We describe a very large improvement of existing hammer-style proof automation over large ITP libraries by combining learning and theorem proving. In particular, we have integrated state-of-the-art machine learners into the E automated theorem prover, and developed methods that allow learning and efficient internal guidance of E over the whole Mizar library. The resulting trained system improves the real-time performance of E on the Mizar library by 70% in a single-strategy setting.

The field of geometric automated theorem provers has a long and rich history, from the early AI approaches of the 1960s, synthetic provers, to today algebraic and synthetic provers. The geometry automated deduction area differs from other areas by the strong connection between the axiomatic theories and its standard models. In many cases the geometric constructions are used to establish the theorems' statements, geometric constructions are, in some provers, used to conduct the proof, used as counter-examples to close some branches of the automatic proof. Synthetic geometry proofs are done using geometric properties, proofs that can have a visual counterpart in the supporting geometric construction. With the growing use of geometry automatic deduction tools as applications in other areas, e.g. in education, the need to evaluate them, using different criteria, is felt. Establishing a ranking among geometric automated theorem provers will be useful for the improvement of the current methods/implementations. Improvements could concern wider scope, better efficiency, proof readability and proof reliability. To achieve the goal of being able to compare geometric automated theorem provers a common test bench is needed: a common language to describe the geometric problems; a comprehensive repository of geometric problems and a set of quality measures.

We extend probabilistic action language pBC+ with the notion of utility as in decision theory. The semantics of the extended pBC+ can be defined as a shorthand notation for a decision-theoretic extension of the probabilistic answer set programming language LPMLN. Alternatively, the semantics of pBC+ can also be defined in terms of Markov Decision Process (MDP), which in turn allows for representing MDP in a succinct and elaboration tolerant way as well as to leverage an MDP solver to compute pBC+. The idea led to the design of the system pbcplus2mdp, which can find an optimal policy of a pBC+ action description using an MDP solver.

Deep learning methods capable of handling relational data have proliferated over the last years. In contrast to traditional relational learning methods that leverage first-order logic for representing such data, these deep learning methods aim at re-representing symbolic relational data in Euclidean spaces. They offer better scalability, but can only numerically approximate relational structures and are less flexible in terms of reasoning tasks supported. This paper introduces a novel framework for relational representation learning that combines the best of both worlds. This framework, inspired by the auto-encoding principle, uses first-order logic as a data representation language, and the mapping between the original and latent representation is done by means of logic programs instead of neural networks. We show how learning can be cast as a constraint optimisation problem for which existing solvers can be used. The use of logic as a representation language makes the proposed framework more accurate (as the representation is exact, rather than approximate), more flexible, and more interpretable than deep learning methods. We experimentally show that these latent representations are indeed beneficial in relational learning tasks.

The traditional ground-and-solve approach to Answer Set Programming (ASP) suffers from the grounding bottleneck, which makes large-scale problem instances unsolvable. Lazy grounding is an alternative approach that interleaves grounding with solving and thus uses space more efficiently. The limited view on the search space in lazy grounding poses unique challenges, however, and can have adverse effects on solving performance. In this paper we present a novel characterization of degrees of laziness in grounding for ASP, i.e. of compromises between lazily grounding as little as possible and the traditional full grounding upfront. We investigate how these degrees of laziness compare to each other formally as well as, by means of an experimental analysis using a number of benchmarks, in terms of their effects on solving performance. Our contributions are the introduction of a range of novel lazy grounding strategies, a formal account on their relationships and their correctness, and an investigation of their effects on solving performance. Experiments show that our approach performs significantly better than state-of-the-art lazy grounding in many cases.

The area of formal ethics is experiencing a shift from a unique or standard approach to normative reasoning, as exemplified by so-called standard deontic logic, to a variety of application-specific theories. However, the adequate handling of normative concepts such as obligation, permission, prohibition, and moral commitment is challenging, as illustrated by the notorious paradoxes of deontic logic. In this article we introduce an approach to design and evaluate theories of normative reasoning. In particular, we present a formal framework based on higher-order logic, a design methodology, and we discuss tool support. Moreover, we illustrate the approach using an example of an implementation, we demonstrate different ways of using it, and we discuss how the design of normative theories is now made accessible to non-specialist users and developers.

Information Retrieval (IR) aims at retrieving documents that are most relevant to a query provided by a user. Traditional techniques rely mostly on syntactic methods. In some cases, however, links at a deeper semantic level must be considered. In this paper, we explore a type of IR task in which documents describe sequences of events, and queries are about the state of the world after such events. In this context, successfully matching documents and query requires considering the events' possibly implicit, uncertain effects and side-effects. We begin by analyzing the problem, then propose an action language based formalization, and finally automate the corresponding IR task using Answer Set Programming.

Artificial Intelligence problems, ranging form planning/scheduling up to game control, include an essential crucial step: describing a model which accurately defines the problem's required data, requirements, allowed transitions and established goals. The ways in which a model can fail are numerous and often lead to a failure of search strategies to provide a quick, optimal, or even any solution. This paper proposes using SMT (Satisfiability Modulo Theories) solvers, such as Z3, to check the validity of a model. We propose two tests: checking whether a final(goal) state exists in the model's described problem space and checking whether the transitions described can provide a path from the identified initial states to any the goal states (meaning a solution has been found). The advantage of using an SMT solver for AI model checking is that they substitute actual search strategies and they work over an abstract representation of the model, that is, a set of logical formulas. Reasoning at an abstract level is not as expensive as exploring the entire solution space. SMT solvers use efficient decision procedures which provide proofs for the logical formulas corresponding to the AI model. A recent addition to Z3 allowed us to describe sequences of transitions as a recursive function, thus we can check if a solution can be found in the defined model.

Weighted model integration (WMI) extends Weighted model counting (WMC) to the integration of functions over mixed discrete-continuous domains. It has shown tremendous promise for solving inference problems in graphical models and probabilistic programming. Yet, state-of-the-art tools for WMI are limited in terms of performance and ignore the independence structure that is crucial to improving efficiency. To address this limitation, we propose an efficient model integration algorithm for theories with tree primal graphs. We exploit the sparse graph structure by using search to performing integration. Our algorithm greatly improves the computational efficiency on such problems and exploits context-specific independence between variables. Experimental results show dramatic speedups compared to existing WMI solvers on problems with tree-shaped dependencies.

A logic program with function symbols is called finitely ground if there is a finite propositional logic program whose stable models are exactly the same as the stable models of this program. Finite groundability is an important property for logic programs with function symbols because it makes feasible to compute such programs' stable models using traditional ASP solvers. In this paper, we introduce new decidable classes of finitely ground programs called poly-bounded and k-EXP-bounded programs, which, to the best of our knowledge, strictly contain all other decidable classes of finitely ground programs discovered so far in the literature. We also study the relevant complexity properties for these classes of programs. We prove that the membership complexities for poly-bounded and k-EXP-bounded programs are EXPTIME-complete and (k+1)-EXPTIME-complete, respectively.