A Sufficient Completeness Checker for Linear Order-Sorted Specifications Modulo Axioms

Hendrix, Joe; Meseguer, José; Ohsaki, Hitoshi

doi:10.1007/11814771_14

Cited by 30 publications

(23 citation statements)

References 9 publications

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“…If however, the PTA does contain an associative symbol that is not commutative, there is a semi-algorithm in [12] that can always show non-emptiness and can often show emptiness using some techniques from machine learning. The algorithms in [12] have been integrated into Maude as part of the Maude Sufficient Completeness Checker (SCC) [10].…”

Section: Propositional Tree Automatamentioning

confidence: 99%

“…We have implemented an algorithm for constructing the tree automaton A CC in Maude automatically from a Maude specification, and have integrated it into the Maude Sufficient Completeness Checker [10]. If we ask the tool to check the µ-canonical completeness of the specification FACTORIAL given in Section 4.1, we are able to verify that it is µ-canonically complete: If we ask the tool to check the INF-LIST specification, the tool generates a counterexample showing that the specification is not µ-canonically complete: …”

Section: Checking µ-Canonical Completenessmentioning

confidence: 99%

“…Given that the CS programs we consider perform rewriting modulo axioms and are order-sorted, our methods are based on Propositional Tree Automata [12], a kind of equational tree automata, that can take into account both sort information and reasoning modulo axioms. These decision procedures have been implemented in an extension of Maude's Sufficient Completeness Checker (SCC) [10], and we use several Maude programs to illustrate both the basic ideas and the use of SCC in verifying CS completeness properties.…”

Section: Introductionmentioning

confidence: 99%

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On the Completeness of Context-Sensitive Order-Sorted Specifications

Hendrix

Meseguer

Lecture Notes in Computer Science

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Abstract. We propose three different notions of completeness for term rewrite specifications supporting order-sorted signatures, deduction modulo axioms, and context-sensitive rewriting relative to a replacement map µ. Our three notions are: (1) an appropriate definition of µ-sufficient completeness with respect to a set of constructor symbols; (2) a definition of µ-canonical completeness under which µ-canonical forms coincide with canonical forms; and (3) a definition of semantic completeness that guarantees that the µ-operational semantics and standard initial algebra semantics are isomorphic. Based on these notions, we use equational tree automata techniques to obtain decision procedures for checking these three kinds of completeness for equational specifications satisfying appropriate requirements such as ground confluence, ground sort-decreasingness, weakly normalization, and left-linearity. Although the general equational tree automata problems are undecidable, our algorithms work modulo any combination of associativity, commutativity, and identity axioms. For all combinations of these axioms except associativity without commutativity, our algorithms are decision procedures. For the associativity without commutativity case, which is undecidable in general, our algorithms use learning techniques that are effective in all practical examples we have considered. We have implemented these algorithms as an extension of the Maude sufficient completeness checker.

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Section: Propositional Tree Automatamentioning

confidence: 99%

Section: Checking µ-Canonical Completenessmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Completeness of Context-Sensitive Order-Sorted Specifications

Hendrix

Meseguer

Lecture Notes in Computer Science

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“…Roughly, the idea of these methods is to check whether ground terms rooted by a defined function symbol and having only constructor terms as proper subterms are either reducible, or constructor terms (which is possible as the root symbol might be subsort overloaded). This is done by describing the respective languages of terms by (propositional) tree automata and then reducing the problem to an emptiness problem for tree automata (we refer to [15] and [16] for more details). The method is suitable for incremental checks following Theorem 6, since it can easily be adapted to consider only terms rooted by defined function symbols of the extending theory instead of all.…”

Section: Theorem 3 (Modularitymentioning

confidence: 99%

“…e.g. [15]). The module TR-MSET-NAT, restricted to equations explicitly defined in the module and particularly not including the ones from the TR-NATURAL module, is sort-decreasing and well-founded recursive as well.…”

Section: (Ms ; Nl) ; L = Ms ; (Nl ; L) Eq U(nil) = Null Eq U(ms) mentioning

confidence: 99%

Incremental checking of well-founded recursive specifications modulo axioms

Schernhammer

Meseguer

2011

Proceedings of the 13th International ACM SIGPLAN Symposium on Principles and Practices of Declarative Programming

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Abstract. We introduce the notion of well-founded recursive order-sorted equational logic (OS) theories modulo axioms. Such theories define functions by well-founded recursion and are inherently terminating. Moreover, for well-founded recursive theories important properties such as confluence and sufficient completeness are modular for so-called fair extensions. This enables us to incrementally check these properties for hierarchies of such theories that occur naturally in modular rule-based functional programs. Well-founded recursive OS theories modulo axioms contain only commutativity and associativity-commutativity axioms. In order to support arbitrary combinations of associativity, commutativity and identity axioms, we show how to eliminate identity and (under certain conditions) associativity (without commutativity) axioms by theory transformations in the last part of the paper.

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