Schematic Cut elimination and the Ordered Pigeonhole Principle [Extended Version]

In previous work, an attempt was made to apply the schematic CERES method [8] to a formal proof with an arbitrary number of {\Pi} 2 cuts (a recursive proof encapsulating the infinitary pigeonhole principle) [5]. However the derived schematic refutation for the characteristic clause set of the proof could not be expressed in the formal language provided in [8]. Without this formalization a Herbrand system cannot be algorithmically extracted. In this work, we provide a restriction of the proof found in [5], the ECA-schema (Eventually Constant Assertion), or ordered infinitary pigeonhole principle, whose analysis can be completely carried out in the framework of [8], this is the first time the framework is used for proof analysis. From the refutation of the clause set and a substitution schema we construct a Herbrand system.Comment: Submitted to IJCAR 2016. Will be a reference for Appendix material in that paper. arXiv admin note: substantial text overlap with arXiv:1503.0855

Recursive First-order Syntactic Unification Modulo Variable Classes

We present a generalization of first-order syntactic unification to a term algebra where variable indexing is part of the object language. Unlike first-order syntactic unification, the number of variables within a given problem is not finitely bound as terms can have self-symmetric subterms (modulo an index shift) allowing the construction of infinitely deep terms containing infinitely many variables, what we refer to as arithmetic progressive terms. Such constructions are related to inductive reasoning. We show that unifiability is decidable for so-called simple linear 1-loops and conjecture decidability for less restricted classes of loops.Comment: pre-prin

Generalisation Through Negation and Predicate Invention

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.Comment: Under peer-revie

A Generic Framework for Higher-Order Generalizations

We consider a generic framework for anti-unification of simply typed lambda terms. It helps to compute generalizations which contain maximally common top part of the input expressions, without nesting generalization variables. The rules of the corresponding anti-unification algorithm are formulated, and their soundness and termination are proved. The algorithm depends on a parameter which decides how to choose terms under generalization variables. Changing the particular values of the parameter, we obtained four new unitary variants of higher-order anti-unification and also showed how the already known pattern generalization fits into the schema

Anti-unification and Generalization: A Survey

Anti-unification (AU), also known as generalization, is a fundamental operation used for inductive inference and is the dual operation to unification, an operation at the foundation of 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, investigations7 often resort to developing application-specific methods that may be covered by existing approaches. We provide the first survey of AU research and its applications, together with a general framework for categorizing existing and future developments.Comment: Accepted at IJCAI 2023 - Survey Trac

Unital Anti-Unification: Type and Algorithms

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

One or Nothing: Anti-unification over the Simply-Typed Lambda Calculus

Generalization techniques have many applications, such as 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 lambda 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.Comment: 12 pages, submitted to Journal (a revised version of the previous submission with more precise proofs

Higher-Order Equational Pattern Anti-Unification

We consider anti-unification for simply typed lambda terms in associative, commutative, and associative-commutative theories and develop a sound and complete algorithm which takes two lambda terms and computes their generalizations in the form of higher-order patterns. The problem is finitary: the minimal complete set of generalizations contains finitely many elements. We define the notion of optimal solution and investigate special fragments of the problem for which the optimal solution can be computed in linear or polynomial time