Anti-unification and the theory of semirings

Cerna, David M.

doi:10.1016/j.tcs.2020.10.020

Cited by 3 publications

(3 citation statements)

References 14 publications

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“…Example 3. From the introduction: Syntactic AU is unitary [22,23], AU over associative (A) and commutative (C) theories is finitary [1], AU over idempotent theories is infinitary [12], and AU with multiple unital equations is nullary [14].…”

Section: Example 1 Consider the Equational Theory Absmentioning

confidence: 99%

“…Concrete examples are the product operation and 0 in number fields and the intersection operation and ∅ in set theory. So far, investigations on antiunification over absorption theories have only considered equational theories defining more elaborate algebraic structures (semirings [12]). In this work, we study pure absorption theories as part of a general study on the anti-unification of subterm-collapsing theories.…”

Section: Anti-unification In Absorption Theoriesmentioning

confidence: 99%

“…For instance, they proved the nullarity of theories containing more than one equational symbol: U >1 , (AU ) >1 (CU ) >1 , (ACU ) 1 , and (AU )(CU ). Also, these authors showed that I, AI, and CI are of type infinitary [13], and Cerna proved that (U I) >1 , (AU I) >1 , (CU I) >1 , (ACU I) >1 , and semirings are of type nullary [12]. This paper extends the state-of-the-art on equational anti-unification by providing an algorithm to solve AU problems in a first-order syntax that includes operators with collapsing symbols, i.e., symbols that are associated with an absorption constant such that f (ε f , x) ≈ ε f ≈ f (x, ε f ).…”

Section: Introductionmentioning

confidence: 99%

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Nominal AC-Matching

Ayala-Rincón,

Fernández,

Silva

et al. 2023

Lecture Notes in Computer Science

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Interest in anti-unification, the dual problem of unification, is on the rise due to applications within the field of software analysis and related areas. For example, anti-unification-based techniques have found uses within clone detection and automatic program repair methods. While syntactic forms of anti-unification are enough for many applications, some aspects of software analysis methods 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., some operators are associated with a special constant satisfying the axiom 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. We also show that the linear variant is finitary.

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Section: Example 1 Consider the Equational Theory Absmentioning

confidence: 99%

Section: Anti-unification In Absorption Theoriesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Nominal AC-Matching

Ayala-Rincón,

Fernández,

Silva

et al. 2023

Lecture Notes in Computer Science

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

Cerna,

Buran

2024

ACM Trans. Comput. Logic

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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.

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Anti-unification and the theory of semirings

Cited by 3 publications

References 14 publications

Nominal AC-Matching

Nominal AC-Matching

Equational Anti-unification over Absorption Theories

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

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