Efficient omission of symmetric solution candidates is essential for combinatorial problem-solving. Most of the existing approaches are instance-specific and focus on the automatic computation of Symmetry Breaking Constraints (SBCs) for each given problem instance. However, the application of such approaches to large-scale instances or advanced problem encodings might be problematic since the computed SBCs are propositional and, therefore, can neither be meaningfully interpreted nor transferred to other instances. As a result, a time-consuming recomputation of SBCs must be done before every invocation of a solver. To overcome these limitations, we introduce a new model-oriented approach for Answer Set Programming that lifts the SBCs of small problem instances into a set of interpretable first-order constraints using the Inductive Logic Programming paradigm. Experiments demonstrate the ability of our framework to learn general constraints from instance-specific SBCs for a collection of combinatorial problems. The obtained results indicate that our approach significantly outperforms a state-of-the-art instance-specific method as well as the direct application of a solver.

Inductive logic programming is a type of machine learning in which logic programs are learned from examples. This learning typically occurs relative to some background knowledge provided as a logic program. This dissertation introduces bottom preprocessing, a method for generating initial constraints on the programs an ILP system must consider. Bottom preprocessing applies ideas from inverse entailment to modern ILP systems. Inverse entailment is an influential early ILP approach introduced with Progol. This dissertation also presents $\bot$-Popper, an implementation of bottom preprocessing for the modern ILP system Popper. It is shown experimentally that bottom preprocessing can reduce learning times of ILP systems on hard problems. This reduction can be especially significant when the amount of background knowledge in the problem is large.

Traditional automated theorem provers for first-order logic depend on speed-optimized search and many handcrafted heuristics that are designed to work best over a wide range of domains. Machine learning approaches in literature either depend on these traditional provers to bootstrap themselves or fall short on reaching comparable performance. In this paper, we propose a general incremental learning algorithm for training domain specific provers for first-order logic without equality, based only on a basic given-clause algorithm, but using a learned clause-scoring function. Clauses are represented as graphs and presented to transformer networks with spectral features. To address the sparsity and the initial lack of training data as well as the lack of a natural curriculum, we adapt hindsight experience replay to theorem proving, so as to be able to learn even when no proof can be found. We show that provers trained this way can match and sometimes surpass state-of-the-art traditional provers on the TPTP dataset in terms of both quantity and quality of the proofs.

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The idea of representing symbolic knowledge in connectionist systems has been a long-standing endeavour which has attracted much attention recently with the objective of combining machine learning and scalable sound reasoning. Early work has shown a correspondence between propositional logic and symmetrical neural networks which nevertheless did not scale well with the number of variables and whose training regime was inefficient. In this paper, we introduce Logical Boltzmann Machines (LBM), a neurosymbolic system that can represent any propositional logic formula in strict disjunctive normal form. We prove equivalence between energy minimization in LBM and logical satisfiability thus showing that LBM is capable of sound reasoning. We evaluate reasoning empirically to show that LBM is capable of finding all satisfying assignments of a class of logical formulae by searching fewer than 0.75% of the possible (approximately 1 billion) assignments. We compare learning in LBM with a symbolic inductive logic programming system, a state-of-the-art neurosymbolic system and a purely neural network-based system, achieving better learning performance in five out of seven data sets.

The two most effective branching strategies LRB and VSIDS perform differently on different types of instances. Generally, LRB is more effective on crafted instances, while VSIDS is more effective on application ones. However, distinguishing the types of instances is difficult. To overcome this drawback, we propose a branching strategy selection approach based on the vivification ratio. This approach uses the LRB branching strategy more to solve the instances with a very low vivification ratio. We tested the instances from the main track of SAT competitions in recent years. The results show that the proposed approach is robust and it significantly increases the number of solved instances. It is worth mentioning that, with the help of our approach, the solver Maple\_CM can solve more than 16 instances for the benchmark from the 2020 SAT competition.

We provide here a dataset for tasks related to natural language understanding and natural language inference. The dataset contains logical puzzles in natural language from three domains: comparing puzzles, knighs and knaves, and zebra puzzles. Each puzzle is associated with the entire set of atomic questions that can be generated based on the relations and individuals occurring in the text. For each question we provide the correct answer: entailment, contradiction or ambiguity. The answer's correctness is verified against theorem provers. Good puzzles have two properties: (i) each piece of information is necessary and (ii) no unnecessary information is provided. These properties make puzzles interesting candidates for machine comprehension tasks.

As a contribution to quantitative set-theoretic inferencing, a translation is proposed of conjunctions of literals of the forms x y \z, x y \ z, and z {x}, where x,y,z stand for variables ranging over the von Neumann universe of sets, into unquantified Boolean formulae of a rather simple conjunctive normal form. The formulae in the target language involve variables ranging over a Boolean ring of sets, along with a difference operator and relators designating equality, non-disjointness and inclusion. Moreover, the result of each translation is a conjunction of literals of the forms x y\z, x y\z and of implications whose antecedents are isolated literals and whose consequents are either inclusions (strict or non-strict) between variables, or equalities between variables. Besides reflecting a simple and natural semantics, which ensures satisfiability-preservation, the proposed translation has quadratic algorithmic time-complexity, and bridges two languages both of which are known to have an NP-complete satisfiability problem. Key words: Satisfiability problem, Computable set theory, Expressibility, Proof verification, NP-completeness, quantitative logical inference.

Inductive logic programming (ILP) (Muggleton 1996) has We propose first-order extensions of LNNs that can been of long-standing interest where the goal is to learn tackle ILP. Since vanilla backpropagation is insufficient for logical rules from labeled data. Since rules are explicitly constraint optimization, we propose flexible learning algorithms symbolic, they provide certain advantages over black box capable of handling a variety of (linear) inequality and models. For instance, learned rules can be inspected, understood equality constraints. We experiment with diverse benchmarks and verified forming a convenient means of storing for ILP including gridworld and knowledge base completion learned knowledge. Consequently, a number of approaches (KBC) that call for learning of different kinds of rules have been proposed to address ILP including, but not limited and show how our approach can tackle both effectively. In to, statistical relational learning (Getoor and Taskar 2007) fact, our KBC results represents a 4-16% relative improvement and more recently, neuro-symbolic methods.

The new computational paradigm of conceptual computation has been introduced in the research program of Artificial Mathematical Intelligence. We provide the explicit artificial generation (or conceptual computation) for the fundamental mathematical notion of topological groups. Specifically, we start with two basic notions belonging to topology and abstract algebra, and we describe recursively formal specifications in the Common Algebraic Specification Language (CASL). The notion of conceptual blending between such conceptual spaces can be materialized computationally in the Heterogeneous Tool Set (HETS). The fundamental notion of topological groups is explicitly generated through three different artificial specifications based on conceptual blending and conceptual identification, starting with the concepts of continuous functions and mathematical groups (described with minimal set-theoretical conditions). This constitutes in additional heuristic evidence for the third pillar of Artificial Mathematical Intelligence.