Term rewriting is a Turing complete model of computation. When taught to students of computer science, key properties of computation as well as techniques to analyze programs on an abstract level are conveyed. This paper gives a swift introduction to term rewriting and presents several automatic tools to analyze term rewrite systems which were developed by the Computational Logic Group at the University of Innsbruck. These include the termination tool TTT2, the confluence prover CSI, the completion tools mkbTT and KBCV, the complexity tool TcT, the strategy tool AutoStrat, as well as FORT, an implementation of the decision procedure for the first-order theory for a decidable class of rewrite systems. Besides its applications in research, this software pool has also proved invaluable for teaching, e.g., in multiple editions of the International Summer School on Rewriting.
This volume contains the proceedings of the First International Workshop on Rewriting Techniques for Real-Time Systems (RTRTS 2010), held in Longyearbyen, Spitsbergen, on April 6-9, 2010. The aim of the workshop is to bring together researchers with an interest in the use of rewriting-based techniques (including rewriting logic) and tools for the modeling, analysis, and/or implementation of real-time and hybrid systems, and to give them the opportunity to present their recent works, discuss future research directions, and exchange ideas. The topics of the workshop comprise, but are not limited…
Rewriting is a common approach to logic optimization based on local transformations. Most commercially available logic synthesis tools include a rewriting engine that may be used multiple times on the same netlist during optimization. This paper presents an And-Inverter graph based rewriting algorithm using 5-input cuts. The best circuits are pre-computed for a subset of NPN classes of 5-variable functions. Cut enumeration and Boolean matching are used to identify replacement candidates. The presented approach is expected to complement existing rewriting approaches which are usually based on 4…
Many automatic theorem-provers rely on rewriting. Using theorems as rewrite rules helps to simplify the subgoals that arise during a proof. LCF is an interactive theorem-prover intended for reasoning about computation. Its implementation of rewriting is presented in detail. LCF provides a family of rewriting functions, and operators to combine them. A succession of functions is described, from pattern matching primitives to the rewriting tool that performs most inferences in LCF proofs. The design is highly modular. Each function performs a basic, specific task, such as recognizing a certain f…
The notion of normal forms is ubiquitous in various equivalent transformations. Confluence (CR), one of the central properties of term rewriting systems (TRSs), concerns uniqueness of normal forms. Yet another such property, which is weaker than confluence, is the property of unique normal forms w.r.t. conversion (UNC). Famous examples having UNC but not CR include the TRSs consisting of S,K,I-rules for the combinatory logic supplemented with various pairing rules (de Vrijer, 1999). Recently, automated confluence proof of TRSs has caught attentions leading to investigations of automatable meth…
We present strategic portgraph rewriting as a basis for the implementation of visual modelling and analysis tools. The goal is to facilitate the specification, analysis and simulation of complex systems, using port graphs. A system is represented by an initial graph and a collection of graph rewriting rules, together with a user-defined strategy to control the application of rules. The strategy language includes constructs to deal with graph traversal and management of rewriting positions in the graph. We give a small-step operational semantics for the language, and describe its implementation…
Automatic and efficient verification of multiplier designs, especially through a provably correct method, is a difficult problem. We show how to utilize a theorem prover, ACL2, to implement an efficient rewriting algorithm for multiplier design verification. Through a basic understanding of the features and data structures of ACL2, we created a verified program that can automatically verify various multiplier designs much faster than the other state-of-the-art tools. Additionally, users of our system have the flexibility to change the specification for the target design to verify variations of…
The theory of finite and infinitary term rewriting is extensively developed for orthogonal rewrite systems, but to a lesser degree for weakly orthogonal rewrite systems. In this note we present some contributions to the latter case of weak orthogonality, where critial pairs are admitted provided they are trivial. We start with a refinement of the by now classical Compression Lemma, as a tool for establishing infinitary confluence, and hence the property of unique infinitary normal forms, for the case of weakly orthogonal TRSs that do not contain collapsing rewrite rules. That this restriction …
We revisit parallel-innermost term rewriting as a model of parallel computation on inductive data structures and provide a corresponding notion of runtime complexity parametric in the size of the start term. We propose automatic techniques to derive both upper and lower bounds on parallel complexity of rewriting that enable a direct reuse of existing techniques for sequential complexity. The applicability and the precision of the method are demonstrated by the relatively light effort in extending the program analysis tool AProVE and by experiments on numerous benchmarks from the literature.
In this paper, we reconsider the unfolding-based technique that we have introduced previously for detecting loops in standard term rewriting. We improve it by guiding the unfolding process, using distinguished positions in the rewrite rules. This results in a depth-first computation of the unfoldings, whereas the original technique was breadth-first. We have implemented this new approach in our tool NTI and compared it to the previous one on a bunch of rewrite systems. The results we get are promising (better times, more successful proofs).
Adding rewriting to a proof assistant based on the Curry-Howard isomorphism, such as Coq, may greatly improve usability of the tool. Unfortunately adding an arbitrary set of rewrite rules may render the underlying formal system undecidable and inconsistent. While ways to ensure termination and confluence, and hence decidability of type-checking, have already been studied to some extent, logical consistency has got little attention so far. In this paper we show that consistency is a consequence of canonicity, which in turn follows from the assumption that all functions defined by rewrite rules …
On Complexity Bounds and Confluence of Parallel Term Rewriting
LIP (UMR CNRS/ENS Lyon/UCB Lyon1/Inria)（LIP（法国国家科学研究中心/里昂高等师范学院/里昂第一大学/法国国家信息与自动化研究所联合研究单位））；Birkbeck, University of London（伦敦大学伯贝克学院）；University Grenoble Alpes（格勒诺布尔阿尔卑斯大学）
Termination of programs, i.e., the absence of infinite computations, ensures the existence of normal forms for all initial expressions, thus providing an essential ingredient for the definition of a normalization semantics for functional programs. In lazy functional languages, though, infinite data structures are often delivered as the outcome of computations. For instance, the list of all prime numbers can be returned as a neverending stream of numerical expressions or data structures. If such streams are allowed, requiring termination is hopeless. In this setting, the notion of productivity …
Rewriting logic is both a flexible semantic framework within which widely different concurrent systems can be naturally specified and a logical framework in which widely different logics can be specified. Maude programs are exactly rewrite theories. Maude has also a formal environment of verification tools. Symbolic computation is a powerful technique for reasoning about the correctness of concurrent systems and for increasing the power of formal tools. We present several new symbolic features of Maude that enhance formal reasoning about Maude programs and the effectiveness of formal tools. Th…
