Maintaining Longest Paths Incrementally

Modeling and programming tools for neighborhood search often support invariants, i.e., data structures specified declaratively and automatically maintained incrementally under changes. This paper considers invariants for longest paths in directed acyclic graphs, a fundamental abstraction for many applications. It presents bounded incremental algorithms for arc insertion and deletion which run in O( + || log||) time and O() time respectively, where || and are measures of the change in the input and output. The paper also shows how to generalize the algorithm to various classes of multiple insertions/deletions encountered in scheduling applications. Preliminary experimental results show that the algorithms behave well in practice.     

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.     

Constructing sets of functions which have a givenF-cardinality

The termF-cardinality of (=F-card()) is introduced whereF: ^{n n} is a partial function and is a set of partial functionsf: ^{n n} . TheF-cardinality yields a lower bound for the worst-case complexity of computingF if only functionsf can be evaluated by the underlying abstract automaton without conditional jumps. This complexity bound isindependent from the oracles available for the abstract machine. Thus it is shown that any automaton which can only apply the four basic arithmetic operations needs (n logn) worst-case time to sortn numbers; this result is even true if conditional jumps witharbitrary conditions are possible. The main result of this paper is the following: Given a total functionF: ^{n n} and a natural numberk, it is almost always possible to construct a set such that itsF-cardinality has the valuek; in addition, can be required to be closed under composition of functionsf,g . Moreover, ifF is continuous, then consists of continuous functions.     

From Conditional Events to Conditional Measures: A New Axiomatic Approach

Our starting point is a definition of conditional event EH which differs from many seemingly similar ones adopted in the relevant literature since 1935, starting with de Finetti. In fact, if we do not assign the same third value u (undetermined) to all conditional events, but make it depend on EH, it turns out that this function t(EH) can be taken as a general conditional uncertainty measure, and we get (through a suitable – in a sense, compulsory – choice of the relevant operations among conditional events) the natural axioms for many different (besides probability) conditional measures.     

Complexity and computability of solutions to linear programming systems

Through key examples and constructs, exact and approximate, complexity, computability, and solution of linear programming systems are reexamined in the light of Khachian's new notion of (approximate) solution. Algorithms, basic theorems, and alternate representations are reviewed. It is shown that the Klee-Minty example hasnever been exponential for (exact) adjacent extreme point algorithms and that the Balinski-Gomory (exact) algorithm continues to be polynomial in cases where (approximate) ellipsoidal centered-cutoff algorithms (Levin, Shor, Khachian, Gacs-Lovasz) are exponential. By model approximation, both the Klee-Minty and the new J. Clausen examples are shown to be trivial (explicitly solvable) interval programming problems. A new notion of computable (approximate) solution is proposed together with ana priori regularization for linear programming systems. New polyhedral constraint contraction algorithms are proposed for approximate solution and the relevance of interval programming for good starts or exact solution is brought forth. It is concluded from all this that the imposed problem ignorance of past complexity research is deleterious to research progress on computability or efficiency of computation.This research was partly supported by Project NR047-071, ONR Contract N00014-80-C-0242, and Project NR047-021, ONR Contract N00014-75-C-0569, with the Center for Cybernetic Studies, The University of Texas at Austin.     

The network complexity and the Turing machine complexity of finite functions

Summary Let L(f) be the network complexity of a Boolean function L(f). For any n-ary Boolean function L(f) let . Hereby p ranges over all relative Turing programs and ranges over all oracles such that given the oracle , the restriction of p to inputs of length n is a program for L(f). p is the number of instructions of p. T _p (n) is the time bound and S _p of the program p relative to the oracle on inputs of length n. Our main results are (1) L(f) O(TC(L(f))), (2) TC(f) O(L(f) ^{2 2+}) for every O.The results of this paper have been reported in a main lecture at the 1975 annual meeting of GAMM, April 2–5, Göttingen     

Sufficiency conditions for existence of an optimal feedback control in stochastic mechanics

In the framework of stochastic mechanics, the following problem is considered: in a set of admissible feedback controls v, with range inE ⁿ , find one minimizing the expectationE _sx { _s ^T L(t, (t), (t, (t)))dt + W _T ((T))} for all (s, x) [0,T) E ⁿ , whereL(t, x, ) = (/12)m ² – U(t, x) is the classical action integrand and is an-dimensional diffusion process in the weak sense, (see Bensoussan, 1982) with drift and diffusion coefficientD constant > 0.W _T andU are given real functions. Sufficiency conditions for the existence of such an optimal feedback control are given. Dedicated to George Leitmann Recommended by G.J. Olsder Presented at the Third Workshop on Control Mechanics in honor of George Leitmann, January 22–24, 1990, University of Southern California, Los Angeles, California (USA).     

Unitary immersions of nonlinear systems

In this paper, we define what we call a unitary immersion of a nonlinear system. We observe that, for classical Hamiltonian systems, this notion contains, in some sense, the concept of quantization. We restrict our attention to degree-zero unitary immersions, where all observation functions must be represented by operators of the type multiplication by a function. We show that the problem of classifying such degree-zero unitary immersions of a given nonlinear system is not obvious. In some cases, we solve this problem.Chargé de Recherche au CNRS.Maître de Conférences.     

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.     

Diagrammatic Analysis of Interval Linear Equations - Part II: The Two-Dimensional Case and Generalization to n Dimensions

Using the results obtained for the one-dimensional case in Part I (Reliable Computing 9(1) (2003), pp. 1–20) of the paper, an analysis of the two-dimensional relational expression a ₁ x ₁ + a ₂ x ₂ b, where {, , , =}, is conducted with the help of a midpoint-radius diagram and other auxiliary diagrams. The solution sets are obtained with a simple boundary-line selection rule derived using these tools, and are characterized by types of one-dimensional cuts through the solution space. A classification of basic possible solution types is provided in detail. The generalization of the approach for n-dimensional interval systems and avenues for further research are also outlined.     

Relativized polynomial hierarchies extending two levels

The existence of relativized polynomial hierarchies extending exactly two levels is shown. More precisely, oraclesX andY are constructed such thatNP(X) ₂ ^{P, X} = ₂ ^{P, X} and ₂ ^{P, Y} ₂ ^{P, Y} = ₂ ^{P, Y} . These results answer a question posed by Baker, Gill, and Solovay ([1]) and generalize their constructions of relativized polynomial hierarchies extending 0 resp. 1 level; i.e.,P(A) = NP(A) resp.P(B) NP(B) = coNP(B) for recursive oraclesA andB.     

Thiele Rational Interpolation for the Numerical Computation of the Reversible Randles–Sevcik Function in Electrochemistry

This paper uses Thiele rational interpolation to derive a simple method for computing the Randles–Sevcik function ^1/2(x), with relative error at most 1.9 × 10^–5 for – < x < . We develop a piecewise approximation method for the numerical computation of ^1/2(x) on the union (–, –10) [–10, 10] (10, ). This approximation is particularly convenient to employ in electrochemical applications where four significant digits of accuracy are usually sufficient. Although this paper is primarily concerned with the approximation of the Randles–Sevcik function, some examples are included that illustrate how Thiele rational interpolation can be employed to generate useful approximations to other functions of interest in scientific work.     

Separating complexity classes related to bounded alternating ω-branching programs

We develop a theory of communication within branching programs that provides exponential lower bounds on the size of branching programs that are bounded alternating. Our theory is based on the algebraic concept of -branching programs, : , a semiring homomorphism, that generalizes ordinary branching programs, -branching programs [M2] andMOD _p-branching programs [DKMW].Due to certain exponential lower and polynomial upper bounds on the size of bounded alternating -branching programs we are able to separate the corresponding complexity classesN _ba ,co-N _{ba ba} , andMOD _p - _ba ,p prime, from each other, and from that classes corresponding to oblivious linear length-bounded branching programs investigated in the past.     

Interval routing schemes

In this paper the problem of routing messages along shortest paths in a distributed network without using complete routing tables is considered. In particular, the complexity of deriving minimum (in terms of number of intervals) interval routing schemes is analyzed under different requirements. For all the cases considered NP-hardness proofs are given, while some approximability results are provided. Moreover, relations among the different cases considered are studied.This work was supported by the EEC ESPRIT II Basic Research Action Program under Contract No. 7141 Algorithms and Complexity II, by the EEC Human Capital and Mobility MAP project, and by the Italian MURST 40% project Algoritmi, Modelli di Calcolo e Strutture Informative.     

A general proof rule for procedures in predicate transformer semantics

Summary A proof rule for the procedure call is derived for procedures with value, result and value-result parameters. It is extended to procedures with unrestricted global variables and to recursive procedures. Like D. Gries's proof rule, it is based on the substitution rule for assignment. However, it is more general and much simpler to apply. Assume that {U} S {V} has been proved about the procedure body S. The proof rule for determining whether a call establishes predicate E is based on finding an adaptation A satisfying AV E, where E is derived from E by some substitution of parameters.     

A Note on the Rational Closure of Knowledge Bases with Both Positive and Negative Knowledge

The notion of the rational closure of a positive knowledge base K of conditional assertions | (standing for if then normally ) was first introduced by Lehmann (1989) and developed by Lehmann and Magidor (1992). Following those authors we would also argue that the rational closure is, in a strong sense, the minimal information, or simplest, rational consequence relation satisfying K. In practice, however, one might expect a knowledge base to consist not just of positive conditional assertions, | , but also negative conditional assertions, _i (standing for not if then normally . Restricting ourselves to a finite language we show that the rational closure still exists for satisfiable knowledge bases containing both positive and negative conditional assertions and has similar properties to those exhibited in Lehmann and Magidor (1992). In particular an algorithm in Lehmann and Magidor (1992) which constructs the rational closure can be adapted to this case and yields, in turn, completeness theorems for the conditional assertions entailed by such a mixed knowledge base.     

A New Implementation of Automath

This paper presents aut, a modern Automath checker. It is a straightforward re-implementation of the Zandleven Automath checker from the seventies. It was implemented about five years ago, in the programming language C. It accepts both the AUT-68 and AUT-QE dialects of Automath. This program was written to restore a damaged version of Jutting's translation of Landau's Grundlagen. Some notable features: It is fast. On a 1 GHz machine it will check the full Jutting formalization (736 K of nonwhitespace Automath source) in 0.6 seconds. Its implementation of -terms does not use named variables or de Bruijn indices (the two common approaches) but instead uses a graph representation. In this representation variables are represented by pointers to a binder. The program can compile an Automath text into one big Automath single line-style -term. It outputs such a term using de Bruijn indices. (These -terms cannot be checked by modern systems like Coq or Agda, because the -typed -calculi of de Bruijn are different from the -typed -calculi of modern type theory.)The source of aut is freely available on the Web at the address .     

The informed neuron: Issues in the use of information theory in the behavioral sciences

The concept of information is virtually ubiquitous in contemporary cognitive science. It is claimed to be processed (in cognitivist theories of perception and comprehension), stored (in cognitivist theories of memory and recognition), and otherwise manipulated and transformed by the human central nervous system. Fred Dretske's extensive philosophical defense of a theory of informational content (semantic information) based upon the Shannon-Weaver formal theory of information is subjected to critical scrutiny. A major difficulty is identified in Dretske's equivocations in the use of the concept of a signal bearing informational content. Gibson's alternative conception of information (construed as analog by Dretske), while avoiding many of the problems located in the conventional use of signal, raises different but equally serious questions. It is proposed that, taken literally, the human CNS does not extract or process information at all; rather, whatever information is construed as locatable in the CNS is information only for an observer-theorist and only for certain purposes.Blood courses through our veins, andinformation through our central nervous system.— A Neuropsychology Textbook.     

The Kreisel length-of-proof problem

A first-order system F has theKreisel length-of-proof property if the following statement is true for all formulas(x): If there is ak1 such that for alln0 there is a proof of(¯n) in F with at mostk lines, then there is a proof of x(x) in F. We consider this property for Parikh systems, which are first-order axiomatic systems that contain a finite number of axiom schemata (including individual axioms) and a finite number of rules of inference. We prove that any usual Parikh system formulation of Peano arithmetic has the Kreisel length-of-proof property if the underlying logic of the system is formulated without a schema for universal instantiation in either one of two ways. (In one way, the formula to be instantiated is built up in steps, and in the other way, the term to be substituted is built up in steps.) Our method of proof uses techniques and ideas from unification theory.