An arithmetic circuit is a labelled, directed, acyclic graph specifying a cascade of arithmetic and logical operations to be performed on sets of non-negative integers. In this paper, we consider the definability of functions from tuples of sets of non-negative integers to sets of non-negative integers by means of arithmetic circuits. We prove two negative results: the first shows, roughly, that a function is not circuit-definable if it has an infinite range and sub-linear growth; the second shows, roughly, that a function is not circuit-definable if it has a finite range and fails to converge on certain 'sparse' chains under inclusion. We observe that various functions of interest fall under these descriptions. © 2009 Springer Berlin Heidelberg.
CITATION STYLE
Pratt-Hartmann, I., & Düntsch, I. (2009). Functions definable by arithmetic circuits. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 5635 LNCS, pp. 409–418). https://doi.org/10.1007/978-3-642-03073-4_42
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