Exponential Gaps Research Articles

Crux and Long Cycles in Graphs

We introduce a notion of the crux of a graph $G$, measuring the order of a smallest dense subgraph in $G$. This simple-looking notion leads to some generalizations of known results about cycles, offering an interesting paradigm of “replacing average degree by crux.” In particular, we prove that every graph contains a cycle of length linear in its crux. Long proved that every subgraph of a hypercube $Q^m$ (resp., discrete torus $C_3^m$) with average degree $d$ contains a path of length $2^{d/2}$ (resp., $2^{d/4}$) and conjectured that there should be a path of length $2^{d}-1$ (resp., $3^{d/2}-1$). As a corollary of our result, together with isoperimetric inequalities, we close these exponential gaps giving asymptotically optimal bounds on long paths in hypercubes, discrete tori, and more generally Hamming graphs. We also consider random subgraphs of $C_4$-free graphs and hypercubes, proving near optimal lower bounds on the lengths of long cycles.

The descriptional power of queue automata of constant length

We consider the notion of a constant length queue automaton—i.e., a traditional queue automaton with a built-in constant limit on the length of its queue—as a formalism for representing regular languages. We show that the descriptional power of constant length queue automata greatly outperforms that of traditional finite state automata, of constant height pushdown automata, and of straight line programs for regular expressions, by providing optimal exponential and double-exponential size gaps. Moreover, we prove that constant height pushdown automata can be simulated by constant length queue automata paying only by a linear size increase, and that removing nondeterminism in constant length queue automata requires an optimal exponential size blow-up, against the optimal double-exponential cost for determinizing constant height pushdown automata. Finally, we investigate the size cost of implementing Boolean language operations on deterministic and nondeterministic constant length queue automata.

Exponential gaps in the length spectrum

We present a separation property for the gaps in the length spectrum of a compact Riemannian manifold with negative curvature. In arbitrary small neighborhoods of the metric for some suitable topology, we show that there are negatively curved metrics with a length spectrum exponentially separated from below. This property was previously known to be false generically.

Periodic Fourier representation of Boolean functions

In this work, we consider a new type of Fourier-like representation of Boolean function f\colon\{+1,-1\}^n\to\{+1,-1\}: f(x) = \cos(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i). This representation, which we call the periodic Fourier representation, of Boolean function is closely related to a certain type of multipartite Bell inequalities and non-adaptive measurement-based quantum computation with linear side-processing NMQCp. The minimum number of non-zero coefficients in the above representation, which we call the periodic Fourier sparsity, is equal to the required number of qubits for the exact computation of f by \NMQCp. Periodic Fourier representations are not unique, and can be directly obtained both from the Fourier representation and the F_2-polynomial representation. In this work, we first show that Boolean functions related to ZZZZ-polynomial have small periodic Fourier sparsities. Second, we show that the periodic Fourier sparsity is at least 2^{\deg_{\mathbb{F}_2}(f)}-1, which means that NMQCp efficiently computes a Boolean function $f$ if and only if F_2-degree of f is small. Furthermore, we show that any symmetric Boolean function, e.g., AND_n, Mod^3_n, Maj_n, etc, can be exactly computed by depth-2 NMQCp using a polynomial number of qubits, that implies exponential gaps between NMQCp and depth-2 NMQCp.

Linear-time limited automata

The time complexity of 1-limited automata is investigated from a descriptional complexity view point. Though the model recognizes regular languages only, it may use quadratic time in the input length. We show that, with a polynomial increase in size and preserving determinism, each 1-limited automaton can be transformed into a linear-time equivalent one. We also obtain polynomial transformations into related models, including weight-reducing Hennie machines (i.e., one-tape Turing machines syntactically forced to operate in linear-time), and we show exponential gaps for the converse transformations in the deterministic case.

Alternation, sparsity and sensitivity: Bounds and exponential gaps

The well-known Sensitivity Conjecture regarding combinatorial complexity measures on Boolean functions states that for any Boolean function f:{0,1}n→{0,1}, block sensitivity of f is polynomially related to sensitivity of f (denoted by s(f)). From the complexity theory side, the Xor Log-Rank Conjecture states that for any Boolean function, f:{0,1}n→{0,1} the communication complexity of a related function f⊕:{0,1}n×{0,1}n→{0,1} (defined as f⊕(x,y)=f(x⊕y)) is bounded by polynomial in logarithm of the sparsity of f (the number of non-zero Fourier coefficients for f, denoted by sparsity(f)). Both the conjectures play a central role in the domains in which they are studied.A recent result of Lin and Zhang (2017) implies that to confirm the above two conjectures it suffices to upper bound alternation of f (denoted by alt(f)) for all Boolean functions f by polynomial in s(f) and logarithm of sparsity(f), respectively. In this context, we show the following results:•We show that there exists a family of Boolean functions for which alt(f) is at least exponential in s(f) and alt(f) is at least exponential in log⁡sparsity(f). En route to the proof, we also show an exponential gap between alt(f) and the decision tree complexity of f, which might be of independent interest.•As our main result, we show that, despite the above exponential gap between alt(f) and log⁡sparsity(f), the Xor Log-Rank Conjecture is true for functions with the alternation upper bounded by poly(log⁡n). It is easy to observe that the Sensitivity Conjecture is also true for this class of functions.•The starting point for the above result is the observation (derived from Lin and Zhang (2017)) that for any Boolean function f, deg(f)≤alt(f)deg2(f)degm(f) where deg(f), deg2(f) and degm(f) are the degrees of f over R, F2 and Zm respectively. We give three further applications of this bound: (1) We show that for Boolean functions f of constant alternation have deg2(f)=Ω(log⁡n). (2) Moreover, these functions also have high sparsity (exp⁡(Ω(deg(f))), thus partially answering a question of Kulkarni and Santha (2013). (3) We observe that our relation upper bounding real degree also improves the upper bound for influence to deg2(f)2⋅alt(f) improving Guo and Komargodski (2017).

Ultrathin Semiconducting Bi2Te2S and Bi2Te2Se with High Electron Mobilities.

High carrier mobility and moderate band gap are two key properties of electronic device applications. Two ultrathin two-dimensional (2D) semiconductors, namely, Bi2Te2S and Bi2Te2Se nanosheets, with novel electronic and optical properties are predicted based on first-principles calculations. The Bi2Te2S and Bi2Te2Se monolayers own moderate band gaps (∼0.7 eV) and high electron mobilities (∼20 000 cm2 V-1 s-1), and they can absorb sunlight efficiently through the whole incident solar spectrum. Meanwhile, layer-dependent exponential decay band gaps are also unveiled. The relatively low interlayer binding energies suggest that these monolayers can be easily exfoliated from bulk structures. Their high dynamical and thermal stabilities are further verified by phonon dispersion calculations and ab initio molecular dynamics simulations. The exceptional properties render Bi2Te2X (X = S, Se) monolayers promising candidates in future high-speed (opto)electronic devices.

On Strong Normalization in Proof-Graphs for Propositional Logic

Traditional proof theory of Propositional Logic deals with proofs whose size can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The use of proof-graphs, instead of trees or lists, for representing proofs is getting popular among proof-theoreticians. Proof-graphs serve as a way to study complexity of propositional proofs and to provide more efficient theorem provers, concerning size of propositional proofs.Fpl-graphs were initially developed for minimal implicational logic representing proofs through references rather than copy. Thus, formulas and sub-deductions preserved in the graph structure, can be shared deleting unnecessary sub-deductions resulting in the reduced proof. In this work, we consider full minimal propositional logic and show how to reduce (eliminating maximal formulas) these representations such that strong normalization theorem can be proved by simply counting the number of maximal formulas in the original derivation. In proof-graphs, the main reason for obtaining the strong normalization property using such simple complexity measure is a direct consequence of the fact that each formula occurs only once in the proof-graph and the case of the hidden maximum formula that usually occurs in the tree-form derivation is already represented in the fpl-graph.

Two-way automata making choices only at the endmarkers

The question of the state-size cost for simulation of two-way nondeterministic automata (2nfas) by two-way deterministic automata (2dfas) was raised in 1978 and, despite many attempts, it is still open. Subsequently, the problem was attacked by restricting the power of 2dfas (e.g., using a restricted input head movement) to the degree for which it was already possible to derive some exponential gaps between the weaker model and the standard 2nfas. Here we use an opposite approach, increasing the power of 2dfas to the degree for which it is still possible to obtain a subexponential conversion from the stronger model to the standard 2dfas. In particular, it turns out that subexponential conversion is possible for two-way automata that make nondeterministic choices only when the input head scans one of the input tape endmarkers. However, there is no restriction on the input head movement. This implies that an exponential gap between 2nfas and 2dfas can be obtained only for unrestricted 2nfas using capabilities beyond the proposed new model.As an additional bonus, conversion into a machine for the complement of the original language is polynomial in this model. The same holds for making such machines self-verifying, halting, or unambiguous. Finally, any superpolynomial lower bound for the simulation of such machines by standard 2dfas would imply L≠NL. In the same way, the alternating version of these machines is related to L=?NL=?P, the classical computational complexity problems.

Proof-graphs for Minimal Implicational Logic

It is well-known that the size of propositional classical proofs can be huge. Proof theoretical studies discovered exponential gaps between normal or cut free proofs and their respective non-normal proofs. The aim of this work is to study how to reduce the weight of propositional deductions. We present the formalism of proof-graphs for purely implicational logic, which are graphs of a specific shape that are intended to capture the logical structure of a deduction. The advantage of this formalism is that formulas can be shared in the reduced proof. In the present paper we give a precise definition of proof-graphs for the minimal implicational logic, together with a normalization procedure for these proof-graphs. In contrast to standard tree-like formalisms, our normalization does not increase the number of nodes, when applied to the corresponding minimal proof-graph representations.

More concise representation of regular languages by automata and regular expressions

We consider two formalisms for representing regular languages: constant height pushdown automata and straight line programs for regular expressions. We constructively prove that their sizes are polynomially related. Comparing them with the sizes of finite state automata and regular expressions, we obtain optimal exponential and double exponential gaps, i.e., a more concise representation of regular languages.

Proof compressions with circuit-structured substitutions

It is well known that the size of propositional classical proofs can be huge. Proof-theoretical studies discovered exponential gaps between cut-free (or normal) proofs and the corresponding (non-normal) proofs with cuts (or modus ponens). The task of automatic theorem proving is, on the other hand, usually based on the construction of cut-free or only atomic-cuts proofs, since this procedure produces less alternative choices. There are familiar tautologies whose cut-free proofs are huge while the non-cut-free ones are small. The aim of this paper is to discuss basic methods of weight and/or size reduction of deductions by switching from traditional tree-structured deductions to circuit-structured deductions. A desired efficiency is achieved by adding the standard weakening rule of inference upgraded by adding suitable (propositional) unifications modulo variable substitutions. We show examples where such a unification provides strong (in fact, exponential) compression of cut-free deductions. Bibliography: 10 titles.

P-immune sets with holes lack self-reducibility properties

No P-immune set having exponential gaps is positive-Turing self-reducible.

Analytic, Microscopic Model of Traffic Flow and Travel Time

A model of disaggregate interactions between individual vehicles was developed that makes explicit the vehicle classes, the class trip rates, and their kinematic parameters (free speed, acceleration rate, length, safety margin). Assuming exponential gaps between vehicles, analytic formulas were derived for the mean value and the variance of the path travel times within each class. The model was successively applied to one-lane roads; two-way, two-lane roads; one-way, two-lane roads; and two-way, three-lane roads.

Some contributions of the study of abstract communication complexity to other areas of computer science

Communication Complexity · 3 —Size of distinct models of branching programs —Depth of decision trees —Data structure problems. To illustrate the progress covered by the above list we mention two specific contributions. The first superlinear lower bound on the size of planar Boolean circuits computing a specific Boolean function and the first superpolylogarithmic lower bounds on the depth of monotone Boolean circuits have been established. The big success of communication complexity application should not to be surprising because we have information transfer in all computing models (for instance, between two parts of input data , between some parts (processors) of a parallel computing model, between two time moments, etc.). So, you can cut hardware, time, or both in your computing model, and then apply lower bounds on the communication complexity of your computing problem. In this way you have a lower bound on the information transfer that must be realized in the computing model considered in order to compute the given task. The appropriate choice of the cut is crucial for obtaining good lower bounds. One of the perspectives is to extend the applications for proving lower bounds for multilective and/or non-oblivious computing models. This is one of the hardest tasks of special importance in complexity theory. The recent results show that using Ramsey theory and communication complexity over overlapping (not disjoint) partitions of inputs one has good chances to achieve progress in this hard topic too. 3. NONDETERMINISTIC AND RANDOMIZED COMPUTATIONS One of the central principal questions of current theoretical computer science is which computational power have nondeterministic and randomized computations, especially in the comparison with the deterministic one. The fundamental questions about polynomial time computations (like P versus NP, P versus ZPP, P versus R) are long-stated open problems. For communication complexity the research has been successful and the relation between determinism, nondeterminism and randomness has been fixed. This has essentially contributed to the understanding of the nature of randomness and nondeterminism. Some of the main results are the following ones: (1) There are exponential gaps between —determinism and Monte Carlo randomness —nondeterminism and bounded error probabilism. (2) Deterministic communication can be bounded by at most twice the product of nondeterministic communication of the language and its complement. This implies an at most quadratic gap between determinism and Las Vegas randomization. A language having this quadratic gap has been found. (3) There is a linear gap between determinism and Las Vegas randomness for oneway communication complexity. (4) O(log n) random bits are sufficient to reach the full power of randomized communication for Las Vegas and Monte Carlo (error-bounded) protocols. (5) In contrast to 4. there exist high thresholds on the amount of nondeterminism (for some computing problems the deterministic communication complexity is

On Unconditional Polynomial Bases in L p and Bergman Spaces

In this paper we consider unconditional bases inL p(T), 1<p<∞,p ≠ 2, consisting of trigonometric polynomials. We give a lower bound for the degree of polynomials in such a basis (Theorem 3.4) and show that this estimate is best possible. This is applied to the Littlewood-Paley-type decompositions. We show that such a decomposition has to contain exponential gaps. We also consider unconditional polynomial bases inH p as bases in Bergman-type spaces and show that they provide explicit isomorphisms between Bergman-type spaces and natural sequences spaces.

On unconditional polynomial bases inL p and bergman spaces

In this paper we consider unconditional bases inL p(T), 1<p<∞,p ≠ 2, consisting of trigonometric polynomials. We give a lower bound for the degree of polynomials in such a basis (Theorem 3.4) and show that this estimate is best possible. This is applied to the Littlewood-Paley-type decompositions. We show that such a decomposition has to contain exponential gaps. We also consider unconditional polynomial bases inH p as bases in Bergman-type spaces and show that they provide explicit isomorphisms between Bergman-type spaces and natural sequences spaces.

On the Expressive Power of OKFDDs

Ordered Decision Diagrams (ODDs) as a means for the representation of Boolean functions are used in many applications in CAD. Depending on the decomposition type, various classes of ODDs have been defined, among them being the Ordered Binary Decision Diagrams (OBDDs), the Ordered Functional Decision Diagrams (OFDDs) and the Ordered Kronecker Functional Decision Diagrams (OKFDDs). Based on a formalization of the concept decomposition type we first investigate all possible decomposition types and prove that already OKFDDs, which result from the application of only three decomposition types, result in the most general class of ODDs. We then show from a (more) theoretical point of view that the generality of OKFDDs is really needed. We prove several exponential gaps between specific classes of ODDs, e.g. between OKFDDs on the one side and OBDDs, OFDDs on the other side. Combining these results it follows that a restriction of the OKFDD concept to subclasses, such as OBDDs and OFDDs as well, results in families of functions which lose their efficient representation.

Information Transfer under Different Sets of Protocols

We study four kinds of protocols in distributed computing. Besides the case of deterministic protocols, we consider random, nondeterministic and probabilistic models. We show a strict containment relationship with exponential gaps. We also explore in some depth the relationship between one-way and two-way communications. It is shown, for example, that the complexity classes of random one-way protocols and of deterministic two-way protocols are incomparable: exponential gaps exist in either direction. On the other hand there is no difference between the complexity of one-way and two-way communications in the nondeterministic case.