Asymptotic properties of minimax trees and game-searching procedures

The model most frequently used for evaluating the behavior of game-searching methods consists of a uniform tree of height h and a branching degree d, where the terminal positions are assigned random, independent and identically distributed values. This paper highlights some curious properties of such trees when h is very large and examines their implications on the complexity of various game-searching methods.If the terminal positions are assigned a WIN-LOSS status with the probabilities P₀ and 1 ? P₀, respectively, then the root node is almost a sure MIN or a sure LOSS, depending on whether P₀ is higher or lower than some fixed-point probability $\text{P1(d)}$. When the terminal positions are assigned continuous real values, the minimax value of the root node converges rapidly to a unique predetermined value $\text{v1}$, which is the $\text{(1 ? P1)-}\text{fractile}$ of the terminal distribution.Exploiting these properties we show that a game with WIN-LOSS terminals can be solved by examining, on the average, $\text{O}\text{[(d)}{}^{\mathrm{<mtext>h</mtext><mtext>2</mtext>}}\text{]}$ terminal positions if positions if $\text{P}{}_0\text{≠ P1}$ and $\text{O}\text{[(}\text{P1}\text{(1 ? P1)}\text{)}{}^{\mathrm{h}}\text{]}$ positions if $\text{P}{}_0\text{= P1}$, the former performance being optimal for all search algorithms. We further show that a game with continuous terminal values can be evaluated by examining an average of $\text{O}\text{[(}\text{P1}\text{(1 ? P1)}\text{)}{}^{\mathrm{h}}\text{]}$ positions, and that this is a lower bound for all directional algorithms. Games with discrete terminal values can, in almost all cases, be evaluated by examining an average of $\text{O}\text{[(d)}{}^{\mathrm{<mtext>h</mtext><mtext>2</mtext>}}\text{]}$ terminal positions. This performance is optimal and is also achieved by the ALPHA-BETA procedure.     

Normal forms for nonautonomous difference equations

We extend Henry Poincaré's normal form theory for autonomous difference equations χ_{k + 1} = f(χ_k) to nonautonomous difference equations χ_{k + 1} = f_k(χ_k). Poincaré's nonresonance condition $\text{α}{}_{\mathrm{j}}\text{-П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{=1α}{}^{\mathrm{qi}}{}_{\mathrm{i}}\text{≠0}$ for eigenvalues is generalized to the new nonresonance condition λ_j∪$\text{α}{}_{\mathrm{j}}\text{⊔П}{}^{\mathrm{n}}{}_{\mathrm{i=1}}\text{α}{}^{\mathrm{qi}}{}_{\mathrm{i}}\text{≠0}$ for spectral intervals.     

Mathematical foundations of Bézier's technique

Bezier's method is one of the most famous in computational geometry. In his book Numerical control Bezier gives excellent expositions of the mathematical foundations of this method. In this paper a new expression of the functions {f_n,i(u)}

$\text{f}{}_{\mathrm{n,i}}\text{(u)=1?}\text{Σ}\text{p=0}\text{i?1}\text{C}{}^{\mathrm{p}}{}_{\mathrm{n}}\text{u}{}^{\mathrm{p}}\text{(1?u)}{}^{\mathrm{n?p}}\text{(i=1,2,…,n)}$

is obtained.Using this formula, we have not only derived some properties of the functions {f_n,i(u)} (for instance $\text{f}{}_{\mathrm{n,n}}\text{(u) < f}{}_{\mathrm{n,n?1}}\text{(u)<...<f}{}_{\mathrm{n,1}}\text{(u)}\text{u}\text{? [0, 1]}$ and functions {f_n,i(u)} increase strictly at [0, 1] etc) but also simplified systematically all the mathematical discussions about Bezier's method.Finally we have proved the plotting theorem completely by matrix calculation.     

Averaging shifted histograms

A method which consists in shifting different histograms of the same spectrum and then taking their average is presented in order to smooth the data and to increase the localization accuracy and separation of the peaks. The statistical properties of this method are investigated. The average of two histograms with shifted bin limits is studied. It is shown that for histograms with random bin limits, distributed according to

$\text{F}{}_{\mathrm{i}}\text{(x)=}\text{∫}\text{?∞}\text{x}\text{?}{}_{\mathrm{i}}\text{(ξ, μ}{}_{\mathrm{i}}\text{, σ)}\text{d}\text{ξ}$

; where the standard deviation σ is very small compared to the difference of the means (μ_i+1 ? μ_i) for ll i the zero order approximation to the variance of this histogram is given by:

$\text{var}\text{∽}\text{(H)=}\text{∑}\text{i=0}\text{m}\text{(A}{}_{\mathrm{i+1}}\text{?a}{}_{\mathrm{i}}\text{)}{}^2\text{F}{}_{\mathrm{i+1}}\text{(x)(1?F}{}_{\mathrm{i+1}}\text{(x))}$

, where

$\text{a}{}_{\mathrm{i}}\text{=}\text{1}\text{x}{}_{\mathrm{i=1}}\text{?x}{}_{\mathrm{i}}\text{∫}\text{x}{}_{\mathrm{i}}\text{x}{}_{\mathrm{i+1}}\text{g(ξ)}\text{d}\text{ξ}$

and g is an unknown function fitted by the histogram. Formula (1) gives also the relation:

$\text{v}\text{a}\text{?}\text{r}\text{(}\text{(H}{}_1\text{+ H}{}_2\text{)}\text{2}\text{) =}\text{1}\text{4}\text{(}\text{v}\text{a}\text{?}\text{r}\text{(H}{}_1\text{(x)) +}\text{v}\text{a}\text{?}\text{r}\text{(H}{}_2\text{(x))}$

, when H₁ and H₂ have stochastically independent bin limits.When the histogram H is considered as a spline function S of order one it is shown that for the minimization criterion with respect to the coefficient of the spline:

$\text{M}{}_1\text{=}\text{min}\text{∫}\text{x}{}_1\text{x}{}_{\mathrm{m+1}}\text{(g(x) ? S}{}_1\text{(x))}{}^2\text{d}\text{x}$

, the following result holds: $\text{M}{}_{\mathrm{a}}\text{?}{}^1{}_2\text{(M}{}_1\text{+ M}{}_2\text{)}$, where $\text{S}{}_{\mathrm{a}}\text{(x) =}{}^1{}_2\text{(S}{}_1\text{(x) + S}{}_2\text{(x))}$. If the number of shifted histograms tends to infinity, then

$\text{S}{}_{\mathrm{\infty }}\text{(x) = [Γ(x + h) + Γ(x ? h) ? 2Γ(x)]/h}{}^2$

, where $\text{Γ(x) =}\text{∫}\text{?∞}\text{x}\text{∫}\text{?∞}\text{η}\text{g(ξ)}\text{d}\text{ξ}\text{d}\text{η}$, and h is a constant bin size. Then

$\text{M}{}_{\mathrm{\infty }}\text{≈}\text{h}{}^4\text{144}\text{∫}\text{x}{}_1\text{x}{}_{\mathrm{m+1}}\text{g″}{}^2\text{(x)}\text{D}\text{x}$

. Extensions to two-dimensional histograms and to higher order (empirical distributions) are presented.     

Crosscorrelation between linearly and nonlinearly distorted versions of a given signal

A given deterministic signal x(.) is distorted by passing it through a linear time-invariant filter and also by subjecting it to the action of an instantaneous nonlinearity. The resulting time crosscorrelation of the two distorted versions of the original signal is expressed by the function

$\text{R}{}_2\text{(s)?∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{?∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{g[x(t)]k(t?t′)x(t?s)dt dt′}$

, where x(.) is the given signal, k(.) is the nonnegative definite impulse response of the linear filter, and g(.) is the output-input characteristic of the zero-memory nonlinear device. The problem considered is that of determining conditions on the pair (x,g) such that R₂(s) ? R₂(0) for all s and any choice of nonnegative definite filter function k; the principal result is the formulation of a necessary and sufficient condition for R₂ to have a global maximum at the origin. In particular, the peak value occurs at the origin if and only if $\text{G}{}_{\mathrm{x}}{}^1\text{(ω)X(ω)}$ is real and nonnegative for all ω ? 0, where G_x(.) and X(.) are the Fourier transforms of g[x(.)] and x(.), respectively. An equivalent condition is that the correlation function

$\text{R}{}_2\text{(s)?∫}{}_{\mathrm{?\infty }}{}^{\mathrm{\infty }}\text{g[x(t)]x(t?s)dt}$

, previously studied by Richardson, be nonnegative definite.Several examples are given, and it is shown that, unlike the case for R₁(.), monotonicity of g(.) is not a sufficient condition for R₂(.) to have a global maximum at s = 0 independently of the choice of filter characteristic k. Certain extensions of these results are given for the case when x(.) is a Gaussian random input.     

Some remarks on a conjecture in parameter adaptive control

A.S. Morse has raised the following question: Do there exist differentiable functions

$\text{f:R}{}^2\text{→ R}\text{and}\text{g:R}{}^2\text{→ R}$

with the property that for every nonzero real number λ and every (x₀, y₀) ∈ $\text{R}$² the solution (x(t),y(t)) of

$\text{x}\text{?}\text{(t) = x(t) + λf(x(t),y(t))}$

,

$\text{y}\text{?}\text{(t) = g(x(t),y(t))}$

,

$\text{x(0) = x}{}_0\text{, y(0) = y}{}_0$

, is defined for all t ? 0 and satisfies

$\text{lim}\text{t → + ∞}$

and y(t) is bounded on [0,∞)? We prove that the answer is yes, and we give explicit real analytic functions f and g which work. However, we prove that if f and g are restricted to be rational functions, the answer is no.     

Improved M-matrix condition for the existence of positive-real weighting

It is proved that, for a given stable transfer matrix G(s), there exists a constant diagonal matrix W which makes WG(s) positive-real if Re g_ii(jω) ≥ 0 and I?? is an M-matrix where ? = (?_jk) is defined by ?_ii = 0 and $\text{?}{}_{\mathrm{jk}}\text{=}\text{sup}{}_{\mathrm{\omega }}\text{|g}{}_{\mathrm{ik}}\text{(gω) |/(}\text{Re}\text{[g}{}_{\mathrm{ii}}\text{(jω)]·}\text{Re}\text{[g}{}_{\mathrm{kk}}\text{(jω])}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$.     

Uniform computational complexity of the derivatives of C∞-functions

We discuss the uniform computational complexity of the derivatives of C^∞-functions in the model of Ko and Friedman (Ko, Complexity Theory of Real Functions, Birkhäuser, Basel, 1991; Ko, Friedman, Theor. Comput. Sci. 20 (1982) 323–352). We construct a polynomial time computable real function g∈C^∞[−1,1] such that the sequence $\text{{|g}{}^{\mathrm{(n)}}\text{(0)|}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is not bounded by any recursive function. On the other hand, we show that if f∈C^∞[−1,1] is polynomial time computable and the sequence of the derivatives of f is uniformly polynomially bounded, i.e., |f⁽ⁿ⁾(x)| is bounded by 2^p(n) for all x∈[−1,1] for some polynomial p, then the sequence $\text{{f}{}^{\mathrm{(n)}}\text{}}{}_{\mathrm{<mtext>n\in </mtext><mtext>N</mtext>}}$ is uniformly polynomial time computable.     

Matrix equations over rings

The main purpose of this work is to establish necessary conditions and sufficient conditions for the existence of a solution of matrix equations whose coefficient matrices have elements belonging to the ring $\text{R=}\text{C}\text{[z}{}_1\text{,z}{}_2\text{,…z}{}_{\mathrm{n}}\text{]}$ of polynomials in n variables with complex coefficients and the ring $\text{R=}\text{R}\text{[z}{}_1\text{,z}{}_2\text{,…z}{}_{\mathrm{n}}\text{]}{}^{\mathrm{n}}$ of rational functions a(z₁,z₂,…z_n)b(z₁,z₂,…,z_n)^?1 with real coefficients and b(z₁,z₂,…,z_n)≠0 for all (z₁,z₂,…,z_n) in Rⁿ. Results obtained are useful in multidimensional systems theory and elsewhere.     

Liquid-liquid solubility and critical phenomena in binary mixtures

By assuming a proportionality exists between a difference in Boltzmann factors and a difference in compositions, the following expression is developed for the mole fraction order parameter in symmetric binary liquid mixtures $\text{Δx = [1 ?}\text{exp}\text{E}\text{RT}{}_{\mathrm{c}}\text{(l?T}{}_{\mathrm{c}}\text{/T)]}{}^{\mathrm{\beta }}$ In the critical limit this expression reduces to the form for simple scaling fx253-fig 1 By treating the original components A and B as if they transfer between the phases in the form of aggregates with the empirical formulas A_mB_1?m and B_n, in which 0<m≤Landn>0, coexistence curves in diverse systems become symmetric and are fitted by the above expressions. Beta exponent values obtained from these fits range in value from 0.312±0.002 to 0.333±0.003.     

On the form equivalence of L-forms

In [5] the notion of an L form F defining a family $\text{L}$_d(F) of languages by means of X-interpretations has been introduced. Here X is one of a number of possible variations of the notion of interpretation originally used in [1] for grammar forms. In this paper it is shown that the questions whether $\text{L}{}_{\mathrm{d}}\text{(F) =}\text{L}{}_{\mathrm{d}}\text{(F}{}_1\text{)}$ for L forms F and F₁ is decidable, if deterministic interpretations of PDOL systems are considered, where L(F) and L(F₁) contain at most one word of length n for any n ? 0, and it is shown that same question is undecidable, if full or uniform interpretations are chosen. In contrast to this, no such results are known for grammar forms at this point.     

The relation between information theory and the differential geometry approach to statistics

?encov has shown that the Riemannian metric on the probability simplex ∑x_i = 1 defined by $\text{(ds)}{}^2\text{=}\text{∑(dx}{}_{\mathrm{i}}\text{)}{}^2\text{x}{}_{\mathrm{i}}$ has an invariance property under certain probabilistically natural mappings. No other Riemannian metric has the same property. The geometry associated with this metric is shown to lead almost automatically to measures of divergence between probability distributions which are associated with Kullback, Bhattacharyya, and Matusita. Certain vector fields are associated in a natural way with random variables. The integral curves of these vector fields yield the maximum entropy or minimum divergence estimates of probabilities. Some other consequences of this geometric view are also explored.     

Determination of a general economic location constant by monte carlo simulation

The problem of estimating the number of markets (or plants) to serve a set of sources in a given geographical area was considered. Markets were located so as to minimize total assembly cost which was considered a linear function of the weighted Euclidean distances between sources and markets. The following predictive function C_m was proposed for estimating the minimum total assembly cost for a given number of markets: $\text{C}{}_{\mathrm{m}}\text{= C}{}_1\text{?(}\text{m?1}\text{m}\text{)}{}^{\mathrm{k}}\text{(}\text{M}\text{M ? 1}\text{)}{}^{\mathrm{k}}\text{(C}{}_1\text{? C}{}_{\mathrm{M}}\text{),m = 1, 2, 3, …, M}$ where m = number of markets being located. M = maximum number of potential market sites. C₁ = minimum assembly cost when only one market is located. C_M = minimum assembly cost when all M markets are located. k = an undetermined constant which specifies the shape of the function.The validity of the C_m function and the range of the k constant were determined by computer Monte Carlo experimentation. The constant k, to a sufficient degree of approximation and ordinary use, was found independent of the number of sources and their distribution. A general economic location co     

Gibbs energies of formation of Mn3C,M(Fe,Mn)3C and Mn23C6 from the ternary phase equilibria in the FeMnC system

The high temperature phase relations in the FeMnC system have been analyzed in light of the recently developed thermodynamic method by the authors to obtain the Gibbs energies of formation of Mn₂₃C₆ and Mn,C. A new thermodyn/amic treatment is outlined and applied to obtain the stability of the ternary carbide M(Fe,Mn)₃C without any a priori assumption of a solution model for the M₃C phase. The recommended Gibbs energies of formation for the Mn carbides, Mn₃C and Mn₂₃C₆ With γ-Mn (graphite) as the Standard states are:

The present method can be extended to obtain a consistent set of thermodynamic data for binary and ternary carbides from various ternary metal-metal-carbon phase relations.     

Unbounded solutions of quasi-linear difference equations

We study positive increasing solutions of the nonlinear difference equation $\text{δ(a}{}_{\mathrm{n}}\text{φ}{}_{\mathrm{p}}\text{(δχ}{}_{\mathrm{n}}\text{))=b}{}_{\mathrm{n}}\text{f(χ}{}_{\mathrm{n+1}}\text{,}\text{φ}{}_{\mathrm{p}}\text{(u)=|u|}{}^{\mathrm{p-2}}\text{u,}\text{p>1}$ where {a_n}, {b_n} are positive real sequences for n ≥ 1, fR → R is continuous with uf(u) > 0 for u ≠ 0. A full characterization of limit behavior of all these solutions in terms of a_n, b_n is established. Examples, showing the essential role of used hypotheses, are also included. The tools used are the Schauder fixed-point theorem and a comparison method based on the reciprocity principle.     

Insertion languages

The operations of insertion ( ← ) and iterated insertion ( ←₁ ) are simple variants of Kleene's operations · and 1 [15] in a manner similar to the operations shuffle and iterated shuffle (see e.g. [13, 23, 20, 14]). Using the operation of iterated insertion, we can generate both the semi-Dyck and the two-sided Dyck languages from certain finite subsets of these languages. Thus the class of languages of the form $\text{S}{}^{\mathrm{\leftarrow }}{}^1$ for finite S forms a natural class of generalized Dyck languages. This class is equivalent to the class of pure unitary languages discussed in [6]. We investigate this class further, by examining for it the problems of equivalence, ambiguity, and determinism, all of which are easily decidable. On the other hand, we show that the problem “$\text{S}{}^{\mathrm{\leftarrow }}\text{1 ∩ T}{}^{\mathrm{\leftarrow }}\text{1 = {λ}?}$” is undecidable for finite, unambiguous S and T. Furthermore, by extending the regular expressions to include the operations ← and $\text{←}{}_1$, we obtain the class of insertion languages which generalizes both the regular languages and the Dyck languages, but is properly contained within the class of context-free languages. Our main result here is that the problem “$\text{L = ∑}{}^1\text{?}$” is undecidable for the class of insertion languages. From this result, it follows that the equivalence problem and the problem “IsL regular?” are also undecidable for this class.     

computer aided intuition in abstract algebra

Two examples are given in which the computer was used to supplement intuition in abstract algebra. In the first example, the computer was used to search Cayley tables of 4 element groupoids to find those which are 5-associative but not 4-associative. (n-associative means that the product of any n elements is independent of the way the factors are grouped by parentheses.) The computer generated examples suggested the existence of n element groupoids which are (2^n?2+1)-associative but not (2^n-2)-associative, for each integer n≧4.In the second example, the computer counted the numbers g₂(m) of invertible 2×2 matrices with entries chosen from the ring Z_mof integers, for m = 2, 3, 4,…, 18. The insight gained from these results led to a proof that there are

invertible n×n matrices over Z_m.Some applications to graduate and undergraduate instruction are indicated.