The Abstraction and Reasoning Corpus (ARC) aims at benchmarking the performance of general artificial intelligence algorithms. The ARC's focus on broad generalization and few-shot learning has made it difficult to solve using pure machine learning. A more promising approach has been to perform program synthesis within an appropriately designed Domain Specific Language (DSL). However, these too have seen limited success. We propose Abstract Reasoning with Graph Abstractions (ARGA), a new object-centric framework that first represents images using graphs and then performs a search for a correct program in a DSL that is based on the abstracted graph space. The complexity of this combinatorial search is tamed through the use of constraint acquisition, state hashing, and Tabu search. An extensive set of experiments demonstrates the promise of ARGA in tackling some of the complicated object-centric tasks of the ARC rather efficiently, producing programs that are correct and easy to understand.

Semantics of various formalisms for knowledge representation can often be described by fixpoints of corresponding operators. For example, in many logics theories of a set of formulas can be seen as fixpoints of the underlying consequence operator [52]. Likewise, in logic programming, default logic or formal argumentation, all the major semantics can be formulated as different types of fixpoints of the same operator (see [22]). Such operators are usually non-monotonic, and so one cannot always be sure whether their fixpoints exist, and how they can be constructed. In order to deal with this'illusive nature' of the fixpoints, Denecker, Marek and Truszczyński [22] introduced a method for approximating each value z of the underlying operator by a pair of elements (x, y). These elements intuitively represent lower and upper bounds on z, and so a corresponding approximation operator for the original, non-monotonic operator, is constructed. If the approximating operator that is obtained is precision-monotonic, intuitively meaning that more precise inputs of the operator give rise to more precise outputs, then by Tarski and Knaster's Fixpoint Theorem the approximating operator has fixpoints that can be constructively computed, and which in turn approximate the fixpoints of the approximated operator, if such fixpoints exist. The usefulness of the algebraic theory that underlies the computation process described above was demonstrated on several knowledge representation formalisms, such as propositional logic programming [20], default logic [23], autoepistemic logic [23], abstract argumentation and abstract dialectical frameworks [50], hybrid MKNF [39], the graph description language SCHACL [11], and active integrity constraints [10], each one of which was shown to be an instantiation of this abstract theory of approximation.

Neural-symbolic computing aims at integrating robust neural learning and sound symbolic reasoning into a single framework, so as to leverage the complementary strengths of both of these, seemingly unrelated (maybe even contradictory) AI paradigms. The central challenge in neural-symbolic computing is to unify the formulation of neural learning and symbolic reasoning into a single framework with common semantics, that is, to seek a joint representation between a neural model and a logical theory that can support the basic grounding learned by the neural model and also stick to the semantics of the logical theory. In this paper, we propose differentiable fuzzy $\mathcal{ALC}$ (DF-$\mathcal{ALC}$) for this role, as a neural-symbolic representation language with the desired semantics. DF-$\mathcal{ALC}$ unifies the description logic $\mathcal{ALC}$ and neural models for symbol grounding; in particular, it infuses an $\mathcal{ALC}$ knowledge base into neural models through differentiable concept and role embeddings. We define a hierarchical loss to the constraint that the grounding learned by neural models must be semantically consistent with $\mathcal{ALC}$ knowledge bases. And we find that capturing the semantics in grounding solely by maximizing satisfiability cannot revise grounding rationally. We further define a rule-based loss for DF adapting to symbol grounding problems. The experiment results show that DF-$\mathcal{ALC}$ with rule-based loss can improve the performance of image object detectors in an unsupervised learning way, even in low-resource situations.

The behaviour of statistical relational representations across differently sized domains has become a focal area of research from both a modelling and a complexity viewpoint.Recently, projectivity of a family of distributions emerged as a key property, ensuring that marginal probabilities are independent of the domain size. However, the formalisation used currently assumes that the domain is characterised only by its size. This contribution extends the notion of projectivity from families of distributions indexed by domain size to functors taking extensional data from a database. This makes projectivity available for the large range of applications taking structured input. We transfer key known results on projective families of distributions to the new setting. This includes a characterisation of projective fragments in different statistical relational formalisms as well as a general representation theorem for projective families of distributions. Furthermore, we prove a correspondence between projectivity and distributions on countably infinite domains, which we use to unify and generalise earlier work on statistical relational representations in infinite domains. Finally, we use the extended notion of projectivity to define a further strengthening, which we call $\sigma$-projectivity, and which allows the use of the same representation in different modes while retaining projectivity.

Epistemic logics typically talk about knowledge of individual agents or groups of explicitly listed agents. Often, however, one wishes to express knowledge of groups of agents specified by a given property, as in `it is common knowledge among economists'. We introduce such a logic of common knowledge, which we term abstract-group epistemic logic (AGEL). That is, AGEL features a common knowledge operator for groups of agents given by concepts in a separate agent logic that we keep generic, with one possible agent logic being ALC. We show that AGEL is EXPTIME-complete, with the lower bound established by reduction from standard group epistemic logic, and the upper bound by a satisfiability-preserving embedding into the full $\mu$-calculus. Further main results include a finite model property (not enjoyed by the full $\mu$-calculus) and a complete axiomatization.

General mathematical reasoning is computationally undecidable, but humans routinely solve new problems. Moreover, discoveries developed over centuries are taught to subsequent generations quickly. What structure enables this, and how might that inform automated mathematical reasoning? We posit that central to both puzzles is the structure of procedural abstractions underlying mathematics. We explore this idea in a case study on 5 sections of beginning algebra on the Khan Academy platform. To define a computational foundation, we introduce Peano, a theorem-proving environment where the set of valid actions at any point is finite. We use Peano to formalize introductory algebra problems and axioms, obtaining well-defined search problems. We observe existing reinforcement learning methods for symbolic reasoning to be insufficient to solve harder problems. Adding the ability to induce reusable abstractions ("tactics") from its own solutions allows an agent to make steady progress, solving all problems. Furthermore, these abstractions induce an order to the problems, seen at random during training. The recovered order has significant agreement with the expert-designed Khan Academy curriculum, and second-generation agents trained on the recovered curriculum learn significantly faster. These results illustrate the synergistic role of abstractions and curricula in the cultural transmission of mathematics.

Reasoning about actions and change (RAC) is essential to understand and interact with the ever-changing environment. Previous AI research has shown the importance of fundamental and indispensable knowledge of actions, i.e., preconditions and effects. However, traditional methods rely on logical formalization which hinders practical applications. With recent transformer-based language models (LMs), reasoning over text is desirable and seemingly feasible, leading to the question of whether LMs can effectively and efficiently learn to solve RAC problems. We propose four essential RAC tasks as a comprehensive textual benchmark and generate problems in a way that minimizes the influence of other linguistic requirements (e.g., grounding) to focus on RAC. The resulting benchmark, TRAC, encompassing problems of various complexities, facilitates a more granular evaluation of LMs, precisely targeting the structural generalization ability much needed for RAC. Experiments with three high-performing transformers indicates that additional efforts are needed to tackle challenges raised by TRAC.

The aim of this course "Introduction to Automata Theory, Languages and Computation" is to give a detailed working explanation regarding each Mathematical model, its corresponding languages, and their provable equivalence. "Theory of Computation" has three major subdivisions namely

Efficient planning in continuous state and action spaces is fundamentally hard, even when the transition model is deterministic and known. One way to alleviate this challenge is to perform bilevel planning with abstractions, where a high-level search for abstract plans is used to guide planning in the original transition space. Previous work has shown that when state abstractions in the form of symbolic predicates are hand-designed, operators and samplers for bilevel planning can be learned from demonstrations. In this work, we propose an algorithm for learning predicates from demonstrations, eliminating the need for manually specified state abstractions. Our key idea is to learn predicates by optimizing a surrogate objective that is tractable but faithful to our real efficient-planning objective. We use this surrogate objective in a hill-climbing search over predicate sets drawn from a grammar. Experimentally, we show across four robotic planning environments that our learned abstractions are able to quickly solve held-out tasks, outperforming six baselines. Code: https://tinyurl.com/predicators-release

We address the problem of modeling, refining, and repairing formal specifications for robotic missions using assume-guarantee contracts. We show how to model mission specifications at various levels of abstraction and implement them using a library of pre-implemented specifications. Suppose the specification cannot be met using components from the library. In that case, we compute a proxy for the best approximation to the specification that can be generated using elements from the library. Afterward, we propose a systematic way to either 1) search for and refine the `missing part' of the specification that the library cannot meet or 2) repair the current specification such that the existing library can refine it. Our methodology for searching and repairing mission requirements leverages the quotient, separation, composition, and merging operations between contracts.