Compression Theorem Research Articles

Honest bounds for complexity classes of recursive functions

Let be the set of recursive functions computable by machines using at most t(x) computation steps on argument x, for all but finitely many inputs x. We call t a name for the complexity class . Suppose we allow our machines to run longer, say h(x, t(x)) steps on argument x, where h is some fixed recursive function. One might expect that, for large enough h, permitting our machines to run longer by an amount h will always allow us to compute new functions, i.e., is a proper subset of . This turns out not to be the case: The “gap theorem” ([2], [3]) implies that for every recursive h there exists a recursive t such that . However, if we restrict our attention to names from a certain subclass of the recursive functions, then we can indeed uniformly increase the size of our -classes. Informally, we call a recursive function t “honest” if some machine computes t(x) in roughly t(x) steps for each argument x. (A precise definition is given in Definition 1 below.) Then according to the “compression theorem” [1], there exists a single recursive function h such that, for every honest t, is a proper subset of . Thus the phenomenon of the gap theorem is avoided by restricting attention to honest functions. It is a surprising consequence of the “honesty theorem” of McCreight and Meyer ([4], [5]) that there is no loss of generality in this restriction. Namely, for any recursive function t there is an honest recursive function t′ such that .

Interionic Potential in Aluminum

It is suggested that, contrary to recent calculations, the interionic potential in aluminum does not have a negative potential well in the vicinity of the first neighbor. Also, it is demonstrated that the calculated Al potential is extremely sensitive to the choice of electron-gas screening and that any screening function which does not satisfy the compressibility theorem accurately gives unreliable results.

Catastrophe in the Random-Phase Approximation: Critique of a Theory of Phase Transitions

A detailed investigation is presented of the theory of phase transitions afforded by the random phase approximation (RPA) and Gaussian fluctuation theory. Arguments are given which indicate that these two theories are equivalent. A rigorous analysis is carried out on the RPA for a simple binary model that should exhibit a phase transition. It is found that when the RPA free energy is used to calculate the thermodynamics for the model a catastrophe region exists in which the system collapses to negatively infinite free energy and thermodynamic stability breaks down before (at lower densities and higher temperature) the region is reached. The catastrophe surface which surrounds the region has been identified in the previous literature as the limit of stability for the model, and the singularities associated with the catastrophe have been incorrectly identified as true critical singularities. The incorrect identification leads to results which contradict the second law of thermodynamics. It is shown that there is no sensible solution to the equations obtained from a Maxwell construction to bridge the catastrophe. The RPA for the binary model is also discussed within the context of the compressibility theorem. When this route to the thermodynamics is taken, it is found that the behavior predicted by the RPA is purely classical. Arguments are presented which indicate that the results found with the binary model are general features of the RPA; the application of the approximation to more general one- and two-component systems yield the same results. Finally, a tentative explanation for the catastrophe is given. The explanation suggests a modified RPA which does not violate the second law of thermodynamics; however, the patched up theory is classical.

Phase transitions in one-dimensional cluster-interaction fluids III. Correlation functions

An explicit expression is derived for the pair correlation function g( r) in the class of one-dimensional many-body cluster interaction models studied previously. It is shown that the product property and the compressibility theorem are satisfied. The asymptotic behavior of g( r) is studied in the simple logarithmic model for pressure and temperature near the vapor pressure curve. Explicit expressions for the nearest neighbor distribution and the cluster length distribution are also derived.

Fluctuation Compressibility Theorem and Its Application to the Pairing Model

A theorem of statistical mechanics relates density fluctuations to compressibility. A new derivation of this is given. The theorem is violated in the BCS model of a superconductor. The difficulty is resolved by those same improvements in the theory which lead to a guage-invariant Meissner effect.