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Nakamoto consensus, the protocol underlying Bitcoin, has the potential to secure a new class of systems which agree on non-mathematical truths. As an example of this capability, we propose a design for a trustless, data availability oracle. This exposition reduces the problem of determining whether or not a registered datum is publicly available to the problem of constructing a network in which either almost all nodes can download a given datum, or almost none of them can.

Bitcoin and Ethereum, whose miners arguably collectively comprise the most powerful computational resource in the history of mankind, offer no more power for processing and verifying transactions than a typical smart phone. The system described herein bypasses this bottleneck and brings scalable computation to Ethereum. Our new system consists of a financial incentive layer atop a dispute resolution layer where the latter takes form of a versatile "verification game." In addition to secure outsourced computation, immediate applications include decentralized mining pools whose operator is an Ethereum smart contract, a cryptocurrency with scalable transaction throughput, and a trustless means for transferring currency between disjoint cryptocurrency systems.

Ethereum has emerged as a dynamic platform for exchanging cryptocurrency tokens. While token crowdsales cannot simultaneously guarantee buyers both certainty of valuation and certainty of participation, we show that if each token buyer specifies a desired purchase quantity at each valuation then everyone can successfully participate. Our implementation introduces smart contract techniques which recruit outside participants in order to circumvent computational complexity barriers.

In December 2015, a bounty emerged to establish both reliable communication and secure transfer of value between the Dogecoin and Ethereum blockchains. This prized "Dogethereum bridge" would allow parties to "lock" a DOGE coin on Dogecoin and in exchange receive a newly minted WOW token in Ethereum. Any subsequent owner of the WOW token could burn it and, in exchange, earn the right to "unlock" a DOGE on Dogecoin. We describe an efficient, trustless, and retrofitting Dogethereum construction which requires no fork but rather employs economic collateral to achieve a "lock" operation in Dogecoin. The protocol relies on bulletproofs, Truebit, and parametrized tokens to efficiently and trustlessly relay events from the "true" Dogecoin blockchain into Ethereum. The present construction not only enables cross-platform exchange but also allows Ethereum smart contracts to trustlessly access Dogecoin. A similar technique adds Ethereum-based smart contracts to Bitcoin and Bitcoin data to Ethereum smart contracts.

We outline a token model for Truebit, a retrofitting, blockchain enhancement which enables secure, community-based computation. The model addresses the challenge of stable task pricing, as raised in the Truebit whitepaper, without appealing to external oracles, exchanges, or hierarchical nodes. The system's sustainable economics and fair market pricing derive from a mintable token format which leverages existing tokens for liquidity. Finally, we introduce a governance layer whose lifecycles culminates with permanent dissolution into utility tokens, thereby tending the network towards autonomous decentralization., Comment: 2 figures

The present paper introduces a practical protocol for provably secure, outsourced computation. Our protocol minimizes overhead for verification by requiring solutions to withstand an interactive game between a prover and challenger. For optimization problems, the best or nearly best of all submitted solutions is expected to be accepted by this approach. Financial incentives and deposits are used in order to overcome the problem of fake participants.

The Ku\v{c}era-G\'acs theorem is a landmark result in algorithmic randomness asserting that every real is computable from a Martin-L\"of random real. If the computation of the first $n$ bits of a sequence requires $n+h(n)$ bits of the random oracle, then $h$ is the redundancy of the computation. Ku\v{c}era implicitly achieved redundancy $n\log n$ while G\'acs used a more elaborate coding procedure which achieves redundancy $\sqrt{n}\log n$. A similar upper bound is implicit in the later proof by Merkle and Mihailovi\'c. In this paper we obtain strict optimal lower bounds on the redundancy in computations from Martin-L\"of random oracles. We show that any nondecreasing computable function $g$ such that $\sum_n 2^{-g(n)}=\infty$ is not a general upper bound on the redundancy in computations from Martin-L\"of random oracles. In fact, there exists a real $X$ such that the redundancy $g$ of any computation of $X$ from a Martin-L\"of random oracle satisfies $\sum_n 2^{-g(n)}<\infty$. Moreover, the class of such reals is comeager and includes a $\Delta^0_2$ real as well as all weakly 2-generic reals. This excludes many slow growing functions such as $\log n$ from bounding the redundancy in computations from random oracles for a large class of reals. On the other hand it was recently shown that if $\sum_n 2^{-g(n)}<\infty$ then $g$ is a general upper bound for the redundancy in computations of any real from some Martin-L\"of random oracle. Our results are obtained as an application of a theory of effective betting strategies with restricted wagers which we develop.

An index $e$ in a numbering of partial-recursive functions is called minimal if every lesser index computes a different function from $e$. Since the 1960's it has been known that, in any reasonable programming language, no effective procedure determines whether or not a given index is minimal. We investigate whether the task of determining minimal indices can be solved in an approximate sense. Our first question, regarding the set of minimal indices, is whether there exists an algorithm which can correctly label 1 out of $k$ indices as either minimal or non-minimal. Our second question, regarding the function which computes minimal indices, is whether one can compute a short list of candidate indices which includes a minimal index for a given program. We give some negative results and leave the possibility of positive results as open questions.

Motivated by an application in network security, we investigate the following "linear" case of Directed Mutlicut. Let $G$ be a directed graph which includes some distinguished vertices $t_1, \ldots, t_k$. What is the size of the smallest edge cut which eliminates all paths from $t_i$ to $t_j$ for all $i < j$? We show that this problem is fixed-parameter tractable when parametrized in the cutset size $p$ via an algorithm running in $O(4^p p n^4)$ time., Comment: 12 pages, 1 figure

Is it possible to find a shortest description for a binary string? The well-known answer is "no, Kolmogorov complexity is not computable." Faced with this barrier, one might instead seek a short list of candidates which includes a laconic description. Remarkably such approximations exist. This paper presents an efficient algorithm which generates a polynomial-size list containing an optimal description for a given input string. Along the way, we employ expander graphs and randomness dispersers to obtain an Explicit Online Matching Theorem for bipartite graphs and a refinement of Muchnik's Conditional Complexity Theorem. Our main result extends recent work by Bauwens, Mahklin, Vereschchagin, and Zimand.

One says that a property $P$ of sets of natural numbers can be made into itself iff there is a numbering $\alpha_0,\alpha_1,\ldots$ of all left-r.e. sets such that the index set $\{e: \alpha_e$ satisfies $P\}$ has the property $P$ as well. For example, the property of being Martin-L\"of random can be made into itself. Herein we characterize those singleton properties which can be made into themselves. A second direction of the present work is the investigation of the structure of left-r.e. sets under inclusion modulo a finite set. In contrast to the corresponding structure for r.e. sets, which has only maximal but no minimal members, both minimal and maximal left-r.e. sets exist. Moreover, our construction of minimal and maximal left-r.e. sets greatly differs from Friedberg's classical construction of maximal r.e. sets. Finally, we investigate whether the properties of minimal and maximal left-r.e. sets can be made into themselves.

We determine the constructive dimension of points in random translates of the Cantor set. The Cantor set "cancels randomness" in the sense that some of its members, when added to Martin-Lof random reals, identify a point with lower constructive dimension than the random itself. In particular, we find the Hausdorff dimension of the set of points in a Cantor set translate with a given constructive dimension.

We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr, and Kurtz random sets, weakly 1-generics and their complementary classes, we find that there exist characterizations of the third and fourth levels of the arithmetic hierarchy purely in terms of these notions. More generally, there exists an equivalence between arithmetic complexity and existence of numberings for classes of left-r.e. sets with shift-persistent elements. While some classes (such as Martin-L\"{o}f randoms and Kurtz non-randoms) have left-r.e. numberings, there is no canonical, or acceptable, left-r.e. numbering for any class of left-r.e. randoms. Finally, we note some fundamental differences between left-r.e. numberings for sets and reals.

In the theory of algorithmic randomness, one of the central notions is that of computable randomness. An infinite binary sequence X is computably random if no recursive martingale (strategy) can win an infinite amount of money by betting on the values of the bits of X. In the classical model, the martingales considered are real-valued, that is, the bets made by the martingale can be arbitrary real numbers. In this paper, we investigate a more restricted model, where only integer-valued martingales are considered, and we study the class of random sequences induced by this model., Comment: Long version of the CiE 2010 paper.

It is possible to enumerate all computer programs. In particular, for every partial computable function, there is a shortest program which computes that function. f-MIN is the set of indices for shortest programs. In 1972, Meyer showed that f-MIN is Turing equivalent to 0'', the halting set with halting set oracle. This paper generalizes the notion of shortest programs, and we use various measures from computability theory to describe the complexity of the resulting "spectral sets." We show that under certain Godel numberings, the spectral sets are exactly the canonical sets 0', 0'', 0''', ... up to Turing equivalence. This is probably not true in general, however we show that spectral sets always contain some useful information. We show that immunity, or "thinness" is a useful characteristic for distinguishing between spectral sets. In the final chapter, we construct a set which neither contains nor is disjoint from any infinite arithmetic set, yet it is 0-majorized and contains a natural spectral set. Thus a pathological set becomes a bit more friendly. Finally, a number of interesting open problems are left for the inspired reader., Comment: PhD thesis, 101 pages

In order to understand the performance of a code under maximum-likelihood (ML) decoding, it is crucial to know the minimal codewords. In the context of linear programming (LP) decoding, it turns out to be necessary to know the minimal pseudo-codewords. This paper studies the minimal codewords and minimal pseudo-codewords of some families of codes derived from projective and Euclidean planes. Although our numerical results are only for codes of very modest length, they suggest that these code families exhibit an interesting property. Namely, all minimal pseudo-codewords that are not multiples of a minimal codeword have an AWGNC pseudo-weight that is strictly larger than the minimum Hamming weight of the code. This observation has positive consequences not only for LP decoding but also for iterative decoding., Comment: To appear in Proc. 2005 IEEE International Symposium on Information Theory, Adelaide, Australia, September 4-9, 2005