Proof Reflection in Coq

We formalize natural deduction for first-order logic in the proof assistant Coq, using de Bruijn indices for variable binding. The main judgment we model is of the form d[:], stating that d is a proof term of formula under hypotheses it can be viewed as a typing relation by the Curry–Howard isomorphism. This relation is proved sound with respect to Coq's native logic and is amenable to the manipulation of formulas and of derivations. As an illustration, we define a reduction relation on proof terms with permutative conversions and prove the property of subject reduction.     

On Skolemization in constrained logics

Quantification in first-order logic always involves all elements of the universe. However, it is often more natural to just quantify over those elements of the universe which satisfy a certain condition. Constrained logics therefore provide restricted quantifiers _X:R F and _X:R F whereX is a set of variables, and which can be read as F holds for all elements satisfying the restrictionR and F holds if there exists an element which satisfiesR. In order to test satisfiability of a set of such formulas by means of an appropriately extended resolution principle, one needs a procedure which transforms them into a set of clauses with constraints. Such a procedure essentially differs from the classical transformation of first-oder formulas into a set of clauses, in particular since quantification over the empty set may occur and since the needed Skolemization procedure has to take the restrictions of restricted quantifiers into account. In the first part of this article we present a procedure that transforms formulas with restricted quantifiers into a set of clauses with constraints while preserving satisfiability. The thus obtained clauses are of the formC R whereC is an ordinary clause andR is a restriction, and can be read as C holds ifR holds. These clauses can now be tested on unsatisfiability via the existingconstrained resolution principle. In the second part we generalize the constrained resolution principle in such a way that it allows for further optimization, and we introduce a combination of unification and constraint solving that can be employed to instantiate this kind of optimization.     

States and the free orthogonality monoid

Let (X, #) be an orthogonality space such that the lattice C(X, #) of closed subsets of (X, #) is orthomodular and let (, ) denote the free orthogonality monoid over (X, #). Let C₀(, ) be the subset of C(, ), consisting of all closures of bounded orthogonal sets. We show that C₀(, ) is a suborthomodular lattice of C(, ) and we provide a necessary and sufficient condition for C₀(, ) to carry a full set of dispersion free states.The work of the second author on this paper was supported by National Science Foundation Grant GP-9005.     

The existence of solutions,stability, and linearization of volterra systems

LetB be a Banach space ofR _n valued continuous functions on [0, ) withfB. Consider the nonlinear Volterra integral equation (^*)x(t)+ _o ^t K(t,s,x(s))ds. We use the implicit function theorem to give sufficient conditions onB andK (t,s,x) for the existence of a unique solutionxB to (^*) for eachf B with f ^B sufficiently small. Moreover, there is a constantM>0 independent off with Mf^B.Part of this work was done while the author was visiting at Wright State University.     

Representation of Algebraic Functions As Power and Fractional Power Series of Special Type

A Maple procedure is described by means of which an algebraic function given by an equation f(x y) = 0 can be expanded into a fractional power series (Puiseux series)

The dynamic clusters method in nonhierarchical clustering

Given a finite setE R ⁿ, the problem is to find clusters (or subsets of similar points inE) and at the same time to find the most typical elements of this set. An original mathematical formulation is given to the problem. The proposed algorithm operates on groups of points, called samplings (samplings may be called multiple centers or cores); these samplings adapt and evolve into interesting clusters. Compared with other clustering algorithms, this algorithm requires less machine time and storage. We provide some propositions about nonprobabilistic convergence and a sufficient condition which ensures the decrease of the criterion. Some computational experiments are presented.     

Tracking primary hyperplane competitors during genetic search

Holland's fundamental theorem of genetic algorithms (the schema theorem) provides a lower bound on the sampling rate of a single hyperplane during genetic search. However, the theorem tracks the change in representation for a single hyperplaneas if its representation is independent of other hyperplanes. Hyperplane samples are clearly interdependent and interactions in the hyperplane samples means that Holland's notion of implicit parallelism does not universally hold when conflicting hyperplanes interact. A set of equations are defined which allows one to model the interaction of strings, or of schemata representing hyperplanes at order-N and hyperplanes less than order-N. These equations do not account for the effects of higher order hyperplanes or co-lateral competitions. Nevertheless, these equations can serve to better describe the interaction of primary hyperplane competitors.     

A New Class of Depth-Size Optimal Parallel Prefix Circuits

Given n values x₁, x₂, ... ,x_n and an associative binary operation o, the prefix problem is to compute x₁ox₂o··· ox_i, 1in. Many combinational circuits for solving the prefix problem, called prefix circuits, have been designed. It has been proved that the size s(D(n)) and the depth d(D(n)) of an n-input prefix circuit D(n) satisfy the inequality d(D(n))+s(D(n))2n–2; thus, a prefix circuit is depth-size optimal if d(D(n))+s(D(n))=2n–2. In this paper, we construct a new depth-size optimal prefix circuit SL(n). In addition, we can build depth-size optimal prefix circuits whose depth can be any integer between d(SL(n)) and n–1. SL(n) has the same maximum fan-out lgn+1 as Snir's SN(n), but the depth of SL(n) is smaller; thus, SL(n) is faster. Compared with another optimal prefix circuit LYD(n), d(LYD(n))+2d(SL(n))d(LYD(n)). However, LYD(n) may have a fan-out of at most 2 lgn–2, and the fan-out of LYD(n) is greater than that of SL(n) for almost all n12. Because an operation node with greater fan-out occupies more chip area and is slower in VLSI implementation, in most cases, SL(n) needs less area and may be faster than LYD(n). Moreover, it is much easier to design SL(n) than LYD(n).     

MAAN: A Multi-Attribute Addressable Network for Grid Information Services

Recent structured Peer-to-Peer (P2P) systems such as Distributed Hash Tables (DHTs) offer scalable key-based lookup for distributed resources. However, they cannot be simply applied to grid information services because grid resources need to be registered and searched using multiple attributes. This paper proposes a Multi-Attribute Addressable Network (MAAN) that extends Chord to support multi-attribute and range queries. MAAN addresses range queries by mapping attribute values to the Chord identifier space via uniform locality preserving hashing. It uses an iterative or single attribute dominated query routing algorithm to resolve multi-attribute based queries. Each node in MAAN only has O(logN) neighbors for N nodes. The number of routing hops to resolve a multi-attribute range query is O(logN+N×s_min), where s_min is the minimum range selectivity on all attributes. When s_min=, it is logarithmic to the number of nodes, which is scalable to a large number of nodes and attributes. We also measured the performance of our MAAN implementation and the experimental results are consistent with our theoretical analysis.     

Generating Boolean \mu-expressions

In this paper, we consider the class of Boolean -functions, which are the Boolean functions definable by -expressions (Boolean expressions in which no variable occurs more than once). We present an algorithm which transforms a Boolean formulaE into an equivalent -expression-if possible-in time linear in E times , where E is the size ofE andn _m is the number of variables that occur more than once inE. As an application, we obtain a polynomial time algorithm for Mundici's problem of recognizing -functions fromk-formulas [17]. Furthermore, we show that recognizing Boolean -functions is co-NP-complete for functions essentially dependent on all variables and we give a bound close to co-NP for the general case.     

The role of “craft language” in learning “Waza”

The role of craft language in the process of teaching (learning) Waza (skill) will be discussed from the perspective of human intelligence.It may be said that the ultimate goal of learning Waza in any Japanese traditional performance is not the perfect reproduction of the teaching (learning) process of Waza. In fact, a special metaphorical language (craft language) is used, which has the effect of encouraging the learner to activate his creative imagination. It is through this activity that the he learns his own habitus (Kata).It is suggested that, in considering the difference of function between natural human intelligence and artificial intelligence, attention should be paid to the imaginative activity of the learner as being an essential factor for mastering Kata.This article is a modified English version of Chapter 5 of my bookWaza kara shiru (Learning from Skill), Tokyo University Press, 1987, pp. 93–105.     

Minimization of norms and logarithmic norms by diagonal similarities

It is a commonly occurring problem to find good norms · or logarithmic norms (·) for a given matrix in the sense that they should be close to respectively the spectral radius (A) and the spectral abscissa (A). Examples may be the certification thatA is convergent, i.e. (A)A<1 or stable, i.e. (A)(A)<0. Often the ordinary norms do not suffice and one would like to try simple modifications of them such as using an ordinary norm for a diagonally transformed matrix. This paper treats this problem for some of the ordinary norms.

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Minimisierung von Normen und Logarithmischen Normen durch Diagonale Transformationen

Zusammenfassung Ein oft vorkommendes praktisches Problem ist die Konstruktion von guten Normen · und logarithmischen Normen (·) für eine gegebene MatrixA. Mit gut wird dann verstanden, daß A den Spektralradius (A)=max |₁| und (A) die Spektralabszisse (A)=max Re _i gut approximieren. Beispiele findet man für konvergente Matrizen wo (A)A<1 gewünscht ist, und für stabile Matrizen wo (A)(A)<0 zu zeigen ist. Wir untersuchen hier, wie weit man mit Diagonaltransformationen und dengewöhnlichsten Normen kommen kann.

     

Cut point languages of markov DTOL systems

Letp satisfy 0 p < 1, then by (p) we denote the family of Markov DTOL languages with cut pointp. In this paper we present a complete classification of the collection of such families (p), 0 p < 1, showing that forms an infinite nondense hierarchy with (0) being its only accumulation point from below. Furthermore it is proved that each language in (p) can be expressed as a finite union of DDTOL languages.Work supported partially by the Natural Sciences and Engineering Research Council of Canada grants Nos. A-3590 and A-7700, and partially by Deutsche Forschungsgemeinschaft.     

Types with intersection: An introduction

This paper is an informal introduction to the theory of types which use a connective for the intersection of two types and a constant for a universal type, besides the usual connective for function-types. This theory was first devised in about 1977 by Coppo, Dezani and Sallé in the context of-calculus and its main development has been by Coppo and Dezani and their collaborators in Turin. With suitable axioms and rules to assign types to-calculus terms, they obtained a system in which (i) the set of types given to a term does not change under-conversion, (ii) some interesting sets of terms, for example the solvable terms and the terms with normal form, can be characterised exactly by the types of their members, and (iii) the type-apparatus is not so complex as polymorphic systems with quantifier-containing types and therefore probably not so expensive to implement mechanically as these systems.There are in fact several variant systems with different detailed properties. This paper defines and motivates the simplest one from which the others are derived, and describes its most basic properties. No proofs are given but the motivation is shown by examples. A comprehensive bibliography is included.     

Jordan surfaces in simply connected digital spaces

Certain tasks in multidimensional digital image analysis, in particular surface detection and volume estimation, lead us to the study of surfaces in the digital environment. It is desirable that these surfaces should have a Jordan property analogous to that of simple closed curves in two dimensions-namely, that they should partition the underlying space into an inside and an outside which are disconnected from each other. A version of this property, called near-Jordanness, has been previously defined and has been shown to be useful within a general theory of surfaces in digital spaces. The definition of a near-Jordan surface is global: it demands that all paths from the interior to the exterior cross the surface. This makes it difficult in many practical applications to check whether surfaces are near-Jordan. The work reported in this paper is motivated by the desire for a local condition, such that if a surface satisfies this condition at each of its elements, then it is guaranteed to be near-Jordan. In the search for such a condition, we were led to a concept of simple connectedness of a digital space, which resembles simple connectedness in ordinary topological spaces. We were then able to formulate the desired local condition for simply connected digital spaces. Many digital spaces, in particular those based on the commonly-studied tessellations of n-dimensional Euclidean space, are shown to be simply connected and thus our theory yields general sufficient conditions for boundaries in binary pictures to be near-Jordan.     

Eine bedingte Form der Integral-Ungleichung von Gronwall,leichte Verallgemeinerung eines Stabilitätssatzes von Perron und überlauffreie Skalierung von Analogrechenschaltungen

Zusammenfassung Es geht in dieser Arbeit in der Hauptsache darum, ein vorgelegtes Differentialgleichungssystem so zu skalieren, daß in der zugehörigen Analogrechnerschaltung die Spannungen an den Ausgängen der Integratoren die durch die Referenzspannung einerseits und durch das Auflösevermögen andererseits gesetzten Schranken nicht über- bzw. unterschreiten. Es werden Abschätzungssätze hergeleitet, die diese Frage im Apriori-Sinn, also ohne die Lösung des Differentialgleichungssystems zu kennen, zu lösen gestatten. Zur Abschätzung werden zunächst Normen, dannKamke-Normen verwendet. Der im Titel erwähnte Satz vonPerron ergibt sich durch spezielle Normengebung und Verzicht auf Abschätzung nach unten. Erschwert werden die Betrachtungen durch die relative Schwäche der Forderung, daß die rechte Seite des Systemsdx/dt=f(x,t) der Bedingung aus xa folgt f(x,t)v(t)x genüge (...:=Norm,a positiv reell). Dadurch scheint es bei Abschätzungen mitKamke-Normen nicht mehr möglich, von den in der Literatur über Existenzbeweise und Abschätzungssätze üblichen Methoden Gebrauch zu machen. Zur Lösung dieser Frage wird eine bedingte Form des bekannten Satzes vonGronwall (auch Satz vonBellman genannt) entwickelt.

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A conditional version of the integral inequality of gronwall, a slight generalization of a stability theorem of perron, and overflow-free scaling of analogue computer set-ups

Summary The main subject of this paper is the scaling of a given set of differential equations in such a way that the output voltages of the integrators of the associated analogue computer set-up do not exceed certain upper and lower bounds imposed by the reference voltage and the limited power of resolution of the elements of the analogue computer. The paper gives a priori bounds on the solution of the differential set. Some of these bounds work with norms, others withKamke-norms.Perron's stability theorem mentioned in the title of this paper results by inserting special norms and neglecting lower bounds. A difficulty arises by the relative weakness of the condition xa implies f(x,t)v(t)x on the right hand side of the setdx/dt=f(x,t), where ... is any norm anda is a positive real constant. As a consequence of this, it seems no longer possible to use the usual techniques known from the literature on existence theorems and bounds for the solution of differential equations. To cope with this situation, a conditional version of the well-known theorem ofGronwall (also known by the name of Lemma ofBellman) will be derived.

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Diese Arbeit ist Teil einer am Institut für Angewandte Mathematik der Technischen Hochschule München unter Anleitung von Herrn o. Prof. Dr. rer. nat. habil.J. Heinhold angefertigten Dissertation.     

Crypt-equivalent algebraic specifications

Summary Equivalence is a fundamental notion for the semantic analysis of algebraic specifications. In this paper the notion of crypt-equivalence is introduced and studied w.r.t. two loose approaches to the semantics of an algebraic specification T: the class of all first-order models of T and the class of all term-generated models of T. Two specifications are called crypt-equivalent if for one specification there exists a predicate logic formula which implicitly defines an expansion (by new functions) of every model of that specification in such a way that the expansion (after forgetting unnecessary functions) is homologous to a model of the other specification, and if vice versa there exists another predicate logic formula with the same properties for the other specification. We speak of first-order crypt-equivalence if this holds for all first-order models, and of inductive crypt-equivalence if this holds for all term-generated models. Characterizations and structural properties of these notions are studied. In particular, it is shown that first order crypt-equivalence is equivalent to the existence of explicit definitions and that in case of positive definability two first-order crypt-equivalent specifications admit the same categories of models and homomorphisms. Similarly, two specifications which are inductively crypt-equivalent via sufficiently complete implicit definitions determine the same associated categories. Moreover, crypt-equivalence is compared with other notions of equivalence for algebraic specifications: in particular, it is shown that first-order cryptequivalence is strictly coarser than abstract semantic equivalence and that inductive crypt-equivalence is strictly finer than inductive simulation equivalence and implementation equivalence.     

Sound and Complete Qualitative Simulation Needs “Quantitative” Filtering

The AI methodology of qualitative reasoning furnishes useful tools to scientists and engineers who need to deal with incomplete system knowledge during design, analysis, or diagnosis tasks. Qualitative simulators have a theoretical soundness guarantee; they cannot overlook any concrete equation implied by their input. On the other hand, the basic qualitative simulation algorithms have been shown to suffer from the incompleteness problem; they may allow non-solutions of the input equation to appear in their output. The question of whether a simulator with purely qualitative input which never predicts spurious behaviors can ever be achieved by adding new filters to the existing algorithm has remained unanswered. In this paper, we show that, if such a sound and complete simulator exists, it will have to be able to handle numerical distinctions with such a high precision that it must contain a component that would better be called a quantitative, rather than qualitative reasoner. This is due to the ability of the pure qualitative format to allow the exact representation of the members of a rich set of numbers.     

Enumeration of Optimal Ternary Constant-Composition Codes

The problem of classification of optimal ternary constant-composition codes is considered. Using a combinatorial-computer method, the number of inequivalent codes is determined for 3 d n 10.     

The way stylized language means: Pattern matching in the child ballads

This paper suggests ways in which the pattern-matching capability of the computer can be used to further our understanding of stylized ballad language. The study is based upon a computer-aided analysis of the entire 595,000- word corpus of Francis James Child'sThe English and Scottish Popular Ballads (1882–1892), a collection of 305 textual traditions, most of which are represented by a variety of texts. The paper focuses on the Mary Hamilton tradition as a means of discussing the function of phatic language in the ballad genre and the significance of textual variation.Cathy Lynn Preston is a Research Associate, Computer Research in the Humanities, at the University of Colorado, Boulder. She is interested in folklore, particularly oral narrative; popular literature of the 18th- and 19th-century, particularly broadside and chapbook; the works of John Gay, Jonathan Swift, Thomas Hardy; Middle English romance and lyric. Her major publications areA KWIC Concordance to Jonathan Swift's A Tale of a Tub, The Battle of the Books, and A Discourse Concerning the Mechanical Operation of the Spirit, A Fragment, (New York: Garland Publishing, 1984) (co-authored with Harold D. Kelling), andA KWIC Concordance to Thomas Hardy's Tess of the d'Urbervilles, (New York: Garland Publishing, 1989).