Rewrite Theories Research Articles

Narrowing and heuristic search for symbolic reachability analysis of concurrent object-oriented systems

A concurrent system specified as a rewrite theory can be analyzed symbolically using narrowing-based reachability analysis. Narrowing-based approaches have been applied to formally analyze cryptographic protocols and parameterized protocols. However, existing narrowing-based analysis methods, based on a breadth-first-search strategy, cannot deal with generic distributed systems with objects and messages due to the symbolic state-space explosion problem and implicit constraints imposed on object-oriented systems. This paper proposes a heuristic search approach for narrowing-based reachability analysis to guide the search for counterexamples with a small number of objects. As a result, our method can effectively find a counterexample if an error state is reachable. In addition, this paper also shows how to encode implicit object-oriented constraints using order-sorted signatures and equational constraints. We demonstrate the effectiveness of our technique using a nontrivial distributed consensus algorithm.

Symbolic analysis and parameter synthesis for networks of parametric timed automata with global variables using Maude and SMT solving

This paper presents a rewriting logic “interpreter” for networks of parametric timed automata with global variables (NPTAVs) in Real-Time Maude style. Since explicit-state analysis is not sound and complete for such dense-time systems, we explain how our real-time rewrite theory can be systematically transformed into a rewrite theory that is amenable to symbolic analysis using the integration of Maude with SMT solving. We show that symbolic reachability analysis using Maude-with-SMT of the resulting rewrite theory is sound and complete for the NPTAV reachability problem. We extend and optimize standard Maude-with-SMT reachability analysis with folding and merging of symbolic states, so that the analysis terminates when the symbolic state space of the NPTAV is finite. These procedures rely on a subsumption relation that requires eliminating existentially quantified SMT variables. We have therefore implemented the Fourier-Motzkin quantifier elimination algorithm, thereby making our rewrite theory executable with any SMT solver connected to Maude. We show how we can provide most reachability and parameter synthesis analysis methods provided by Imitator, a state-of-the-art tool for NPTAVs. We compare the performance of our analysis methods with those of Imitator on a wide range of benchmarks, with the expected outcome. The practical contributions are two-fold: providing new analysis methods for NPTAVs—e.g., allowing more general state properties and supporting reachability analysis combined with user-defined execution strategies—not supported by Imitator, and developing symbolic analysis methods for both real-time rewrite theories and rewrite theories in general.

Business processes resource management using rewriting logic and deep-learning-based predictive monitoring

A significant task in business process optimization is concerned with streamlining the allocation and sharing of resources. This paper presents an approach for analyzing business process provisioning under a resource prediction strategy based on deep learning. A timed and probabilistic rewrite theory specification formalizes the semantics of business processes. It is integrated with an external oracle in the form of a long short-term memory neural network that can be queried to predict how traces of the process may advance within a time frame. Comparison of execution time and resource occupancy under different parameters is included for several case studies, as well as details on the construction of the deep learning model and its integration with Maude.

An efficient canonical narrowing implementation with irreducibility and SMT constraints for generic symbolic protocol analysis

Narrowing and unification are very useful tools for symbolic analysis of rewrite theories, and thus for any model that can be specified in that way. A very clear example of their application is the field of formal cryptographic protocol analysis, which is why narrowing and unification are used in tools such as Maude-NPA, Tamarin and Akiss. In this work we present the implementation of a canonical narrowing algorithm, which improves the standard narrowing algorithm, extended to be able to process rewrite theories with conditional rules. The conditions of the rules will contain SMT constraints, which will be carried throughout the execution of the algorithm to determine if the solutions have associated satisfiable or unsatisfiable constraints, and in the latter case, discard them.

Capturing constrained constructor patterns in matching logic

Reachability logic for rewrite theories consists of a specification of system states that are given by constrained constructor patterns, a transition relation that is given by a rewrite theory, and reachability properties expressed as pairs of state specifications. Matching logic has been recently proposed as a unifying foundation for programming languages, specification and verification. It is known that reachability properties can be naturally expressed in matching logic. In this paper, we show that constrained constructor patterns can be faithfully specified as a matching logic theory. As a result, we obtain a full encoding of reachability logic for rewrite theories as matching logic theories, by combining the two encodings. We also show that the main properties of constrained constructor patterns can be specified and proved within matching logic, using the existing proof system.

String diagram rewrite theory II: Rewriting with symmetric monoidal structure

AbstractSymmetric monoidal theories (SMTs) generalise algebraic theories in a way that make them suitable to express resource-sensitive systems, in which variables cannot be copied or discarded at will. In SMTs, traditional tree-like terms are replaced by string diagrams, topological entities that can be intuitively thought of as diagrams of wires and boxes. Recently, string diagrams have become increasingly popular as a graphical syntax to reason about computational models across diverse fields, including programming language semantics, circuit theory, quantum mechanics, linguistics, and control theory. In applications, it is often convenient to implement the equations appearing in SMTs as rewriting rules. This poses the challenge of extending the traditional theory of term rewriting, which has been developed for algebraic theories, to string diagrams. In this paper, we develop a mathematical theory of string diagram rewriting for SMTs. Our approach exploits the correspondence between string diagram rewriting and double pushout (DPO) rewriting of certain graphs, introduced in the first paper of this series. Such a correspondence is only sound when the SMT includes a Frobenius algebra structure. In the present work, we show how an analogous correspondence may be established for arbitrary SMTs, once an appropriate notion of DPO rewriting (which we call convex) is identified. As proof of concept, we use our approach to show termination of two SMTs of interest: Frobenius semi-algebras and bialgebras.

Confluence of left-linear higher-order rewrite theories by checking their nested critical pairs

AbstractUser-defined higher-order rewrite rules are becoming a standard in proof assistants based on intuitionistic type theory. This raises the question of proving that they preserve the properties of beta-reductions for the corresponding type systems. In a series of papers, we develop techniques based on van Oostrom’s decreasing diagrams that reduce confluence proofs to the checking of various forms of critical pairs for higher-order rewrite rules extending beta-reduction on pure lambda-terms. As shown in a previous paper of the two middle authors, confluence of a terminating set of left-linear rewrite rules is obtained when their critical pairs are joinable, beta-rewrite steps being disallowed. The present paper concentrates on the case where arbitrary beta-rewrite steps are allowed for joining critical pairs. The rewrite relation used for analyzing confluence may rewrite arbitrarily many non-overlapping redexes in a single step. This relation gives rise to critical pairs that overlap both horizontally, as with parallel rewriting, but also vertically, forming chains of successive overlaps. Practical examples of use of this technique are analyzed.

String Diagram Rewrite Theory I: Rewriting with Frobenius Structure

String diagrams are a powerful and intuitive graphical syntax, originating in theoretical physics and later formalised in the context of symmetric monoidal categories. In recent years, they have found application in the modelling of various computational structures, in fields as diverse as Computer Science, Physics, Control Theory, Linguistics, and Biology. In several of these proposals, transformations of systems are modelled as rewrite rules of diagrams. These developments require a mathematical foundation for string diagram rewriting: whereas rewrite theory for terms is well-understood, the two-dimensional nature of string diagrams poses quite a few additional challenges. This work systematises and expands a series of recent conference papers, laying down such a foundation. As a first step, we focus on the case of rewrite systems for string diagrammatic theories that feature a Frobenius algebra. This common structure provides a more permissive notion of composition than the usual one available in monoidal categories, and has found many applications in areas such as concurrency, quantum theory, and electrical circuits. Notably, this structure provides an exact correspondence between the syntactic notion of string diagrams modulo Frobenius structure and the combinatorial structure of hypergraphs. Our work introduces a combinatorial interpretation of string diagram rewriting modulo Frobenius structures in terms of double-pushout hypergraph rewriting. We prove this interpretation to be sound and complete and we also show that the approach can be generalised to rewriting modulo multiple Frobenius structures. As a proof of concept, we show how to derive from these results a termination strategy for Interacting Bialgebras, an important rewrite theory in the study of quantum circuits and signal flow graphs.

Symbolic Specialization of Rewriting Logic Theories withPresto

AbstractThis paper introduces$\tt{{Presto}}$, a symbolic partial evaluator for Maude’s rewriting logic theories that can improve system analysis and verification. In$\tt{{Presto}}$, the automated optimization of a conditional rewrite theory$\mathcal{R}$(whose rules define the concurrent transitions of a system) is achieved by partially evaluating, with respect to the rules of$\mathcal{R}$, an underlying, companion equational logic theory$\mathcal{E}$that specifies the algebraic structure of the system states of$\mathcal{R}$. This can be particularly useful for specializing an overly general equational theory$\mathcal{E}$whose operators may obey complex combinations of associativity, commutativity, and/or identity axioms, when being plugged into a host rewrite theory$\mathcal{R}$as happens, for instance, in protocol analysis, where sophisticated equational theories for cryptography are used.$\tt{{Presto}}$implements different unfolding operators that are based onfolding variant narrowing(the symbolic engine of Maude’s equational theories). When combined with an appropriate abstraction algorithm, they allow the specialization to be adapted to the theory termination behavior and bring significant improvement while ensuring strong correctness and termination of the specialization. We demonstrate the effectiveness of$\tt{{Presto}}$in several examples of protocol analysis where it achieves a significant speed-up. Actually, the transformation provided by$\tt{{Presto}}$may cut down an infinite folding variant narrowing space to a finite one, and moreover, some of the costly algebraic axioms and rule conditions may be eliminated as well. As far as we know, this is the first partial evaluator for Maude that respects the semantics of functional, logic, concurrent, and object-oriented computations.

A partial evaluation methodology for optimizing rewrite theories incrementally

Partial evaluation (PE) is a branch of computer science that achieves code optimization via specialization. This article describes a PE methodology for optimizing rewrite theories that encode concurrent as well as nondeterministic systems by means of the Maude language. The main advantages of the proposed methodology can be summarized as follows:•An automatic program optimization technique for rewrite theories featuring several PE criteria that support the specialization of a broad class of rewrite theories.•An incremental partial evaluation modality that allows the key specialization components to be encapsulated at the desired granularity level to facilitate progressive refinements of the specialization.•All executability theory requirements are preserved by the PE transformation. Also the transformation ensures the semantic equivalence between the original rewrite theory and the specialized theory under rather mild conditions.

Formal Verification of Multitask Hybrid Systems by the OTS/CafeOBJ Method

Hybrid systems combine both continuous and discrete behaviors, which occur frequently in safety-critical applications in various domains including Internet-of-Things (IoT) and Cyber-Physical Systems (CPS) applications such as health care, transportation, and robotics. For safe and reliable information society with IoT and CPS technologies, it is important to establish a way to specify and verify hybrid systems formally. Formal descriptions of hybrid systems may help us to verify desired properties of a given system formally with computer supports. We propose a way to describe a formal specification of a given multitask hybrid system as an observational transition system (OTS) in CafeOBJ algebraic specification language. OTSs are models where systems behaviors are described through observations. CafeOBJ supports specification execution based on a rewrite theory. We verify that OTS/CafeOBJ specifications of hybrid systems satisfy desired property by the proof score method based on equational reasoning implemented in CafeOBJ interpreter. In this paper, we specify a signal control system with an arbitrary number of vehicles by our proposed method, and verify the system satisfies a safety property by the proof score method.

Optimization of rewrite theories by equational partial evaluation

In this paper, we develop an automated optimization framework for rewrite theories that supports sorts, subsort overloading, equations and algebraic axioms with free/non-free constructors, and rewrite rules modeling concurrent system transitions whose state structure is defined by means of the equations. The main idea of the framework is to make the system computations more efficient by partially evaluating the equations to the specific calls that are required by the transition rules. This can be particularly useful for automatically optimizing rewrite theories that contain overly general equational theories which perform unnecessary and costly computations involving pattern matching and/or unification modulo equations and axioms. The transformation is based on a suitable unfolding operator parameter that relies on the symbolic operational engine of Maude's equational theories, called folding variant narrowing, together with a generic abstraction operator. Depending on the properties of the rewrite theory, the unfolding and abstraction operators must be fine-tuned to achieve the biggest optimization possible while ensuring termination and total correctness of the transformation. We formalize two instances of our scheme for the case when the rewrite theory either has an infinite number of most general variants or a finite number of most general variants. Finally, we discuss some experimental results which demonstrate that the proposed optimization technique pays off in practice.

Simulating and model checking membrane systems using strategies in Maude

Membrane systems are a biologically-inspired computational model based on the structure of biological cells and the way chemicals interact and traverse their membranes. Although their dynamics are described by rules, encoding membrane systems into rewriting logic is not straightforward due to its complex control mechanisms. Multiple alternatives have been proposed in the literature and implemented in the Maude specification language. The recent release of the Maude strategy language and its associated strategy-aware model checker [29] allow specifying these systems more easily, so that they become executable and verifiable for free. An easily-extensible interactive environment transforms membrane specifications into rewrite theories controlled by appropriate strategies, and allows simulating and verifying membrane computations by means of them.

Order-sorted Homeomorphic Embedding Modulo Combinations of Associativity and/or Commutativity Axioms*

The Homeomorphic Embedding relation has been amply used for defining termination criteria of symbolic methods for program analysis, transformation, and verification. However, homeomorphic embedding has never been investigated in the context of order-sorted rewrite theories that support symbolic execution methods modulo equational axioms. This paper generalizes the symbolic homeomorphic embedding relation to order–sorted rewrite theories that may contain various combinations of associativity and/or commutativity axioms for different binary operators. We systematically measure the performance of different, increasingly efficient formulations of the homeomorphic embedding relation modulo axioms that we implement in Maude. Our experimental results show that the most efficient version indeed pays off in practice.

A Constructor-Based Reachability Logic for Rewrite Theories

Reachability logic has been applied to \(\mathbb {K}\) rewrite-rule-based language definitions as a language-generic logic of programs. To be able to verify not just code but also distributed system designs, a new rewrite-theory-generic reachability logic is presented and proved sound for a wide class of rewrite theories. Constructor-based semantic unification, matching, and satisfiability procedures greatly increase the range of decidable background theories that can be used in reachability logic proofs. New methods for proving invariants of possibly never terminating distributed systems are developed, and experiments with a prototype implementation illustrating the new proof methods are presented.

A partial evaluation framework for order-sorted equational programs modulo axioms

Partial evaluation is a powerful and general program optimization technique with many successful applications. Existing PE schemes do not apply to expressive rule-based languages like Maude, CafeOBJ, OBJ, ASF+SDF, and ELAN, which support: 1) rich type structures with sorts, subsorts, and overloading; and 2) equational rewriting modulo various combinations of axioms such as associativity, commutativity, and identity. In this paper, we develop the new foundations needed and illustrate the key concepts by showing how they apply to partial evaluation of expressive programs written in Maude. Our partial evaluation scheme is based on an automatic unfolding algorithm that computes term variants and relies on high-performance order-sorted equational least general generalization and order-sorted equational homeomorphic embedding algorithms for ensuring termination. We show that our partial evaluation technique is sound and complete for convergent rewrite theories that may contain various combinations of associativity, commutativity, and/or identity axioms for different binary operators. We demonstrate the effectiveness of Maude's automatic partial evaluator, Victoria, on several examples where it shows significant speed-ups.

Symbolic Analysis of Maude Theories with Narval

AbstractConcurrent functional languages that are endowed with symbolic reasoning capabilities such as Maude offer a high-level, elegant, and efficient approach to programming and analyzing complex, highly nondeterministic software systems. Maude’s symbolic capabilities are based on equational unification and narrowing in rewrite theories, and provide Maude with advanced logic programming capabilities such as unification modulo user-definable equational theories and symbolic reachability analysis in rewrite theories. Intricate computing problems may be effectively and naturally solved in Maude thanks to the synergy of these recently developed symbolic capabilities and classical Maude features, such as: (i) rich type structures with sorts (types), subsorts, and overloading; (ii) equational rewriting modulo various combinations of axioms such as associativity, commutativity, and identity; and (iii) classical reachability analysis in rewrite theories. However, the combination of all of these features may hinder the understanding of Maude symbolic computations for non-experienced developers. The purpose of this article is to describe how programming and analysis of Maude rewrite theories can be made easier by providing a sophisticated graphical tool called Narval that supports the fine-grained inspection of Maude symbolic computations.

Generalized rewrite theories, coherence completion, and symbolic methods

A new notion of generalized rewrite theory suitable for symbolic reasoning and generalizing the standard notion in [19] is motivated and defined. Also, new requirements for symbolic executability of generalized rewrite theories that extend those in [33] for standard rewrite theories, including a generalized notion of coherence, are given. Symbolic executability, including coherence, is both ensured and made available for a wide class of such theories by automatable theory transformations. Using these foundations, several symbolic reasoning methods using generalized rewrite theories are studied, including: (i) symbolic description of sets of terms by pattern predicates; (ii) reasoning about universal reachability properties by generalized rewriting; (iii) reasoning about existential reachability properties by constrained narrowing; and (iv) symbolic verification of safety properties such as invariants and stability properties.

Static correction of Maude programs with assertions

In this paper, we present a novel transformation method for Maude programs featuring both automatic program diagnosis and correction. The input of our method is a reference specification A of the program behavior that is given in the form of assertions together with an overly general program R whose execution might violate the assertions. Our correction technique translates R into a refined program R′ in which every computation is also a computation in R that satisfies the assertions of A. The technique is first formalized for topmost rewrite theories, and then we generalize it to larger classes of rewrite theories that support nested structured configurations. Our technique copes with infinite space states and does not require the knowledge of any failing run. We report experiments that assess the effectiveness of assertion-driven correction.

Reducing Total Correctness to Partial Correctness by a Transformation of the Language Semantics

We give a language-parametric solution to the problem of total correctness, by automatically reducing it to the problem of partial correctness, under the assumption that an expression whose value decreases with each program step in a well-founded order is provided. Our approach assumes that the programming language semantics is given as a rewrite theory. We implement a prototype on top of the RMT tool and we show that it works in practice on a number of examples.