Maude: specification and programming in rewriting logic

Maude is a high-level language and a high-performance system supporting executable specification and declarative programming in rewriting logic. Since rewriting logic contains equational logic, Maude also supports equational specification and programming in its sublanguage of functional modules and theories. The underlying equational logic chosen for Maude is membership equational logic, that has sorts, subsorts, operator overloading, and partiality definable by membership and equality conditions. Rewriting logic is reflective, in the sense of being able to express its own metalevel at the object level. Reflection is systematically exploited in Maude endowing the language with powerful metaprogramming capabilities, including both user-definable module operations and declarative strategies to guide the deduction process. This paper explains and illustrates with examples the main concepts of Maude's language design, including its underlying logic, functional, system and object-oriented modules, as well as parameterized modules, theories, and views. We also explain how Maude supports reflection, metaprogramming and internal strategies. The paper outlines the principles underlying the Maude system implementation, including its semicompilation techniques. We conclude with some remarks about applications, work on a formal environment for Maude, and a mobile language extension of Maude

Order-sorted algebra I: equational deduction for multiple inheritance, overloading, exceptions and partial operations

AbstractThis paper generalizes many-sorted algebra (MSA) to order-sorted algebra (OSA) by allowing a partial ordering relation on the set of sorts. This supports abstract data types with multiple inheritance (in roughly the sense of object-oriented programming), several forms of polymorphism and overloading, partial operations (as total on equationally defined subsorts), exception handling, and an operational semantics based on term rewriting. We give the basic algebraic constructions for OSA, including quotient, image, product and term algebra, and we prove their basic properties, including quotient, homomorphism, and initiality theorems. The paper's major mathematical results include a notion of OSA deduction, a completeness theorem for it, and an OSA Birkhoff variety theorem. We also develop conditional OSA, including initiality, completeness, and McKinsey-Malcev quasivariety theorems, and we reduce OSA to (conditional) MSA, which allows lifting many known MSA results to OSA. Retracts, which intuitively are left inverses to subsort inclusions, provide relatively inexpensive run-time error handling. We show that it is safe to add retracts to any OSA signature, in the sense that it gives rise to a conservative extension. A final section compares and contrasts many different approaches to OSA. This paper also includes several examples demonstrating the flexibility and applicability of OSA, including some standard benchmarks like stack and list, as well as a much more substantial example, the number hierarchy from the naturals up to the quaternions

Termination of Order-sorted Rewriting

International audienceIn this paper, the problem of termination of rewriting in order-sorted algebras is addressed for the first time. Our goal is to perform termination proofs of programs for executable specification languages like OBJ3. An extension of Lexicographic Path Ordering is proposed, that gives a termination proof for order-sorted rewrite systems, that would not terminate in the unsorted case. We mention also, that this extension provides a termination tool for unsorted terminating systems, that usual orderings cannot handle

Maude: specification and programming in rewriting logic

AbstractMaude is a high-level language and a high-performance system supporting executable specification and declarative programming in rewriting logic. Since rewriting logic contains equational logic, Maude also supports equational specification and programming in its sublanguage of functional modules and theories. The underlying equational logic chosen for Maude is membership equational logic, that has sorts, subsorts, operator overloading, and partiality definable by membership and equality conditions. Rewriting logic is reflective, in the sense of being able to express its own metalevel at the object level. Reflection is systematically exploited in Maude endowing the language with powerful metaprogramming capabilities, including both user-definable module operations and declarative strategies to guide the deduction process. This paper explains and illustrates with examples the main concepts of Maude's language design, including its underlying logic, functional, system and object-oriented modules, as well as parameterized modules, theories, and views. We also explain how Maude supports reflection, metaprogramming and internal strategies. The paper outlines the principles underlying the Maude system implementation, including its semicompilation techniques. We conclude with some remarks about applications, work on a formal environment for Maude, and a mobile language extension of Maude

Specification and Verification of Distributed Embedded Systems: A Traffic Intersection Product Family

Distributed embedded systems (DESs) are no longer the exception; they are the rule in many application areas such as avionics, the automotive industry, traffic systems, sensor networks, and medical devices. Formal DES specification and verification is challenging due to state space explosion and the need to support real-time features. This paper reports on an extensive industry-based case study involving a DES product family for a pedestrian and car 4-way traffic intersection in which autonomous devices communicate by asynchronous message passing without a centralized controller. All the safety requirements and a liveness requirement informally specified in the requirements document have been formally verified using Real-Time Maude and its model checking features.Comment: In Proceedings RTRTS 2010, arXiv:1009.398

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

[EN] 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.M. Alpuente and S. Escobar have been partially supported by the EU (FEDER) and the Spanish MCIU under grant RTI2018-094403-B-C32, by the Spanish Generalitat Valenciana under grant PROMETEO/2019/098, and by the European Union's Horizon 2020 research and innovation programme under grant agreement No. 952215 (TAILOR). J. Meseguer has been supported by NRL under contract number N00173-17-1-G002. A. Cuenca-Ortega has been supported by the SENESCYT, Ecuador (scholarship program 2013).Alpuente Frasnedo, M.; Cuenca-Ortega, A.; Escobar Román, S.; Meseguer, J. (2020). Order-sorted Homeomorphic Embedding modulo Combinations of Associativity and/or Commutativity Axioms. Fundamenta Informaticae. 177(3-4):297-329. https://doi.org/10.3233/FI-2020-1991S2973291773-

Executable Structural Operational Semantics in Maude

This paper describes in detail how to bridge the gap between theory and practice when implementing in Maude structural operational semantics described in rewriting logic, where transitions become rewrites and inference rules become conditional rewrite rules with rewrites in the conditions, as made possible by the new features in Maude 2.0. We validate this technique using it in several case studies: a functional language Fpl (evaluation and computation semantics, including an abstract machine), imperative languages WhileL (evaluation and computation semantics) and GuardL with nondeterminism (computation semantics), Kahn’s functional language Mini-ML (evaluation or natural semantics), Milner’s CCS (with strong and weak transitions), and Full LOTOS (including ACT ONE data type specifications). In addition, on top of CCS we develop an implementation of the Hennessy-Milner modal logic for describing local capabilities of processes, and for LOTOS we build an entire tool where Full LOTOS specifications can be entered and executed (without user knowledge of the underlying implementation of the semantics). We also compare this method based on transitions as rewrites with another one based on transitions as judgements

Reconstructing Z3 Proofs With KeY

KeY dient zur formalen Verifikation spezifizierter Eigenschaften von Java-Programmen.Dafür werden aus der formalen Spezifikation sowie dem Programm-Code Beweisverpflichtungen generiert. Diese werden dann Schritt für Schritt in eine Menge von Formeln der Prädikatenlogik erster Stufe überführt. Da diese allerdings unentscheidbar ist, ist es eine große Herausforderung, einen Beweis für diese Formelmenge zu finden. Moderne SMT-Solver, wie zum Beispiel Z3, sind genau auf diesen Anwendungszweck hin optimiert. Daher ist schon lange in KeY die Möglichkeit eingebaut, (Teil-)Probleme für SMT-Solver zu übersetzen. Das Ergebnis bei diesem Vorgehen ist ein partieller Beweis in KeY, der von (möglicherweise mehreren) SMT-Antworten komplettiert wird. Da Z3 aber auch Beweise für seine Antworten liefern kann, gibt es hier Verbesserungspotential: In dieser Thesis wird eine Technik zum Nachspielen der Z3 Beweise in KeY vorgestellt, sodass man als Ergebnis einen geschlossenen Beweis in KeY erhält und die SMT-Antworten verworfen werden können. Herausforderungen sowohl systematischer als auch technischer Natur werden identifiziert and Lösungen dafür vogestellt. Schließlich wird auch eine prototypische Implementierung der Technik zum Nachspielen der Beweise zur Verfügung gestellt. Im Evaluations-Teil der Arbeit wird die Leistungsfähigkeit dieser Implementierung sowie die zukünfigen Möglichkeiten erörtert