Rewrite Theories Research Articles

Combining Techniques to Reduce State Space and Prove Strong Properties

An on-the-fly symmetry reduction technique that exploits the lexicographic order on metarepresentations of Maude terms, and a technique that uses auxiliary data to verify strong properties that are not directly expressible in propositional temporal logic are presented. Both are implemented by simple transformations of rewrite theories. They are applied in the verification of a strong-consistency property of a client-server protocol, a simplification of the Chain-Replication protocol.

Equational Abstractions for Reducing the State Space of Rewrite Theories

The combinatorial explosion of state spaces is the biggest problem in applying model checking methods to concurrent systems. In this paper we present a new state-space reduction technique that is tailored to system specifications in Rewriting Logic, a unified semantic framework for concurrency which is based on conditional term rewriting modulo equational theories. The idea is to hide “unimportant” details of the system's behavior (such as internal computations) in the equations, and to represent only “interesting” state changes (such as communication operations) by explicit transitions. We show how this optimization can be implemented by transforming the Rewriting Logic specification, avoiding the construction of the full state space. Moreover we establish the correctness of our technique by proving that the original and the reduced system are weakly bisimilar, and demonstrate its usability by applying it to the concurrent functional programming language Erlang.

Probabilistic Modeling and Analysis of DoS Protection for the ASV Protocol

The Adaptive Selective Verification (ASV) protocol was recently proposed as an effective and efficient DoS countermeasure within the shared channel model, in which clients and attackers probabilistically share communication bandwidth with the server. ASV has been manually shown to satisfy some desirable availability and bandwidth consumption properties. Due to the probabilistic nature of the protocol and its underlying attacker model, it is intrinsically difficult to build a faithful model of the protocol with which one may automatically verify its properties. This paper fills the gap between manual analysis and simulation-based experimental analysis of ASV, through automated formal analysis. We describe a formal model of ASV using probabilistic rewrite theories, implemented in a probabilistic extension of Maude, and show how it can be used to formally verify various characteristics of ASV through automated statistical quantitative model checking analysis techniques. In particular, we formally verify ASV's connection confidence theorem and a slightly more general bandwidth consumption theorem of ASV. This is followed by a statistical comparison of ASV with non-adaptive selective verification protocols. We conclude with remarks on possible further development and future work.

Inductive proof search modulo

We present an original narrowing-based proof search method for inductive theorems in equational rewrite theories given by a rewrite system $\mathcal{R}$ and a set E of equalities. It has the specificity to be grounded on deduction modulo and to rely on narrowing to provide both induction variables and instantiation schemas. Whenever the equational rewrite system $(\mathcal{R},E)$ has good properties of termination, sufficient completeness, and when E is constructor and variable preserving, narrowing at defined-innermost positions leads to consider only unifiers which are constructor substitutions. This is especially interesting for associative and associative-commutative theories for which the general proof search system is refined. The method is shown to be sound and refutationally correct and complete. A major feature of our approach is to provide a constructive proof in deduction modulo for each successful instance of the proof search procedure.

Algebraic Stuttering Simulations

Rewrite theories and their associated Kripke structures constitute a flexible and executable framework in which a wide range of systems can be studied. We present a general notion of simulation between Kripke structures, study its categorical aspects, and propose rewriting logic as a framework in which these simulations can be represented. Several representability results showing that rewriting logic is indeed a suitable framework for this purpose are given, and we illustrate its use with two examples.

Narrowing and Rewriting Logic: from Foundations to Applications

Narrowing was originally introduced to solve equational E-unification problems. It has also been recognized as a key mechanism to unify functional and logic programming. In both cases, narrowing supports equational reasoning and assumes confluent equations. The main goal of this work is to show that narrowing can be greatly generalized, so as to support a much wider range of applications, when it is performed with rewrite theories (Σ, E, R), where (Σ, E) is an equational theory, and R is a collection of rewrite rules with no restrictions. Such theories axiomatize concurrent systems, whose states are equivalence classes of terms modulo E, and whose transitions are specified by R. In this context, narrowing is generalized from an equational reasoning technique to a symbolic model checking technique for reachability analysis of a, typically infinite, concurrent system. We survey the foundations of this approach, suitable narrowing strategies, and various applications to security protocol verification, theorem proving, and programming languages.

The measurement calculus

Measurement-based quantum computation has emerged from the physics community as a new approach to quantum computation where the notion of measurement is the main driving force of computation. This is in contrast with the more traditional circuit model that is based on unitary operations. Among measurement-based quantum computation methods, the recently introduced one-way quantum computer [Raussendorf and Briegel 2001] stands out as fundamental. We develop a rigorous mathematical model underlying the one-way quantum computer and present a concrete syntax and operational semantics for programs, which we call patterns , and an algebra of these patterns derived from a denotational semantics. More importantly, we present a calculus for reasoning locally and compositionally about these patterns. We present a rewrite theory and prove a general standardization theorem which allows all patterns to be put in a semantically equivalent standard form. Standardization has far-reaching consequences: a new physical architecture based on performing all the entanglement in the beginning, parallelization by exposing the dependency structure of measurements and expressiveness theorems. Furthermore we formalize several other measurement-based models, for example, Teleportation, Phase and Pauli models and present compositional embeddings of them into and from the one-way model. This allows us to transfer all the theory we develop for the one-way model to these models. This shows that the framework we have developed has a general impact on measurement-based computation and is not just particular to the one-way quantum computer.

Coding Mobile Synchronizing Petri Nets into Rewriting Logic

Mobile Synchronizing Petri Nets (MSPN's) are a model for mobility and coordination based on coloured Petri Nets, in which systems are composed of a collection of (possibly mobile) hardware devices and mobile agents, both modelled homogenously. In this paper we approach their verification, for which we have chosen to code MSPN's into rewriting logic. In order to obtain a representation of MSPN systems by means of a rewrite theory, we develop a class of them, that we call ν-Abstract Petri nets (ν-APN's), which are easily representable in that framework. Moreover, the obtained representation provides a local mechanism for fresh name generation. Then we prove that, even if ν-APN's are a particular class of MSPN systems, they are strong enough to capture the behaviour of any MSPN system. We have chosen Maude to implement ν-APN's, as well as the translation from MSPN's to ν-APN's, for which we make intensive use of its reflective features.

Symbolic reachability analysis using narrowing and its application to verification of cryptographic protocols

Narrowing was introduced, and has traditionally been used, to solve equations in initial and free algebras modulo a set of equations E. This paper proposes a generalization of narrowing which can be used to solve reachability goals in initial and free models of a rewrite theory ?. We show that narrowing is sound and weakly complete (i.e., complete for normalized solutions) under appropriate executability assumptions about ?. We also show that in general narrowing is not strongly complete, that is, not complete when some solutions can be further rewritten by ?. We then identify several large classes of rewrite theories, covering many practical applications, for which narrowing is strongly complete. Finally, we illustrate an application of narrowing to analysis of cryptographic protocols.

Semantic foundations for generalized rewrite theories

Rewriting logic (RL) is a logic of actions whose models are concurrent systems. Rewrite theories involve the specification of equational theories of data and state structures together with a set of rewrite rules that model the dynamics of concurrent systems. Since its introduction, more than one decade ago, RL has attracted the interest of both theorists and practitioners, who have contributed in showing its generality as a semantic and logical framework and also as a programming paradigm. The experimentation conducted in these years has suggested that some significant extensions to the original definition of the logic would be very useful in practice. These extensions may develop along several dimensions, like the choice of the underlying equational logic, the kind of side conditions allowed in rewrite rules and operational concerns for the execution of certain rewrites. In particular, the Maude system now supports subsorting and conditional sentences in the equational logic for data, and also frozen arguments to block undesired nested rewrites; moreover, it allows equality and membership assertions in rule conditions. In this paper, we give a detailed presentation of the inference rules, model theory, and completeness of such generalized rewrite theories. Our results provide a mathematical semantics for Maude, and a foundation for formal reasoning about Maude specifications.

The Rewriting Logic Semantics Project

Rewriting logic is a flexible and expressive logical framework that unifies denotational semantics and SOS in a novel way, avoiding their respective limitations and allowing very succinct semantic definitions. The fact that a rewrite theory's axioms include both equations and rewrite rules provides a very useful “abstraction knob” to find the right balance between abstraction and observability in semantic definitions. Such semantic definitions are directly executable as interpreters in a rewriting logic language such as Maude, whose generic formal tools can be used to endow those interpreters with powerful program analysis capabilities.

Proving Modal Properties of Rewrite Theories Using Maude's Metalevel

Rewriting logic is a very expressive formalism for the specification of concurrent and distributed systems; more generally, it is a logic of change. In contrast, VLRL is a modal logic built on top of rewriting logic to reason precisely about that change. Here we present a technique to mechanically prove VLRL properties of rewrite theories using the reflective capability of rewriting logic through its Maude implementation.

A Verification Logic for Rewriting Logic

This paper proposes the development of a logic for verifying properties of programs in rewriting logic. Rewriting logic is primarily a logic of change, in which deduction corresponds directly to computation, and not a logic to talk about change in a more indirect and global manner, such as the different modal and temporal logics that can be found in the literature. We start by defining a modal action logic (VLRL) in which rewrite rules are captured as actions. The main novelty of this logic is a topological modality associated with state constructors that allows us to reason about the structure of states, stating that the current state can be decomposed into regions satisfying certain properties. Then, on top of the modal logic, we define a temporal logic for reasoning about properties of the computations generated from rewrite theories, and demonstrate its potential by means of several examples.

Surgical scenario for laparoscopic surgery with timed automata

In this paper we show that the satisfaction of timed CTL (TCTL) formulas under the natural continuous semantics for both discrete-time and dense-time timed Kripke structures can be reduced to a model-checking problem in the pointwise semantics for a large class of timed Kripke structures, which includes many discrete-event systems. We then present a TCTL model checking algorithm for the pointwise case. An important consequence of our results is that they together describe a sound and complete TCTL model checking procedure for time-robust real-time rewrite theories also for dense time domains. We have implemented such a TCTL model checker for Real-Time Maude. Our model checker provides for free a sound and complete TCTL model checker for subsets of modeling languages, such as Ptolemy II and (Synchronous) AADL, which have Real-Time Maude analysis integrated into their tool environments.

Modular Rewriting Semantics in Practice

We present a general method to achieve modularity of semantic definitions of programming languages specified as rewrite theories, so that semantic rules do not have to be redefined in language extensions. We illustrate the practical use of this method by means of two language case studies: two different semantics for CCS, and three different semantics for the GNU bc language.

Symbolic Reachability Analysis Using Narrowing and its Application to Verification of Cryptographic Protocols

Narrowing was introduced, and has traditionally been used, to solve equations in initial and free algebras modulo a set of equations E. This paper proposes a generalization of narrowing which can be used to solve reachability goals in initial and free models of a rewrite theory R. We show that narrowing is sound and weakly complete (i.e., complete for normalized solutions) under reasonable executability assumptions about R. We also show that in general narrowing is not strongly com- plete, that is, not complete when some solutions can be further rewritten by R. We then identify several large classes of rewrite theories, covering many practical applications, for which narrowing is strongly complete. Finally, we illustrate an application of narrowing to analysis of cryptographic protocols.

Proving VLRL Action Properties with the Maude Model Checker

The Verification Logic for Rewriting Logic (VLRL) is a modal action logic in which rewrite rules are captured as actions. This paper studies a possible representation of the VLRL action formulae using the Next and the Until operators of Linear Temporal Logic (LTL). In particular, it studies the use of the Maude model checker to prove VLRL action formulae. Action modalities of VLRL fix the transition that will take place in a state and the context in which it will be applied, while LTL operators do not. Thus, to represent action modalities in LTL it is necessary to transform the initial rewrite theory into a new one in which the states carry the information about the transitions used and the context in which they have taken place. VLRL properties are then studied in the transformed theory by translating VLRL formulae into equivalent LTL formulae.

μ-Calculus Model Checking in Maude

In this paper, a rewrite theory for checking μ-calculus properties is developed. We use the same framework proposed in [Steven Eker, José Meseguer, and Ambarish Sridharanarayanan. The Maude LTL model checker. In Proceedings of the Fourth International Workshop on Rewriting Logic, volume 71 of Electronic Notes in Theoretical Computer Science. Elsevier, 2002] and demonstrate how rewriting logic can be used as a unified formalism from model specification to verification algorithm implementation. Furthermore, since μ-calculus is more expressive than LTL, this work can be seen as an extension to [Steven Eker, José Meseguer, and Ambarish Sridharanarayanan. The Maude LTL model checker. In Proceedings of the Fourth International Workshop on Rewriting Logic, volume 71 of Electronic Notes in Theoretical Computer Science. Elsevier, 2002] in theory. We also develop a CTL to μ-calculus translator to help users write CTL specifications more easily. However, the corresponding LTL to μ-calculus translator is missing. The LTL model checker in [Steven Eker, José Meseguer, and Ambarish Sridharanarayanan. The Maude LTL model checker. In Proceedings of the Fourth International Workshop on Rewriting Logic, volume 71 of Electronic Notes in Theoretical Computer Science. Elsevier, 2002] is still preferred in practice.

The Maude LTL Model Checker

The Maude LTL model checker supports on-the-fly explicit-state model checking of concurrent systems expressed as rewrite theories with performance comparable to that of current tools of that kind, such as SPIN. This greatly expands the range of applications amenable to model checking analysis. Besides traditional areas well supported by current tools, such as hardware and communication protocols, many new applications in areas such as rewriting logic models of cell biology, or next-generation reflective distributed systems can be easily specified and model checked with our tool.

Tiling Transactions in Rewriting Logic

We propose a modular high-level approach to the specification of transactions in rewriting logic, where the operational and the abstract views are related by suitable adjunctions between categories of tile theories and of rewrite theories.