The goal of inductive logic programming (ILP) is to find a set of logical rules that generalises training examples and background knowledge. We introduce an ILP approach that identifies pointless rules. A rule is pointless if it contains a redundant literal or cannot discriminate against negative examples. We show that ignoring pointless rules allows an ILP system to soundly prune the hypothesis space. Our experiments on multiple domains, including visual reasoning and game playing, show that our approach can reduce learning times by 99% whilst maintaining predictive accuracies.

The goal of inductive logic programming (ILP) is to search for a hypothesis that generalises training data and background knowledge. The challenge is searching vast hypothesis spaces, which is exacerbated because many logically equivalent hypotheses exist. To address this challenge, we introduce a method to break symmetries in the hypothesis space. We implement our idea in answer set programming. Our experiments on multiple domains, including visual reasoning and game playing, show that our approach can reduce solving times from over an hour to just 17 seconds.

Inductive logic programming (ILP) is a form of logical machine learning. The goal is to search a hypothesis space for a hypothesis that generalises training examples and background knowledge. We introduce an approach that shrinks the hypothesis space before an ILP system searches it. Our approach uses background knowledge to find rules that cannot be in an optimal hypothesis regardless of the training examples. For instance, our approach discovers relationships such as even numbers cannot be odd and prime numbers greater than 2 are odd. It then removes violating rules from the hypothesis space. We implement our approach using answer set programming and use it to shrink the hypothesis space of a constraint-based ILP system. Our experiments on multiple domains, including visual reasoning and game playing, show that our approach can substantially reduce learning times whilst maintaining predictive accuracies. For instance, given just 10 seconds of preprocessing time, our approach can reduce learning times from over 10 hours to only 2 seconds.

We introduce a fragment of second-order unification, referred to as \emph{Second-Order Ground Unification (SOGU)}, with the following properties: (i) only one second-order variable is allowed, and (ii) first-order variables do not occur. We study an equational variant of SOGU where the signature contains \textit{associative} binary function symbols (ASOGU) and show that Hilbert's 10$^{th}$ problem is reducible to ASOGU unifiability, thus proving undecidability. Our reduction provides a new lower bound for the undecidability of second-order unification, as previous results required first-order variable occurrences, multiple second-order variables, and/or equational theories involving \textit{length-reducing} rewrite systems. Furthermore, our reduction holds even in the case when associativity of the binary function symbol is restricted to \emph{power associative}, i.e. f(f(x,x),x)= f(x,f(x,x)), as our construction requires a single constant.

Knowledge refactoring compresses logic programs by replacing them with new rules. Current approaches struggle to scale to large programs. To overcome this limitation, we introduce a constrained optimisation refactoring approach. Our first key idea is to encode the problem with decision variables based on literals rather than rules. Our second key idea is to focus on linear invented rules. Our empirical results on multiple domains show that our approach can refactor programs quicker and with more compression than the previous state-of-the-art approach, sometimes by 60%.

We look at the generalization problem modulo some equational theories. This problem is dual to the unification problem: given two input terms, we want to find a common term whose respective two instances are equivalent to the original terms modulo the theory. There exist algorithms for finding generalizations over various equational theories. We focus on modular construction of equational generalization algorithms for the union of signature-disjoint theories. Specifically, we consider the class of regular and collapse-free theories, showing how to combine existing generalization algorithms to produce specific solutions in these cases. Additionally, we identify a class of theories that admit a generalization algorithm based on the application of axioms to resolve the problem. To define this class, we rely on the notion of syntactic theories, a concept originally introduced to develop unification procedures similar to the one known for syntactic unification. We demonstrate that syntactic theories are also helpful in developing generalization procedures similar to those used for syntactic generalization.

The ability to generalise from a small number of examples is a fundamental challenge in machine learning. To tackle this challenge, we introduce an inductive logic programming (ILP) approach that combines negation and predicate invention. Combining these two features allows an ILP system to generalise better by learning rules with universally quantified body-only variables. We implement our idea in NOPI, which can learn normal logic programs with predicate invention, including Datalog programs with stratified negation. Our experimental results on multiple domains show that our approach can improve predictive accuracies and learning times.

Abstract Interest in anti-unification, the dual problem of unification, is rising due to various new applications. For example, anti-unification-based techniques have been used recently in software analysis and related areas such as clone detection and automatic program repair. While syntactic forms of anti-unification have found many interesting uses, some aspects of modern applications are more appropriately modeled by reasoning modulo an equational theory. Thus, extending existing anti-unification methods to deal with important equational theories is the natural step forward. This paper considers anti-unification modulo pure absorption theories, i.e., where some function symbols are associated with a special constant satisfying the axiom $$f(x,\varepsilon _{f}) \,\approx \, f(\varepsilon _{f},x) \,\approx \, \varepsilon _{f}$$ f ( x , ε f ) ≈ f ( ε f , x ) ≈ ε f . We provide a sound and complete rule-based algorithm for such theories. Furthermore, we show that anti-unification modulo absorption is infinitary. Despite this, our algorithm terminates and produces a finitary algorithmic representation of the minimal complete set of solutions.

Generalization techniques have many applications, including template construction, argument generalization, and indexing. Modern interactive provers can exploit advancement in generalization methods over expressive type theories to further develop proof generalization techniques and other transformations. So far, investigations concerned with anti-unification (AU) over λ-terms and similar type theories have focused on developing algorithms for well-studied variants. These variants forbid the nesting of generalization variables, restrict the structure of their arguments, and are unitary . Extending these methods to more expressive variants is important to applications. We consider the case of nested generalization variables and show that the AU problem is nullary (using capture-avoiding substitutions), even when the arguments to free variables are severely restricted.

Anti-unification (AU) is a fundamental operation for generalization computation used for inductive inference. It is the dual operation to unification, an operation at the foundation of automated theorem proving. Interest in AU from the AI and related communities is growing, but without a systematic study of the concept nor surveys of existing work, investigations often resort to developing application-specific methods that existing approaches may cover. We provide the first survey of AU research and its applications and a general framework for categorizing existing and future developments.

Learning complex programs through inductive logic programming (ILP) remains a formidable challenge. Existing higher-order enabled ILP systems show improved accuracy and learning performance, though remain hampered by the limitations of the underlying learning mechanism. Experimental results show that our extension of the versatile Learning From Failures paradigm by higher-order definitions significantly improves learning performance without the burdensome human guidance required by existing systems. Our theoretical framework captures a class of higher-order definitions preserving soundness of existing subsumption-based pruning methods.

Abstract Proof schemata are infinite sequences of proofs which are defined inductively. In this paper we present a general framework for schemata of terms, formulas and unifiers and define a resolution calculus for schemata of quantifier-free formulas. The new calculus generalizes and improves former approaches to schematic deduction. As an application of the method we present a schematic refutation formalizing a proof of a weak form of the pigeon hole principle.

Unital equational theories are defined by axioms that assert the existence of the unit element for some function symbols. We study anti-unification (AU) in unital theories and address the problems of establishing generalization type and designing anti-unification algorithms. First, we prove that when the term signature contains at least two unital functions, anti-unification is of the nullary type by showing that there exists an AU problem, which does not have a minimal complete set of generalizations. Next, we consider two special cases: the linear variant and the fragment with only one unital symbol, and design AU algorithms for them. The algorithms are terminating, sound, complete, and return tree grammars from which the set of generalizations can be constructed. Anti-unification for both special cases is finitary. Further, the algorithm for the one-unital fragment is extended to the unrestricted case. It terminates and returns a tree grammar which produces an infinite set of generalizations. At the end, we discuss how the nullary type of unital anti-unification might affect the anti-unification problem in some combined theories, and list some open questions.

In this paper, we share our experiences concerning the introduction of the Android-based self-study app AXolotl within the first-semester logic course offered at our university. This course is mandatory for students majoring in Computer Science and Artificial Intelligence. AXolotl was used as part of an optional lab assignment bridging clausal reasoning and SAT solving with classical reasoning, proof construction, and first-order logic. The app provides an intuitive interface for proof construction in various logical calculi and aids the students through rule application. The goal of the lab assignment was to help students make a smoother transition from clausal and decompositional reasoning used earlier in the course to inferential and contextual reasoning required for proof construction and first-order logic. We observed that the lab had a positive influence on students' understanding and end the paper with a discussion of these results.

Abstract We consider anti-unification for simply typed lambda terms in theories defined by associativity, commutativity, identity (unit element) axioms and their combinations and develop a sound and complete algorithm which takes two lambda terms and computes their equational generalizations in the form of higher-order patterns. The problem is finitary: the minimal complete set of such generalizations contains finitely many elements. We define the notion of optimal solution and investigate special restrictions of the problem for which the optimal solution can be computed in linear or polynomial time.

In this article, we address two problems related to idempotent anti-unification. First, we show that there exists an anti-unification problem with a single idempotent symbol that has an infinite minimal complete set of generalizations. It means that anti-unification with a single idempotent symbol has infinitary or nullary generalization type, similar to anti-unification with two idempotent symbols, shown earlier by Loïc Pottier. Next, we develop an algorithm that takes an arbitrary idempotent anti-unification problem and computes a representation of its solution set in the form of a regular tree grammar. The algorithm does not depend on the number of idempotent function symbols in the input terms. The language generated by the grammar is the minimal complete set of generalizations of the given anti-unification problem, which implies that idempotent anti-unification is infinitary.