Rewriting with extensional polymorphic λ-calculus

Term rewriting has proven to be an important technique in theorem proving. In this paper, we illustrate that rewrite systems and strategies for higher-order term rewriting, which includes the usual notion of first-order rewriting, can be naturally specified and implemented in a higher-order logic programming language. We adopt a notion of higherorder rewrite system which uses the simply typed λ-calculus as the language for expressing rules, with a restriction on the occurrences of free variables on the left hand sides of rules so that matching of terms with rewrite templates is decidable. The logic programming language contains an implementation of the simply-typed lambda calculus including Βη-conversion and higher-order unification. In addition, universal quantification in queries and the bodies of clauses is permitted. For higher-order rewriting, we show how these operations implemented at the meta-level provide elegant mechanisms for the object-level operations of descending through terms and matching terms with rewrite templates. We discuss tactic style theorem proving in this environment and illustrate how term rewriting strategies can be expressed as tactic-style search.

Though higher-order unification is in general undecidable, there are expressive and decidable variants. Several interesting special cases and variants stem from restricting algorithms to search only unifiers where the number of bound variables is restricted. The intention of this paper is to summarize results in this area and to shed some light on the connections between context unification, decidable variants of higher-order, second order unification and string unification. Since this paper is intended to appear in a volume in celebrating Jorg Siekmann’s 60th birthday, we will take the opportunity to give hints on how the motivating power of Jorg Siekmann has contributed to and put forward the research in unification.

This paper concerns the tracking of automotive radar targets by a radar using a high-resolution 2D spectral estimator in order to detect the radar targets. We show how to implement the tracking, discuss the measurement variances, and show how we can exploit the high resolution in order to make a better tracking filter. Existing automotive radar systems are affected by a limited field of view, poor angular accuracy, or low measurement update rate. By using a combination of a waveform solving, the target ambiguity problem within a short measurement time and a conformal array design, we eliminate these problems and by the use of the 2D APES estimator, we achieve a high angular resolution, which is used in order to improve the tracking. The simulation results show that we are able to track a laterally moving vehicle very accurately and that the new improved tracking filter achieves more robust and accurate tracking along with a short initialization time. These attributes can be very important for safety critical applications.

A mechanical proof of the Knuth---Bendix Critical Pair Theorem in the higher-order language of the theorem prover PVS is described. This well-known theorem states that a Term Rewriting System is locally confluent if and only if all its critical pairs are joinable. The formalization of this theorem follows Huet's well-known structure of proof in which the restriction on strong normalization or Noetherian was dropped and the result presented as a lemma. In order to formalize the Knuth---Bendix Critical Pair Theorem we rely on previously developed PVS theories for abstract reduction systems, named ars, and term rewriting systems, named trs, which were built upon the PVS libraries for finite sequences and sets. On the one hand, the theory trs is composed of subtheories for dealing with the structure of terms, for replacements of subterms and substitutions and jointly with the theory ars it allows for adequate specifications of elaborate notions of term rewriting systems such as the one of critical pairs. On the other hand, ars specifies basic definitions and notions of abstract reduction systems such as reduction, termination, normal forms, and confluence as well as non basic concepts such as strong normalization.

Given a term rewriting system R, a term t is inductive reducible if every ground instance ts of it is reducible. A pair (t, s) of terms is inductive co-reducible if every ground instance (ts, ss) of it for which ts and ss are distinct, is reducible. Inductive (co)-reducibility has been proved to be the fundamental tool for mechanizing inductive proofs, together with Knuth-Bendix completion (Jouannaud and Kounalis [JK, 86 and 89]).In [JK, 86 and 89] an algorithm for testing inductive reducibility is also presented which is tractable in pratical cases, but restricted to left-linear term rewriting systems. The solution of the inductive-(co)-reducibility problem, for the general case, turned out to be surprisingly complicated. Decidability of inductive reducibility for arbitrary term rewriting systems has been first proved by Plaisted [PLA, 85] and next by Kapur, Narendran, and Zhang [KNZ, 87]. However Plaisted's and Kapur, Narendran, and Zhang's algorithms amount to intractable computation, even in very simple cases.We present here a new algorithm for the general case which outperforms Plaisted's and Kapur, Narendran and Zhang's algorithms and even our previous algorithm in case of a left-linear term rewriting system. We then show how to adapt it to check for inductive co-reducibility.KeywordsEquationsTerm rewriting systemsProof by consistencyTest SetsGround terms in normal form

On abstract normalisation beyond neededness

This paper generalizes the Huet and Lévy theory of normalization by neededness to an abstract setting. We define Stable Deterministic Residual Structures (SDRS) and Deterministic Family Structures (DFS) by axiomatizing some properties of the residual relation and the family relation on redexes in an Abstract Rewriting System. We present two proofs of the Relative Normalization Theorem, one for SDRSs for regular stable sets, and another for DFSs for all stable sets of desirable ‘normal forms’. We further prove the Relative Optimality Theorem for DFSs. We extend this result to deterministic Computation Structures which are deterministic Event Structures with an extra relation expressing self-essentiality.

Past, present, and forthcoming planetary rover missions to Mars and other planetary bodies are equipped with a large number of scientific cameras. The very large number of images resulting from this, combined with tight time constraints for navigation, measurements, and analyses, pose a major challenge for the mission teams in terms of scientific target evaluation. Shatter cones are the only macroscopic evidence for impact‐induced shock metamorphism and therefore impact craters on Earth. The typical features of shatter cones, such as striations and horsetail structures, are particularly suitable for machine learning methods. The necessary training images do not exist for such a case; therefore, we pursued the approach of producing them artificially. Using PRo3D, a viewer developed for the interactive exploration and geologic analysis of high‐resolution planetary surface reconstructions, we virtually placed shatter cones in 3‐D background scenes processed from true Mars rover imagery. We use PRo3D‐rendered images of such scenes as training data for machine learning architectures. Terrestrial analog studies in Ethiopia supported our lab work and were used to test the resulting neural network of this feasibility study. The result showed that our approach with shatter cones in artificial Mars rover scenes is suitable to train neural networks for automatic detection of shatter cones. In addition, we have identified several aspects that can be used to improve the training of the neural network and increase the recognition rate. For example, using background data with a higher resolution in order to have equal resolution of object (shatter cone) and Martian background and increase the number of objects that can be placed in the training data set. Also using better lighting reconstructions and a better radiometric adaption between object and Martian background would further improve the results.

Validation of a global 3D heliospheric model with observations for the May 12, 1997 CME event

The calculation of current densities in voxel-based high-resolution anatomy models allows insight in the current distribution inside the human body. The geometric complexity of the human body requires a very high resolution in order to properly resolve small organs. We present a local grid refinement scheme for low-frequency current calculations, allowing to finely resolve areas of particular interest. Using a Lagrange-multiplier approach, the whole model including the fine and coarse grid domains can be computed with only one solution of an algebraic system of equations. The presented approach is validated and tested on numerical examples.

Geochemical characteristics of Holocene loess-paleosol sequences in central Chinese Loess Plateau and their implications for East Asian monsoon evolution

A PVS Theory for Term Rewriting Systems

Polymorphic Rewriting Conserves Algebraic Confluence

Unique normal forms for lambda calculus with surjective pairing

A random context production has a permitting and forbidding context. A symbol can be rewritten using such a production if all the permitting context symbols and no forbidding context symbol appear in the sentential string. In this paper we limit the effect of forbidding context symbols to be within a certain distance from the symbol to be rewritten. Outside this distance the forbidding context symbols do not influence the rewriting of a symbol. This restriction strictly increases the generating power of the rewriting system. A further result of this paper is a “negative parallel” version of Penttonen's normal form.