Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied

Pattern matching in compilers

In this thesis we develop tools for effective and flexible pattern matching. We introduce a new pattern matching system called amethyst. Amethyst is not only a generator of parsers of programming languages, but can also serve as an alternative to tools for matching regular expressions. Our framework also produces dynamic parsers. Its intended use is in the context of IDE (accurate syntax highlighting and error detection on the fly). Amethyst offers pattern matching of general data structures. This makes it a useful tool for implementing compiler optimizations such as constant folding, instruction scheduling, and dataflow analysis in general. The parsers produced are essentially top-down parsers. Linear time complexity is obtained by introducing the novel notion of structured grammars and regularized regular expressions. Amethyst uses techniques known from compiler optimizations to produce effective parsers.Comment: master thesi

A theory and its metatheory in FS 0

Feferman has proposed FS0, a theory of finitary inductive systems, as a framework theory suitable for various purposes, including reasoning both in and about encoded theories. I look here at how practical FS0 really is. I formalise of a sequent calculus presentation of classical propositional logic in FS0 and show this can be used for work in both the theory and the metatheory. the latter is illustrated with a discussion of a proof of Gentzen's Hauptsatz

(HO)RPO Revisited

The notion of computability closure has been introduced for proving the termination of the combination of higher-order rewriting and beta-reduction. It is also used for strengthening the higher-order recursive path ordering. In the present paper, we study in more details the relations between the computability closure and the (higher-order) recursive path ordering. We show that the first-order recursive path ordering is equal to an ordering naturally defined from the computability closure. In the higher-order case, we get an ordering containing the higher-order recursive path ordering whose well-foundedness relies on the correctness of the computability closure. This provides a simple way to extend the higher-order recursive path ordering to richer type systems

Enhanced Realizability Interpretation for Program Extraction

This thesis presents Intuitionistic Fixed Point Logic (IFP), a schema for formal systems aimed to work with program extraction from proofs. IFP in its basic form allows proof construction based on natural deduction inference rules, extended by induction and coinduction. The corresponding system RIFP (IFP with realiz-ers) enables transforming logical proofs into programs utilizing the enhanced re-alizability interpretation. The theoretical research is put into practice in PRAWF1, a Haskell-based proof assistant for program extraction

Structural Studies of Three Factors That Affect the Prokaryotic Transcription Cycle; Microcin J25, LAMBDA Q and T4 GP33

This thesis examines three factors that affect the prokaryotic transcription cycle; Microcin J25, a peptidyl inhibitor of RNAP; Q antiterminator protein, a bacteriophage protein that modifies RNAP to readthrough termination signals in phage late genes; and Gp33, a T4 phage protein that is a co-activator of T4 phage late gene transcription. We determined that the Microcin J25 peptide (MccJ25) has a lassoed-tail structure using mass spectrometry and NMR. The 21 amino acid MccJ25 peptide is produced from a 58 amino acid pro-peptide, McjA, that is truncated and post-translationally modified by two maturase enzymes. We synthesized McjA and constructed a protein-A tagged biosynthesis cassette to be used in the analysis of MccJ25 maturation. Escherichia coli (Ec) RNAP has not, thus far, been amenable to crystallization despite considerable effort. One potential hindrance to crystallization may be the carboxy terminal domain of the alpha subunit (α-CTD, residues 249-329) that is connected to an amino-terminal domain (α-NTD, residues 8-235) by a 14-residue linker. We successfully expressed, purified and crystallized a version of Ec RNAP without the α-CTD. The Q antiterminator proteins from lambdoid phage, modify Ec RNAP to read through termination sites within the phage late gene. A structure of Q protein could reveal how Q antitermination occurs and may provide insight into the mechanism of termination. We developed an improved expression and purification protocol for λQ, and performed biophysical characterization on both λQ and the related 82Q. The T4 phage protein Gp33 is a co-activator that interacts with both Ec RNAP and T4 DNA replication machinery to upregulate transcription from T4 late promoters. Gp33 is thought to have an analogous function to E. coli σ704 in that it binds the beta flap of RNAP and helps recruit RNAP to promoters. Ec RNAP bound to Gp33 is recruited to the promoter by a unique mechanism through a protein-protein interaction between Gp33 and the DNA-bound sliding clamp of the T4 DNA replisome, Gp45. We obtained crystals of Gp33 bound to the Ec Beta flap that diffracted to 3.0 Ångstroms. The preliminary structure shows Gp33 as a mainly helical protein that wraps around the Beta flap

Finite temperature quantum simulation of stabilizer Hamiltonians

We present a scheme for robust finite temperature quantum simulation of stabilizer Hamiltonians. The scheme is designed for realization in a physical system consisting of a finite set of neutral atoms trapped in an addressable optical lattice that are controllable via 1- and 2-body operations together with dissipative 1-body operations such as optical pumping. We show that these minimal physical constraints suffice for design of a quantum simulation scheme for any stabilizer Hamiltonian at either finite or zero temperature. We demonstrate the approach with application to the abelian and non-abelian toric codes.Comment: 13 pages, 2 figure