The Polylog-Time Hierarchy Captured by Restricted Second-Order Logic

Abstract

Let SO plog denote the restriction of second-order logic, where second-order quantification ranges over relations of size at most poly-logarithmic in the size of the structure. In this article we investigate the problem, which Turing machine complexity class is captured by Boolean queries over ordered relational structures that can be expressed in this logic. For this we define a hierarchy of fragments Σ plog m (and Π plog m ) defined by formulae with alternating blocks of existential and universal second-order quantifiers in quantifier-prenex normal form. We first show that the existential fragment Σ plog 1 captures NPolyLogTime, i.e. the class of Boolean queries that can be accepted by a non-deterministic Turing machine with random access to the input in time O((log n) k ) for some k ≥ 0. Using alternating Turing machines with random access input allows us to characterise also the fragments Σ plog m (and Π plog m ) as those Boolean queries with at most m alternating blocks of second-order quantifiers that are accepted by an alternating Turing machine. Consequently, SO plog captures the whole poly-logarithmic time hierarchy. We demonstrate the relevance of this logic and complexity class by several problems in database theory. ACM Subject Classification Theory of computation → Finite Model TheoryAccording to Immerman, the credo of descriptive complexity theory is that "the computational complexity of all problems in Computer Science can be understood via the complexity of their logical descriptions" [17, p.5]. Starting from Fagin's fundamental result [8] that the existential fragment SO∃ of second-order logic over finite relational structures captures all decision problems that are accepted by a non-deterministic Turing machine in polynomial 23:3 deterministic Turing machine with random access to the input in time O((log n) k ) for some k ≥ 0. Using alternating Turing machines with random access input allows us to characterise also the fragments Σ plog m (and Π plog m ) as those Boolean queries with at most m alternating blocks of second-order quantifiers that are accepted by an alternating Turing machine. That is, we obtain Σ plog m =Σ plog m (and Π plog m =Π plog m ). Consequently, SO plog captures the whole poly-logarithmic time hierarchy PLH.1 The result in [2] is actually more general, allowing any set of Boolean functions F of n O(1) inputs complying with a padding property and containing the functions OR and AND. The restricted secondorder logic is defined by extending first-order logic with a second-order quantifier Q f for each f ∈ F which range over relations on the sub-domain {1, . . . , log n}, where n is the size of the interpreting structure. The case related to our result is when F = {OR, AND}, which gives raise to restricted existential and universal second-order quantifiers.

Proper Hierarchies in Polylogarithmic Time and Absence of Complete Problems

“…The following result confirms that the problems that can be described in the index logic are in DP olylogT ime and vice versa. Regarding nondeterministic polylogarithmic time, the restricted second-order logic SO plog defined in [3,4,5] captures the polylogarithmic-time hierarchy, with its quantifier prenex fragments Σ plog m and Π plog m capturing the corresponding levelsΣ plog m andΠ plog m of this hierarchy, respectively. SO plog is a fragment of second-order logic where second-order quantification range over relations of polylogarithmic size and first-order quantification is restricted to the existential fragment of first-order logic plus universal quantification over variables under the scope of a second-order variable.…”

“…universal) block. Note that by Lemma 3 in [4], for every SO plog formula ϕ there is an equivalent formula ϕ that is in quantifier prenex normal form. In the following we will assume that the reader is familiar with the techniques that can be applied to transform arbitrary SO plog formulae into equivalent formulae in Skolem normal form.…”

“…SUCCk (ȳ,z) can be expressed by a quantifierfree SO plog formula (for details refer to SUCC k in Section 4 in [4]). In turn, |X| = |Y | can be expressed by an existential SO plog formula using second order variables of arity k + 1 and exponent k (for details refer to Section 3.1 in [4]). Finally, SEQ(X) can be expressed in SO plog as follows:…”

“…In [3] we started a deeper investigation of sub-linear time computations emphasising complexity classes DPolyLogTime and NPolyLogTime of decision problems that can be solved deterministically or non-deterministically with a time complexity in O(log k n) for some k, where n is as usual the size of the input. We extended these complexity classes to a complete hierarchy, the polylogarithmic time hierarchy, analogous to the polynomial time hierarchy, and for each class Σ plog m or Π plog m (m ∈ N) in the hierarchy we defined a fragment of semantically restricted second-order logic capturing it [5,6].…”

Descriptive Complexity of Deterministic Polylogarithmic Time

“…On the other hand, nondeterministic polylogarithmic time complexity classes, defined in terms of alternating random-access Turing machines and related families of circuits, have received some attention [13,14]. Recently, a theorem analogous to Fagin's famous theorem [4], was proven for nondeterministic polylogarithmic time [14].…”