A Rewriting Calculus for Cyclic Higher-order Term Graphs

Introduced at the end of the nineties, the Rewriting Calculus (ρ-calculus, for short) is a simple calculus that fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the obtained structured results are first class objects of the calculus. The evaluation mechanism, generalizing beta-reduction, strongly relies on term matching in various theories.In this paper we propose an extension of the ρ-calculus, handling graph like structures rather than simple terms. The transformations are performed by explicit application of rewrite rules as first class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.The calculus over terms is naturally generalized by using unification constraints in addition to the standard ρ-calculus matching constraints. This therefore provides us with the basics for a natural extension of an explicit substitution calculus to term graphs. Several examples illustrating the introduced concepts are given.     

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.     

On Term Graphs as an Adhesive Category

The recent interest in bisimulation congruences for reduction systems, stimulated by the research on general (often graphical) frameworks for nominal calculi, has brought forward many proposals for categorical formalisms where relevant properties of observational equivalences could be auto- matically verified.Interestingly, some of these formalisms also identified suitable categories where the standard tools and techniques developed for the double-pushout approach to graph transformation [A. Corradini, U. Montanari, F. Rossi, H. Ehrig, R. Heckel and M. Löwe, Algebraic approaches to graph transformation I: Basic concepts and double pushout approach, in: G. Rozenberg, editor, Handbook of Graph Grammars and Computing by Graph Transformation, I: Foundations, World Scientific, 1997, pp. 163–246] could be recast, thus providing a valid alternative to the High-Level Replacement Systems paradigm [H. Ehrig, A. Habel, H.-J. Kreowski and F. Parisi-Presicce, Parallelism and concurrency in highlevel replacement systems, Mathematical Structures in Computer Science 1 (1991) 361–404].In this paper we consider the category of term graphs, and we prove that it (partly) fits in the general framework for adhesive categories, developed in [S. Lack, and P. Sobociński, Adhesive categories, in: I. Walukiewicz, editor, Foundations of Software Science and Computation Structures, Lect. Notes in Comp. Sci. 2987 (2004), pp. 273–288, P. Sobociński, “Deriving bisimulation congruences from reduction systems”, Ph.D. thesis, BRICS, Department of Computer Science, University of Aaurhus (2004)], extended in [H. Ehrig, A. Habel, J. Padberg and U. Prange, Adhesive high-level replacement categories and systems, in: G. Engels and F. Parisi-Presicce, editors, Graph Transformation, Lect. Notes in Comp. Sci. (2004)] and applied to reduction systems in [V. Sassone, and P. Sobociński, Congruences for contextual graph-rewriting, Technical Report RS-04-11, BRICS, Department of Computer Science, University of Aarhus (2004)]. The main technical achievement concerns the proof that the category of term graphs is actually quasi-adhesive, obtained by proving the existence of suitable Van Kampen squares.     

项重写的图实现

图重写能够有效地实现项重写。文章从项重写的图实现的角度出发，研究了图重写模拟项重写的正确性和完备性：在无环出现的情况下，图重写对一切项重写下正确；在无环出现的条件下，图重写对左线性合流的项重写是完备的。     

Conditional Term Graph Rewriting with Indirect Sharing

For reasons of efficiency, term rewriting is usually implemented by term graph rewriting. In term rewriting, expressions are represented as terms, whereas in term graph rewriting these are represented as directed graphs. Unlike terms, graphs allow a sharing of common subexpressions. In previous work, we have shown that conditional term graph rewriting is a sound and complete implementation for a certain class of CTRSs with strict equality, provided that a minimal structure sharing scheme is used. In this paper, we will show that this is also true for two different extensions of normal CTRSs. In contrast to the previous work, however, a non-minimal structure sharing scheme can be used. That is, the amount of sharing is increased.     

Modeling Pointer Redirection as Cyclic Term-graph Rewriting

We tackle the problem of data-structure rewriting including global and local pointer redirections. Each basic rewrite step may perform three kinds of actions: (i) Local redirection, the aim of which is to redirect specific pointers determined by means of a pattern; (ii) Replacement, that may add new information to data-structures; (iii) Global redirection, which is aimed at redirecting all pointers targeting a node towards another one. We define a new framework, following the double-pushout approach, where graph rewrite rules may mix these three kinds of actions in a row. We define first the category of graphs we consider and then we define rewrite rules as pairs of graph homomorphisms of the form L←K→R. In our setting, graph K is not arbitrary, it is used to encode pointer redirection. Furthermore, pushouts do not always exist and complement pushouts, when they exist, are not unique. Despite these concerns, our definition of rewriting steps is such that a rewrite rule can always be fired, once a matching is found.     

From Chemical Rules to Term Rewriting

In this paper, rule-based programming is explored in the field of automated generation of chemical reaction mechanisms. We explore a class of graphs and a graph rewriting relation where vertices are preserved and only edges are changed. We show how to represent cyclic labeled graphs by decorated labeled trees or forests, then how to transform trees into terms. A graph rewriting relation is defined, then simulated by a tree rewriting relation, which can be in turn simulated by a rewriting relation on equivalence classes of terms. As a consequence, this kind of graph rewriting can be implemented using term rewriting. This study is motivated by the design of the GasEl system for the generation of kinetics reactions mechanisms. In GasEl, chemical reactions correspond to graph rewrite rules and are implemented by conditional rewriting rules in ELAN. The control of their application is done through the ELAN strategy language.     

The Better Bubbling Lemma

BCD [Barendregt, Henk, Mario Coppo and Mariangiola Dezani-Ciancaglini, A filter lambda model and the completeness of type assignment, JSL 48 (1983), 931–940] relies for its modeling of λ calculus in intersection type filters on a key theorem which I call BL (for the Bubbling Lemma, following someone). This lemma has been extended in [Castagna, Giuseppe, and Alain Frisch. A gentle introduction to semantic subtyping. In PPDP05, ACM Press (full version) and ICALP05, LNCS volume 3580, Springer-Verlag (summary), 2005. Joint ICALP-PPDP keynote talk; Dezani-Ciancaglini, M., A. Frisch, E. Giovannetti and Y. Motohama, 2002. The Relevance of Semantic Subtyping, Electronic Notes in Theoretical Computer Science 70 No. 1, 15pp] to encompass Boolean structure, including specifically union types; this extended lemma I call BBL (the Better Bubbling Lemma). There are resonances, explored in [Dezani-Ciancaglini, M., R.K. Meyer and Y. Motohama, The semantics of entailment omega, NDJFL 43 (2002), 129–145] and [Dezani-Ciancaglini, M., A. Frisch, E. Giovannetti and Y. Motohama, 2002. The Relevance of Semantic Subtyping, Electronic Notes in Theoretical Computer Science 70 No. 1, 15pp], between intersection and union type theories and the already existing minimal positive relevant logic B+ of [Routley, Richard, and Robert K. Meyer, The semantics of entailment III, JPL 1 (1972), 192–208]. (Indeed [Pal, Koushik, and Robert K. Meyer. 2005. Basic relevant theories for combinators at levels I and II, AJL 3, http://www.philosophy.unimelb.edu.au/2005/2003_2.pdf, 19 pages] applies BL and BBL to get further results linking combinators to relevant theories and propositions.) On these resonances the filters of algebra become the theories of logic. The semantics of [Meyer, Robert K., Yoko Motohama and Viviana Bono, Truth translations of relevant logics, in B. Brown and F. Lepage, eds., “Truth and Probability: Essays in Honour of Hugues Leblanc,” pages 59–84, College Publications, King's College, London, 2005] yields here a new and short proof of BBL, which takes account of full Boolean structure by encompassing not only B+ but also its conservative Boolean extension CB [Routley, Richard, and Robert K. Meyer, The semantics of entailment III, JPL 1 (1972), 192–208; Meyer, Robert K., 1995. Types and the Boolean system CB+, cslab.anu.edu.au/ftp/techreports/SRS/1995/TR-SRS-5-95/meyer.ps.gz; Meyer, Robert K., Yoko Motohama and Viviana Bono, Truth translations of relevant logics, in B. Brown and F. Lepage, eds., “Truth and Probability: Essays in Honour of Hugues Leblanc,” pages 59–84, College Publications, King's College, London, 2005.].     

Autowrite: A Tool for Term Rewrite Systems and Tree Automata

Autowrite is an experimental software tool written in Common Lisp Oriented System (CLOS) which handles term rewrite systems and bottom-up tree automata. A graphical interface written using McCLIM, (the free implementation of the CLIM specification) frees the user of any Lisp knowledge. Software and documentation can be found at http://dept-info.labri.u-bordeaux.fr/~idurand/autowrite. Autowrite was initially designed to check call-by-need properties of term rewrite systems. For this purpose, it implements the tree automata constructions used in [F. Jacquemard. Decidable approximations of term rewriting systems. In Proc. 7th RTA, volume 1103 of LNCS, pages 362–376, 1996; I. Durand and A. Middeldorp. Decidable call by need computations in term rewriting (extended abstract). In Proc. 14th CADE, volume 1249 of LNAI, pages 4–18, 1997; Irène Durand and Aart Middeldorp. On the complexity of deciding call-by-need. Technical Report 1196–98, LaBRI, 1998; T. Nagaya and Y. Toyama. Decidability for left-linear growing term rewriting systems. Information and Computation, 178(2):499–514, 2002] and many useful operations on terms, term rewrite systems and tree automata.     

Bisimilarity in Term Graph Rewriting

We present a survey of confluence properties of (acyclic) term graph rewriting. Results and counterexamples are given for different kinds of term graph rewriting; besides plain applications of rewrite rules, extensions with the operations of collapsing and copying, and both operations together are considered. Collapsing and copying together constitute bisimilarity of term graphs. We establish sufficient conditions for—and counterexamples to—confluence, confluence modulo bisimilarity, and the Church–Rosser property modulo bisimilarity. Moreover, we address rewriting modulo bisimilarity, that is, rewriting of bisimilarity classes of term graphs.     

Building Interpreters with Rewriting Strategies

Programming language semantics based on pure rewrite rules suffers from the gap between the rewriting strategy implemented in rewriting engines and the intended evaluation strategy. This paper shows how programmable rewriting strategies can be used to implement interpreters for programming languages based on rewrite rules. The advantage of this approach is that reduction rules are first class entities that can be reused in different strategies, even in other kinds of program transformations such as optimizers. The approach is illustrated with several interpreters for the lambda calculus based on implicit and explicit (parallel) substitution, different strategies including normalization, eager evaluation, lazy evaluation, and lazy evaluation with updates. An extension with pattern matching and choice shows that such interpreters can easily be extended.     

Lower Bounds for Runtime Complexity of Term Rewriting

We present the first approach to deduce lower bounds for (worst-case) runtime complexity of term rewrite systems (TRSs) automatically. Inferring lower runtime bounds is useful to detect bugs and to complement existing methods that compute upper complexity bounds. Our approach is based on two techniques: the induction technique generates suitable families of rewrite sequences and uses induction proofs to find a relation between the length of a rewrite sequence and the size of the first term in the sequence. The loop detection technique searches for “decreasing loops”. Decreasing loops generalize the notion of loops for TRSs, and allow us to detect families of rewrite sequences with linear, exponential, or infinite length. We implemented our approach in the tool AProVE and evaluated it by extensive experiments.     

Decidability for Left-Linear Growing Term Rewriting Systems

A term rewriting system is called growing if each variable occurring on both the left-hand side and the right-hand side of a rewrite rule occurs at depth zero or one in the left-hand side. Jacquemard showed that the reachability and the sequentiality of linear (i.e., left-right-linear) growing term rewriting systems are decidable. In this paper we show that Jacquemard's result can be extended to left-linear growing rewriting systems that may have right-nonlinear rewrite rules. This implies that the reachability and the joinability of some class of right-linear term rewriting systems are decidable, which improves the results for right-ground term rewriting systems by Oyamaguchi. Our result extends the class of left-linear term rewriting systems having a decidable call-by-need normalizing strategy. Moreover, we prove that the termination property is decidable for almost orthogonal growing term rewriting systems.     

On the Jacopini Technique

The general concern of the Jacopini technique is the question: “Is it consistent to extend a given lambda calculus with certain equations?” The technique was introduced by Jacopini in 1975 in his proof that in the untyped lambda calculusΩis easy, i.e.,Ωcan be assumed equal to any other (closed) term without violating the consistency of the lambda calculus. The presentations of the Jacopini technique that are known from the literature are difficult to understand and hard to generalise. In this paper we generalise the Jacopini technique for arbitrary lambda calculi. We introduce the concept ofproof-replaceabilityby which the structure of the technique is simplified considerably. We illustrate the simplicity and generality of our formulation of the technique with some examples. We apply the Jacopini technique to theλμ-calculus, and we prove a general theorem concerning the consistency of extensions of theλμ-calculus of a certain form. Many well known examples (e.g., the easiness ofΩ) are immediate consequences of this general theorem.     

Inequational Deduction as Term Graph Rewriting

Multi-algebras allow to model nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results.We propose a simple inequational deduction system, based on term graphs, for inferring inclusions of derived relations in a multi-algebra, and we show that term graph rewriting provides a sound and complete implementation of it.     

Matrix Interpretations for Proving Termination of Term Rewriting

We present a new method for automatically proving termination of term rewriting. It is based on the well-known idea of interpretation of terms where every rewrite step causes a decrease, but instead of the usual natural numbers we use vectors of natural numbers, ordered by a particular nontotal well-founded ordering. Function symbols are interpreted by linear mappings represented by matrices. This method allows us to prove termination and relative termination. A modification of the latter, in which strict steps are only allowed at the top, turns out to be helpful in combination with the dependency pair transformation. By bounding the dimension and the matrix coefficients, the search problem becomes finite. Our implementation transforms it to a Boolean satisfiability problem (SAT), to be solved by a state-of-the-art SAT solver.     

A Higher-Order Calculus for Graph Transformation

This paper presents a formalism for defining higher-order systems based on the notion of graph transformation (by rewriting or interaction). The syntax is inspired by the Combinatory Reduction Systems of Klop. The rewrite rules can be used to define first-order systems, such as graph or term-graph rewriting systems, Lafont's interaction nets, the interaction systems of Asperti and Laneve, the non-deterministic nets of Alexiev, or a process calculus. They can also be used to specify higher-order systems such as hierarchical graphs and proof nets of Linear Logic, or to specify the operational semantics of graph-based languages.     

Linear-Time Self-Interpretation of the Pure Lambda Calculus

We show that linear-time self-interpretation of the pure untyped lambda calculus is possible, in the sense that interpretation has a constant overhead compared to direct execution under various execution models. The present paper shows this result for reduction to weak head normal form under call-by-name, call-by-value and call-by-need.We use a self-interpreter based on previous work on self-interpretation and partial evaluation of the pure untyped lambda calculus.We use operational semantics to define each reduction strategy. For each of these we show a simulation lemma that states that each inference step in the evaluation of a term by the operational semantics is simulated by a sequence of steps in evaluation of the self-interpreter applied to the term (using the same operational semantics).By assigning costs to the inference rules in the operational semantics, we can compare the cost of normal evaluation and self-interpretation. Three different cost-measures are used: number of beta-reductions, cost of a substitution-based implementation (similar to graph reduction) and cost of an environment-based implementation.For call-by-need we use a non-deterministic semantics, which simplifies the proof considerably.     

Formal Proofs About Rewriting Using ACL2

We present an application of the ACL2 theorem prover to reason about rewrite systems theory. We describe the formalization and representation aspects of our work using the first-order, quantifier-free logic of ACL2 and we sketch some of the main points of the proof effort. First, we present a formalization of abstract reduction systems and then we show how this abstraction can be instantiated to establish results about term rewriting. The main theorems we mechanically proved are Newman's lemma (for abstract reductions) and Knuth–Bendix critical pair theorem (for term rewriting).     

Towards Lambda Calculus Order-Incompleteness

After Scott, mathematical models of the type-free lambda calculus are constructed by order theoretic methods and classified into semantics according to the nature of their representable functions. Selinger [48] asked if there is a lambda theory that is not induced by any non-trivially partially ordered model (order-incompleteness problem). In terms of Alexandroff topology (the strongest topology whose specialization order is the order of the considered model) the problem of order-incompleteness can be also characterized as follows: a lambda theory T is order-incomplete if, and only if, every partially ordered model of T is partitioned by the Alexandroff topology in an infinite number of connected components (= minimal upper and lower sets), each one containing exactly one element of the model. Towards an answer to the order-incompleteness problem, we give a topological proof of the following result: there exists a lambda theory whose partially ordered models are partitioned by the Alexandroff topology in an infinite number of connected components, each one containing at most one λ-term denotation. This result implies the incompleteness of every semantics of lambda calculus given in terms of partially ordered models whose Alexandroff topology has a finite number of connected components (e.g.the Alexandroff topology of the models of the continuous, stable and strongly stable semantics is connected).