Extending Context-Sensitivity in Term Rewriting

We propose a generalized version of context-sensitivity in term rewriting based on the notion of "forbidden patterns". The basic idea is that a rewrite step should be forbidden if the redex to be contracted has a certain shape and appears in a certain context. This shape and context is expressed through forbidden patterns. In particular we analyze the relationships among this novel approach and the commonly used notion of context-sensitivity in term rewriting, as well as the feasibility of rewriting with forbidden patterns from a computational point of view. The latter feasibility is characterized by demanding that restricting a rewrite relation yields an improved termination behaviour while still being powerful enough to compute meaningful results. Sufficient criteria for both kinds of properties in certain classes of rewrite systems with forbidden patterns are presented

An Algebra of Hierarchical Graphs

We define an algebraic theory of hierarchical graphs, whose axioms characterise graph isomorphism: two terms are equated exactly when they represent the same graph. Our algebra can be understood as a high-level language for describing graphs with a node-sharing, embedding structure, and it is then well suited for defining graphical representations of software models where nesting and linking are key aspects

Applications and extensions of context-sensitive rewriting

[EN] Context-sensitive rewriting is a restriction of term rewriting which is obtained by imposing replacement restrictions on the arguments of function symbols. It has proven useful to analyze computational properties of programs written in sophisticated rewriting-based programming languages such asCafeOBJ, Haskell, Maude, OBJ*, etc. Also, a number of extensions(e.g., to conditional rewritingor constrained equational systems) and generalizations(e.g., controlled rewritingor forbidden patterns) of context-sensitive rewriting have been proposed. In this paper, we provide an overview of these applications and related issues. (C) 2021 Elsevier Inc. All rights reserved.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32 and PROMETEO/2019/098.Lucas Alba, S. (2021). Applications and extensions of context-sensitive rewriting. Journal of Logical and Algebraic Methods in Programming. 121:1-33. https://doi.org/10.1016/j.jlamp.2021.10068013312

Rewriting and Well-Definedness within a Proof System

Term rewriting has a significant presence in various areas, not least in automated theorem proving where it is used as a proof technique. Many theorem provers employ specialised proof tactics for rewriting. This results in an interleaving between deduction and computation (i.e., rewriting) steps. If the logic of reasoning supports partial functions, it is necessary that rewriting copes with potentially ill-defined terms. In this paper, we provide a basis for integrating rewriting with a deductive proof system that deals with well-definedness. The definitions and theorems presented in this paper are the theoretical foundations for an extensible rewriting-based prover that has been implemented for the set theoretical formalism Event-B.Comment: In Proceedings PAR 2010, arXiv:1012.455

Analytical learning and term-rewriting systems

Analytical learning is a set of machine learning techniques for revising the representation of a theory based on a small set of examples of that theory. When the representation of the theory is correct and complete but perhaps inefficient, an important objective of such analysis is to improve the computational efficiency of the representation. Several algorithms with this purpose have been suggested, most of which are closely tied to a first order logical language and are variants of goal regression, such as the familiar explanation based generalization (EBG) procedure. But because predicate calculus is a poor representation for some domains, these learning algorithms are extended to apply to other computational models. It is shown that the goal regression technique applies to a large family of programming languages, all based on a kind of term rewriting system. Included in this family are three language families of importance to artificial intelligence: logic programming, such as Prolog; lambda calculus, such as LISP; and combinatorial based languages, such as FP. A new analytical learning algorithm, AL-2, is exhibited that learns from success but is otherwise quite different from EBG. These results suggest that term rewriting systems are a good framework for analytical learning research in general, and that further research should be directed toward developing new techniques

VerdictDB: Universalizing Approximate Query Processing

Despite 25 years of research in academia, approximate query processing (AQP) has had little industrial adoption. One of the major causes of this slow adoption is the reluctance of traditional vendors to make radical changes to their legacy codebases, and the preoccupation of newer vendors (e.g., SQL-on-Hadoop products) with implementing standard features. Additionally, the few AQP engines that are available are each tied to a specific platform and require users to completely abandon their existing databases---an unrealistic expectation given the infancy of the AQP technology. Therefore, we argue that a universal solution is needed: a database-agnostic approximation engine that will widen the reach of this emerging technology across various platforms. Our proposal, called VerdictDB, uses a middleware architecture that requires no changes to the backend database, and thus, can work with all off-the-shelf engines. Operating at the driver-level, VerdictDB intercepts analytical queries issued to the database and rewrites them into another query that, if executed by any standard relational engine, will yield sufficient information for computing an approximate answer. VerdictDB uses the returned result set to compute an approximate answer and error estimates, which are then passed on to the user or application. However, lack of access to the query execution layer introduces significant challenges in terms of generality, correctness, and efficiency. This paper shows how VerdictDB overcomes these challenges and delivers up to 171$\times$ speedup (18.45$\times$ on average) for a variety of existing engines, such as Impala, Spark SQL, and Amazon Redshift, while incurring less than 2.6% relative error. VerdictDB is open-sourced under Apache License.Comment: Extended technical report of the paper that appeared in Proceedings of the 2018 International Conference on Management of Data, pp. 1461-1476. ACM, 201

Modularity of Convergence and Strong Convergence in Infinitary Rewriting

Properties of Term Rewriting Systems are called modular iff they are preserved under (and reflected by) disjoint union, i.e. when combining two Term Rewriting Systems with disjoint signatures. Convergence is the property of Infinitary Term Rewriting Systems that all reduction sequences converge to a limit. Strong Convergence requires in addition that redex positions in a reduction sequence move arbitrarily deep. In this paper it is shown that both Convergence and Strong Convergence are modular properties of non-collapsing Infinitary Term Rewriting Systems, provided (for convergence) that the term metrics are granular. This generalises known modularity results beyond metric \infty