Fast Sampling and Counting k -SAT Solutions in the Local Lemma Regime

We give new algorithms based on Markov chains to sample and approximately count satisfying assignments to k -uniform CNF formulas where each variable appears at most d times. For any k and d satisfying kd < n o(1) and k ≥ 20 log k + 20 log d + 60, the new sampling algorithm runs in close to linear time, and the counting algorithm runs in close to quadratic time. Our approach is inspired by Moitra (JACM, 2019), which remarkably utilizes the Lovász local lemma in approximate counting. Our main technical contribution is to use the local lemma to bypass the connectivity barrier in traditional Markov chain approaches, which makes the well-developed MCMC method applicable on disconnected state spaces such as SAT solutions. The benefit of our approach is to avoid the enumeration of local structures and obtain fixed polynomial running times, even if k = ω (1) or d = ω (1).

Human-Centric Functional Modeling and Evolutionary Biology

The newly emerging science of Human-Centric Functional Modeling provides an approach towards modeling biological and other systems that is hypothesized to maximize human capacity to understand and navigate complexity in those systems. This paper provide an overview exploring how Human-Centric Functional Modeling might be applied in evolutionary biology, and how this increase in capacity to understand the complexity that organisms have evolved into might be achieved. The broader usefulness of Human-Centric Functional Modeling is that it provides a simple mathematical definition of what constitutes a biological system, defines the problem-solving domain of any biological system in terms of abstract mathematical spaces, and provides an expression defining general problem-solving ability in any such domain. This enables it to be seen that all systems with general problem-solving ability in their own domain are potentially an abstraction of a single mathematical pattern of adaptive problem-solving that might apply to all domains. From this perspective nature has already potentially solved problems in biological organisms that can be represented in some abstract functional state spaces as the same general problem that must be solved to address problems in a wide range of other systems, including existential challenges from poverty to climate change, where Human-Centric Functional Modeling enables it to be seen that not only can nature’s solutions be copied, but that nature has demonstrated its solutions to have worked for hundreds of millions of years.

Functional State Spaces and their Formation in Systems from Biological Organisms to the Physical Universe

This paper explores how the emerging science of Human-Centric Functional Modeling or HCFM provides a universal approach to modeling systems that is hypothesized to maximize human capacity to understand and navigate the complexity of systems, and how it facilitates a kind of biomimicry in which the human organism is represented in terms of abstract mathematical spaces that can be used to define simple expressions to represent properties like “complexity” for human systems like cognition, where the same spaces can be used to represent other systems, including the entire physical universe, so that the underlying equivalence of the representations allows the same mathematical expressions to define the same properties where applicable for these very different systems, and therefore allows deep insights to potentially be gained about these systems through looking inward to observe how one’s own cognition functions from one’s first person experience. This paper explores how from this Human-Centric Functional Modeling perspective the properties governing the evolution of life in its functional state space might also govern the formation of the universe in its own functional state space. Human-Centric Functional Modeling also has other significant benefits, one is that in defining behavior in terms of mathematical spaces it enables all the mathematical disciplines that apply to such spaces (e.g. functor theory, category theory, process theory) to be used to understand and navigate the relationships between concepts described in those spaces. Another is that in providing a self-contained representation of the human meaning of any entity, including of any region in the physical universe, Human-Centric Functional Modeling potentially defines the first complete semantic representation of concepts, physical objects, or any other entities represented in a functional state space. When applied to the physical universe this implies that all theoretical or experimental data can be stored in that single model and all theories tested against it to increase capacity to impact a research question. When applied to other systems semantic modeling has equally important implications. Another benefit of Human-Centric Functional Modeling is that it is also a human-centric expression of “constructor theory”, which in the case of physical systems enables accurate predictions to be made about their physical behavior simply from observations of their functions, without needing to understand the specific physics through which the functions are implemented in those systems.

A Survey of Algorithms for Black-Box Safety Validation of Cyber-Physical Systems

Autonomous cyber-physical systems (CPS) can improve safety and efficiency for safety-critical applications, but require rigorous testing before deployment. The complexity of these systems often precludes the use of formal verification and real-world testing can be too dangerous during development. Therefore, simulation-based techniques have been developed that treat the system under test as a black box operating in a simulated environment. Safety validation tasks include finding disturbances in the environment that cause the system to fail (falsification), finding the most-likely failure, and estimating the probability that the system fails. Motivated by the prevalence of safety-critical artificial intelligence, this work provides a survey of state-of-the-art safety validation techniques for CPS with a focus on applied algorithms and their modifications for the safety validation problem. We present and discuss algorithms in the domains of optimization, path planning, reinforcement learning, and importance sampling. Problem decomposition techniques are presented to help scale algorithms to large state spaces, which are common for CPS. A brief overview of safety-critical applications is given, including autonomous vehicles and aircraft collision avoidance systems. Finally, we present a survey of existing academic and commercially available safety validation tools.

Infinite-state graph transformation systems under adverse conditions

We present an approach for modeling adverse conditions by graph transformation systems. To this end, we introduce joint graph transformation systems which involve a system, an interfering environment, and an automaton modeling their interaction. For joint graph transformation systems, we present notions of correctness under adverse conditions. Some instances of correctness are expressible in LTL (linear temporal logic), or in CTL (computation tree logic), respectively. In these cases, verification of joint graph transformation systems is reduced to temporal model checking. To handle infinite state spaces, we incorporate the concept of well-structuredness. We discuss ideas for the verification of joint graph transformation systems using results based on well-structuredness.

Equilibria Existence in Bayesian Games: Climbing the Countable Borel Equivalence Relation Hierarchy

The solution concept of a Bayesian equilibrium of a Bayesian game is inherently an interim concept. The corresponding ex ante solution concept has been termed a Harsányi equilibrium; examples have appeared in the literature showing that there are Bayesian games with uncountable state spaces that have no Bayesian approximate equilibria but do admit a Harsányi approximate equilibrium, thus exhibiting divergent behaviour in the ex ante and interim stages. Smoothness, a concept from descriptive set theory, has been shown in previous works to guarantee the existence of Bayesian equilibria. We show here that higher rungs in the countable Borel equivalence relation hierarchy can also shed light on equilibrium existence. In particular, hyperfiniteness, the next step above smoothness, is a sufficient condition for the existence of Harsányi approximate equilibria in purely atomic Bayesian games.