We study the *refuter* problems for proof complexity lower bounds. Suppose ϕ is a hard tautology that does not admit any length-s proof in some proof system P. In the corresponding refuter problem, we are given (query access to) a purported length-s proof π in P that claims to have proved ϕ, and our goal is to find an invalid derivation step within π. As suggested by witnessing theorems in bounded arithmetic, the *computational complexity* of these refuter problems is closely tied to the *metamathematics* of the underlying lower bounds.

Abstract. We study the problem of finding a Tarski fixed point over the [Formula: see text]-dimensional grid [Formula: see text]. We give a black-box reduction from the Tarski problem to the same problem with an additional promise that the input function has a unique fixed point. It implies that the Tarski problem and the unique Tarski problem have exactly the same query complexity. Our reduction is based on a novel notion of partial-information functions which we use to fool algorithms for the unique Tarski problem as if they were working on a monotone function with a unique fixed point.

Stackelberg equilibria, as functions of the players' payoffs, can inversely reveal information about the players' incentives. In this paper, we study to what extent one can learn about the leader's incentives by actively querying the leader's optimal commitments against strategically designed followers. We show that, by using polynomially many queries and operations, one can learn a payoff function that is strategically equivalent to the leader's, in the sense that: 1) it preserves the leader's preference over almost all strategy profiles; and 2) it preserves the set of all possible (strong) Stackelberg equilibria the leader may engage in, considering all possible follower types. As an application, we show that the information acquired by our algorithm is sufficient for a follower to induce the best possible Stackelberg equilibrium by imitating a different follower type. To the best of our knowledge, we are the first to demonstrate that this is possible without knowing the leader's payoffs beforehand.

In the ever-evolving blockchain ecosystem, decentralized exchanges (DEXs) have seen significant growth, which, however, has also brought challenges of Maximal Extractable Value (MEV). DEXs offer a decentralized platform for cryptocurrency trading. Such trading mechanisms primarily include Constant Function Market Makers (CFMMs) and batch auctions.

The standard model of Boolean function property testing is not well suited for testing sparse functions which have few satisfying assignments, since every such function is close (in the usual Hamming distance metric) to the constant-0 function. In this work we propose and investigate a new model for property testing of Boolean functions, called relative-error testing, which provides a natural framework for testing sparse functions.

We give an algorithm for finding an ε-fixed point of a contraction map f : [0, 1] k \(\mapsto\) [0, 1] k under the \(\ell _\infty\) -norm with query complexity O ( k log (1/ε).

Inspired by the classic problem of Boolean function monotonicity testing, we investigate the testability of other well-studied properties of combinatorial finite set systems, specifically \emph{intersecting} families and \emph{union-closed} families. A function $f: \{0,1\}^n \to \{0,1\}$ is intersecting (respectively, union-closed) if its set of satisfying assignments corresponds to an intersecting family (respectively, a union-closed family) of subsets of $[n]$. Our main results are that -- in sharp contrast with the property of being a monotone set system -- the property of being an intersecting set system, and the property of being a union-closed set system, both turn out to be information-theoretically difficult to test. We show that: $\bullet$ For $ε\geq Ω(1/\sqrt{n})$, any non-adaptive two-sided $ε$-tester for intersectingness must make $2^{Ω(n^{1/4}/\sqrtε)}$ queries. We also give a $2^{Ω(\sqrt{n \log(1/ε)})}$-query lower bound for non-adaptive one-sided $ε$-testers for intersectingness. $\bullet$ For $ε\geq 1/2^{Ω(n^{0.49})}$, any non-adaptive two-sided $ε$-tester for union-closedness must make $n^{Ω(\log(1/ε))}$ queries. Thus, neither intersectingness nor union-closedness shares the $\mathrm{poly}(n,1/ε)$-query non-adaptive testability that is enjoyed by monotonicity. To complement our lower bounds, we also give a simple $\mathrm{poly}(n^{\sqrt{n\log(1/ε)}},1/ε)$-query, one-sided, non-adaptive algorithm for $ε$-testing each of these properties (intersectingness and union-closedness). We thus achieve nearly tight upper and lower bounds for two-sided testing of intersectingness when $ε= Θ(1/\sqrt{n})$, and for one-sided testing of intersectingness when $ε=Θ(1).$

We give the first super-polynomial (in fact, mildly exponential) lower bounds for tolerant testing (equivalently, distance estimation) of monotonicity, unateness, and juntas with a constant separation between the “yes” and “no” cases. Specifically, we give

We give an algorithm for finding an є-fixed point of a contraction map f:[0,1]k↦[0,1]k under the ℓ∞-norm with query complexity O (k2log(1/є ) ).

We study the query complexity of finding a Tarski fixed point over the k-dimensional grid {1,...,n}k. Improving on the previous best upper bound of O(log⌈2k/3⌉n)[7], we give a new algorithm with query complexity O(log⌈(k+1)/2⌉n). This is based on a novel decomposition theorem about a weaker variant of the Tarski fixed point problem, where the input consists of a monotone function f:[n]k→[n]k and a monotone sign function b:[n]k→ {-1,0,1} and the goal is to find a point x ∈ [n]k that satisfies either f(x) ≼ x and b(x) ≤ 0 or f(x) ≽ x and b(x) ≥ 0.

The Internet-scale peer-to-peer (P2P) systems usually build their success on distributed protocols. For example, the well-known BitTorrent network for resource exchange is based on the proportional response protocol, where each participant exchanges its resources with its neighbors in proportion to what it has received in the previous round. The dynamics of such a protocol has been proved to converge to a market equilibrium. On the other hand, it requires thorough incentive analysis to show the robustness of such protocol, as the distributed agents may strategically manipulate the system once they are able to benefit. Recent studies have developed strategyproofness results of the proportional response protocol against agent deviations in the forms of weight cheating and edge deleting. However, the protocol is not truthful against Sybil attacks, under which an agent may create several fictitious identities and control these fictitious identities to exchange resources with others. In this paper, we apply the concept of incentive ratio to measure how much the utility of a strategic agent in a market equilibrium can be improved by playing Sybil attacks. We prove a tight incentive ratio of two for any agent launching Sybil attacks over general networks. The tight incentive ratio of two closes an open problem modeling the successful tit-for-tat protocol for Internet resource exchanging and also presents a complete picture in this line of theoretical studies with real applications.

Fundamental issues in resource sharing over large scale networks have gained much attention from the research community, in response to the growth of sharing economy over the Internet and mobile networks. We are particularly interested in the fundamental file sharing and subsequently P2P network bandwidth sharing developed by BitTorrent and later formalized by Wu and Zhang [15] as the proportional response protocol. It is of practical importance in the design to provide agent incentives to follow the distributed protocol out of their own rationality. We study the robustness of the distributed protocol in this incentive issue against a Sybil attack, a common type of grave threat in P2P network. For the resource sharing on rings, and we characterize the utility gain from a Sybil attack in the concept of incentive ratio. Previous works proved the incentive ratio is lower bounded by two and upper bounded by four, and later the upper bound is improved to three. It has been listed in [5] and [9] as an open problem to tighten them. In this paper, we completely resolve this open problem with a better understanding on the influence from different class agents to the resource allocation under the distributed protocol.