Exponential Time Hierarchy Research Articles

Second-Order Quantified Boolean Logic

Second-order quantified Boolean formulas (SOQBFs) generalize quantified Boolean formulas (QBFs) by admitting second-order quantifiers on function variables in addition to first-order quantifiers on atomic variables. Recent endeavors establish that the complexity of SOQBF satisfiability corresponds to the exponential-time hierarchy (EXPH), similar to that of QBF satisfiability corresponding to the polynomial-time hierarchy (PH). This fact reveals the succinct expression power of SOQBFs in encoding decision problems not efficiently doable by QBFs. In this paper, we investigate the second-order quantified Boolean logic with the following main results: First, we present a procedure of quantifier elimination converting SOQBFs to QBFs and a game interpretation of SOQBF semantics. Second, we devise a sound and complete refutation-proof system for SOQBF. Third, we develop an algorithm for countermodel extraction from a refutation proof. Finally, we show potential applications of SOQBFs in system design and multi-agent planning. With these advances, we anticipate practical tools for development.

Capturing Bisimulation-Invariant Exponential-Time Complexity Classes

Otto's Theorem characterises the bisimulation-invariant PTIME queries over graphs as exactly those that can be formulated in the polyadic mu-calculus, hinging on the Immerman-Vardi Theorem which characterises PTIME (over ordered structures) by First-Order Logic with least fixpoints. This connection has been extended to characterise bisimulation-invariant EXPTIME by an extension of the polyadic mu-calculus with functions on predicates, making use of Immerman's characterisation of EXPTIME by Second-Order Logic with least fixpoints. In this paper we show that the bisimulation-invariant versions of all classes in the exponential time hierarchy have logical counterparts which arise as extensions of the polyadic mu-calculus by higher-order functions. This makes use of the characterisation of k-EXPTIME by Higher-Order Logic (of order k+1) with least fixpoints, due to Freire and Martins.

On the expressivity of elementary linear logic: Characterizing Ptime and an exponential time hierarchy

Elementary linear logic is a simple variant of linear logic due to Girard and which characterizes in the proofs-as-programs approach the class of elementary functions, that is to say functions computable in time bounded by a tower of exponentials of fixed height. Other systems like light and soft linear logics have then been defined to characterize in a similar way the more interesting complexity class of polynomial time functions, but at the price of either a more complicated syntax or of more sophisticated encodings. These logical systems can serve as the basis of type systems for λ-calculus ensuring polynomial time complexity bounds on well-typed terms.This paper aims at reviving interest in elementary linear logic by showing that, despite its simplicity, it can capture smaller complexity classes than that of elementary functions. For that we carry a detailed analysis of its normalization procedure, and study the complexity of functions represented by a given type. We then show that by considering a slight variant of this system, with type fixpoints and free weakening (elementary affine logic with fixpoints) we can characterize the complexity of functions of type !W⊸!k+2B, where W and B are respectively types for binary words and booleans. The key point is a sharper study of the normalization bounds. We characterize in this way the class P of polynomial time predicates, and more generally the hierarchy of classes k-EXP, for k≥0, where k-EXP is the union of DTIME(2kni), for i≥1.

Model-checking hierarchical structures

Hierarchical graph definitions allow a modular description of structures using modules for the specification of repeated substructures. Beside this modularity, hierarchical graph definitions allow us to specify structures of exponential size using polynomial size descriptions. In many cases, this succinctness increases the computational complexity of decision problems when input structures are defined hierarchically. In this paper, the model-checking problem for first-order logic (FO), monadic second-order logic (MSO), and second-order logic (SO) on hierarchically defined input structures is investigated. It is shown that in general these model-checking problems are exponentially harder than their non-hierarchical counterparts, where the input structures are given explicitly. As a consequence, several new complete problems for the levels of the polynomial time hierarchy and the exponential time hierarchy are obtained. Based on classical results of Gaifman and Courcelle, two restrictions on the structure of hierarchical graph definitions that lead to more efficient model-checking algorithms are presented.

The Descriptive Complexity of the Deterministic Exponential Time Hierarchy

In Descriptive Complexity, we investigate the use of logics to characterize computational complexity classes. Since 1974, when Fagin proved that the class NP is captured by existential second-order logic, considered the first result in this area, other relations between logics and complexity classes have been established. Well-known results usually involve first-order logic and its extensions, and complexity classes in polynomial time or space. Some examples are that the first-order logic extended by the least fixed-point operator captures the class P and the second-order logic extended by the transitive closure operator captures the class PSPACE. In this paper, we will analyze the combined use of higher-order logics of order i, HOi, for i⩾2, extended by the least fixed-point operator, and we will prove that each level of this hierarchy captures each level of the deterministic exponential time hierarchy. As a corollary, we will prove that the hierarchy of HOi(LFP), for i⩾2, does not collapse, that is, HOi(LFP)⊂HOi+1(LFP).

Expressive power and abstraction in Essence

Development of languages for specifying or modelling problems is an important direction in constraint modelling. To provide greater abstraction and modelling convenience, these languages are becoming more syntactically rich, leading to a variety of questions about their expressive power. In this paper, we consider the expressiveness of Essence, a specification language with a rich variety of syntactic features. We identify natural fragments of Essence that capture the complexity classes P, NP, all levels \(\Sigma_i^p\) of the polynomial-time hierarchy, and all levels k-NEXP of the nondeterministic exponential-time hierarchy. The union of these classes is the very large complexity class ELEMENTARY. One goal is to begin to understand which features play a role in the high expressive power of the language and which are purely features of convenience. We also discuss the formalization of arithmetic in Essence and related languages, a notion of capturing NP-search which is slightly different than that of capturing NP, and a conjectured limit to the expressive power of Essence. Our study is an application of descriptive complexity theory, and illustrates the value of taking a logic-based view of modelling and specification languages.

Third-Order Computation and Bounded Arithmetic

We describe a natural generalization of ordinary computation to a third-order (i.e. three-sorted) setting. We give a function calculus with nice properties and recursion-theoretic characterizations of several large complexity classes, including classes of functions and predicates from the levels of the exponential-time hierarchy. We then present a number of third-order theories of bounded arithmetic whose definable functions are the classes of the exponential time hierarchy in the third-order setting.

Polylogarithmic-round interactive proofs for coNP collapse the exponential hierarchy

If every language in coNP has a constant-round interactive proof system, then the polynomial-time hierarchy collapses [R.B. Boppana, J. Håstad, S. Zachos, Does co-NP have short interactive proofs? Information Processing Letters 25 (2) (1987) 127–132]. On the other hand, the well-known LFKN protocol gives O ( n ) -round interactive proof systems for all languages in coNP [C. Lund, L. Fortnow, H. Karloff, N. Nisan, Algebraic methods for interactive proof systems, Journal of the Association for Computing Machinery 39 (4) (1992) 859–868]. We consider the question of whether it is possible for coNP to have interactive proof systems with polylogarithmic-round complexity. We show that this is unlikely by proving that if a coNP-complete set has a polylogarithmic-round interactive proof system, then the exponential-time hierarchy collapses. We also consider exponential versions of the Karp–Lipton theorem and Yap’s theorem.

Dimension, entropy rates, and compression

This paper develops new relationships between resource-bounded dimension, entropy rates, and compression. New tools for calculating dimensions are given and used to improve previous results about circuit-size complexity classes. Approximate counting of SpanP functions is used to prove that the NP-entropy rate is an upper bound for dimension in Δ 3 E , the third level of the exponential-time hierarchy. This general result is applied to simultaneously improve the results of Mayordomo [E. Mayordomo, Contributions to the study of resource-bounded measure, PhD thesis, Universitat Politècnica de Catalunya, 1994] on the measure on P / poly in Δ 3 E and of Lutz [J.H. Lutz, Dimension in complexity classes, SIAM J. Comput. 32 (5) (2003) 1236–1259] on the dimension of exponential-size circuit complexity classes in ESPACE. Entropy rates of efficiently rankable sets, sets that are optimally compressible, are studied in conjunction with time-bounded dimension. It is shown that rankable entropy rates give upper bounds for time-bounded dimensions. We use this to improve results of Lutz [J.H. Lutz, Almost everywhere high nonuniform complexity, J. Comput. System Sci. 44 (2) (1992) 220–258] about polynomial-size circuit complexity classes from resource-bounded measure to dimension. Exact characterizations of the effective dimensions in terms of Kolmogorov complexity rates at the polynomial-space and higher levels have been established, but in the time-bounded setting no such equivalence is known. We introduce the concept of polynomial-time superranking as an extension of ranking. We show that superranking provides an equivalent definition of polynomial-time dimension. From this superranking characterization we show that polynomial-time Kolmogorov complexity rates give a lower bound on polynomial-time dimension.

Separating Complexity Classes Using Autoreducibility

A set is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate the polynomial-time hierarchy from exponential space by showing that all Turing complete sets for certain levels of the exponential-time hierarchy are autoreducible but there exists some Turing complete set for doubly exponential space that is not. Although we already knew how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's program to complexity theory. We feel such techniques may prove unknown separations in the future. In particular, if we could settle the question as to whether all Turing complete sets for doubly exponential time are autoreducible, we would separate either polynomial time from polynomial space, and nondeterministic logarithmic space from nondeterministic polynomial time, or else the polynomial-time hierarchy from exponential time. We also look at the autoreducibility of complete sets under nonadaptive, bounded query, probabilistic, and nonuniform reductions. We show how settling some of these autoreducibility questions will also lead to new complexity class separations.

Separating classes in the exponential-time hierarchy from classes in PH

We are interested in separating classes in the exponential-time hierarchy, EXPH, from classes in the polynomial-time hierarchy, PH. In this paper we show that, for any fixed integer c, the class of sets accepted in deterministic polynomial time using at most O( n c ) queries to an NP oracle, P NP[ O( n c )] , is a proper subset of NEXP. This improves a previous result by Fu et al. [7]. Further, we generalize this separation to related levels of PH and EXPH showing that, for any fixed integer c and i ⩾ 1, Δ i P[O(n c)] ⊊ ∑ i − 1 EXP . This improves the long standing separations which result from the relativization of the time hierarchy theorem [9, 6, 17, 3, 1].

Lowness Properties of Sets in the Exponential-Time Hierarchy

The notion of “lowness” was introduced in computational complexity theory by Schöning [J, Comput. Systems Sci., 27 (1983), pp. 14–28] who studied sets in the class NP. This notion may be interpreted as setting an upper bound on the amount of information that can be encoded by a set. Here ideas from previous studies are incorporated in order to capture the notion of a set being exponentially low. The main result asserts the existence of a sparse set E such that ${\operatorname{DEXT}}(E) = {\operatorname{DEXT}}$, i.e., E is “exponentially low,” but E is not in the class P. In contrast, any set with small generalized Kolmogorov complexity that is exponentially low must be in the class P. In addition, we show that for each $k \geqq 2$, any sparse set S that is low with respect to the class $\Sigma _k^E $ of the exponential-time hierarchy (i.e., $\Sigma _k^E {(S) = \Sigma _k^E } $ ) must be in the class $\Sigma _k^P $ of the polynomial-time hierarchy. Similarly, for each $k \geqq 4$, any set with polynomial-size circuits that is low with respect to the class $\Sigma _k^E $ must be in the class $\Sigma _k^P $ .