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The well-known proof of termination of reduction in simply typed calculi is adapted to a monomorphically typed lambda-calculus with case and constructors and recursive data types. The proof differs at several places from the standard proof. Perhaps it is useful and can be extended also to more complex calculi.
Performance and storage requirements of topology-conserving maps for robot manipulator control
(1989)
A new programming paradigm for the control of a robot manipulator by learning the mapping between the Cartesian space and the joint space (inverse Kinematic) is discussed. It is based on a Neural Network model of optimal mapping between two high-dimensional spaces by Kohonen. This paper describes the approach and presents the optimal mapping, based on the principle of maximal information gain. It is shown that Kohonens mapping in the 2-dimensional case is optimal in this sense. Furthermore, the principal control error made by the learned mapping is evaluated for the example of the commonly used PUMA robot, the trade-off between storage resources and positional error is discussed and an optimal position encoding resolution is proposed.
The selection of features for classification, clustering and approximation is an important task in pattern recognition, data mining and soft computing. For real-valued features, this contribution shows how feature selection for a high number of features can be implemented using mutual in-formation. Especially, the common problem for mutual information computation of computing joint probabilities for many dimensions using only a few samples is treated by using the Rènyi mutual information of order two as computational base. For this, the Grassberger-Takens corre-lation integral is used which was developed for estimating probability densities in chaos theory. Additionally, an adaptive procedure for computing the hypercube size is introduced and for real world applications, the treatment of missing values is included. The computation procedure is accelerated by exploiting the ranking of the set of real feature values especially for the example of time series. As example, a small blackbox-glassbox example shows how the relevant features and their time lags are determined in the time series even if the input feature time series determine nonlinearly the output. A more realistic example from chemical industry shows that this enables a better ap-proximation of the input-output mapping than the best neural network approach developed for an international contest. By the computationally efficient implementation, mutual information becomes an attractive tool for feature selection even for a high number of real-valued features.
It is well known that artificial neural nets can be used as approximators of any continous functions to any desired degree. Nevertheless, for a given application and a given network architecture the non-trivial task rests to determine the necessary number of neurons and the necessary accuracy (number of bits) per weight for a satisfactory operation. In this paper the problem is treated by an information theoretic approach. The values for the weights and thresholds in the approximator network are determined analytically. Furthermore, the accuracy of the weights and the number of neurons are seen as general system parameters which determine the the maximal output information (i.e. the approximation error) by the absolute amount and the relative distribution of information contained in the network. A new principle of optimal information distribution is proposed and the conditions for the optimal system parameters are derived. For the simple, instructive example of a linear approximation of a non-linear, quadratic function, the principle of optimal information distribution gives the the optimal system parameters, i.e. the number of neurons and the different resolutions of the variables.
Towards correctness of program transformations through unification and critical pair computation
(2010)
Correctness of program transformations in extended lambda-calculi with a contextual semantics is usually based on reasoning about the operational semantics which is a rewrite semantics. A successful approach is the combination of a context lemma with the computation of overlaps between program transformations and the reduction rules, which results in so-called complete sets of diagrams. The method is similar to the computation of critical pairs for the completion of term rewriting systems. We explore cases where the computation of these overlaps can be done in a first order way by variants of critical pair computation that use unification algorithms. As a case study of an application we describe a finitary and decidable unification algorithm for the combination of the equational theory of left-commutativity modelling multi-sets, context variables and many-sorted unification. Sets of equations are restricted to be almost linear, i.e. every variable and context variable occurs at most once, where we allow one exception: variables of a sort without ground terms may occur several times. Every context variable must have an argument-sort in the free part of the signature. We also extend the unification algorithm by the treatment of binding-chains in let- and letrec-environments and by context-classes. This results in a unification algorithm that can be applied to all overlaps of normal-order reductions and transformations in an extended lambda calculus with letrec that we use as a case study.
This paper shows the equivalence of applicative similarity and contextual approximation, and hence also of bisimilarity and contextual equivalence, in the deterministic call-by-need lambda calculus with letrec. Bisimilarity simplifies equivalence proofs in the calculus and opens a way for more convenient correctness proofs for program transformations. Although this property may be a natural one to expect, to the best of our knowledge, this paper is the first one providing a proof. The proof technique is to transfer the contextual approximation into Abramsky's lazy lambda calculus by a fully abstract and surjective translation. This also shows that the natural embedding of Abramsky's lazy lambda calculus into the call-by-need lambda calculus with letrec is an isomorphism between the respective term-models.We show that the equivalence property proven in this paper transfers to a call-by-need letrec calculus developed by Ariola and Felleisen.
This note shows that in non-deterministic extended lambda calculi with letrec, the tool of applicative (bi)simulation is in general not usable for contextual equivalence, by giving a counterexample adapted from data flow analysis. It also shown that there is a flaw in a lemma and a theorem concerning finite simulation in a conference paper by the first two authors.
A logical framework consisting of a polymorphic call-by-value functional language and a first-order logic on the values is presented, which is a reconstruction of the logic of the verification system VeriFun. The reconstruction uses contextual semantics to define the logical value of equations. It equates undefinedness and non-termination, which is a standard semantical approach. The main results of this paper are: Meta-theorems about the globality of several classes of theorems in the logic, and proofs of global correctness of transformations and deduction rules. The deduction rules of VeriFun are globally correct if rules depending on termination are appropriately formulated. The reconstruction also gives hints on generalizations of the VeriFun framework: reasoning on nonterminating expressions and functions, mutual recursive functions and abstractions in the data values, and formulas with arbitrary quantifier prefix could be allowed.
Iterative arrays (IAs) are a, parallel computational model with a sequential processing of the input. They are one-dimensional arrays of interacting identical deterministic finite automata. In this note, realtime-lAs with sublinear space bounds are used to accept formal languages. The existence of a proper hierarchy of space complexity classes between logarithmic anel linear space bounds is proved. Furthermore, an optimal spacc lower bound for non-regular language recognition is shown. Key words: Iterative arrays, cellular automata, space bounded computations, decidability questions, formal languages, theory of computation
It is shown that between one-turn pushdown automata (1-turn PDAs) and deterministic finite automata (DFAs) there will be savings concerning the size of description not bounded by any recursive function, so-called non-recursive tradeoffs. Considering the number of turns of the stack height as a consumable resource of PDAs, we can show the existence of non-recursive trade-offs between PDAs performing k+ 1 turns and k turns for k >= 1. Furthermore, non-recursive trade-offs are shown between arbitrary PDAs and PDAs which perform only a finite number of turns. Finally, several decidability questions are shown to be undecidable and not semidecidable.