Autonomous and semi-autonomous systems for safety-critical applications require rigorous testing before deployment. Due to the complexity of these systems, formal verification may be impossible and real-world testing may be dangerous during development. Therefore, simulation-based techniques have been developed that treat the system under test as a black box during testing. Safety validation tasks include finding disturbances to the system that cause it 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 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, and 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.

Learning the correct use of mathematical language frequently poses a challenge for beginner students. At the same time, it is a basic skill, required both for understanding mathematical texts and for presenting one's own work. In mathematical lectures and typical textbooks, this is rarely explictly discusses, though some offer a brief discussion, along with some formalization exercises (see, e.g., [Ha]). In this note, we present two pieces of software that pursue the goal to support beginner students in learning the use of formal language. The first one, called "Math Dictations" (a word that we learned from M. Junk, who used the concept (but no automatization thereof) in introductory courses at the university of Konstanz), challenges students to translate a proposition given in natural language, such as "the real function f is strictly increasing" into a quantifier formula such as x y(x y f(x) f(y)). It is similar to the formalization exercises that form part of the "Mathematical Logic Tutor" by A. Moreno (see [BM]), but goes beyond this in (i) allowing first-order logic rather than propositional logic and (ii) using a restricted automated theorem prover for evaluating solutions, so that many solutions, rather than a single one, are recognized as correct answers. After the "Math Dictations" had been implemented and the first version of this article had been posted, we were made aware of the fact that this kind of formalization exercise is available in the Edukera system

The Apperception Engine is an unsupervised learning system. Given a sequence of sensory inputs, it constructs a symbolic causal theory that both explains the sensory sequence and also satisfies a set of unity conditions. The unity conditions insist that the constituents of the theory - objects, properties, and laws - must be integrated into a coherent whole. Once a theory has been constructed, it can be applied to predict future sensor readings, retrodict earlier readings, or impute missing readings. In this paper, we evaluate the Apperception Engine in a diverse variety of domains, including cellular automata, rhythms and simple nursery tunes, multi-modal binding problems, occlusion tasks, and sequence induction intelligence tests. In each domain, we test our engine's ability to predict future sensor values, retrodict earlier sensor values, and impute missing sensory data. The engine performs well in all these domains, significantly outperforming neural net baselines and state of the art inductive logic programming systems. These results are significant because neural nets typically struggle to solve the binding problem (where information from different modalities must somehow be combined together into different aspects of one unified object) and fail to solve occlusion tasks (in which objects are sometimes visible and sometimes obscured from view). We note in particular that in the sequence induction intelligence tests, our system achieved human-level performance. This is notable because our system is not a bespoke system designed specifically to solve intelligence tests, but a general-purpose system that was designed to make sense of any sensory sequence.

Program synthesis techniques construct or infer programs from user-provided specifications, such as input-output examples. Yet most specifications, especially those given by end-users, leave the synthesis problem radically ill-posed, because many programs may simultaneously satisfy the specification. Prior work resolves this ambiguity by using various inductive biases, such as a preference for simpler programs. This work introduces a new inductive bias derived by modeling the program synthesis task as rational communication, drawing insights from recursive reasoning models of pragmatics. Given a specification, we score a candidate program both on its consistency with the specification, and also whether a rational speaker would chose this particular specification to communicate that program. We develop efficient algorithms for such an approach when learning from input-output examples, and build a pragmatic program synthesizer over a simple grid-like layout domain. A user study finds that end-user participants communicate more effectively with the pragmatic program synthesizer over a non-pragmatic one.

It is well-know that deciding consistency for normal answer set programs (ASP) is NP-complete, thus, as hard as the satisfaction problem for classical propositional logic (SAT). The best algorithms to solve these problems take exponential time in the worst case. The exponential time hypothesis (ETH) implies that this result is tight for SAT, that is, SAT cannot be solved in subexponential time. This immediately establishes that the result is also tight for the consistency problem for ASP. However, accounting for the treewidth of the problem, the consistency problem for ASP is slightly harder than SAT: while SAT can be solved by an algorithm that runs in exponential time in the treewidth k, it was recently shown that ASP requires exponential time in k \cdot log(k). This extra cost is due checking that there are no self-supported true atoms due to positive cycles in the program. In this paper, we refine the above result and show that the consistency problem for ASP can be solved in exponential time in k \cdot log({\lambda}) where {\lambda} is the minimum between the treewidth and the size of the largest strongly-connected component in the positive dependency graph of the program. We provide a dynamic programming algorithm that solves the problem and a treewidth-aware reduction from ASP to SAT that adhere to the above limit.

Modern SAT solvers routinely operate at scales that make it impractical to query a neural network for every branching decision. NeuroCore, proposed by Selsam and Bjorner, offered a proof-of-concept that neural networks can still accelerate SAT solvers by only periodically refocusing a score-based branching heuristic. However, that work suffered from several limitations: their modified solvers require GPU acceleration, further ablations showed that they were no better than a random baseline on the SATCOMP 2018 benchmark, and their training target of unsat cores required an expensive data pipeline which only labels relatively easy unsatisfiable problems. We address all these limitations, using a simpler network architecture allowing CPU inference for even large industrial problems with millions of clauses, and training instead to predict {\em glue variables}---a target for which it is easier to generate labelled data, and which can also be formulated as a reinforcement learning task. We demonstrate the effectiveness of our approach by modifying the state-of-the-art SAT solver CaDiCaL, improving its performance on SATCOMP 2018 and SATRACE 2019 with supervised learning and its performance on a dataset of SHA-1 preimage attacks with reinforcement learning.

Actual causality is increasingly well understood. Recent formal approaches, proposed by Halpern and Pearl, have made this concept mature enough to be amenable to automated reasoning. Actual causality is especially vital for building accountable, explainable systems. Among other reasons, causality reasoning is computationally hard due to the requirements of counterfactuality and the minimality of causes. Previous approaches presented either inefficient or restricted, and domain-specific, solutions to the problem of automating causality reasoning. In this paper, we present a novel approach to formulate different notions of causal reasoning, over binary acyclic models, as optimization problems, based on quantifiable notions within counterfactual computations. We contribute and compare two compact, non-trivial, and sound integer linear programming (ILP) and Maximum Satisfiability (MaxSAT) encodings to check causality. Given a candidate cause, both approaches identify what a minimal cause is. Also, we present an ILP encoding to infer causality without requiring a candidate cause. We show that both notions are efficiently automated. Using models with more than $8000$ variables, checking is computed in a matter of seconds, with MaxSAT outperforming ILP in many cases. In contrast, inference is computed in a matter of minutes.

In learning-assisted theorem proving, one of the most critical challenges is to generalize to theorems unlike those seen at training time. In this paper, we introduce INT, an INequality Theorem proving benchmark, specifically designed to test agents' generalization ability. INT is based on a procedure for generating theorems and proofs; this procedure's knobs allow us to measure 6 different types of generalization, each reflecting a distinct challenge characteristic of automated theorem proving. In addition, unlike prior benchmarks for learning-assisted theorem proving, INT provides a lightweight and user-friendly theorem proving environment with fast simulations, conducive to performing learning-based and search-based research. We introduce learning-based baselines and evaluate them across 6 dimensions of generalization with the benchmark. We then evaluate the same agents augmented with Monte Carlo Tree Search (MCTS) at test time, and show that MCTS can help to prove new theorems.

We study the separation of positive and negative data examples in terms of description logic (DL) concepts and formulas of decidable FO fragments, in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols from the data and ontology that can be used for separation. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated. Our main results are that (a projective form of) the weak version is decidable in $\mathcal{ALCI}$ while it is undecidable in the guarded fragment GF, the guarded negation fragment GNF, and the DL $\mathcal{ALCFIO}$, and that strong separability is decidable in $\mathcal{ALCI}$, GF, and GNF. We also provide (mostly tight) complexity bounds.

The difference between object-language and metalanguage is crucial for logical analysis, but has yet not been examined for the field of computer science. In this paper the difference is examined with regard to inferential relations. It is argued that inferential relations in a metalanguage (like a calculus for propositional logic) cannot represent conceptual relations of natural language. Inferential relations govern our concept use and understanding. Several approaches in the field of Natural Language Understanding (NLU) and Natural Language Inference (NLI) take this insight in account, but do not consider, how an inference can be assessed as a good inference. I present a logical analysis that can assesss the normative dimension of inferences, which is a crucial part of logical understanding and goes beyond formal understanding of metalanguages.