF: [I2420-N31]) and the Research Foundation Flanders (FWO: [G0G6516N]). It was further supported by the Austrian Research Promotion Agency (FFG) through the COMET funding for the Software Competence Center Hagenberg.
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- 1.
The term random access refers to the manner how random-access memory (RAM) is read and written. In contrast to sequential memory, the time it takes to read or write using RAM is almost independent of the physical location of the data in the memory. We want to emphasise that there is nothing random in random access.
- 2.
This ensures that \(F^\mathbf{A}_{\varphi , \bar{\mathtt {x}}, X}\) is monotonous and thus that the least fixed point \(\mathrm {lfp}(F^\mathbf{A}_{\varphi , \bar{\mathtt {x}}, X})\) is guaranteed to exists.
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Ferrarotti, F., González, S., Turull Torres, J.M., Van den Bussche, J., Virtema, J. (2019). Descriptive Complexity of Deterministic Polylogarithmic Time. In: Iemhoff, R., Moortgat, M., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2019. Lecture Notes in Computer Science(), vol 11541. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-59533-6_13
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