First Order Extensions of Residue Classes and Uniform Circuit Complexity

OUP accepted manuscript

“…The first significant lower bounds were given by Furst, Saxe and Sipser in [12] and Ajtai in [1], with further improvement by Hastad [15], demonstrating that the PARITY function, which can be decided with MOD 2 gates, can not be decided in AC 0 . Therefore, AC 0 = ACC (2). Another bound given by Smolensky in [22], proves that if p and q are distinct primes then ACC(p) = ACC(q).…”

“…We define Res(0, +, ×)+MOD = Res(0, +, ×)+ q>0 MOD(q) and Res(0, +, ×, <) In the presence of a built-in order relation it is logically indistinct to work with the standard finite models A m or with the finite residue classes Z m . This is the contents of the following theorem whose proof can be found in [2].…”

“…A first order logic for finite residue classes, denoted as Res(0, +, ×) with + and × being modular addition and multiplication, and extensions of this logic with certain generalized quantifiers, were shown by us in [2] to coincide with the first order logic, and corresponding extensions, for the standard finite models with arithmetic operations as considered in [16]. Therefore, from the descriptive complexity perspective, the computational complexity classes that can be described with these logics for finite residue classes are the DLOGTIME-uniform circuit complexity classes, consisting of circuits of constant depth and polynomial size for which a description can be efficiently computed from the size of their inputs.…”

“…The perspective of finite residue classes as instances of problems in these circuit complexity classes allow us to leverage the algebraic machinery proper of finite rings and fields of integer polynomials, algebraic number theory and, in general, from class field theory to study expressibility problems in these logics. In [2] we gave a brief account of the use of some of these algebraic tools to distinguish the expressive power of the logic of finite residue classes from its extensions with generalized quantifiers and in the addition of a built-in order. In this paper, we review those tools in more detail and expand our methods and results.…”

“…The paper is organized as follows. After some necessary preliminaries in circuit complexity and descriptive complexity (Section 2), we define in Section 3 the logic of finite residue classes, and state without proof our result from [2] on capturing circuit complexity classes by these logics. In Section 4 we review the notion of the prime spectrum of a sentence, its characterization as the spectrum of polynomial equations and the properties of the Boolean algebra of this spectra, derived from the seminal results by Ax [3,4].…”