1 Introduction
The class [22] is the class of functions that count the number of solutions to problems in , e.g. #Sat is the function that on input a formula returns the number of satisfying assignments of
Equivalently, functions in count accepting paths of nondeterministic polynomial time Turing machines (NPTMs).
complete problems are hard to count, but it is not the case that problems in are easy to count as well. When we consider counting, nontrivial facts hold. First of all there exist complete problems, that have decision version in , e.g. #Dnf. Moreover, some of them can be approximated, e.g. the Permanent [12] and #Dnf [14], while others cannot, e.g. #Is [13]. The class of problems in with decision version in is called , and a subclass of is , which contains all selfreducible problems in [17]. Their significance will be apparent in what follows.
Since many counting problems cannot be exactly computed in polynomial time, the interest of the community has turned to the complexity of approximating them. On one side, there is an enormous literature on approximation algorithms and inapproximability results for individual problems in [13, 9, 12, 14, 22]
. On the other hand, there have been attempts to classify counting problems with respect to their approximability
[2, 3, 8, 19].1.0.1 Related work.
From a unifying point of view, the most important results regarding approximability are the following. Every function in either admits an fpras, or does not admit any polynomial approximation ratio [20]; we will therefore call the latter inapproximable. For selfreducible problems in , fpras is equivalent to almost uniform sampling [20]. With respect to approximation preserving reductions, there are three main classes of functions in [8]: (a) the class of functions that admit an fpras, (b) the class of functions that are interreducible with #Sat, and (c) the class of problems that are interreducible with #Bis. Problems in the second class do not admit an fpras unless =, while the approximability status of problems in the third class is unknown and the conjecture is that they are neither interreducible with #Sat, nor they admit an fpras. We will denote the class of problems that admit an fpras.
Several works have attempted to provide a structural characterization that exactly captures , in terms of path counting [4, 17], interval size functions [5], or descriptive complexity [3]. Since counting problems with complete decision version are inapproximable unless [8], those that admit fpras should be found among those with easy decision version (i.e., in or even in ). Even more specifically, in search of a logical characterization of a class that exactly captures , Arenas et al. [3] show that subclasses of are contained in , and they implicitly propose to study subclasses of with certain additional properties in order to come up with approximable problems. Notably, most problems proven so far to admit an fpras belong to , and several counting complexity classes proven to admit an fpras, namely , [19], , [3] and [2], are subclasses of .
Counting problems in have also been studied in terms of descriptive complexity [3, 6, 7, 8, 19]. Arenas et al. [3] raised the question of defining classes in terms of descriptive complexity that capture either or robust subclasses of , as one of the most important open questions in the area. A robust class of counting problems needs either to have a natural complete problem or to be closed under addition, multiplication and subtraction by one [3]. In particular, satisfies both of the above properties [3, 4].
1.0.2 Our contribution.
In the first part of the paper we focus on the exploration of the structure of through descriptive complexity. In particular, we define two subclasses of , namely and , via logical characterizations; for both these classes we show robustness by providing natural complete problems for them. Namely, we prove that the problem #Disj2Sat of computing the number of satisfying assignments to disjunctions of 2SAT formulae is complete for under parsimonious reductions. This reveals that problems hard for under parsimonious reductions cannot admit an fpras unless . We also prove that #MonotoneSat is complete for under product reductions. Our result is the first completeness result for #MonotoneSat under reductions stronger than Turing. Notably, the complexity of #MonotoneSat has been investigated in [11, 5] and it is still open whether it is complete for , or for a subclass of under reductions for which the class is downwards closed. Although, is not known to be downwards closed under product reductions, our result is a step towards understanding the exact complexity of #MonotoneSat.
In the second part of this paper we examine the relationship between the class and . As we already mentioned, most (if not all) problems proven so far to admit fpras belong to , so we would like to examine whether . Of course, problems in have decision version in [10], so if we assume
this is probably not the case. Therefore, a more realistic goal is to determine assumptions under which the conjecture
might be true. The world so far is depicted in Figure 1, where denotes the class of problems in with decision version in .In this work we refine this picture by proving that (a) unless =, which means that proving would be at least as hard as proving , (b) if and only if , (c) lies between two classes that can be seen as counting versions of and , and (d) , which is the subclass of with zero error probability when the function value is zero, lies between two classes that we introduce here, that can both be seen as counting versions of , and which surprisingly do not coincide unless =. Finally, we give a complete picture of inclusions among all the classes defined or discussed in this paper with respect to different conjectures about the vs. vs. questions.
2 Two robust subclasses of
In this section we give the logical characterization of two robust subclasses of . Each one of them has a natural complete problem. Two kinds of reductions will be used for the completeness results; parsimonious and product reductions. Note that both of them preserve approximations of multiplicative error [8, 19].
Definition 1
Let , be two counting functions.
(a) We say that there is a parsimonious (or Karp) reduction from to , symb. , if there is a polynomialtime computable function , such that for every it holds that .
(b) We say that there is a product reduction from to , symb. , if there are polynomialtime computable functions such that for every it holds that .
The formal definitions of the classes , , and follow.
Definition 2
(a) [22] is the class of functions for which there exists a polynomialtime decidable binary relation and a polynomial such that for all , .
Equivalently, .
(b) is the class of functions in that are computable in polynomial time.
(c) [17] , where is the decision version of the function .
(d) [17] , where all computation paths of on input .
2.1 The class
In order to define the first class we make use of the framework of Quantitative SecondOrder Logics (QSO) defined in [3].
Given a relational vocabulary , the set of FirstOrder logic formulae over is given by the grammar:
where are firstorder variables, , is a tuple of first order variables, represents a tautology, and represents the negation of a tautology.
We define a literal to be either of the form or , where is a secondorder variable and is a tuple of firstorder variables. A 2SAT clause over is a formula of the form , where each of the ’s, , can be either a literal or a firstorder formula over . In addition, at least one of them is a firstorder formula. The set of 2SAT formulae over are given by:
where are tuples of firstorder variables, and are 2SAT clauses for every .
The set of formulae over is given by the following grammar:
where is a 2SAT formula, , is a firstorder variable and is a secondorder variable. The syntax of formulae includes the counting operators of addition , , . Specifically, , are called firstorder and secondorder quantitative quantifiers respectively.
Let be a relational vocabulary, a structure with universe , a firstorder assignment for and a secondorder assignment for . Then the evaluation of a formula over is defined as a function that on input returns a number in . The function is recursively defined in Table 1. A formula is said to be a sentence if it does not have any free variable, that is, every variable in is under the scope of a usual quantifier (, ) or a quantitative quantifier. It is important to notice that if is a sentence over a vocabulary , then for every structure , firstorder assignments , for and secondorder assignments , for , it holds that . Thus, in such a case we use the term to denote for some arbitrary firstorder assignment and some arbitrary secondorder assignment for .
At this point it is clear that for any formula , a function is defined. In the rest of the paper we will use the same notation, namely , both for the set of formulae and the set of corresponding counting functions.^{1}^{1}1Moreover, we will use the terms ‘(counting) problem’ and ‘(counting) function’ interchangeably throughout the paper.
The following inclusion holds between the class [8] and the class defined presently.
Proposition 1
Proof
A function is in the class if it can be expressed in the form , where is an unquantified CNF formula in which each clause has at most one occurrence of an unnegated variable from , and at most one occurrence of a negated variable from . Alternatively, the function can be expressed in the form The RestrictedHorn formula is also a 2SAT formula.
Therefore, .
The class contains problems that are tractable, such as #2Col, which is known to be computable in polynomial time [9]. It also contains all the problems in , such as #Bis, #1P1NSat, #Downsets [8]. These three problems are complete for under approximation preserving reductions and are not believed to have an fpras. At last, the problem #Is [8], which is interriducible with #Sat under approximation preserving reductions, belongs to as well.
We next show that a generalization of #2Sat, which we will call #Disj2Sat, is complete for under parsimonious reductions.
2.1.1 Membership of #Disj2Sat in
In propositional logic, a 2SAT formula is a conjunction of clauses that contain at most two literals. Suppose we are given a propositional formula , which is a disjunction of 2SAT formulae, then #Disj2Sat on input equals the number of satisfying assignments of .
In this subsection we assume that 2SAT formulae consist of clauses which contain exactly two literals since we can rewrite a clause of the form as , for any literal .
Theorem 2.1
Proof
Consider the vocabulary where , , are ternary relations and is a binary relation. This vocabulary can encode any formula which is a disjunction of 2SAT formulae. More precisely, iff clause is of the form , iff is , iff is , iff is and iff clause appears in the “disjunct” .
Let be an input to #Disj2Sat encoded by an ordered structure , where the universe consists of elements representing variables, clauses and “disjuncts”. Then, it holds that the number of satisfying assignments of is equal to , where
Thus, is defined by which is in .
2.1.2 Hardness of #Disj2Sat
Suppose we have a formula in and an input structure over a vocabulary . We describe a polynomialtime reduction that given and , it returns a propositional formula which is a disjunction of 2SAT formulae and it holds that . The reduction is a parsimonious reduction, i.e. it preserves the values of the functions involved.
Theorem 2.2
#Disj2Sat is hard for under parsimonious reductions.
Proof
By Proposition 5.1 of [3], can be written in the form
, where each is a sequence of secondorder variables and each is a 2SAT clause. Each term of the sum can be replaced by where is the union of all . Now we have expressed in the following form
.
The next step is to expand the firstorder quantifiers and sum operators and replace their variables with firstorder constants from the universe .
In this way, we obtain . Each firstorder subformula of has no freevariables and is either satisfied or not satisfied by , so we can replace it by or respectively. Also, after grouping the sums and the conjunctions, we get . The formulae are conjunctions of clauses that consist of , and at most two literals of the form or for some secondorder variable and some tuple of firstorder constants . We can eliminate the clauses that contain a and remove from the clauses that contain it. After this simplification, some combinations of variableconstants may not appear in the remaining formula. For any such combination , we add a clause , since can have any truth value.
So, we have reformulated the above formula and we get . After replacing every appearance of by a propositional variable , the part becomes a disjunction of 2SAT formulae. Finally, we introduce new propositional variables and define
. The formula is a disjunction of 2SAT formulae and the number of its satisfying assignments is equal to . Moreover, every transformation we made requires polynomial time in the size of the input structure .
It is known that #2Sat has no fpras unless , since it is equivalent to counting all independent sets in a graph [8]. Thus, problems hard for under parsimonious reductions also cannot admit an fpras unless .
2.1.3 Inclusion of in
Several problems in , like #1P1NSat, #Is, #2Col, and #2Sat, are also in . We next prove that this is not a coincidence.
Theorem 2.3
Proof
Since is exactly the Karp closure of selfreducible functions of [17], it suffices to show that the complete problem #Disj2Sat is such a function.
First of all, Disj2Sat belongs to . Thus #Disj2Sat .
Secondly, every counting function associated with the problem of counting satisfying assignments for a propositional formula is selfreducible ^{2}^{2}2 contains all selfreducible problems in , with decision version in . Intuitively, selfreducibility means that counting the number of solutions to an instance of a problem, can be performed recursively by computing the number of solutions to some other instances of the same problem. For example, #Sat is selfreducible: the number of satisfying assignments of a formula is equal to the sum of the number of satisfying assignments of and , where is with its first variable fixed to .. So #Disj2Sat has this property as well.
Therefore, any formula defines a function that belongs to .
Corollary 1
2.2 The class
To define the second class , we make use of the framework presented in [19].
We say that a counting problem belongs to the class if for any ordered structure over a vocabulary , which is an input to , it holds that . The formula is of the form , where is a firstorder formula over and is a positive appearance of a secondorder variable. We call the formula a variable, since it contains only one secondorder variable. Moreover, we allow counting only the assignments to the secondorder variable under which the structure satisfies .
Proposition 2
, where #Vc is the problem of counting the vertex covers of all sizes in a graph.
Proof
An input graph to #Vc can be encoded as a finite structure using the vocabulary , where is the edge relation and is a binary relation. The universe is the set of all vertices and all edges. iff vertex is an endpoint of edge . Then, . Therefore, .
2.2.1 Completeness of #MonotoneSat for
Given a propositional formula in conjunctive normal form, where all the literals are positive, #MonotoneSat on input equals the number of satisfying assignments of .
Theorem 2.4
Proof
Consider the vocabulary with the binary relation to indicate that the variable appears in the clause . Given a structure that encodes a formula , which is an input to #MonotoneSat, it holds that #MonotoneSat=.
Therefore, #MonotoneSat .
Theorem 2.5
#MonotoneSat is hard for under product reductions.
Proof
We show that there is a polynomialtime product reduction from any to #MonotoneSat. This means that there are polynomialtime computable functions and , such that for every strucrure that is an input to we have .
Suppose we have a problem and a structure . Then, there exists a formula of the form such that .
The formula can be written in the form
By substituting firstorder subformulae by or and simplifying, we obtain , where each is a tuple of firstorder constants. To define , we have simplified the subformulae containing and . As a result, there may be some combinations of the secondorder variable and firstorder constants that do not appear in . Let be the number of these combinations. The last transformation consists of replacing every with a propositional variable , so we get the output of the function , which is . This formula has no negated variables, so it can be an input to #MonotoneSat. Finally, since the missing variables can have any truth value, we have .
2.2.2 Inclusion of in
Theorem 2.6
Proof
It is easy to prove that and that is closed under product reductions. Thus, the above results imply that every counting problem in belongs to .
3 On vs.
In this section we study the relationship between the classes and . First of all we give some definitions and facts that will be needed.
Theorem 3.1
[17] (a) . The inclusions are proper unless .
(b) is the Karp closure of selfreducible functions.
We consider to be the class of functions in that admit fpras, and we also introduce an ancillary class . Formally:
Definition 3
A function belongs to if and there exists a randomized algorithm that on input returns a value such that
in time poly().
We further say that a function belongs to if whenever the returned value equals with probability 1.
We begin with the following observation.^{3}^{3}3The following theorem is probably wellknown among counting complexity researchers. However, since we have not been able to find a proof in the literature we provide one here for the sake of completeness.
Theorem 3.2
if and only if =.
Proof
For the one direction we observe that if then there are functions in , that are not in . For example, #Is belongs to , and does not admit an fpras unless = [13].
The other direction derives from a Stockmeyer’s well known theorem [21]. By Stockmeyer’s theorem there exists an fpras, with access to a oracle, for any problem in . If = then [23]. Finally it is easy to see that an fpras with access to a oracle, can be replaced by another fpras, that simulates the oracle calls itself.
Corollary 2
if and only if if and only if .
Proof
iff = is an immediate corollary of the proof of Theorem 3.2 along with the observations that and .
We prove that iff . Suppose that and let be a function in . Then . Now we can modify the fpras for so that it returns the correct value of with probability 1 if . We can do this since we can decide if in polynomial time. So, .
The other direction is trivial by the inclusion .
Now we examine the opposite inclusion, i.e. whether is a subset of . To this end we introduce two classes that contain counting problems with decision in .
Recall that if a counting function admits an fpras, then its decision version, i.e. deciding whether , is in . In a similar way, if a counting function belongs to , then its decision version is in . So we need to define the subclass of with decision in . Clearly, if for a problem in the corresponding counting machine has an behavior (i.e., either a majority of paths are accepting or all paths are rejecting) then the decision version is naturally in . However, this seems to be a quite restrictive requirement. Therefore we will examine two subclasses of .
For that we need the following definition of the set of Turing Machines associated to problems in .
Definition 4
Let be an NPTM. We denote by the polynomial such that on inputs of size , makes nondeterministic choices.
either or
Definition 5
=
Definition 6
=
Note that , although restrictive, contains counting versions of some of the most representative problems in , for which no deterministic algorithms are known. For example consider the polynomial identity testing problem (Pit ^{4}^{4}4Determining the computational complexity of polynomial identity testing is considered one of the most important open problems in the mathematical field of Algebraic Computing Complexity.): Given an arithmetic circuit of degree that computes a polynomial in a field, determine whether the polynomial is not equal to the zero polynomial. A probabilistic solution to it is to evaluate it on a random point (from a sufficiently large subset of the field). If the polynomial is zero then all points will be evaluated to else the probability of getting is at most . A counting analogue of Pit is to count the number of elements in that evaluate to nonzero values; clearly this problem belongs to . Another problem in is to count the number of compositeness witnesses (as defined by the MillerRabin primality test) on input an integer ; although in this case the decision problem is in (a prime number has no such witnesses and this can be checked deterministically by AKS algorithm [1]), for a composite number at least half of the integers in are MillerRabin witnesses, hence there exists a NPTM that has as many accepting paths as the number of witnesses.
contains natural counting problems as well. Two examples in are #Exact Matchings and #BlueRed Matchings, which are counting versions of Exact Matching [18] and BlueRed Matching [16], respectively, both of which belong to RP (in fact in RNC) as shown in [15, 16]; however, it is still open so far whether they can be solved in polynomial time. Therefore it is also open whether #Exact Matchings and #BlueRed Matchings belong to .
We will now focus on relationships among the aforementioned classes. We start by presenting some unconditional inclusions and then we explore possible inclusions under the condition that either or holds.
3.1 Unconditional inclusions
Theorem 3.3
. Also .
Proof
Let . We will show that . We will construct an NPTM s.t. on input , . Let . We construct that computes and then it computes s.t. . makes nondeterministic choices . Each such determines a path, in particular, corresponds to the st path (since is the first path). returns yes iff , so . Since
The other inclusions are immediate by definitions.
Theorem 3.4
Proof
For the first inclusion, let Let . There exists an s.t. , Let be the number of nondeterministic choices of Let
. We can compute an estimate
of by choosing paths uniformly at random. Then we can compute .To proceed with the proof we need the following lemma.
Lemma 1
(Unbiased estimator.) Let
be two finite sets, and let . Suppose we take samples from uniformly at random, and let a be the number of them that belong to . Then is an unbiased estimator of , and it suffices in order to haveIf , then so by the unbiased estimator of lemma 1, satisfies the definition of fpras. If then , so the estimated value is with probability 1.
For the second inclusion, let , we will show that the decision version of , i.e. deciding if , is in . On input we run the fpras for with e.g. We return yes iff
By the definition of , if then the fpras returns , so we return yes with probability . If , then with probability at least , so we return yes with the same probability.
Corollary 3
.
Corollary 4
If then =.
Proof
If , then , and then for all , . So if via then , and thus . Thus =.
Corollary 5
If then =.
Proof
If = then they are both equal to , thus . Therefore, = by Corollary 2.
3.2 Conditional inclusions / Possible worlds
Now we will explore further relationships between the above mentioned classes, and we will present two possible worlds inside , with respect to vs. vs. .
Theorem 3.5
The inclusions depicted in Figure 3 hold under the corresponding assumptions on top of each subfigure.
Proof
First note that intersections between any of the above classes are nonempty, because is a subclass of all of them. For the rest of the inclusions, we have the following.

In the case of .

By definitions, =. Therefore,

By Theorem 3.1, the inclusions

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