Descriptive complexity of #P functions: A new perspective

Abstract

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 $\mathrm{\#}{\text{AC}}^0$ appear as classes of this hierarchy. In this way, we unconditionally place $\mathrm{\#}{\text{AC}}^0$ 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 $\mathrm{\#}{\text{AC}}^0$ which can be descriptively characterized as a class in our framework.

Introduction

The complexity of arithmetic computations is a current focal topic in complexity theory. Most prominent is Valiant's class #P of all functions that count accepting paths of nondeterministic polynomial-time Turing machines. This class has interesting complete problems like counting the number of satisfying assignments of propositional formulae or counting the number of perfect matchings of bipartite graphs (which is equivalent to computing the permanent of the adjacency matrix of such graphs [1]).

The class #P has been characterized in a model-theoretic way by Saluja, Subrahmanyam and Thakur in [2]. Their characterization is a natural generalization of Fagin's Theorem: Given a first-order formula with a free relational variable, instead of asking if there exists an assignment to this variable that makes the formula true (NP = ESO), we now ask to count how many such assignments there are. In this way, the class #P is characterized: $\mathrm{\#}\text{P}=\mathrm{\#}{\text{FO}}^{\mathrm{rel}}$. We use the superscript rel to denote that we are counting assignments to relational variables. The decision version PP of #P has been characterized from a descriptive complexity point of view in [3].

From another point of view, the class #P can be seen as the class of those functions that can be computed by arithmetic circuits of polynomial size, i.e., circuits with plus and times gates instead of the usual Boolean gates (cf., e.g., [4]). This is why here we speak of arithmetic computations. In the following, all circuit complexity classes we are referring to will be FO-uniform classes, which means that there are FO-formulae describing the circuits for all input lengths (a formal definition will be given).

It is very natural to restrict the resource bounds of such arithmetic circuits. An important class defined in this way is the class $\mathrm{\#}{\text{AC}}^0$ of all functions computed by polynomial-size bounded-depth arithmetic circuits. It is interesting to note that $\mathrm{\#}{\text{AC}}^0$ and all analogous classes defined by arithmetic circuits, i.e., plus-times circuits, can also be defined making use of a suitable counting process for Boolean circuits in which negations only occur in the form of input gates being labeled as negated: A witness that such a Boolean circuit accepts its input is a so-called proof tree of the circuit, i.e., a subtree of the circuit unwound into a tree containing the output gate and for each contained gate a minimum number of its inputs that allow to deduce that it evaluates to 1. That also means that all contained input gates have to evaluate to 1. Then the arithmetic class $\mathrm{\#}{\text{AC}}^0$, restricted to binary inputs, can be characterized as the counting class of all functions that count proof trees of (Boolean) ${\text{AC}}^0$ circuits. The correspondence between arithmetic computations and counting classes is explored in [5]. In this paper, we are mainly interested in these counting classes, and without further mention we use the notation $\mathrm{\#}{\text{AC}}^0$ in this vein.

There was no model-theoretic characterization of $\mathrm{\#}{\text{AC}}^0$, until it was recently shown in [6] that $\mathrm{\#}{\text{AC}}^0=\mathrm{\#}{\mathrm{\Pi }}_{\mathrm{1}}^{\mathrm{Skolem}}$, where $\mathrm{\#}{\mathrm{\Pi }}_{\mathrm{1}}^{\mathrm{Skolem}}$ means counting of possible Skolem functions for FO-formulae.

The aim of this paper is to compare the model-theoretic characterization obtained in [2] to that from [6] in order to get a unified view of both arithmetic circuit classes, $\mathrm{\#}{\text{AC}}^0$ and #P. This is done by noticing that the number of Skolem functions of an FO-formula can be counted as satisfying assignments to free function variables in a ${\mathrm{\Pi }}_1$-formula. This gives rise to the idea to restate the result by Saluja et al. counting functions instead of relations. We call our class where we count assignments to function variables #FO, in contrast to Saluja et al.'s $\mathrm{\#}{\text{FO}}^{\mathrm{rel}}$. In this setting, we get $\mathrm{\#}\text{P}=\mathrm{\#}\text{FO}=\mathrm{\#}{\mathrm{\Pi }}_{\mathrm{1}}$, which places both classes within $\mathrm{\#}{\mathrm{\Pi }}_{\mathrm{1}}$.

Furthermore, we show that $\mathrm{\#}{\text{AC}}^0$ actually corresponds to a syntactic fragment $\mathrm{\#}{\mathrm{\Pi }}_{\mathrm{1}}^{\mathrm{prefix}}$ of $\mathrm{\#}{\mathrm{\Pi }}_{\mathrm{1}}$ and, considering further syntactic subclasses of #FO defined by quantifier alternations, we get the inclusions

Thus we establish (unconditionally, i.e., under no complexity theoretic assumptions) the complete structure of the alternation hierarchy within #FO and show where $\mathrm{\#}{\text{AC}}^0$ is located in this hierarchy.

Once we know that only universal quantifiers suffice to obtain the full class, i.e., $\mathrm{\#}{\mathrm{\Pi }}_{\mathrm{1}}=\mathrm{\#}\text{P}$, it is a natural question to ask how many universal quantifiers are needed to express certain functions. We obtain the result that the hierarchy based on the number of universal variables is infinite; however, a possible connection to the depth hierarchy within $\mathrm{\#}{\text{AC}}^0$ remains open.

We also study the question of which of the classes in our hierarchy are tractable. The main result we obtain is that all functions in ${\mathrm{\Sigma }}_1$ posses a fully polynomial-time randomized approximation scheme (see e.g. [7]). For this, we consider a generalization of disjunctive normal-form we call “pseudo-DNF”. A pseudo-DNF has identities of the form $f(\bar{a})=b$ or $f(\bar{a})\ne b$ as literals, where f is a function symbol, $\bar{a}\in {\mathbb{N}}^k$ for some $k\in \mathbb{N}$ and $b\in \mathbb{N}$. These are then evaluated over assignments to the function symbols. The main technical result we obtain in this area is an FPRAS for counting satisfying assignments for pseudo-k-DNF formulae. Then we show that every problem in $\mathrm{\#}{\mathrm{\Sigma }}_1$ can be reduced to such a problem. The concept of pseudo-DNF and the approximability may be of independent interest.

We see that counting assignments to free function variables instead of relation variables in first-order formulae leads us to a hierarchy of arithmetic classes suitable for a study of the power and complexity of the class $\mathrm{\#}{\text{AC}}^0$. We show that the hierarchy introduced by Saluja et al. [2] is not suitable for such a goal, see Section 7.

This paper is an extended full version of an earlier conference paper.¹ In the meantime, another paper appeared [8] that extended the framework by Saluja et al. In that paper, a so called quantitative logic is introduced, where arithmetic operators are introduced into second-order logic. This framework allowed for characterizations of the classes FP, #P and FPSPACE. Furthermore, in a recent paper [9], the study of descriptive complexity of arithmetic circuit classes was extended from $\mathrm{\#}{\text{AC}}^0$ to other small arithmetic circuit classes: Inspired by a work of Compton and Laflamme [10], variants of a certain recursion on predicates was introduced into FO. This yields characterizations of the Boolean classes ${\text{NC}}^1$, ${\text{SAC}}^1$ and ${\text{AC}}^1$ which extend to the respective counting classes. Another recent paper [11] studied functions counting assignments to free relational variables in ${\mathrm{\Sigma }}_1^1$ formulae, including interesting connections to counting problems in team-based logics. In this way, a characterization of the class $\mathrm{\#}\cdot \text{NP}$ was obtained.

This paper is organized as follows: In the next section, we introduce relevant concepts from finite model theory. Here, we also introduce the Saluja et al. hierarchy, and we explain the model-theoretic characterization of $\mathrm{\#}{\text{AC}}^0$. In Sect. 3 we introduce our new framework and the class #FO and its subclasses. In Sect. 4 we determine the full structure of the alternation hierarchy within #FO and place $\mathrm{\#}{\text{AC}}^0$ in this hierarchy. In Sect. 5 we study the class $\mathrm{\#}{\mathrm{\Sigma }}_1$ with respect to efficient approximability and show that every function in this class admits an FPRAS. Finally, we conclude in Sect. 8 with some open questions. In Sect. 6 we study the hierarchy defined by the number of universal variables in the $\mathrm{\#}{\mathrm{\Pi }}_{\mathrm{1}}$-fragment. In Sect. 7 we turn to the hierarchy defined by Saluja et al. and show that the arithmetic class $\mathrm{\#}{\text{AC}}^0$ is incomparable to all except the level-0 class and the full class of this hierarchy.

Our proofs make use of a number of different results and techniques, some stemming from computational complexity theory (such as separation of Boolean circuit classes or the time hierarchy theorem for nondeterministic RAMs), some from model theory (like closure of certain fragments of first-order logic under extensions or taking substructures) or descriptive complexity (correspondence between time-bounded NRAMs and fragments of existential second-order logic). Most techniques have to be adapted to work in our very low complexity setting (new counting reductions, use of the right set of built-in relations, etc.). Our paper sits right in the intersection of finite model theory and computational complexity theory.