Automated Conjecturing II: Chomp and Reasoned Game Play

We demonstrate the use of a program that generates conjectures about positions of the combinatorial game Chomp—explanations of why certain moves are bad. These could be used in the design of a Chomp-playing program that gives reasons for its moves. We prove one of these Chomp conjectures—demonstrating that our conjecturing program can produce genuine Chomp knowledge. The conjectures are generated by a general purpose conjecturing program that was previously and successfully used to generate mathematical conjectures. Our program is initialized with Chomp invariants and example game boards—the conjectures take the form of invariant-relation statements interpreted to be true for all board positions of a certain kind. The conjectures describe a theory of Chomp positions. The program uses limited, natural input and suggests how theories generated on-the-fly might be used in a variety of situations where decisions—based on reasons—are required.

On physical scattering density fluctuations of amorphous samples

Using the rigorous results obtained by Wiener [Acta Math. (1930), 30, 118–242] on the Fourier integral of a bounded function and the condition that small-angle scattering intensities of amorphous samples are almost everywhere continuous, the conditions that must be obeyed by a function η(r) for this to be considered a physical scattering density fluctuation are obtained. These conditions can be recast in the following form: the V → ∞ limit of the modulus of the Fourier transform of η(r), evaluated over a cubic box of volume V and divided by V 1/2, exists and its square obeys the Porod invariant relation. Some examples of one-dimensional scattering density functions obeying the aforesaid condition are numerically illustrated.

Elastoplastic Invariant Relation for Deformation of Alkali-Halide Crystals

Plastic strain localization patterns in compression-strained alkali halide (NaCl, KCl, and LiF) crystals have been studied using a double-exposure speckle photography technique. The main parameters of strain localization autowaves at the linear stages of deformation hardening in alkali halide crystals have been determined. A quantitative relationship between the macroscopic parameters of plastic flow localization and microscopic parameters of strained alkali halide crystals has been established.

On an extension of Mycielski’s theorem and invariant scrambled sets

Let $(X,f)$ be a dynamical system, where $X$ is a perfect Polish space and $f:X\rightarrow X$ is a continuous map. In this paper we study the invariant dependent sets of a given relation string ${\it\alpha}=\{R_{1},R_{2},\ldots \}$ on $X$. To do so, we need the relation string ${\it\alpha}$ to satisfy some dynamical properties, and we say that ${\it\alpha}$ is $f$-invariant (see Definition 3.1). We show that if ${\it\alpha}=\{R_{1},R_{2},\ldots \}$ is an $f$-invariant relation string and $R_{n}\subset X^{n}$ is a residual subset for each $n\geq 1$, then there exists a dense Mycielski subset $B\subset X$ such that the invariant subset $\bigcup _{i=0}^{\infty }f^{i}B$ is a dependent set of $R_{n}$ for each $n\geq 1$ (see Theorems 5.4 and 5.5). This result extends Mycielski’s theorem (see Theorem A) when $X$ is a perfect Polish space (see Corollary 5.6). Furthermore, in two applications of the main results, we simplify the proofs of known results on chaotic sets in an elegant way.

Physical determinants of the shape of the psychophysical curve relating tactile roughness to raised-dot spacing: implications for neuronal coding of roughness

There are conflicting reports as to whether the shape of the psychometric relation between perceived roughness and tactile element spacing [spatial period (SP)] follows an inverted U-shape or a monotonic linear increase. This is a critical issue because the former result has been used to assess neuronal codes for roughness. We tested the hypothesis that the relation's shape is critically dependent on tactile element height (raised dots). Subjects rated the roughness of low (0.36 mm)- and high (1.8 mm)-raised-dot surfaces displaced under their fingertip. Inverted U-shaped curves were obtained as the SP of low-dot surfaces was increased (1.3–6.2 mm, tetragonal arrays); a monotonic increase was observed for high-dot surfaces. We hypothesized that roughness is not a single sensory continuum across the tested SPs of low-dot surfaces, predicting that roughness discrimination would show deviations from the invariant relation between threshold (ΔS) and the value of the standard (S) surface (Weber fraction, ΔS/S) expected for a single continuum. The results showed that Weber fractions were increased for SPs on the descending limb of the inverted U-shaped curve. There was also an increase in the Weber fraction for high-dot surfaces but only at the peak (3 mm), corresponding to the SP at which the slope of the psychometric function showed a modest decline. Together the results indicate that tactile roughness is not a continuum across low-dot SPs of 1.3–6.2 mm. These findings suggest that correlating the inverted U-shaped function with neuronal codes is of questionable validity. A simple intensive code may well contribute to tactile roughness.

Denotational Aspects of Untyped Normalization by Evaluation

We show that the standard normalization-by-evaluation construction for the simply-typed lambda_{beta eta}-calculus has a natural counterpart for the untyped lambda_beta-calculus, with the central type-indexed logical relation replaced by a "recursively defined'' <em>invariant relation</em>, in the style of Pitts. In fact, the construction can be seen as generalizing a computational-adequacy argument for an untyped, call-by-name language to normalization instead of evaluation.<br /> <br />In the untyped setting, not all terms have normal forms, so the normalization function is necessarily partial. We establish its correctness in the senses of <em>soundness</em> (the output term, if any, is in normal form and beta-equivalent to the input term); <em>identification</em> ( beta-equivalent terms are mapped to the same result); and <em>completeness</em> (the function is defined for all terms that do have normal forms). We also show how the semantic construction enables a simple yet formal correctness proof for the normalization algorithm, expressed as a functional program in an ML-like call-by-value language.<br /> <br />Finally, we generalize the construction to produce an infinitary variant of normal forms, namely <em>Böhm trees</em>. We show that the three-part characterization of correctness, as well as the proofs, extend naturally to this generalization.