Defining substitution for a language with binders like the simply typed $\lambda$-calculus requires repetition, defining substitution and renaming separately. To verify the categorical properties of this calculus, we must repeat the same argument many times. We present a lightweight method that avoids repetition and that gives rise to a simply typed category with families (CwF) isomorphic to the initial simply typed CwF. Our paper is a literate Agda script.

Setchain has been proposed to increase blockchain scalability by relaxing the strict total order requirement among transactions. Setchain organizes elements into a sequence of sets, referred to as epochs, so that elements within each epoch are unordered. In this paper, we propose and evaluate three distinct Setchain algorithms, that leverage an underlying block-based ledger. Vanilla is a basic implementation that serves as a reference point. Compresschain aggregates elements into batches, and compresses these batches before appending them as epochs in the ledger. Hashchain converts batches into fixed-length hashes which are appended as epochs in the ledger. This requires Hashchain to use a distributed service to obtain the batch contents from its hash. To allow light clients to safely interact with only one server, the proposed algorithms maintain, as part of the Setchain, proofs for the epochs. An epoch-proof is the hash of the epoch, cryptographically signed by a server. A client can verify the correctness of an epoch with $f+1$ epoch-proofs (where $f$ is the maximum number of Byzantine servers assumed). All three Setchain algorithms are implemented on top of the CometBFT blockchain application platform. We conducted performance evaluations across various configurations, using clusters of four, seven, and ten servers. Our results show that the Setchain algorithms reach orders of magnitude higher throughput than the underlying blockchain, and achieve finality with latency below 4 seconds.

Defining substitution for a language with binders like the simply typed $\lambda$-calculus requires repetition, defining substitution and renaming separately. To verify the categorical properties of this calculus, we must repeat the same argument many times. We present a lightweight method that avoids repetition and that gives rise to a simply typed category with families (CwF) isomorphic to the initial simply typed CwF. Our paper is a literate Agda script.

Fair allocation of indivisible goods has attracted extensive attention over the last two decades, yielding numerous elegant algorithmic results and producing challenging open questions. The problem becomes much harder in the presence of strategic agents. Ideally, one would want to design truthful mechanisms that produce allocations with fairness guarantees. However, in the standard setting without monetary transfers, it is generally impossible to have truthful mechanisms that provide non-trivial fairness guarantees. Recently, Amanatidis et al. [2021] suggested the study of mechanisms that produce fair allocations in their equilibria. Specifically, when the agents have additive valuation functions, the simple Round-Robin algorithm always has pure Nash equilibria and the corresponding allocations are envy-free up to one good (EF1) with respect to the agents' true valuation functions. Following this agenda, we show that this outstanding property of the Round-Robin mechanism extends much beyond the above default assumption of additivity. In particular, we prove that for agents with cancelable valuation functions (a natural class that contains, e.g., additive and budget-additive functions), this simple mechanism always has equilibria and even its approximate equilibria correspond to approximately EF1 allocations with respect to the agents' true valuation functions. Further, we show that the approximate EF1 fairness of approximate equilibria surprisingly holds for the important class of submodular valuation functions as well, even though exact equilibria fail to exist!