Standard Rewriting Research Articles

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.

Rewriting modulo SMT and open system analysis

This paper proposes rewriting modulo SMT, a new technique that combines the power of SMT solving, rewriting modulo theories, and model checking. Rewriting modulo SMT is ideally suited to model and analyze reachability properties of infinite-state open systems, i.e., systems that interact with a nondeterministic environment. Such systems exhibit both internal nondeterminism, which is proper to the system, and external nondeterminism, which is due to the environment. In a reflective formalism, such as rewriting logic, rewriting modulo SMT can be reduced to standard rewriting. Hence, rewriting modulo SMT naturally extends rewriting-based reachability analysis techniques, which are available for closed systems, to open systems. Furthermore, a single state expression with symbolic constraints can now denote an infinite set of concrete states. The proposed technique is illustrated with the formal analysis of: (i) a real-time system that is beyond the scope of timed-automata methods and (ii) automatic detection of reachability violations in a synchronous language developed to support autonomous spacecraft operations.

CYCLICALLY PRESENTED GROUPS AND RESULTANTS

We consider the family of irreducible cyclically presented groups on n generators whose generating word (in the standard rewrite) has length at most 15. Using the software packages KBMAG, quotpic and MAGMA, together with group and number theoretic methods, we show that if 6 ≤ n ≤ 100 then the group is non-trivial. In an appendix we list the 47 cases within 2 ≤ n ≤ 100 for which we know the group to be trivial, and 27 further cases for which triviality has yet to be determined.

A new decidability technique for ground term rewriting systems with applications

Programming language interpreters, proving equations (e.g. x 3 = x implies the ring is Abelian), abstract data types, program transformation and optimization, and even computation itself (e.g., turing machine) can all be specified by a set of rules, called a rewrite system. Two fundamental properties of a rewrite system are the confluence or Church--Rosser property and the unique normalization property. In this article, we develop a standard form for ground rewrite systems and the concept of standard rewriting. These concepts are then used to: prove a pumping lemma for them, and to derive a new and direct decidability technique for decision problems of ground rewrite systems. To illustrate the usefulness of these concepts, we apply them to prove: (i) polynomial size bounds for witnesses to violations of unique normalization and confluence for ground rewrite systems containing unary symbols and constants, and (ii) polynomial height bounds for witnesses to violations of unique normalization and confluence for arbitrary ground systems. Apart from the fact that our technique is direct in contrast to previous decidability results for both problems, which were indirectly obtained using tree automata techniques, this approach also yields tighter bounds for rewrite systems with unary symbols than the ones that can be derived with the indirect approach. Finally, as part of our results, we give a polynomial-time algorithm for checking whether a rewrite system has the unique normalization property for all subterms in the rules of the system.

Modular and Incremental Automated Termination Proofs

We propose a modular approach of term rewriting systems, making the best of their hierarchical structure. We definerewriting modules and then provide a new method to prove termination incrementally. We obtain new and powerful termination criteria for standard rewriting, thanks to the combination of dependency pairs and Open image in new window . Taking benefit of the generality of the module approach while restraining the notion of termination itself (thus relaxing constraints over hierarchies components), we can easily express previous results and methods the premises of which either include restrictions over unions or make a particular reduction strategy compulsory. We describe our implementation of the modular approach. Proofs are fully automated and performed incrementally. Since convenient orderings are simpler, we observe a dramatic speedup in the finding of the proof.

Some interesting connections between the slow growing hierarchy and the Ackermann function

AbstractIt is shown that the so called slow growing hierarchy depends non trivially on the choice of its underlying structure of ordinals. To this end we investigate the growth rate behaviour of the slow growing hierarchy along natural subsets of notations for Γ0. Let T be the set-theoretic ordinal notation system for Γ0 and Ttree the tree ordinal representation for Γ0. It is shown in this paper that (Gα)α∈T matches up with the class of functions which are elementary recursive in the Ackermann function as does (by folklore). By thinning out terms in which the addition function symbol occurs we single out subsystems T* ⊆ T and Ttree* ⊆ Ttree (both of order type not exceeding ε0) and prove that still matches up with but now consists of elementary recursive functions only. We discuss the relationship between these results and the Γ0-based termination proof for the standard rewrite system for the Ackermann function.

Behavioral and Coinductive Rewriting (invited talk)

Behavioral rewriting differs from standard rewriting in taking account of the weaker inference rules of behavioral logic, but it shares much with standard rewriting, including notions like termination and confluence. We describe an efficient implementation of behavioral rewriting that uses standard rewriting. Circular coinductive rewriting combines behavioral rewriting with circular coinduction, giving a surprisingly powerful proof method for behavioral properties; it is implemented in the BOBJ system, which is used in our examples. These include several lazy functional stream program equivalences and a behavioral refinement.

A general theory of sharing graphs

Sharing graphs are the structures introduced by Lamping for the implementation of optimal reductions of lambda-calculus. Gonthier's reformulation of Lamping's technique inside Geometry of Interaction and Asperti and Laneve's work on Interaction Systems have shown that sharing graphs implement a wide class of calculi. We give a semantical characterization of sharing graphs independent of the calculus to be implemented. By means of an algebraic interpretation of sharing graphs, we define a subclass of them, the so-called proper sharing graphs, on which the usual notion of graph unfolding gives a lower semi-lattice. The least-shared-instance of a proper sharing graph is its maximal proper unfolding, that is, the unique proper unshared graph that the unfolding partial order associates to it. Exploiting a simulation property between the reductions of a proper sharing graph and the reductions of its least-shared-instance, we prove that the read-back of a proper sharing graph can be computed via an unfolding or read-back reduction. Proper sharing graphs implement in a distributed and local way any graph calculus with a global reduction in the style of the beta-rule of lambda-calculus. In fact, correctness of the sharing implementation requires the so-called box nesting property only, or equivalently, it is proved under the only assumption that two redexes never partially overlap. Thus, sharing graphs constitute an abstract machine, say the sharing graph machine, that seems to be the most natural low-level computational model for functional languages. Moreover, Lévy's optimal reductions correspond to lazy reductions of that sharing machine. We stress on the proof strategy followed in the paper: it rests on an amazing interplay between standard rewriting system properties (strong normalization, confluence, and unique normal form) and algebraic properties definable via the techniques of Geometry of Interaction.