776 research outputs found
Canonized Rewriting and Ground AC Completion Modulo Shostak Theories : Design and Implementation
AC-completion efficiently handles equality modulo associative and commutative
function symbols. When the input is ground, the procedure terminates and
provides a decision algorithm for the word problem. In this paper, we present a
modular extension of ground AC-completion for deciding formulas in the
combination of the theory of equality with user-defined AC symbols,
uninterpreted symbols and an arbitrary signature disjoint Shostak theory X. Our
algorithm, called AC(X), is obtained by augmenting in a modular way ground
AC-completion with the canonizer and solver present for the theory X. This
integration rests on canonized rewriting, a new relation reminiscent to
normalized rewriting, which integrates canonizers in rewriting steps. AC(X) is
proved sound, complete and terminating, and is implemented to extend the core
of the Alt-Ergo theorem prover.Comment: 30 pages, full version of the paper TACAS'11 paper "Canonized
Rewriting and Ground AC-Completion Modulo Shostak Theories" accepted for
publication by LMCS (Logical Methods in Computer Science
Termination of rewrite relations on -terms based on Girard's notion of reducibility
In this paper, we show how to extend the notion of reducibility introduced by
Girard for proving the termination of -reduction in the polymorphic
-calculus, to prove the termination of various kinds of rewrite
relations on -terms, including rewriting modulo some equational theory
and rewriting with matching modulo , by using the notion of
computability closure. This provides a powerful termination criterion for
various higher-order rewriting frameworks, including Klop's Combinatory
Reductions Systems with simple types and Nipkow's Higher-order Rewrite Systems
Constrained completion: Theory, implementation, and results
The Knuth-Bendix completion procedure produces complete sets of reductions but can not handle certain rewrite rules such as commutativity. In order to handle such theories, completion procedure were created to find complete sets of reductions modulo an equational theory. The major problem with this method is that it requires a specialized unification algorithm for the equational theory. Although this method works well when such an algorithm exists, these algorithms are not always available and thus alternative methods are needed to attack problems. A way of doing this is to use a completion procedure which finds complete sets of constrained reductions. This type of completion procedure neither requires specialized unification algorithms nor will it fail due to unorientable identities.
We present a look at complete sets of reductions with constraints, developed by Gerald Peterson, and the implementation of such a completion procedure for use with HIPER - a fast completion system. The completion procedure code is given and shown correct along with the various support procedures which are needed by the constrained system. These support procedures include a procedure to find constraints using the lexicographic path ordering and a normal form procedure for constraints.
The procedure has been implemented for use under the fast HIPER system, developed by Jim Christian, and thus is quick. We apply this new system, HIPER- extension, to attack a variety of word problems. Implementation alternatives are discussed, developed, and compared with each other as well as with the HIPER system.
Finally, we look at the problem of finding a complete set of reductions for a ternary boolean algebra. Given are alternatives to attacking this problem and the already known solution along with its run in the HIPER-extension system --Abstract, page iii
A Reduction-Preserving Completion for Proving Confluence of Non-Terminating Term Rewriting Systems
We give a method to prove confluence of term rewriting systems that contain
non-terminating rewrite rules such as commutativity and associativity. Usually,
confluence of term rewriting systems containing such rules is proved by
treating them as equational term rewriting systems and considering E-critical
pairs and/or termination modulo E. In contrast, our method is based solely on
usual critical pairs and it also (partially) works even if the system is not
terminating modulo E. We first present confluence criteria for term rewriting
systems whose rewrite rules can be partitioned into a terminating part and a
possibly non-terminating part. We then give a reduction-preserving completion
procedure so that the applicability of the criteria is enhanced. In contrast to
the well-known Knuth-Bendix completion procedure which preserves the
equivalence relation of the system, our completion procedure preserves the
reduction relation of the system, by which confluence of the original system is
inferred from that of the completed system
The Role of Term Symmetry in E-Unification and E-Completion
A major portion of the work and time involved in completing an incomplete set of reductions using an E-completion procedure such as the one described by Knuth and Bendix [070] or its extension to associative-commutative equational theories as described by Peterson and Stickel [PS81] is spent calculating critical pairs and subsequently testing them for coherence. A pruning technique which removes from consideration those critical pairs that represent redundant or superfluous information, either before, during, or after their calculation, can therefore make a marked difference in the run time and efficiency of an E-completion procedure to which it is applied.
The exploitation of term symmetry is one such pruning technique. The calculation of redundant critical pairs can be avoided by detecting the term symmetries that can occur between the subterms of the left-hand side of the major reduction being used, and later between the unifiers of these subterms with the left-hand side of the minor reduction. After calculation, and even after reduction to normal form, the observation of term symmetries can lead to significant savings.
The results in this paper were achieved through the development and use of a flexible E-unification algorithm which is currently written to process pairs of terms which may contain any combination of Null-E, C (Commutative), AC (Associative-Commutative) and ACI (Associative-Commutative with Identity) operators. One characteristic of this E-unification algorithm that we have not observed in any other to date is the ability to process a pair of terms which have different ACI top-level operators. In addition, the algorithm is a modular design which is a variation of the Yelick model [Ye85], and is easily extended to process terms containing operators of additional equational theories by simply plugging in a unification module for the new theory
State space c-reductions for concurrent systems in rewriting logic
We present c-reductions, a state space reduction technique.
The rough idea is to exploit some equivalence relation on states (possibly capturing system regularities) that preserves behavioral properties, and explore the induced quotient system. This is done by means of a canonizer
function, which maps each state into a (non necessarily unique) canonical representative of its equivalence class. The approach exploits the expressiveness of rewriting logic and its realization in Maude to enjoy several advantages over similar approaches: exibility and simplicity in
the definition of the reductions (supporting not only traditional symmetry reductions, but also name reuse and name abstraction); reasoning support for checking and proving correctness of the reductions; and automatization
of the reduction infrastructure via Maude's meta-programming
features. The approach has been validated over a set of representative case studies, exhibiting comparable results with respect to other tools
Meta SOS - A Maude Based SOS Meta-Theory Framework
Meta SOS is a software framework designed to integrate the results from the
meta-theory of structural operational semantics (SOS). These results include
deriving semantic properties of language constructs just by syntactically
analyzing their rule-based definition, as well as automatically deriving sound
and ground-complete axiomatizations for languages, when considering a notion of
behavioural equivalence. This paper describes the Meta SOS framework by
blending aspects from the meta-theory of SOS, details on their implementation
in Maude, and running examples.Comment: In Proceedings EXPRESS/SOS 2013, arXiv:1307.690
Rewriting Logic Semantics of a Plan Execution Language
The Plan Execution Interchange Language (PLEXIL) is a synchronous language
developed by NASA to support autonomous spacecraft operations. In this paper,
we propose a rewriting logic semantics of PLEXIL in Maude, a high-performance
logical engine. The rewriting logic semantics is by itself a formal interpreter
of the language and can be used as a semantic benchmark for the implementation
of PLEXIL executives. The implementation in Maude has the additional benefit of
making available to PLEXIL designers and developers all the formal analysis and
verification tools provided by Maude. The formalization of the PLEXIL semantics
in rewriting logic poses an interesting challenge due to the synchronous nature
of the language and the prioritized rules defining its semantics. To overcome
this difficulty, we propose a general procedure for simulating synchronous set
relations in rewriting logic that is sound and, for deterministic relations,
complete. We also report on two issues at the design level of the original
PLEXIL semantics that were identified with the help of the executable
specification in Maude
On Isomorphism of "Functional" Intersection and Union Types
Type isomorphism is useful for retrieving library components, since a
function in a library can have a type different from, but isomorphic to, the
one expected by the user. Moreover type isomorphism gives for free the coercion
required to include the function in the user program with the right type. The
present paper faces the problem of type isomorphism in a system with
intersection and union types. In the presence of intersection and union,
isomorphism is not a congruence and cannot be characterised in an equational
way. A characterisation can still be given, quite complicated by the
interference between functional and non functional types. This drawback is
faced in the paper by interpreting each atomic type as the set of functions
mapping any argument into the interpretation of the type itself. This choice
has been suggested by the initial projection of Scott's inverse limit
lambda-model. The main result of this paper is a condition assuring type
isomorphism, based on an isomorphism preserving reduction.Comment: In Proceedings ITRS 2014, arXiv:1503.0437
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