A Logical Characterization of Constant-Depth Circuits over the Reals

In this paper we give an Immerman's Theorem for real-valued computation. We define circuits operating over real numbers and show that families of such circuits of polynomial size and constant depth decide exactly those sets of vectors of reals that can be defined in first-order logic on R-structures in the sense of Cucker and Meer. Our characterization holds both non-uniformily as well as for many natural uniformity conditions.Comment: 24 pages, submitted to WoLLIC 202

On the expressive power of planar perfect matching and permanents of bounded treewidth matrices

Valiant introduced some 25 years ago an algebraic model of computation along with the complexity classes VP and VNP, which can be viewed as analogues of the classical classes P and NP. They are defined using non-uniform sequences of arithmetic circuits and provides a framework to study the complexity for sequences of polynomials. Prominent examples of difficult (that is, VNP-complete) problems in this model includes the permanent and hamiltonian polynomials. While the permanent and hamiltonian polynomials in general are difficult to evaluate, there have been research on which special cases of these polynomials admits efficient evaluation. For instance, Barvinok has shown that if the underlying matrix has bounded rank, both the permanent and the hamiltonian polynomials can be evaluated in polynomial time, and thus are in VP. Courcelle, Makowsky and Rotics have shown that for matrices of bounded treewidth several difficult problems (including evaluating the permanent and hamiltonian polynomials) can be solved efficiently. An earlier result of this flavour is Kasteleyn's theorem which states that the sum of weights of perfect matchings of a planar graph can be computed in polynomial time, and thus is in VP also. For general graphs this problem is VNP-complete. In this paper we investigate the expressive power of the above results. We show that the permanent and hamiltonian polynomials for matrices of bounded treewidth both are equivalent to arithmetic formulas. Also, arithmetic weakly skew circuits are shown to be equivalent to the sum of weights of perfect matchings of planar graphs.Comment: 14 page

Descriptive Complexity of #AC^0 Functions

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC^0 appear as classes of this hierarchy. In this way, we unconditionally place #AC^0 properly in a strict hierarchy of arithmetic classes within #P. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al. We argue that our approach is better suited to study arithmetic circuit classes such as #AC^0 which can be descriptively characterized as a class in our framework

Descriptive complexity of #P functions : A new perspective

We introduce a new framework for a descriptive complexity approach to arithmetic computations. We define a hierarchy of classes based on the idea of counting assignments to free function variables in first-order formulae. We completely determine the inclusion structure and show that #P and #AC0 appear as classes of this hierarchy. In this way, we unconditionally place #AC0 properly in a strict hierarchy of arithmetic classes within #P. Furthermore, we show that some of our classes admit efficient approximation in the sense of FPRAS. We compare our classes with a hierarchy within #P defined in a model-theoretic way by Saluja et al. and argue that our approach is better suited to study arithmetic circuit classes such as #AC0 which can be descriptively characterized as a class in our framework.Peer reviewe