Derandomizing from Random Strings

In this paper we show that BPP is truth-table reducible to the set of Kolmogorov random strings R_K. It was previously known that PSPACE, and hence BPP is Turing-reducible to R_K. The earlier proof relied on the adaptivity of the Turing-reduction to find a Kolmogorov-random string of polynomial length using the set R_K as oracle. Our new non-adaptive result relies on a new fundamental fact about the set R_K, namely each initial segment of the characteristic sequence of R_K is not compressible by recursive means. As a partial converse to our claim we show that strings of high Kolmogorov-complexity when used as advice are not much more useful than randomly chosen strings

Complexity, BioComplexity, the Connectionist Conjecture and Ontology of Complexity\ud

This paper develops and integrates major ideas and concepts on complexity and biocomplexity - the connectionist conjecture, universal ontology of complexity, irreducible complexity of totality & inherent randomness, perpetual evolution of information, emergence of criticality and equivalence of symmetry & complexity. This paper introduces the Connectionist Conjecture which states that the one and only representation of Totality is the connectionist one i.e. in terms of nodes and edges. This paper also introduces an idea of Universal Ontology of Complexity and develops concepts in that direction. The paper also develops ideas and concepts on the perpetual evolution of information, irreducibility and computability of totality, all in the context of the Connectionist Conjecture. The paper indicates that the control and communication are the prime functionals that are responsible for the symmetry and complexity of complex phenomenon. The paper takes the stand that the phenomenon of life (including its evolution) is probably the nearest to what we can describe with the term “complexity”. The paper also assumes that signaling and communication within the living world and of the living world with the environment creates the connectionist structure of the biocomplexity. With life and its evolution as the substrate, the paper develops ideas towards the ontology of complexity. The paper introduces new complexity theoretic interpretations of fundamental biomolecular parameters. The paper also develops ideas on the methodology to determine the complexity of “true” complex phenomena.\u

Dimension Extractors and Optimal Decompression

A *dimension extractor* is an algorithm designed to increase the effective dimension -- i.e., the amount of computational randomness -- of an infinite binary sequence, in order to turn a "partially random" sequence into a "more random" sequence. Extractors are exhibited for various effective dimensions, including constructive, computable, space-bounded, time-bounded, and finite-state dimension. Using similar techniques, the Kucera-Gacs theorem is examined from the perspective of decompression, by showing that every infinite sequence S is Turing reducible to a Martin-Loef random sequence R such that the asymptotic number of bits of R needed to compute n bits of S, divided by n, is precisely the constructive dimension of S, which is shown to be the optimal ratio of query bits to computed bits achievable with Turing reductions. The extractors and decompressors that are developed lead directly to new characterizations of some effective dimensions in terms of optimal decompression by Turing reductions.Comment: This report was combined with a different conference paper "Every Sequence is Decompressible from a Random One" (cs.IT/0511074, at http://dx.doi.org/10.1007/11780342_17), and both titles were changed, with the conference paper incorporated as section 5 of this new combined paper. The combined paper was accepted to the journal Theory of Computing Systems, as part of a special issue of invited papers from the second conference on Computability in Europe, 200

Rational Proofs with Multiple Provers

Interactive proofs (IP) model a world where a verifier delegates computation to an untrustworthy prover, verifying the prover's claims before accepting them. IP protocols have applications in areas such as verifiable computation outsourcing, computation delegation, cloud computing. In these applications, the verifier may pay the prover based on the quality of his work. Rational interactive proofs (RIP), introduced by Azar and Micali (2012), are an interactive-proof system with payments, in which the prover is rational rather than untrustworthy---he may lie, but only to increase his payment. Rational proofs leverage the provers' rationality to obtain simple and efficient protocols. Azar and Micali show that RIP=IP(=PSAPCE). They leave the question of whether multiple provers are more powerful than a single prover for rational and classical proofs as an open problem. In this paper, we introduce multi-prover rational interactive proofs (MRIP). Here, a verifier cross-checks the provers' answers with each other and pays them according to the messages exchanged. The provers are cooperative and maximize their total expected payment if and only if the verifier learns the correct answer to the problem. We further refine the model of MRIP to incorporate utility gap, which is the loss in payment suffered by provers who mislead the verifier to the wrong answer. We define the class of MRIP protocols with constant, noticeable and negligible utility gaps. We give tight characterization for all three MRIP classes. We show that under standard complexity-theoretic assumptions, MRIP is more powerful than both RIP and MIP ; and this is true even the utility gap is required to be constant. Furthermore the full power of each MRIP class can be achieved using only two provers and three rounds. (A preliminary version of this paper appeared at ITCS 2016. This is the full version that contains new results.)Comment: Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science. ACM, 201