Proof compressions with circuit-structured substitutions

It is well known that the size of propositional classical proofs can be huge. Proof-theoretical studies discovered exponential gaps between cut-free (or normal) proofs and the corresponding (non-normal) proofs with cuts (or modus ponens). The task of automatic theorem proving is, on the other hand, usually based on the construction of cut-free or only atomic-cuts proofs, since this procedure produces less alternative choices. There are familiar tautologies whose cut-free proofs are huge while the non-cut-free ones are small. The aim of this paper is to discuss basic methods of weight and/or size reduction of deductions by switching from traditional tree-structured deductions to circuit-structured deductions. A desired efficiency is achieved by adding the standard weakening rule of inference upgraded by adding suitable (propositional) unifications modulo variable substitutions. We show examples where such a unification provides strong (in fact, exponential) compression of cut-free deductions. Bibliography: 10 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 358, 2008, pp. 77–99.     

Synthesis of S,N-ketenacetals from β-dicarbonyl compounds and methyl thiocyanate

Translated from Izvestiya Akademii Nauk SSSR, Seriya Khimicheskaya, No. 4, pp. 941–942, April, 1988.     

Reflectometer for studying the surface of liquid systems

To study the free surface of liquids, a Reverans reflectometer with a vertical neutron scattering plane has been designed and is fabricated at Petersburg Nuclear Physics Institute. This reflectometer is intended to investigate surfaces and interfaces in liquid magnetic or nonmagnetic systems. The principles of reflectometer operation are described. The general view of the reflectometer with ready units is presented and its main technical parameters are given. The results of preliminary tests are reported.     

Fracture toughness and crystallographic characteristics of Li6GdB3O9 single crystals

Characteristics of the structure of single crystals of binary lithium-gadolinium borate Li₆GdB₃O₉ with mixed-type bonding (ionic and covalent) are studied. The system of cleavage planes in Li₆GdB₃O₉ single crystals is associated with the break of the Gd-O bridging bonds and the longest Li-O bonds in the Li five-vertex polyhedra. The fracture toughness in the cleavage planes is determined by microindentation. It is shown that the character of resistance to deformation and fracture toughness are determined by the orientation of the cleavage planes with respect to the faces where boron-oxygen triangles with directional covalent bonds are located.     

Study of the Thermal Behavior of Wedge-Shaped Samples of HfB<Subscript>2</Subscript>–45 vol % SiC Ultra-High-Temperature Composite in a High-Enthalpy Air Flow

By spark plasma sintering, HfB₂–45 vol % SiC ultra-high-temperature ceramic was prepared, from which wedge-shaped samples were cut. The behavior of the samples was examined in a flow of dissociated air produced by an induction plasmatron, where the surface temperature of the leading edges of the samples reached ~2700°C. The dependence of the temperature distribution gradient on the distance from the leading edges of the samples was experimentally investigated. For the samples after the experiments, the elemental and phase compositions were determined, and features of the surface microstructure in various regions of the sample and on its polished surface were studied.     

μSR-study of reactivity of substituent ions and point-like defects in quartz monocrystals

SR-method has been used for studying reactions of muonium as an analogue of a light hydrogen isotope with impurity atoms and point-like defects. The relaxation rate of muonium polarization _Mu was measured for 11 samples with different concentrations of impurity atoms and defects. The rate constants of reactions of muonium with Fe, Al, Li, Ba, Cu, Mn, Ti and Pb atoms in quartz were defined.