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

Faithful (meta-)encodings of programmable strategies into term rewriting systems

Rewriting is a formalism widely used in computer science and mathematical logic. When using rewriting as a programming or modeling paradigm, the rewrite rules describe the transformations one wants to operate and rewriting strategies are used to con- trol their application. The operational semantics of these strategies are generally accepted and approaches for analyzing the termination of specific strategies have been studied. We propose in this paper a generic encoding of classic control and traversal strategies used in rewrite based languages such as Maude, Stratego and Tom into a plain term rewriting system. The encoding is proven sound and complete and, as a direct consequence, estab- lished termination methods used for term rewriting systems can be applied to analyze the termination of strategy controlled term rewriting systems. We show that the encoding of strategies into term rewriting systems can be easily adapted to handle many-sorted signa- tures and we use a meta-level representation of terms to reduce the size of the encodings. The corresponding implementation in Tom generates term rewriting systems compatible with the syntax of termination tools such as AProVE and TTT2, tools which turned out to be very effective in (dis)proving the termination of the generated term rewriting systems. The approach can also be seen as a generic strategy compiler which can be integrated into languages providing pattern matching primitives; experiments in Tom show that applying our encoding leads to performances comparable to the native Tom strategies

Reflection in conditional rewriting logic

AbstractWe recall general metalogical axioms for a reflective logic based on the notion of a universal theory, that is, a theory that can simulate the deductions of all other theories in a class of theories of interest, including itself. We then show that conditional rewriting logic is reflective, generalizing in two stages: first to the unsorted conditional case, and then to the many-sorted conditional case, the already known result for unconditional and unsorted rewriting logic (Reflection in Rewriting Logic: Metalogical Foundations and Metaprogramming Applications. CSLI Publications, 2000). This work should be seen as providing foundations for many useful applications of rewriting logic reflection. The results presented here have greatly influenced the design of the Maude language, which implements rewriting logic and supports its reflective capabilities, and have been used as a theoretical foundation for applications such as internal rewrite strategies, reflective design of theorem proving tools, module algebra and metaprogramming, and metareasoning in metalogical frameworks

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

A Reflective Theorem Prover for the Connection Calculus

Rewriting logic can be used to prototype systems for automated deduction. In this paper, we illustrate how this approach allows experiments with deduction strategies in a flexible and conceptually satisfying way. This is achieved by exploiting the reflective property of rewriting logic. By specifying a theorem prover in this way one quickly obtains a readable, reliable and reasonably efficient system which can be used both as a platform for tactic experiments and as a basis for an optimized implementation. The approach is illustrated by specifying a calculus for the connection method in rewriting logic which clearly separates rules from tactics

The Structure of First-Order Causality

Game semantics describe the interactive behavior of proofs by interpreting formulas as games on which proofs induce strategies. Such a semantics is introduced here for capturing dependencies induced by quantifications in first-order propositional logic. One of the main difficulties that has to be faced during the elaboration of this kind of semantics is to characterize definable strategies, that is strategies which actually behave like a proof. This is usually done by restricting the model to strategies satisfying subtle combinatorial conditions, whose preservation under composition is often difficult to show. Here, we present an original methodology to achieve this task, which requires to combine advanced tools from game semantics, rewriting theory and categorical algebra. We introduce a diagrammatic presentation of the monoidal category of definable strategies of our model, by the means of generators and relations: those strategies can be generated from a finite set of atomic strategies and the equality between strategies admits a finite axiomatization, this equational structure corresponding to a polarized variation of the notion of bialgebra. This work thus bridges algebra and denotational semantics in order to reveal the structure of dependencies induced by first-order quantifiers, and lays the foundations for a mechanized analysis of causality in programming languages

Rewriting Systems for Reachability in Vector Addition Systems with Pairs

15 pagesInternational audienceWe adapt hypergraph rewriting system to a generalization of Vector Addition Systems with States (VASS) that we call vector addition systems with pairs (VASP). We give rewriting systems and strategies, that allow us to obtain reachability equivalence results between some classes of VASP and VASS. Reachability for the later is well known be equivalent to reachability in Petri nets. VASP generalize also Branching Extension of VASS (BVASS) for which it is unknown if they are more expressive than VASS. We consider here a more restricted notion of reachability for VASP than that for BVASS. However the reachability decision problem corresponding is already equivalent to decidability of the provability in Multiplicative and Exponential Linear Logic (MELL), a question left open for more than 20 years

Controlling Rewriting: study and implementation of a strategy formalism (Abstract)

AbstractThis paper summarizes my PhD thesis devoted to an introduction of a new strategy formalism for the first-order rewrite system, called ELAN. Goals of my PhD thesis are proposing and studying different constructions expressing the control of rewriting at the level of rules and strategies, studying a strategy-directed cooperation of procedures (i.e. solvers), and finally, exploring certain reflexive aspects of computational systems to be able to express their transformations by computational systems. The principal goal is a design of a declarative, strictly typed and extensible strategy language based on rewriting logic within the existing framework ELAN. A programming style of the strategy language, different language constructions and extensions (e.g. high-level or polymorphic strategies) and several used implementation techniques (e.g. partial evaluation, or compilation) are also studied in this thesis. This paper outlines the principal problems attacked in this thesis, highlights several new ideas and proposed solutions

Combining Runtime Checking and Slicing to Improve Maude Error Diagnosis

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-319-23165-5_3This paper introduces the idea of using assertion checking for enhancing the dynamic slicing of Maude computation traces. Since trace slicing can greatly simplify the size and complexity of the analyzed traces, our methodology can be useful for improving the diagnosis of erroneous Maude programs. The proposed methodology is based on (i) a logical notation for specifying two types of user-defined assertions that are imposed on execution runs: functional assertions and system assertions; (ii) a runtime checking technique that dynamically tests the assertions and is provably safe in the sense that all errors flagged are definite violations of the specifications; and (iii) a mechanism based on equational least general generalization that automatically derives accurate criteria for slicing from falsified assertions.This work has been partially supported by the EU (FEDER) and the Spanish MINECO project ref. TIN2013-45732-C4-01 (DAMAS), and by Generalitat Valenciana ref. PROMETEOII/2015/013 (SmartLogic). F. Frechina was supported by FPU-ME grant AP2010-5681, and J. Sapiña was supported by FPI-UPV grant SP2013-0083.Alpuente Frasnedo, M.; Ballis, D.; Frechina Navarro, F.; Sapiña Sanchis, J. (2015). Combining Runtime Checking and Slicing to Improve Maude Error Diagnosis. En Logic, Rewriting, and Concurrency. Essays Dedicated to José Meseguer on the Occasion of His 65th Birthday. 72-96. https://doi.org/10.1007/978-3-319-23165-5_3S7296Alpuente, M., Ballis, D., Espert, J., Romero, D.: Backward trace slicing for rewriting logic theories. In: Bjørner, N., Sofronie-Stokkermans, V. (eds.) CADE 2011. LNCS, vol. 6803, pp. 34–48. Springer, Heidelberg (2011)Alpuente, M., Ballis, D., Frechina, F., Romero, D.: Backward trace slicing for conditional rewrite theories. In: Bjørner, N., Voronkov, A. (eds.) LPAR-18 2012. LNCS, vol. 7180, pp. 62–76. Springer, Heidelberg (2012)Alpuente, M., Ballis, D., Frechina, F., Romero, D.: Julienne: a trace slicer for conditional rewrite theories. In: Giannakopoulou, D., Méry, D. (eds.) FM 2012. LNCS, vol. 7436, pp. 28–32. Springer, Heidelberg (2012)Alpuente, M., Ballis, D., Frechina, F., Romero, D.: Using conditional trace slicing for improving Maude programs. Sci. Comput. Program. 80, Part B:385–415 (2014)Alpuente, M., Ballis, D., Frechina, F., Sapiña, J.: Slicing-based trace analysis of rewriting logic specifications with $$I$$ Julienne. In: Felleisen, M., Gardner, P. (eds.) ESOP 2013. LNCS, vol. 7792, pp. 121–124. Springer, Heidelberg (2013)Alpuente, M., Ballis, D., Frechina, F., Sapiña, J.: Inspecting rewriting logic computations (in a Parametric and Stepwise Way). In: Iida, S., Meseguer, J., Ogata, K. (eds.) Specification, Algebra, and Software. LNCS, vol. 8373, pp. 229–255. Springer, Heidelberg (2014)Alpuente, M., Ballis, D., Frechina, F., Sapiña, J.: Debugging Maude programs via runtime assertion checking and trace slicing. Technical report, Department of Computer Systems and Computation, Universitat Politècnica de València (2015). http://safe-tools.dsic.upv.es/abets/abets-tr.pdfAlpuente, M., Ballis, D., Frechina, F., Sapiña, J.: Exploring conditional rewriting logic computations. J. 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