The Complexity of Membership Problems for Circuits over Sets of Integers

Emptiness problems for integer circuits

“…We prove that PIT is logspace many-one equivalent to MC(∩, +, ×) studied in [22], MC Z (+, ×), MC Z (∩, +, ×) studied in [32], and EQ(+, ×) studied in [11]. This characterizes the complexity of these problems and shows that the question for improved bounds is equivalent to a well-studied, open problem in algebraic computing complexity.…”

“…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. Travers [32] and Breunig [6] considered membership problems for circuits over integers (MC Z ) and positive integers (MC N + ), respectively. Glaßer et al [11] investigated equivalence problems for circuits over sets of natural numbers (EQ), i.e., the problem of deciding whether two given circuits compute the same set.…”

“…The integer variants MC Z (O), EC Z (O), and Σ 1 -EC Z (O) are defined analogously (assigned and unassigned inputs are from Z, the complement is defined with respect to Z). A systematic study of the membership problems MC Z (O) was done by Travers [32]. We use the following abbreviations: we write n for the singleton {n}; we write C for I(C), where C is a circuit; we write MC(∪, ∩, , +, ×) for MC({∪, ∩, , +, ×}) and the like.…”

“…Then we provide arguments that suggest the difficulty of proving the decidability of EC( , +, ×) and EC(∪, ∩, , +, ×, ). Finally we draw connections to polynomial identity testing (PIT) and show that the open questions for the complexities of MC(∩, +, ×) [22], MC Z (∩, +, ×), MC Z (+, ×) [32], and EQ(+, ×) [11] are equivalent to the well-studied, open question for the complexity of PIT.…”

Circuit satisfiability and constraint satisfaction around Skolem Arithmetic

“…Apart from the research on circuit problems mentioned above there has been work on other variants like circuits over integers [26] and positive natural numbers [27], equivalence problems for circuits [28], functions computed by circuits [29], and equations over sets of natural numbers [30,31]. Typically, the complexity of membership of circuits is similar to the corresponding equivalence of circuits problem, though the latter may be slightly higher and belies some imperfect bounds 2 .…”