Abstract
We study the computational complexity of emptiness problems for circuits over sets of natural numbers with the operations union, intersection, complement, addition, and multiplication. For most settings of allowed operations we precisely characterize the complexity in terms of completeness for classes like NL, NP, and PSPACE. The case where intersection, addition, and multiplication is allowed turns out to be equivalent to the complement of polynomial identity testing (PIT).Our results imply the following improvements and insights on problems studied in earlier papers. We improve the bounds for the membership problem MC(∪,∩,x‾,+,×) studied by McKenzie and Wagner 2007 and for the equivalence problem EQ(∪,∩,x‾,+,×) studied by Glaßer et al. 2010. Moreover, it turns out that the following problems are equivalent to PIT, which shows that the challenge to improve their bounds is just a reformulation of a well-studied, major open problem in algebraic computing complexity:•membership problem MC(∩,+,×) studied by McKenzie and Wagner 2007•integer membership problems MCZ(+,×), MCZ(∩,+,×) studied by Travers 2006•equivalence problem EQ(+,×) studied by Glaßer et al. 2010.
Highlights
Stockmeyer and Meyer [31] investigated membership and equivalence problems for integer expressions, which are built up from single natural numbers using set operations (, ∪, ∩) and pairwise addition (+)
It turns out that the following problems are equivalent to polynomial identity testing (PIT), which shows that the challenge to improve their bounds is just a reformulation of a major open problem in algebraic computing complexity: membership problem MC(∩, +, ×) studied by McKenzie and Wagner 2007 integer membership problems MCZ(+, ×) and MCZ(∩, +, ×) studied by Travers 2006 equivalence problem EQ(+, ×) studied by Glaßer et al 2010
The membership problem for expressions is the question of whether the set described by a given expression contains some given natural number
Summary
Stockmeyer and Meyer [31] investigated membership and equivalence problems for integer expressions, which are built up from single natural numbers using set operations ( , ∪, ∩) and pairwise addition (+). Wagner [33], Yang [34], and McKenzie and Wagner [22] studied the complexity of membership problems for circuits over natural numbers (MC): Here, for a given circuit C with numbers assigned to the input gates, one has to decide whether a given number b belongs to the set described by the circuit. Satisfiability problems for circuits over sets of natural numbers, studied by Glaßer et al [13], are a generalization of the membership problems investigated by McKenzie and Wagner [22]: Here the circuits can have unassigned input gates. We show that EQ(∩, +, ×) is ≤lmog-complete for the complement of the second level of the Boolean hierarchy over PIT This characterizes the complexity of this equivalence problem and explains the difficulty of improving the known upper bound [11]. A comprehensive presentation is provided in the technical report [4]
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