Reinhard Bündgen

Reduce the Redex -> ReDuX

The ReDuX-system is a work-bench for programming and experimenting with term rewriting systems. It is focused towards the implementation of completion procedures with special emphasis on inductive completion. From the programmers point of view ReDuX provides a large library of data types and algorithms (over 450) which allows for high level programming. The experimentalist also finds a collection of ready-to-run programs (see Table 1, and [WB91]). ReDuX has been developed as an extension of the TC- and IC-sytems [Kuc82a, Bun87] and has been used as a research tool over the last years. For the last two years it has also been employed as a tutorial system for courses on term rewriting systems at the University of Tubingen.

A fine-grained parallel completion procedure

We present a parallel Knuth-Bendix completion algorithm where the inner loop, deriving the consequences of adding a new rule to the system, is multi-threaded. The selection of the best new rule in the outer loop, and hence the completion strategy, is exactly the same as for the sequential algorithm. Our implementation, which is within the PARSAC-2 parallel symbolic computation system, exhibits good parallel speedups on a standard multi-processor workstation.

Buchberger's algorithm: the term rewriter's point of view

Abstract We analyse the relations between completion procedures for polynomials and terms and thereby show how Buchbergers algorithm for multivariate polynomials over finite fields and over the rationals can be simulated using term completion modulo AC. To specify the rational numbers an infinite term rewriting system is needed. However, for the simulation of each particular ideal completion a finite approximation of the infinite rule set is sufficient. This approximation can be constructed during the completion. The division operation needed in Buchbergers algorithm reduces to a narrowing procedure which becomes part of the critical pair computation process.

Computing Ground Reducability and Inductively Complete Positions

We provide the extended ground-reducibility test which is essential for induction with term-rewriting systems based on [Kuc89]: Given a term, determine at which sets of positions it is ground-reducible by which subsets of rules. The core of our method is a new parallel cover algorithm based on recursive decomposition. From this we obtain a separation algorithm which determines constructors and defined function symbols in a term-algebra presented by a rewrite system. We then reduce our main problem of extended ground-reducibility to separation and cover. Furthermore, using the knowledge of algebra separation, we refine the bounds of [JK86] for the size of ground reduction test-sets. Both our cover algorithm and our extended ground-reducibility test are engineered to be adaptive to actual problem structure, i.e., to allow for lower than the worst case bounds for test-sets on well conditioned problems, including well conditioned subproblems of difficult cases.

Strategy compliant multi-threaded term completion

Abstract We report on the design, implementation, and performance,of the parallel term-rewriting system PaReDuX.We discuss the parallelization of three term completion procedures: Knuth-Bendix completion, completion modulo AC, and unfailing completion. Our parallelization is strategy-compliant, i.e., the parallel code performs exactly the same work as the sequential code, but the work load is shared by many processors. PaReDuX is designed for shared memory parallel architectures, such as multi-processor workstations, where it shows good performance on a variety of examples.

Simulating Buchberger's algorithm by Knuth-Bendix completion

We present a canonical term rewriting system whose initial model is isomorphic to GF(q)[x1,...,x n ]. Using this set of rewrite rules and additional ground equations specifying an ideal we can simulate Buchbergers algorithm for polynomials over finite fields using Knuth-Bendix term completion modulo AC. In order to simplify our proofs we exhibit a critical pair criterion which transforms critical pairs into simpler ones.

Completion of integral polynomials by AC-term completion

We present a canonical term rewriting system RX whose ground normal forms can directly be mapped to integral polynomials in distributive normal form. Completing RX and a set of ground equations simulates the Grobner base computation for the ideal presented by the ground equations. With our approach, we can clearly show the correspondences of the key features of algebraic completion procedures for integral polynomial ideals ([Lau76], [Buc84]) and their simulation in a term rewriting environment.

A Master-Slave Approach to Parallel Term Rewriting on a Hierarchical Multiprocessor

We report on a parallel implementation of an unfailing term completion procedure on a network of multiprocessor workstations. Our parallelization concept integrates distributed search parallelism on the network, based on a master-slave approach, with the parallel execution of each search on a multiprocessor. Both levels of parallelism are realized by a uniform fork /join paradigm using multi-threading. In many of our examples we are able to combine the benefits of distributed and shared-memory approaches for superior overall speed-ups.

Combining Computer Algebra and Rule Based Reasoning

We present extended term rewriting systems as a means to describe a simplification relation for an equational specification with a built-in domain of external objects. Even if the extended term rewriting system is canonical, the combined relation including built-in computations of ‘ground terms’ needs neither be terminating nor confluent. We investigate restrictions on the extended term rewriting systems and the built-in domains under which these properties hold. A very important property of extended term rewriting systems is decomposition freedom. Among others decomposition free extended term rewriting systems allow for efficient simplifications. Some interesting algebraic applications of canonical simplification relations are presented.

Verification of the Sparrow processor

We present a new gate-level hardware verification method based on term rewriting systems. As an application, we formally verify the Sparrow microprocessor with the term rewriting theorem prover ReDuX. Our designs are given as net-lists in BLIF format. We mechanically compile the net-lists into the formal axiomatization of Sparrow as a term rewriting system. ReDuX can then emulate Sparrow symbolically. We manually produce verification conditions from the user-level processor specification and verify each one of them. Our axiomatization corresponds directly to net-lists, and thus is intuitive and close to the hardware. Except for simple equations no higher concept of logic is involved.