On the alleged simplicity of impure proof

Roughly, a proof of a theorem, is “pure” if it draws only on what is “close” or “intrinsic” to that theorem. Mathematicians employ a variety of terms to identify pure proofs, saying that a pure proof is one that avoids what is “extrinsic,” “extraneous,” “distant,” “remote,” “alien,” or “foreign” to the problem or theorem under investigation. In the background of these attributions is the view that there is a distance measure (or a variety of such measures) between mathematical statements and proofs. Mathematicians have paid little attention to specifying such distance measures precisely because in practice certain methods of proof have seemed self- evidently impure by design: think for instance of analytic geometry and analytic number theory. By contrast, mathematicians have paid considerable attention to whether such impurities are a good thing or to be avoided, and some have claimed that they are valuable because generally impure proofs are simpler than pure proofs. This article is an investigation of this claim, formulated more precisely by proof- theoretic means. After assembling evidence from proof theory that may be thought to support this claim, we will argue that on the contrary this evidence does not support the claim

Functional Characterization of EngAMS, a P-Loop GTPase of Mycobacterium smegmatis

Bacterial P-loop GTPases belong to a family of proteins that selectively hydrolyze a small molecule guanosine tri-phosphate (GTP) to guanosine di-phosphate (GDP) and inorganic phosphate, and regulate several essential cellular activities such as cell division, chromosomal segregation and ribosomal assembly. A comparative genome sequence analysis of different mycobacterial species indicates the presence of multiple P-loop GTPases that exhibit highly conserved motifs. However, an exact function of most of these GTPases in mycobacteria remains elusive. In the present study we characterized the function of a P-loop GTPase in mycobacteria by employing an EngA homologue from Mycobacterium smegmatis, encoded by an open reading frame, designated as MSMEG_3738. Amino acid sequence alignment and phylogenetic analysis suggest that MSMEG_3738 (termed as EngAMS) is highly conserved in mycobacteria. Homology modeling of EngAMS reveals a cloverleaf structure comprising of α/β fold typical to EngA family of GTPases. Recombinant EngAMS purified from E. coli exhibits a GTP hydrolysis activity which is inhibited by the presence of GDP. Interestingly, the EngAMS protein is co-eluted with 16S and 23S ribosomal RNA during purification and exhibits association with 30S, 50S and 70S ribosomal subunits. Further studies demonstrate that GTP is essential for interaction of EngAMS with 50S subunit of ribosome and specifically C-terminal domains of EngAMS are required to facilitate this interaction. Moreover, EngAMS devoid of N-terminal region interacts well with 50S even in the absence of GTP, indicating a regulatory role of the N-terminal domain in EngAMS-50S interaction

Limiting programs for induction in artificial intelligence

This thesis examines a novel induction-based frameworkfor logic programming. Limiting programs are logic programs distinguishedby two features, in general they contain an infinite data streamover which induction will be performed, and in general it is not possiblefor a system to know when a solution for any program is correct. Thesefacts are characteristic of some problems involving induction in artificialintelligence, and several problems in knowledge representation andlogic programming have exactly these properties. This thesis presentsa specification language for problems with an inductive nature, limitingprograms, and a resolution based system, limiting resolution, for solvingthese problems. This framework has properties which guaranteethat the system will converge upon a particular answer in the limit.Solutions to problems which have such an inductive property bynature can be implemented using the language, and solved with thesolver. For instance, many classification problems are inductive bynature. Some generalized planning problems also have the inductiveproperty. For a class of generalized planning problems, we show thatidentifying a collection of domains where a plan reaches a goal is equivalentto producing a plan. This thesis gives examples of both.Limiting resolution works by a generate-and-test strategy, creatinga potential solution and iteratively looking for a contradiction with thegrowing stream of data provided. Limiting resolution can be implementedby modifying conventional PROLOG technology. The generateand-test strategy has some inherent inefficiencies. Two improvementshave arisen from this work; the first is a tabling strategy which recordspreviously failed attempts to produce a solution and thereby avoids redundanttest steps. The second is based on the heuristic observationthat for some problems the size of the test step is proportional to thecloseness of the generated potential-solution to the real solution, in asuitable metric. The observation can be used to improve the performanceof limiting resolution.Thus this thesis describes, from theoretical foundations to implementation,a coherent methodology for incorporating induction intoexisting general A.I. programming techniques, along with examples ofhow to perform such tasks

Polygenic risk scores for prediction of breast cancer and breast cancer subtypes

Abstract Stratification of women according to their risk of breast cancer based on polygenic risk scores (PRSs) could improve screening and prevention strategies. Our aim was to develop PRSs, optimized for prediction of estrogen receptor (ER)-specific disease, from the largest available genome-wide association dataset and to empirically validate the PRSs in prospective studies. The development dataset comprised 94,075 case subjects and 75,017 control subjects of European ancestry from 69 studies, divided into training and validation sets. Samples were genotyped using genome-wide arrays, and single-nucleotide polymorphisms (SNPs) were selected by stepwise regression or lasso penalized regression. The best performing PRSs were validated in an independent test set comprising 11,428 case subjects and 18,323 control subjects from 10 prospective studies and 190,040 women from UK Biobank (3,215 incident breast cancers). For the best PRSs (313 SNPs), the odds ratio for overall disease per 1 standard deviation in ten prospective studies was 1.61 (95%CI: 1.57–1.65) with area under receiver-operator curve (AUC) = 0.630 (95%CI: 0.628–0.651). The lifetime risk of overall breast cancer in the top centile of the PRSs was 32.6%. Compared with women in the middle quintile, those in the highest 1% of risk had 4.37- and 2.78-fold risks, and those in the lowest 1% of risk had 0.16- and 0.27-fold risks, of developing ER-positive and ER-negative disease, respectively. Goodness-of-fit tests indicated that this PRS was well calibrated and predicts disease risk accurately in the tails of the distribution. This PRS is a powerful and reliable predictor of breast cancer risk that may improve breast cancer prevention programs