240 research outputs found
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
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