Related papers: Robust Simulations and Significant Separations
For a complexity class $C$ and language $L$, a constructive separation of $L \notin C$ gives an efficient algorithm (also called a refuter) to find counterexamples (bad inputs) for every $C$-algorithm attempting to decide $L$. We study the…
Formal explainability guarantees the rigor of computed explanations, and so it is paramount in domains where rigor is critical, including those deemed high-risk. Unfortunately, since its inception formal explainability has been hampered by…
This paper builds model-theoretic tools to detect changes in complexity among the simple theories. We develop a generalization of dividing, called shearing, which depends on a so-called context c. This leads to defining c-superstability, a…
We develop a complexity theory for approximate real computations. We first produce a theory for exact computations but with condition numbers. The input size depends on a condition number, which is not assumed known by the machine. The…
We propose a decision-theoretic framework for computational complexity, complementary to classical theory: moving from syntactic exactness (Turing / Shannon) to semantic simulability (Le Cam). While classical theory classifies problems by…
Probabilistic independence is a useful concept for describing the result of random sampling---a basic operation in all probabilistic languages---and for reasoning about groups of random variables. Nevertheless, existing verification methods…
We continue the study of statistical/computational tradeoffs in learning robust classifiers, following the recent work of Bubeck, Lee, Price and Razenshteyn who showed examples of classification tasks where (a) an efficient robust…
Probabilistic separation logic offers an approach to reasoning about imperative probabilistic programs in which a separating conjunction is used as a mechanism for expressing independence properties. Crucial to the effectiveness of the…
We study the computational complexity theory of smooth, finite-dimensional dynamical systems. Building off of previous work, we give definitions for what it means for a smooth dynamical system to simulate a Turing machine. We then show that…
Separation logic is a substructural logic which has proved to have numerous and fruitful applications to the verification of programs working on dynamic data structures. Recently, Barthe, Hsu and Liao have proposed a new way of giving…
We present a study of the notion of coalgebraic simulation introduced by Hughes and Jacobs. Although in their original paper they allow any functorial order in their definition of coalgebraic simulation, for the simulation relations to have…
We consider the sample complexity of learning with adversarial robustness. Most prior theoretical results for this problem have considered a setting where different classes in the data are close together or overlapping. Motivated by some…
In the well-known complexity class NP are combinatorial problems, whose optimization counterparts are important for many practical settings. These problems typically consider full knowledge about the input. In practical settings, however,…
The problem of non-iterative one-shot and non-destructive correction of unavoidable mistakes arises in all Artificial Intelligence applications in the real world. Its solution requires robust separation of samples with errors from samples…
We investigate the possibility to separate the bisimulation-invariant fragment of P from that of NP, resp. PSPACE. We build on Otto's Theorem stating that the bisimulation-invariant queries in P are exactly those that are definable in the…
We propose and investigate probabilistic guarantees for the adversarial robustness of classification algorithms. While traditional formal verification approaches for robustness are intractable and sampling-based approaches do not provide…
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in…
We revisit the classical concept of near-decomposability in complex systems, introduced by Herbert Simon in his foundational article The Architecture of Complexity, by developing an explicit quantitative analysis based on singular…
This paper establishes the separation of complexity classes $\mathbf{P}$ and $\mathbf{NP}$ through a novel homological algebraic approach grounded in category theory. We construct the computational category $\mathbf{Comp}$, embedding…
Approximate simulation, an extension of simulation relations from formal methods to continuous systems, is a powerful tool for hierarchical control of complex systems. Finding an approximate simulation relation between the full "concrete"…