Generalization ALgorithms and Applications [GALA]

@techreport{RISC6080,
author = {David M. Cerna and Temur Kutsia},
title = {{Unital Anti-Unification: Type and Algorithms}},
language = {english},
abstract = {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 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. },
number = {20-02},
year = {2020},
month = {Febrary},
howpublished = {RISC Report},
keywords = {Anti-unification, tree grammars, unital theories, collapse theories},
length = {19},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

@techreport{RISC6074,
author = {Temur Kutsia},
title = {{Unification modulo alpha-equivalence in a mathematical assistant system}},
language = {english},
number = {20-01},
year = {2020},
length = {21},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

@techreport{RISC5918,
author = {David M. Cerna and Temur Kutsia},
title = {{Higher-Order Pattern Generalization Modulo Equational Theories}},
language = {english},
abstract = {In this paper we address Three problems related to unital anti-unification. First, we develop a generalalgorithm based on a tree grammar representation of the set of computed generalizations. Secondlywe show that restricting the algorithm to computing linear generalizations only or to term signaturescontaining a single unital function results in a procedure which is minimal complete and Finitary.Thirdly, we show that when the term signature contains two unital functions, unital anti-unification isNullary.The algorithm does not depend on the number of idempotent function symbols in the input terms. Thelanguage generated by the grammar is the minimal complete set of generalizations of the givenanti-unification problem, which implies that idempotent anti-unification is infinitary.},
year = {2019},
length = {40},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

@techreport{RISC5939,
author = {Temur Kutsia and Cleo Pau},
title = {{Computing All Maximal Clique Partitions in a Graph}},
language = {english},
number = {19-04},
year = {2019},
length = {9},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

@techreport{RISC5950,
author = {Temur Kutsia and Cleo Pau},
title = {{Solving Proximity Constraints}},
language = {english},
number = {19-06},
year = {2019},
length = {22},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

@techreport{RISC5953,
author = {Temur Kutsia and Cleo Pau},
title = {{Matching and Generalization Modulo Proximity and Tolerance}},
language = {english},
number = {19-07},
year = {2019},
length = {5},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

@techreport{RISC5549,
author = {Alexander Baumgartner and Temur Kutsia and Jordi Levy and Mateu Villaret},
title = {{Term-Graph Anti-Unification}},
language = {english},
number = {18-02},
year = {2018},
length = {19},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}

@techreport{RISC5982,
author = {David M. Cerna},
title = {{A General Recursive Construction for Schematic Resolution Derivations}},
language = {english},
abstract = {Proof schemata provide an alternative formalism for handling inductive argumentation, while non-trivially extending {\em Herbrand's theorem} to a fragment of arithmetic and thus allowing the construction of {\em Herbrand sequents} and {\em expansion trees}. Existing proof analysis methods for proof schemata extract an unsatisfiable characteristic formula representing the cut structure of the proof and from its refutation construct a Herbrand sequent. Unfortunately, constructing the refutation is a task which is highly non-trivial. An automated method for constructing such refutations exists, but it only works for a very weak fragment of arithmetic and is hard to use interactively. More expressive yet interactive methods for the formalization of recursive resolution refutation are complex, hard to work with, and still limited to an undesirably weak class of recursion. In this work we note a particular problem with previous methods, namely they mix the recursive structure with the calculus of refutation. Also we present a modular recursive structure independent of the resolution formalism and proof construction. We illustrate the expressive power of the so called {\em finite saturated tree} formalism by formalizing the Non-injectivity Assertion's schematic refutation (a variant of the infinitary Pigeonhole principle). None of the previously developed formalism are able to formalize this refutation.},
year = {2018},
length = {42},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Schloss Hagenberg, 4232 Hagenberg, Austria}
}