Abstraction is a commonly used process to represent some low-level system by a more coarse specification with the goal to omit unnecessary details while preserving important aspects. While recent work on abstraction in the situation calculus has focused on non-probabilistic domains, we describe an approach to abstraction of probabilistic and dynamic systems. Based on a variant of the situation calculus with probabilistic belief, we define a notion of bisimulation that allows to abstract a detailed probabilistic basic action theory with noisy actuators and sensors by a possibly non-stochastic basic action theory. By doing so, we obtain abstract Golog programs that omit unnecessary details and which can be translated back to a detailed program for actual execution. This simplifies the implementation of noisy robot programs, opens up the possibility of using non-stochastic reasoning methods (e.g., planning) on probabilistic problems, and provides domain descriptions that are more easily understandable and explainable.

We propose a logic of east and west (LEW ) for points in 1D Euclidean space. It formalises primitive direction relations: east (E), west (W) and indeterminate east/west (Iew). It has a parameter τ ∈ N>1, which is referred to as the level of indeterminacy in directions. For every τ ∈ N>1, we provide a sound and complete axiomatisation of LEW , and prove that its satisfiability problem is NP-complete. In addition, we show that the finite axiomatisability of LEW depends on τ : if τ = 2 or τ = 3, then there exists a finite sound and complete axiomatisation; if τ > 3, then the logic is not finitely axiomatisable. LEW can be easily extended to higher-dimensional Euclidean spaces. Extending LEW to 2D Euclidean space makes it suitable for reasoning about not perfectly aligned representations of the same spatial objects in different datasets, for example, in crowd-sourced digital maps.

Program synthesis strives to generate a computer program as a solution to a given problem specification, expressed with input-output examples or natural language descriptions. The prevalence of large language models advances the state-of-the-art for program synthesis, though limited training resources and data impede open access to such models. To democratize this, we train and release a family of large language models up to 16.1B parameters, called CODEGEN, on natural language and programming language data, and open source the training library JAXFORMER. We show the utility of the trained model by demonstrating that it is competitive with the previous state-of-the-art on zero-shot Python code generation on HumanEval. We further investigate the multi-step paradigm for program synthesis, where a single program is factorized into multiple prompts specifying subproblems. To this end, we construct an open benchmark, Multi-Turn Programming Benchmark (MTPB), consisting of 115 diverse problem sets that are factorized into multi-turn prompts. Our analysis on MTPB shows that the same intent provided to CODEGEN in multi-turn fashion significantly improves program synthesis over that provided as a single turn. We make the training library JAXFORMER and model checkpoints available as open source contribution: https://github.com/salesforce/CodeGen.

Cylindrical Algebraic Decomposition (CAD) is a key proof technique for formal verification of cyber-physical systems. CAD is computationally expensive, with worst-case doubly-exponential complexity. Selecting an optimal variable ordering is paramount to efficient use of CAD. Prior work has demonstrated that machine learning can be useful in determining efficient variable orderings. Much of this work has been driven by CAD problems extracted from applications of the MetiTarski theorem prover. In this paper, we revisit this prior work and consider issues of bias in existing training and test data. We observe that the classical MetiTarski benchmarks are heavily biased towards particular variable orderings. To address this, we apply symmetries to create a new dataset containing more than 41K MetiTarski challenges designed to remove bias. Furthermore, we evaluate issues of information leakage, and test the generalizability of our models on the new dataset.

In 1969, Per Lindstrom proved his celebrated theorem characterising the first-order logic and established criteria for the first-order definability of formal theories for discrete structures. K. J. Barwise, S. Shelah, J. Vaananen and others extended Lindstrom's characterizability program to classes of infinitary logic systems, including a recent paper by M. Dzamonja and J. Vaananen on Karp's chain logic, which satisfies interpolation, undefinability of well-order, and is maximal in the class of logic systems with these properties. The novelty of the chain logic is in its new definition of satisfability. In our paper, we give a framework for Lindstrom's type characterizability of predicate logic systems interpreted semantically in models with objects based on measures (analytic structures). In particular, Hajek's Logic of Integral is redefined as an abstract logic with a new type of Hajek's satisfiability and constitutes a maximal logic in the class of logic systems for describing analytic structures with Lebesgue integrals and satisfying compactness, elementary chain condition, and weak negation.

Moshe Vardi's horizons are big when it comes to computer science. Celebrated for his contributions to formal methods and computational complexity, he is also well known for his role with ACM, and, in particular, with this publication. Vardi took over as Editor-in-Chief of Communications of the ACM in 2008. Though he stepped down to become a Senior Editor in 2017, he continues to publish regularly on topics from corporate ethics to post-graduate education. Here, he talks about socioeconomics, social media, and the impact of computing on society.

We introduce ProofNet, a benchmark for autoformalization and formal proving of undergraduate-level mathematics. The ProofNet benchmarks consists of 371 examples, each consisting of a formal theorem statement in Lean 3, a natural language theorem statement, and a natural language proof. The problems are primarily drawn from popular undergraduate pure mathematics textbooks and cover topics such as real and complex analysis, linear algebra, abstract algebra, and topology. We intend for ProofNet to be a challenging benchmark that will drive progress in autoformalization and automatic theorem proving. We report baseline results on statement autoformalization via in-context learning. Moreover, we introduce two novel statement autoformalization methods: prompt retrieval and distilled backtranslation.

Over the past three decades, the logic programming paradigm has been successfully expanded to support probabilistic modeling, inference and learning. The resulting paradigm of probabilistic logic programming (PLP) and its programming languages owes much of its success to a declarative semantics, the so-called distribution semantics. However, the distribution semantics is limited to discrete random variables only. While PLP has been extended in various ways for supporting hybrid, that is, mixed discrete and continuous random variables, we are still lacking a declarative semantics for hybrid PLP that not only generalizes the distribution semantics and the modeling language but also the standard inference algorithm that is based on knowledge compilation. We contribute the hybrid distribution semantics together with the hybrid PLP language DC-ProbLog and its inference engine infinitesimal algebraic likelihood weighting (IALW). These have the original distribution semantics, standard PLP languages such as ProbLog, and standard inference engines for PLP based on knowledge compilation as special cases. Thus, we generalize the state-of-the-art of PLP towards hybrid PLP in three different aspects: semantics, language and inference. Furthermore, IALW is the first inference algorithm for hybrid probabilistic programming based on knowledge compilation.

The formalization of existing mathematical proofs is a notoriously difficult process. Despite decades of research on automation and proof assistants, writing formal proofs remains arduous and only accessible to a few experts. While previous studies to automate formalization focused on powerful search algorithms, no attempts were made to take advantage of available informal proofs. In this work, we introduce Draft, Sketch, and Prove (DSP), a method that maps informal proofs to formal proof sketches, and uses the sketches to guide an automated prover by directing its search to easier sub-problems. We investigate two relevant setups where informal proofs are either written by humans or generated by a language model. Our experiments and ablation studies show that large language models are able to produce well-structured formal sketches that follow the same reasoning steps as the informal proofs. Guiding an automated prover with these sketches enhances its performance from 20.9% to 39.3% on a collection of mathematical competition problems.

Transformer architectures have achieved great success in solving natural language tasks, which learn strong language representations from large-scale unlabeled texts. In this paper, we seek to go further beyond and explore a new logical inductive bias for better language representation learning. Logic reasoning is known as a formal methodology to reach answers from given knowledge and facts. Inspired by such a view, we develop a novel neural architecture named FOLNet (First-Order Logic Network), to encode this new inductive bias. We construct a set of neural logic operators as learnable Horn clauses, which are further forward-chained into a fully differentiable neural architecture (FOLNet). Interestingly, we find that the self-attention module in transformers can be composed by two of our neural logic operators, which probably explains their strong reasoning performance. Our proposed FOLNet has the same input and output interfaces as other pretrained models and thus could be pretrained/finetuned by using similar losses. It also allows FOLNet to be used in a plug-and-play manner when replacing other pretrained models. With our logical inductive bias, the same set of ``logic deduction skills'' learned through pretraining are expected to be equally capable of solving diverse downstream tasks. For this reason, FOLNet learns language representations that have much stronger transfer capabilities. Experimental results on several language understanding tasks show that our pretrained FOLNet model outperforms the existing strong transformer-based approaches.