On the Power of Unambiguity in Logspace

We report progress on the \NL vs \UL problem. [-] We show unconditionally that the complexity class \ReachFewL\subseteq\UL. This improves on the earlier known upper bound \ReachFewL \subseteq \FewL. [-] We investigate the complexity of min-uniqueness - a central notion in studying the \NL vs \UL problem. We show that min-uniqueness is necessary and sufficient for showing \NL =\UL. We revisit the class \OptL[\log n] and show that {\sc ShortestPathLength} - computing the length of the shortest path in a DAG, is complete for \OptL[\log n]. We introduce \UOptL[\log n], an unambiguous version of \OptL[\log n], and show that (a) \NL =\UL if and only if \OptL[\log n] = \UOptL[\log n], (b) \LogFew \leq \UOptL[\log n] \leq \SPL. [-] We show that the reachability problem over graphs embedded on 3 pages is complete for \NL. This contrasts with the reachability problem over graphs embedded on 2 pages which is logspace equivalent to the reachability problem in planar graphs and hence is in \UL.Comment: 14 pages, 3 figure

On Modal Logics of Partial Recursive Functions

The classical propositional logic is known to be sound and complete with respect to the set semantics that interprets connectives as set operations. The paper extends propositional language by a new binary modality that corresponds to partial recursive function type constructor under the above interpretation. The cases of deterministic and non-deterministic functions are considered and for both of them semantically complete modal logics are described and decidability of these logics is established

The Consequences of Eliminating NP Solutions

Given a function based on the computation of an NP machine, can one in general eliminate some solutions? That is, can one in general decrease the ambiguity? This simple question remains, even after extensive study by many researchers over many years, mostly unanswered. However, complexity-theoretic consequences and enabling conditions are known. In this tutorial-style article we look at some of those, focusing on the most natural framings: reducing the number of solutions of NP functions, refining the solutions of NP functions, and subtracting from or otherwise shrinking #P functions. We will see how small advice strings are important here, but we also will see how increasing advice size to achieve robustness is central to the proof of a key ambiguity-reduction result for NP functions

Computability vs. Nondeterministic and P vs. NP

This paper demonstrates the relativity of Computability and Nondeterministic; the nondeterministic is just Turing's undecidable Decision rather than the Nondeterministic Polynomial time. Based on analysis about TM, UM, DTM, NTM, Turing Reducible, beta-reduction, P-reducible, isomorph, tautology, semi-decidable, checking relation, the oracle and NP-completeness, etc., it reinterprets The Church-Turing Thesis that is equivalent of the Polynomial time and actual time; it redefines the NTM based on its undecidable set of its internal state. It comes to the conclusions: The P-reducible is misdirected from the Turing Reducible with its oracle; The NP-completeness is a reversal to The Church-Turing Thesis; The Cook-Levin theorem is an equipollent of two uncertains. This paper brings forth new concepts: NP (nondeterministic problem) and NP-algorithm (defined as the optimal algorithm to get the best fit approximation value of NP). P versus NP is the relativity of Computability and Nondeterministic, P/=NP. The NP-algorithm is effective approximate way to NP by TM

Generic case completeness

In this note we introduce a notion of a generically (strongly generically) NP-complete problem and show that the randomized bounded version of the halting problem is strongly generically NP-complete

Credimus

We believe that economic design and computational complexity---while already important to each other---should become even more important to each other with each passing year. But for that to happen, experts in on the one hand such areas as social choice, economics, and political science and on the other hand computational complexity will have to better understand each other's worldviews. This article, written by two complexity theorists who also work in computational social choice theory, focuses on one direction of that process by presenting a brief overview of how most computational complexity theorists view the world. Although our immediate motivation is to make the lens through which complexity theorists see the world be better understood by those in the social sciences, we also feel that even within computer science it is very important for nontheoreticians to understand how theoreticians think, just as it is equally important within computer science for theoreticians to understand how nontheoreticians think

Turing-equivalent automata using a fixed-size quantum memory

In this paper, we introduce a new public quantum interactive proof system and the first quantum alternating Turing machine: qAM proof system and qATM, respectively. Both are obtained from their classical counterparts (Arthur-Merlin proof system and alternating Turing machine, respectively,) by augmenting them with a fixed-size quantum register. We focus on space-bounded computation, and obtain the following surprising results: Both of them with constant-space are Turing-equivalent. More specifically, we show that for any Turing-recognizable language, there exists a constant-space weak-qAM system, (the nonmembers do not need to be rejected with high probability), and we show that any Turing-recognizable language can be recognized by a constant-space qATM even with one-way input head. For strong proof systems, where the nonmembers must be rejected with high probability, we show that the known space-bounded classical private protocols can also be simulated by our public qAM system with the same space bound. Besides, we introduce a strong version of qATM: The qATM that must halt in every computation path. Then, we show that strong qATMs (similar to private ATMs) can simulate deterministic space with exponentially less space. This leads to shifting the deterministic space hierarchy exactly by one-level. The method behind the main results is a new public protocol cleverly using its fixed-size quantum register. Interestingly, the quantum part of this public protocol cannot be simulated by any space-bounded classical protocol in some cases.Comment: 28 page

Beautiful Structures: An Appreciation of the Contributions of Alan Selman

Professor Alan Selman has been a giant in the field of computational complexity for the past forty years. This article is an appreciation, on the occasion of his retirement, of some of the most lovely concepts and results that Alan has contributed to the field.Comment: This article will appear, in slightly different form, in the Complexity Theory Column of the September 2014 issue of SIGACT New

Satisfiability is quasilinear complete in NQL

Considered are the classes QL (quasilinear) and NQL (nondet quasllmear) of all those problems that can be solved by deterministic (nondetermlnlsttc, respectively) Turmg machines in time O(n(log n) ~) for some k Effloent algorithms have time bounds of th~s type, it is argued. Many of the "exhausUve search" type problems such as satlsflablhty and colorabdlty are complete in NQL with respect to reductions that take O(n(log n) k) steps This lmphes that QL = NQL iff satisfiabdlty is m QL CR CATEGORIES: 5.2

Different Approaches to Proof Systems

The classical approach to proof complexity perceives proof systems as deterministic, uniform, surjective, polynomial-time computable functions that map strings to (propositional) tautologies. This approach has been intensively studied since the late 70’s and a lot of progress has been made. During the last years research was started investigating alternative notions of proof systems. There are interesting results stemming from dropping the uniformity requirement, allowing oracle access, using quantum computations, or employing probabilism. These lead to different notions of proof systems for which we survey recent results in this paper