Unital Anti-Unification: Type and Algorithms

Cerna, David M.; Kutsia, Temur

doi:10.4230/lipics.fscd.2020.26

Cited by 4 publications

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“…For instance, for the case of a set B of equations consisting of any combination of associativity and commutativity axioms for different function symbols, the set of B-lggs is finite. However, as shown in [10], this may not be the case when B contains identity axioms, i.e., ACU, AU, CU, and U.…”

Section: Introductionmentioning

confidence: 99%

“…The generalization type of an equational theory is defined similarly (but dually) to the unification types, i.e., based on the existence and cardinality of a minimal and complete set of B-lggs [10]: Unitary (type 1) Any generalization problem in the theory has one single B-lgg. Finitary (type ω) Any generalization problem in the theory has a finite, minimal, and complete set of B-lggs whose cardinality is greater than one for at least one problem.…”

Section: Introductionmentioning

confidence: 99%

“…In the motivating example of Section 2 of [10], it was shown that our inference system in [4] is incomplete for the case of the identity.…”

Section: Introductionmentioning

confidence: 99%

“…Given three extra constants a, b, and c, and the generalization problem g(f (a, c), a) w g (c, b), where w stands for a variable denoting the computed generalizer, the algorithm of [4] computes the generalizer given by {w → g(f (x, c), y)}, where x and y are new variables. However, the algorithm of [10] computes the more specific, non-linear solution g(f (x, c), f (x, y)), where x and y are new variables, which is actually the (only) least general B-generalizer for the input problem, proving that our original algorithm was generally incomplete. Nevertheless, completeness holds for linear generalization problems as shown in [10].…”

Section: Introductionmentioning

confidence: 99%

“…However, the algorithm of [10] computes the more specific, non-linear solution g(f (x, c), f (x, y)), where x and y are new variables, which is actually the (only) least general B-generalizer for the input problem, proving that our original algorithm was generally incomplete. Nevertheless, completeness holds for linear generalization problems as shown in [10].…”

Section: Introductionmentioning

confidence: 99%

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Order-sorted equational generalization algorithm revisited

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Generalization, also called anti-unification, is the dual of unification. A generalizer of two terms t and t is a term t of which t and t are substitution instances. The dual of most general equational unifiers is that of least general equational generalizers, i.e., most specific anti-instances modulo equations. In a previous work, we extended the classical untyped generalization algorithm to: (1) an order-sorted typed setting with sorts, subsorts, and subtype polymorphism; (2) work modulo equational theories, where function symbols can obey any combination of associativity, commutativity, and identity axioms (including the empty set of such axioms); and (3) the combination of both, which results in a modular, order-sorted equational generalization algorithm. However, Cerna and Kutsia showed that our algorithm is generally incomplete for the case of identity axioms and a counterexample was given. Furthermore, they proved that, in theories with two identity elements or more, generalization with identity axioms is generally nullary, yet it is finitary for both the linear and one-unital fragments, i.e., either solutions with repeated variables are disregarded or the considered theories are restricted to having just one function symbol with an identity or unit element. In this work, we show how we can easily extend our original inference system to cope with the non-linear fragment and identify a more general class than one-unit theories where generalization with identity axioms is finitary.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…In the motivating example of Section 2 of [10], it was shown that our inference system in [4] is incomplete for the case of the identity.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Order-sorted equational generalization algorithm revisited

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Variant-Based Equational Anti-unification

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Equational Anti-unification over Absorption Theories

Ayala-Rincón,

Cerna,

Barragán

et al. 2024

Automated Reasoning

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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.

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Unital Anti-Unification: Type and Algorithms

Cited by 4 publications

References 0 publications

Order-sorted equational generalization algorithm revisited

Order-sorted equational generalization algorithm revisited

Variant-Based Equational Anti-unification

Equational Anti-unification over Absorption Theories

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