Scott Smith

Microcrystallography, high-pressure cryocooling and BioSAXS at MacCHESS

Three research initiatives pursued by the Macromolecular Diffraction Facility at the Cornell High Energy Synchrotron Source (MacCHESS) are presented.

Computational foundations of basic recursive function theory

The theory of computability often called basic recursive function theory is usually motivated and developed using Churchs thesis. It is shown that there is an alternative computability theory in which some of the basic results on unsolvability become more absolute. Results on completeness become simpler, and many of the central concepts become more abstract. In this approach computations are viewed as mathematical objects, and the major theorems in recursion theory may be classified according to which axioms about computation are needed to prove them. The theory is a typed theory of functions over the natural numbers, and there are unsolvable problems in this setting independent of the existence of indexings. The unsolvability results are interpreted to show that the partial function concept serves to distinguish between classical and constructive type theories.<<ETX>>

HYBRID PARTIAL-TOTAL TYPE THEORY

In this paper a hybrid type theory HTT is defined which combines the programming language notion of partial type and the logical notion of total type into a single theory. A new partial type constructor is added to the type theory: objects in may diverge, but if they converge, they must be members of the type A. A fixed point typing rule is given to allow for typing of fixed points. The underlying theory is based on ideas from Martin-Lof’s Intuitionistic Type Theory and Feferman’s Class Theory. The extraction paradigm of constructive type theory is extended to allow direct extraction of arbitrary fixed points. Important features of general programming logics such as LCF are preserved, including the typing of all partial recursive functions, a partial ordering on computations, and a fixed point induction principle. The resulting theory is thus intended as a general-purpose programming logic. Rules are presented and soundness of the theory established.

Adventures in scarcity: collecting, processing and understanding sparse data in serial microcrystallography

There is a growing interest in pursuing serial microcrystallography (SMX) experiments at existing storage ring (SR) sources. For very small crystals, radiation damage occurs before sufficient diffraction is recorded to determine the orientation of the crystal. The challenge is to merge data from a large number of such “sparse” frames in order to measure the full reciprocal space intensity. With the EMC algorithm, we show that the diffracted intensity of a crystal can still be reconstructed even without knowledge of the orientation of the crystal in any sparse frame. Recent results show that EMC-based SMX experiments should be feasible at SR sources.

In-line SEC-SAXS and MALS/DLS/RI for the Analysis of Polydisperse Macromolecules

Small Angle X-ray Scattering (SAXS) is a powerful tool for the structural analysis of biological macromolecules in solution and has seen a surge in popularity amongst structural biologists in the past decade. In part, this is because SAXS benefits greatly from the sensitivity and throughput that can be achieved at modern high brightness synchrotron sources. However, the critical need for highly monodisperse samples in SAXS analysis can be a challenge, and as such a number of labs have moved to develop in-line Size Exclusion Chromatography (SEC) at the beamline. Real-time SAXS on elution profiles not only improves monodispersity of samples and provides information on possible oligomeric states, but it also offers new modes of data analysis that can take advantage of the inherent concentration profiles underlying elution peaks and distributions of partially resolved species. Efforts to extend the synergy between SEC and SAXS to other biophysical methods are ongoing. The newly commissioned G1 BioSAXS facility at MacCHESS now offers the option of combining real-time SEC-SAXS with multi-angle static (MALS) and dynamic (DLS) light scattering along with refractive index (RI) detection. In this talk we give a brief overview of the performance and capabilities of the new BioSAXS station at MacCHESS with emphasis on detection limits and signal quality. We then discuss how the complementary light scattering techniques can be combined to offer new insights for complex inhomogeneous samples in terms of biological information and data quality assessment. We also discuss the limitations and possible future developments of these approaches as biologists seek to investigate more dynamic systems as well as shorter time scales.

Reflective semantics of constructive type theory

It is well-known that the proof theory of many sufficiently powerful logics may be represented internally by G6delization. Here we show that for Constructive Type Theory, it is furthermore possible to define a semantics of the types in the type theory itself, and that this proce.dure results in new reemoning principles for type theory. Paradoxes are avoided by stratifying the definition in layers. 1 I n t r o d u c t i o n Given a sufficiently powerful logical theory L such as Peano Arithmethic, it is well-known that the proof theory of L may be expressed internally via GSdelization. This is accomplished by defining a metafunctlon [A] that encodes formulas A as data, and a predicate ProvableL( [A] ) which is true just when formula A is provable. This gives L knowledge of its own proof theory, but it doesnt know that it knows it: the embedded proof theory could just as well be for some different logic. What is needed then are principles of self-knowledge that connect the provability predicate with the actual proof theory. Feferman [Fef62] for one has studied adding such principles; the principle most relevant to this work is Given a theory Lk, extend it to give a theory Lk+l with added reflection axiom ~L~§ P~o~ableL~([A]) ~ A. In other wordsr if we prove that we can prove it, we can prove it. Note that this is not a true self-knowledge principle, because Lk does not have self-knowledge of the reflection axiom. Given a theory L1, this principle induces a hierarchy of theories L1, L2, L3,. . . , which we call the reflected proof hierarchy. Logicians have studied this hierarchy to understand its proof-theoreric strength; here we are not directly interested in this issue. We are interested in how computer scientists have found it applicable for automated theorem proving systems. Suppose there was an assumption A from which we wished to prove B. If the theorem-proving system knew of a way of always proving B from A, not by having assumption A ~ B but by observing some syntactic property of A, the system would still have to construct each step of the proof. Given a reflected proof system, this construction may be avoided. Suppose we were to prove the theorem V [A], [B 1 . [A] is of appropriate syntactic form & ProvableL ( [A] ) ~ ProvableL ( [B 91 ) *emcil scott@cs.jhu.edu, phone (410) 516-5299, fax (410) 516-6134.