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Pure bigraphs: structure and dynamics
, 2005
"... Bigraphs are graphs whose nodes may be nested, representing locality, independently of the edges connecting them. They may be equipped with reaction rules, forming a bigraphical reactive system (Brs) in which bigraphs can reconfigure themselves. Following an earlier paper describing link graphs, a c ..."
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Cited by 50 (5 self)
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Bigraphs are graphs whose nodes may be nested, representing locality, independently of the edges connecting them. They may be equipped with reaction rules, forming a bigraphical reactive system (Brs) in which bigraphs can reconfigure themselves. Following an earlier paper describing link graphs, a constituent of bigraphs, this paper is a devoted to pure bigraphs, which in turn underlie various more refined forms. Elsewhere it is shown that behavioural analysis for Petri nets, πcalculus and mobile ambients can all be recovered in the uniform framework of bigraphs. The paper first develops the dynamic theory of an abstract structure, a wide reactive system (Wrs), of which a Brs is an instance. In this context, labelled transitions are defined in such a way that the induced bisimilarity is a congruence. This work is then specialised to Brss, whose graphical structure allows many refinements of the theory. The latter part of the paper emphasizes bigraphical theory that is relevant to the treatment of dynamics via labelled transitions. As a running example, the theory is applied to finite pure CCS, whose resulting transition system and bisimilarity are analysed in detail. The paper also mentions briefly the use of bigraphs to model pervasive computing and
A RewritingBased Inference System for the NRL Protocol Analyzer and its MetaLogical Properties
, 2005
"... The NRL Protocol Analyzer (NPA) is a tool for the formal specification and analysis of cryptographic protocols that has been used with great effect on a number of complex reallife protocols. One of the most interesting of its features is that it can be used to reason about security in face of attem ..."
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Cited by 22 (14 self)
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The NRL Protocol Analyzer (NPA) is a tool for the formal specification and analysis of cryptographic protocols that has been used with great effect on a number of complex reallife protocols. One of the most interesting of its features is that it can be used to reason about security in face of attempted attacks on lowlevel algebraic properties of the functions used in a protocol. Indeed, it has been used successfully to either reproduce or discover a number of such attacks. In this paper we give for the first time a precise formal specification of the main features of the NPA inference system: its grammarbased techniques for invariant generation and its backwards reachability analysis method. This formal specification is given within the wellknown rewriting framework so that the inference system is specified as a set of rewrite rules modulo an equational theory describing the behavior of the cryptographic algorithms involved. We then use this formalization to prove some important metalogical properties about the NPA inference system, including the soundness and completeness of the search algorithm and soundness of the grammar generation algorithm. The formalization and soundness and completeness theorems not only provide also a better understanding of the NPA as it currently operates, but provide a modular basis which can be used as a starting point for increasing the types of equational theories it can handle.
Automated reasoning in Kleene algebra
 CADE 2007, LNCS 4603
, 2007
"... Abstract. It has often been claimed that model checking, special purpose automated deduction or interactive theorem proving are needed for formal program development. We demonstrate that offtheshelf automated proof and counterexample search is an interesting alternative if combined with the right ..."
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Cited by 19 (10 self)
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Abstract. It has often been claimed that model checking, special purpose automated deduction or interactive theorem proving are needed for formal program development. We demonstrate that offtheshelf automated proof and counterexample search is an interesting alternative if combined with the right domain model. We implement variants of Kleene algebras axiomatically in Prover9/Mace4 and perform proof experiments about Hoare, dynamic, temporal logics, concurrency control and termination analysis. They confirm that a simple automated analysis of some important program properties is possible. Particular benefits of this novel approach include “soft ” model checking in a firstorder setting, crosstheory reasoning between standard formalisms and full automation of some (co)inductive arguments. Kleene algebras might therefore provide lightweight formal methods with heavyweight automation. 1
Formal design and verification of operational transformation algorithms for copies convergence
 Theoretical Computer Science
, 2005
"... algorithms for copies convergence ..."
Symbolic Model Checking of InfiniteState Systems Using Narrowing
"... Rewriting is a general and expressive way of specifying concurrent systems, where concurrent transitions are axiomatized by rewrite rules. Narrowing is a complete symbolic method for model checking reachability properties. We show that this method can be reinterpreted as a lifting simulation relatin ..."
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Cited by 12 (7 self)
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Rewriting is a general and expressive way of specifying concurrent systems, where concurrent transitions are axiomatized by rewrite rules. Narrowing is a complete symbolic method for model checking reachability properties. We show that this method can be reinterpreted as a lifting simulation relating the original system and the symbolic system associated to the narrowing transitions. Since the narrowing graph can be infinite, this lifting simulation only gives us a semidecision procedure for the failure of invariants. However, we propose new methods for folding the narrowing tree that can in practice result in finite systems that symbolically simulate the original system and can be used to algorithmically verify its properties. We also show how both narrowing and folding can be used to symbolically model check systems which, in addition, have state predicates, and therefore correspond to Kripke structures on which ACTL∗ and LTL formulas can be algorithmically verified using such finite symbolic abstractions.
Natural narrowing for general term rewriting systems
 Proc. of 16th International Conference on Rewriting Techniques and Applications, RTA’05, Lecture Notes in Computer Science
, 2005
"... Abstract. For narrowing to be an efficient evaluation mechanism, several lazy narrowing strategies have been proposed, although typically for the restricted case of leftlinear constructor systems. These assumptions, while reasonable for functional programming applications, are too restrictive for a ..."
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Cited by 8 (5 self)
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Abstract. For narrowing to be an efficient evaluation mechanism, several lazy narrowing strategies have been proposed, although typically for the restricted case of leftlinear constructor systems. These assumptions, while reasonable for functional programming applications, are too restrictive for a much broader range of applications to which narrowing can be fruitfully applied, including applications where rules have a nonequational meaning either as transitions in a concurrent system or as inferences in a logical system. In this paper, we propose an efficient lazy narrowing strategy called natural narrowing which can be applied to general term rewriting systems with no restrictions whatsoever. An important consequence of this generalization is the wide range of applications that can now be efficiently supported by narrowing. We highlight a few such applications including symbolic model checking, theorem proving, programming languages, and partial evaluation. What thus emerges is a general and efficient unified mechanism based on narrowing, that seamlessly integrates a very wide range of applications in programming and proving. 1
Implementing Natural Rewriting and Narrowing Efficiently
"... Outermostneeded rewriting/narrowing is a sound and complete optimal demanddriven strategy for the class of inductively sequential constructor systems. Its parallel extension, known as weakly, deals with noninductively sequential constructor systems. Recently, refinements of (weakly) outermostnee ..."
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Cited by 8 (5 self)
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Outermostneeded rewriting/narrowing is a sound and complete optimal demanddriven strategy for the class of inductively sequential constructor systems. Its parallel extension, known as weakly, deals with noninductively sequential constructor systems. Recently, refinements of (weakly) outermostneeded rewriting and narrowing have been obtained. These new strategies are called natural rewriting and natural narrowing, respectively, and incorporate a better treatment of demandedness. In this paper, we address the problem of how to implement natural rewriting and narrowing eciently by using a refinement of the notion of definitional tree, which we call matching definitional tree. We also show how to compile...
Partial Order Infinitary Term Rewriting and Böhm Trees
, 2010
"... We study an alternative model of in nitary term rewriting. Instead of a metric on terms, a partial order on partial terms is employed to formalise convergence of reductions. We consider both a weak and a strong notion of convergence and show that the metric model of convergence coincides with the p ..."
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Cited by 7 (6 self)
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We study an alternative model of in nitary term rewriting. Instead of a metric on terms, a partial order on partial terms is employed to formalise convergence of reductions. We consider both a weak and a strong notion of convergence and show that the metric model of convergence coincides with the partial model restricted to total terms. Hence, partial order convergence constitutes a conservative extension of metric convergence that additionally offers a finegrained distinction between di erent levels of divergence. In the second part, we focus our investigation on strong convergence of orthogonal systems. The main result is that the gap between the metric model and the partial order model can be bridged by simply extending the term rewriting system by additional rules. These extensions are the wellknown Böhm extensions. Based on this result, we are able to establish that contrary to the metric setting orthogonal systems are both infinitarily confluent and infinitarily normalising in the partial order setting. The unique infinitary normal forms that the partial order model admits are Böhm trees.
Variant Narrowing and Equational Unification
 In Proc. of WRLA 2008, ENTCS
, 2009
"... Abstract. Narrowing is a wellknown complete procedure for equational Eunification when E can be decomposed as a union E = ∆ ⊎ B with B a set of axioms for which a finitary unification algorithm exists, and ∆ a set of confluent, terminating, and Bcoherent rewrite rules. However, when B ̸ = ∅, ef ..."
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Cited by 7 (5 self)
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Abstract. Narrowing is a wellknown complete procedure for equational Eunification when E can be decomposed as a union E = ∆ ⊎ B with B a set of axioms for which a finitary unification algorithm exists, and ∆ a set of confluent, terminating, and Bcoherent rewrite rules. However, when B ̸ = ∅, efficient narrowing strategies such as basic narrowing easily fail to be complete and cannot be used. This poses two challenges to narrowingbased equational unification: (i) finding efficient narrowing strategies that are complete modulo B under mild assumptions on B, and (ii) finding sufficient conditions under which such narrowing strategies yield finitary Eunification algorithms. Inspired by Comon and Delaune’s notion of Evariant for a term, we propose a new narrowing strategy called variant narrowing that has a search space potentially much smaller than full narrowing, is complete, and yields a finitary Eunification algorithm when E has the finite variant property. We furthermore identify a class of equational theories for which the finite bound ensuring the finite variant property can be effectively computed by a generic algorithm. We also discuss applications to the formal analysis of cryptographic protocols modulo the algebraic properties of the underlying cryptographic functions. 1
Infinitary Normalization
 We Will Show Them: Essays in Honour of Dov Gabbay
, 2005
"... abstract. In infinitary orthogonal firstorder term rewriting the properties confluence (CR), Uniqueness of Normal forms (UN), Parallel Moves Lemma (PML) have been generalized to their infinitary versions CR ∞ , UN ∞ , PML ∞ , and so on. Several relations between these properties have been establish ..."
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Cited by 6 (1 self)
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abstract. In infinitary orthogonal firstorder term rewriting the properties confluence (CR), Uniqueness of Normal forms (UN), Parallel Moves Lemma (PML) have been generalized to their infinitary versions CR ∞ , UN ∞ , PML ∞ , and so on. Several relations between these properties have been established in the literature. Generalization of the termination properties, Strong Normalization (SN) and Weak Normalization (WN) to SN ∞ and WN ∞ is less straightforward. We present and explain the definitions of these infinitary normalization notions, and establish that as a global property of orthogonal TRSs they coincide, so at that level there is just one notion of infinitary normalization. Locally, at the level of individual terms, the notions are still different. In the setting of orthogonal term rewriting we also provide an elementary proof of UN ∞ , the infinitary Unique Normal form property. 12