Probabilistically checkable proofs (PCPs) can be verified based only on a constant amount of random queries, such that any correct claim has a proof that is always accepted, and incorrect claims are rejected with high probability (regardless of the given alleged proof). We consider two possible features of PCPs:• A PCP is strong if it rejects an alleged proof of a correct claim with probability proportional to its distance from some correct proof of that claim. • A PCP is smooth if each location in a proof is queried with equal probability. We prove that all sets in N P have PCPs that are both smooth and strong, are of polynomial length and can be verified based on a constant number of queries. This is achieved by following the proof of the PCP theorem of Arora et al. (JACM 45(3):501-555, 1998), providing a stronger analysis of the Hadamard and Reed-Muller based PCPs and a refined PCP composition theorem. In fact, we show that any set in N P has a smooth strong canonical PCP of Proximity (PCPP), meaning that there is an efficiently computable bijection of N P witnesses to correct proofs. This improves on the recent construction of Dinur et al. (in: Blum (ed) 10th innovations in theoretical computer science conference, ITCS, San Diego, 2019) of PCPPs that are strong canonical but inherently non-smooth. Our result implies the hardness of approximating the satisfiability of "stable" 3CNF formulae with bounded variable occurrence, where stable means that the number of clauses violated by an assignment is proportional to its distance from a satisfying assignment (in the relative Hamming metric). This proves a hypothesis used in the work of Friggstad, Khodamoradi and Salavatipour (in: Chan (ed) Proceedings of the 30th annual ACM-SIAM symposium on discrete algorithms, SODA, San Diego, 2019), suggesting a connection between the hardness of these instances and other stable optimization problems.

We introduce a variant of PCPs, that we refer to as rectangular PCPs, wherein proofs are thought of as square matrices, and the random coins used by the verifier can be partitioned into two disjoint sets, one determining the row of each query and the other determining the column.We construct PCPs that are efficient, short, smooth and (almost-)rectangular. As a key application, we show that proofs for hard languages in NTIME(2 n ), when viewed as matrices, are rigid infinitely often. This strengthens and considerably simplifies a recent result of Alman and Chen [FOCS, 2019] constructing explicit rigid matrices in FNP. Namely, we prove the following theorem:• There is a constant δ ∈ (0, 1) such that there is an FNP-machine that, for infinitely many N , on input 1 N outputs N × N matrices with entries in F2 that are δN 2 -far (in Hamming distance) from matrices of rank at most 2 log N/Ω(log log N) .Our construction of rectangular PCPs starts with an analysis of how randomness yields queries in the Reed-Muller-based outer PCP of Ben-Sasson, Goldreich, Harsha, Sudan and Vadhan [SICOMP, 2006; CCC, 2005]. We then show how to preserve rectangularity under PCP composition and a smoothnessinducing transformation. This warrants refined and stronger notions of rectangularity, which we prove for the outer PCP and its transforms.

Randomly masking and predicting word tokens has been a successful approach in pre-training language models for a variety of downstream tasks. In this work, we observe that the same idea also applies naturally to sequential decision making, where many well-studied tasks like behavior cloning, offline RL, inverse dynamics, and waypoint conditioning correspond to different sequence maskings over a sequence of states, actions, and returns. We introduce the FlexiBiT framework, which provides a unified way to specify models which can be trained on many different sequential decision making tasks. We show that a single FlexiBiT model is simultaneously capable of carrying out many tasks with performance similar to or better than specialized models. Additionally, we show that performance can be further improved by fine-tuning our general model on specific tasks of interest.

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