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.
|Name||Lecture notes in computer science|
|Conference||5th Conference on Computability in Europe, CiE 2009|
|Period||1/07/09 → …|
- Arithmetic circuit
- Complex algebra
- Expressive power
- Integer expression