Deriving the full-reducing Krivine machine from the small-step operational semantics of normal order

We derive by program transformation Pierre Crégut s full-reducing Krivine machine KN from the structural operational semantics of the normal order reduction strategy in a closure-converted pure lambda calculus. We thus establish the correspondence between the strategy and the machine, and showcase our technique for deriving full-reducing abstract machines. Actually, the machine we obtain is a slightly optimised version that can work with open terms and may be used in implementations of proof assistants.

A syntactic and functional correspondence between reduction semantics and reduction-free full normalisers

Olivier Danvy and others have shown the syntactic correspondence between reduction semantics (a small-step semantics) and abstract machines, as well as the functional correspondence between reduction-free normalisers (a big-step semantics) and abstract machines. The correspondences are established by program transformation (so-called interderivation) techniques. A reduction semantics and a reduction-free normaliser are interderivable when the abstract machine obtained from them is the same. However, the correspondences fail when the underlying reduction strategy is hybrid, i.e., relies on another sub-strategy. Hybridisation is an essential structural property of full-reducing and complete strategies. Hybridisation is unproblematic in the functional correspondence. But in the syntactic correspondence the refocusing and inlining-of-iterate-function steps become context sensitive, preventing the refunctionalisation of the abstract machine. We show how to solve the problem and showcase the interderivation of normalisers for normal order, the standard, full-reducing and complete strategy of the pure lambda calculus. Our solution makes it possible to interderive, rather than contrive, full-reducing abstract machines. As expected, the machine we obtain is a variant of Pierre Crégut s full Krivine machine KN.

Canonical simplification of finite objects, well quasi-ordered by tree embedding /

"UILU-ENG 79 1727."

Modeling obligations with event-calculus

Time plays an important role in norms. In this paper we start from our previously proposed classification of obligations, and point out some shortcomings of Event Calculus (EC) to represent obligations. We proposed an extension of EC that avoids such shortcomings and we show how to use it to model the various types of obligations.

Insight into the fractional calculus via a spreadsheet

Many students of calculus are not aware that the calculus they have learned is a special case (integer order) of fractional calculus. Fractional calculus is the study of arbitrary order derivatives and integrals and their applications. The article begins by stating a naive question from a student in a paper by Larson (1974) and establishes, for polynomials and exponential functions, that they can be deformed into their derivative using the μ-th order fractional derivatives for 0<μ<1. Through the power of Excel we illustrate the continuous deformations dynamically through conditional formatting. Some applications are discussed and a connection made to mathematics education.

Permissions in Deontic Event-Calculus

Permissions are special case of deontic effects and play important role compliance. Essentially they are used to determine the obligations or prohibitions to contrary. A formal language e.g., temporal logic, event-calculus et., not able to represent permissions is doomed to be unable to represent most of the real-life legal norms. In this paper we address this issue and extend deontic-event-calculus (DEC) with new predicates for modelling permissions enabling it to elegantly capture the intuition of real-life cases of permissions.

Solution of a System of Generalized Abel Integral Equations Using Fractional Calculus

A method is presented for obtaining useful closed form solution of a system of generalized Abel integral equations by using the ideas of fractional integral operators and their applications. This system appears in solving certain mixed boundary value problems arising in the classical theory of elasticity.

A computational algorithm for the verification of tautologies in propositional calculus

A computational algorithm (based on Smullyan's analytic tableau method) that varifies whether a given well-formed formula in propositional calculus is a tautology or not has been implemented on a DEC system 10. The stepwise refinement approch of program development used for this implementation forms the subject matter of this paper. The top-down design has resulted in a modular and reliable program package. This computational algoritlhm compares favourably with the algorithm based on the well-known resolution principle used in theorem provers.

Phage lambda beta protein, a component of general recombination, is associated with host ribosomal S1 protein

We have purified phage lambda beta protein produced by a recombinant plasmid carrying bet gene and confirm that it forms a complex with a protein of relative molecular mass 70 kDa. Therefore, beta protein, a component of general genetic recombination, is associated with two functionally diverse complexes; one containing exonuclease and the other 70 kDa protein. Using a number of independent methods, we show that 70 kDa protein is the ribosomal S1 protein of E. coli. Further, the association of 70 kDa protein with beta protein is biologically significant, as the former inhibits joining of the terminal ends of lambda chromosome and renaturation of complementary single stranded DNA promoted by the latter. More importantly, these findings initiate an understanding of an important mode of host- virus interaction in general with specific implication(s) in homologous genetic recombination.

Formation of linear plasmid multimers promoted by the phage lambda Red-system in lon mutants of Escherichia coli

We report here the formation of plasmid linear multimers promoted by the Red-system of phage lambda using a multicopy plasmid comprised of lambda red alpha and red beta genes, under the control of the lambda cI857 repressor. Our observations have revealed that the multimerization of plasmid DNA is dependent on the red beta and recA genes, suggesting a concerted role for these functions in the formation of plasmid multimers. The formation of multimers occurred in a recBCD+ sbcB+ xthA+ lon genetic background at a higher frequency than in the isogenic lon+ host cells. The multimers comprised tandem repeats of monomer plasmid DNA. Treatment of purified plasmid DNA with exonuclease III revealed the presence of free double-chain ends in the molecules. Determination of the size of multimeric DNA, by pulse field gel electrophoresis, revealed that the bulk of the DNA was in the range 50-240 kb, representing approximately 5-24 unit lengths of monomeric plasmid DNA. We provide a conceptual framework for Red-system-promoted formation and enhanced accumulation of plasmid linear multimers in lon mutants of E. coli.

The homologous recombination system of phage lambda. Pairing activities of beta protein

The red genes of phage lambda specify two proteins, exonuclease and beta protein, which are essential for its general genetic recombination in recA- cells. These proteins seem to occur in vivo as an equimolar complex. In addition, beta protein forms a complex with another polypeptide, probably of phage origin, of Mr 70,000. The 70-kDa protein appears to be neither a precursor nor an aggregated form of either exonuclease or beta protein, since antibodies directed against the latter two proteins failed to react with 70-kDa protein on Ouchterlony double diffusion analysis. beta protein promotes Mg2+-dependent renaturation of complementary strands (Kmiec, E., and Holloman, W. K. (1981) J. Biol. Chem. 256, 12636-12639). To look for other pairing activities of beta protein, we developed methods of purification to free it of associated exonuclease. Exonuclease-free beta protein appeared unable to cause the pairing of a single strand with duplex DNA; however, like Escherichia coli single strand binding protein (SSB), beta protein stimulated formation of joint molecules by recA protein from linear duplex DNA and homologous circular single strands. Like recA protein, but unlike SSB, beta protein promoted the joining of the complementary single-stranded ends of phage lambda DNA. beta protein specifically protected single-stranded DNA from digestion by pancreatic DNase. The half-time for renaturation catalyzed by beta protein was independent of DNA concentration, unlike renaturation promoted by SSB and spontaneous renaturation, which are second order reactions. Thus, beta protein resembles recA protein in its ability to bring single-stranded DNA molecules together and resembles SSB in its ability to reduce secondary structure in single-stranded DNA.

Syntheses and structural studies of new bicyclic 1,3-diphenyl-1,3,2.lambda.3,4.lambda.3-diazadiphosphetidines

Reactions of cis-[(C6H5N)PC1]z(1 ) with the difunctional reagents HO(CH2)20H,H (CH3)N(CHz)zN(CH3)HH, (CH3)N(CH& OH, and HO(CHz)30Hi n the presence of triethylamine yield the new bicyclic 1,3,2X3,4h3-diazadiphosphetidines[( C6H5- N)PIZ[-O(CHZ)Zo-l (2), [(C6H5N)PlZ[-(CH3)N(CHZ)ZN(CH3)-l (319 [(C6H~N)PlZ~-(CH3)N(cHZ)20 (4), and [(C6H5 N)P],[-Q(CH2),0-] (5), respectively. The products have been characterized by elemental analyses and IR and NMR spectroscopic data. The structures of 4 and 5 have been determined by single-crystal X-ray analysis. Crystal data for 4: monoclinic, P2,/c, a = 9.823 (2) A, b = 8.608 (1) A, c = 18.423 (3) A, i3 = 90.55 (1)O, Z = 4. Crystal data for 5 monoclinic, P2,/c, a = 9.727 (2) A, b = 8.064 (2) A, c = 19.702 (4) A, @ =I 91.31 (l)', 2 = 4. The structures have been solved by direct methods and refined to R = 0.028 for 4 and R = 0.050 for 5. Compound 4 is the first example of an aminoalkoxy-l,3,2X3,4X3-diazadiphosphetidine. The PzNz ring is slightly puckered in both 4 and 5 and the puckering occurs in a manner opposite to that observed for cis-[(RN)PX],structures.

Syntheses, spectroscopy, structures, and conformations of .lambda.3-cyclotriphosphazanes: role of negative hyperconjugation

The reactions of As-chlorocyclotriphosphazane [EtNPCl], with phenols or trifluoroethanol yield the respective aryloxy- or trifluoroethoxy-containingX 3-cyclotriphosphazanes [EtNP(OR)]3 (R = C6H4Br-4 (2),C 6H5 (3C,6 H3-Mez-3,5 (4), C6H3Mez-2,6 (5), CH2CF3 (6)) as their cis-transisomericmixtures. The products have beencharacterized by IRand NMRspectroscopy. Thecrystalstructuresofboth thecis (2a) and trans(2b) isomer_softhep-bromophenoxy derivative have been determined by X-ray diffraction. Crystal data for 2a: triclinic, P1, a = 9.872(4) A, b = 13.438(6) A, c = 13.548(8) A, CY = 117.02(5)', 0 = 96.00(6)', y = 105.38(4)O, Z = 2, final R = 0.080. Crystal data for 2b: monoclinic, P21/n, a = 12.721(6) A, b = 13.468(7) A, c = 17.882(5) A, /3 = 101.62(3)O, Z = 4, final R = 0.066. The cis isomer exhibits a chair-triaxial conformation and the trans isomer a boat-triaxial conformation. Conformational preferences of X3-cyclotriphosphazanes have been probed by both MNDO and ab initio calculations on model systems [HNPXIp (X = H, F). In addition to vicinal lone pair repulsions, negative hyperconjugative interactions involving the nitrogen lone pairs and adjacent P-X Q* orbitals are found to be important (especially when X is an electronegative substituent) in determining the conformational preferences of X3-cyclotriphosphazanes. The calculations also show that the axial - equatorial conversion at phosphorus has a large activation barrier in these systems