Transducer degrees: atoms, infima and suprema

Although finite state transducers are very natural and simple devices, surprisingly little is known about the transducibility relation they induce on streams (infinite words). We collect some intriguing problems that have been unsolved since several years. The transducibility relation arising from finite state transduction induces a partial order of stream degrees, which we call Transducer degrees, analogous to the well-known Turing degrees or degrees of unsolvability. We show that there are pairs of degrees without supremum and without infimum. The former result is somewhat surprising since every finite set of degrees has a supremum if we strengthen the machine model to Turing machines, but also if we weaken it to Mealy machines

Degrees of infinite words, polynomials and atoms

Our objects of study are finite state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount i

Decreasing Diagrams for Confluence and Commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf's theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

Decreasing diagrams with two labels are complete for confluence of countable systems

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract reduction systems, it is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps ? with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract reduction systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels su ce for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. We also show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Finally, as a background theme, we discuss the logical issue of first-order definability of the notion of confluence

Degrees of Infinite Words, Polynomials and Atoms

We study finite-state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount importance in a wide range of fields, from formal languages to pure mathematics and physics. While finite automata for recognising and transforming languages are well-understood, very little is known about the power of automata to transform infinite words.The word transformation realised by finite-state transducers gives rise to a complexity comparison of words and thereby induces equivalence classes, called (transducer) degrees, and a partial order on these degrees. The ensuing hierarchy of degrees is analogous to the recursion-theoretic degrees of unsolvability, also known as Turing degrees, where the transformational devices are Turing machines. However, as a complexity measure, Turing machines are too strong: they trivialise the classification problem by identifying all computable words. Finite-state transducers give rise to a much more fine-grained, discriminating hierarchy. In contrast to Turing degrees, hardly anything is known about transducer degrees, in spite of their naturality.We use methods from linear algebra and analysis to show that there are infinitely many atoms in the transducer degrees, that is, minimal non-trivial degrees

Decreasing diagrams for confluence and commutation

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract rewrite systems. It is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps with labels from a wellfounded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract rewrite systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. Secondly, we show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Thirdly, investigating the possibility of a confluence hierarchy, we determine the first-order (non-)definability of the notion of confluence and related properties, using techniques from finite model theory. We find that in particular Hanf ’s theorem is fruitful for elegant proofs of undefinability of properties of abstract rewrite systems

Mechanisms of transversion mutation are dependent on sequence context and nucleotide paucity during antibody somatic hypermutation

Somatic hypermutation of antibodies during humoral immune responses depends on expression of Activation Induced Deaminase (AID) in antibody-producing B cells. AID initiates somatic hypermutation by converting cytosine (C) residues in antibody genes into uracil (U) residues, by deamination. Alone, conversion of cytosine into uracil can only produce C:G to T:A transition mutations, by replication across U (phase 1A mutation). Processing of C deaminations by base excision repair (BER) or mismatch repair (MMR) diversifies mutation, predominantly at C:G (phase 1B mutation) and A:T (phase 2 mutation), respectively. Mutations at C along the Ig variable region are not equally distributed. AID de-aminates C more often if they occur as part of WRCY motif (A/T,A/G,C,C/T). WRCY sequences are concentrated in hypervariable regions of Ig genes, where nucleotide substitutions are likely to be effective at generating useful amino acid substitutions to optimize affinity maturation. Of all WRCY motifs, AGCT and AACT are the most mutated hotspots. AGCT is also enriched in switch regions and facilitates CSR. In Chapter three, using large datasets of a transgenic mouse model, I compared Igh hypermutation between SWHEL B cells, SWHEL B cells deficient for UNG2 via retroviral expression of the uracil glycosylase inhibitor (ugi), SWHEL B cells deficient for MutSα by crossing Msh2ko alleles into SWHEL mice and SWHEL B cells deficient for both UNG2 and MutSα. I found that phase 1B mutations occur by distinct MMR-independent or MMR dependent pathways. At or in proximity to AGCW motifs, phase 1B mutations were driven by UNG2 without requirement for mismatch repair. Deaminations in AGCW were refractive both to processing by UNG2 and to high-fidelity base excision repair (BER) downstream of UNG2, regardless of mismatch repair activity. Outside AGCW motifs, transversions at C:G are co-dependent on UNG2 and MMR. Classically, MMR mediates high fidelity repair of mismatches introduced during replication. The reasons for the profound differences in repair accuracy between classical and AID-induced MMR have not been elicited. During S-phase of the cell replication cycle, when classical post-replication MMR occurs, nucleotide triphosphate (dNTP) levels are optimal for DNA replication, while in G1-phase dNTP levels are lower. Since there is evidence that AID is active in G1-phase, we hypothesized that low dNTP levels may be the cause of low fidelity MMR. Two enzymes are the major determinant of dNTP pools: ribonucleotide reductase (RNR), which converts ribonucleotides into deoxyribonucleotides predominantly during S-phase, and SAMHD1, which degrades dNTPs predominantly outside of S-phase. In Chapters four and five, I quantified antibody hypermutation in B cells lacking SAMHD1 and/or over-expressing RNR. I observed a 2-fold decrease in mutations at A:T bases in cells lacking SAMHD1. This decrease was comparable to the decrease induced by RNR over-expression and was consistent with our hypothesis. Unexpectedly, loss of SAMHD1 also decreased transversion mutations at C:G by about 70%, and almost doubled transition mutations at C:G bases. RNR over-expression had no obvious impact on transversion mutations at C:G, but increased transition mutations at C:G bases similarly to loss of SAMHD1. Furthermore, loss of SAMHD1 decreased AID/BER-dependent antibody class switch recombination, while RNR over-expression did not. These findings indicate that dNTPs play a role in MMR-mediated antibody mutation, as predicted by our hypothesis, but they also indicate a major role for SAMHD1 in AID-induced BER that was not predicted by our hypothesis or by current models of antibody hypermutation. This important finding warrants further investigation to identify the mechanism

Decreasing Diagrams with Two Labels Are Complete for Confluence of Countable Systems

Like termination, confluence is a central property of rewrite systems. Unlike for termination, however, there exists no known complexity hierarchy for confluence. In this paper we investigate whether the decreasing diagrams technique can be used to obtain such a hierarchy. The decreasing diagrams technique is one of the strongest and most versatile methods for proving confluence of abstract reduction systems, it is complete for countable systems, and it has many well-known confluence criteria as corollaries. So what makes decreasing diagrams so powerful? In contrast to other confluence techniques, decreasing diagrams employ a labelling of the steps -> with labels from a well-founded order in order to conclude confluence of the underlying unlabelled relation. Hence it is natural to ask how the size of the label set influences the strength of the technique. In particular, what class of abstract reduction systems can be proven confluent using decreasing diagrams restricted to 1 label, 2 labels, 3 labels, and so on? Surprisingly, we find that two labels suffice for proving confluence for every abstract rewrite system having the cofinality property, thus in particular for every confluent, countable system. We also show that this result stands in sharp contrast to the situation for commutation of rewrite relations, where the hierarchy does not collapse. Finally, as a background theme, we discuss the logical issue of first-order definability of the notion of confluence

Taking Gene Therapy into the Clinic

Gene therapy represents a promising novel treatment strategy for colorectal cancer. Preclinical data has been encouraging and several clinical trials are underway. Many phase 1 trials have proven the safety of the reagents but have yet to demonstrate significant therapeutic benefit. Ongoing efforts are being made to improve the efficiency of gene delivery and accuracy of gene targeting with the aim of enhancing antitumor potency. It is envisaged that gene therapy will be used in combination with other therapies including surgery, chemotherapy, and radiotherapy to facilitate the improvements in cancer treatments in the future