Theory Morphisms in Church's Type Theory with Quotation and Evaluation

${\rm CTT}_{\rm qe}$ is a version of Church's type theory with global quotation and evaluation operators that is engineered to reason about the interplay of syntax and semantics and to formalize syntax-based mathematical algorithms. ${\rm CTT}_{\rm uqe}$ is a variant of ${\rm CTT}_{\rm qe}$ that admits undefined expressions, partial functions, and multiple base types of individuals. It is better suited than ${\rm CTT}_{\rm qe}$ as a logic for building networks of theories connected by theory morphisms. This paper presents the syntax and semantics of ${\rm CTT}_{\rm uqe}$, defines a notion of a theory morphism from one ${\rm CTT}_{\rm uqe}$ theory to another, and gives two simple examples that illustrate the use of theory morphisms in ${\rm CTT}_{\rm uqe}$.Comment: 17 page

Formalizing Mathematical Knowledge as a Biform Theory Graph: A Case Study

A biform theory is a combination of an axiomatic theory and an algorithmic theory that supports the integration of reasoning and computation. These are ideal for formalizing algorithms that manipulate mathematical expressions. A theory graph is a network of theories connected by meaning-preserving theory morphisms that map the formulas of one theory to the formulas of another theory. Theory graphs are in turn well suited for formalizing mathematical knowledge at the most convenient level of abstraction using the most convenient vocabulary. We are interested in the problem of whether a body of mathematical knowledge can be effectively formalized as a theory graph of biform theories. As a test case, we look at the graph of theories encoding natural number arithmetic. We used two different formalisms to do this, which we describe and compare. The first is realized in ${\rm CTT}_{\rm uqe}$, a version of Church's type theory with quotation and evaluation, and the second is realized in Agda, a dependently typed programming language.Comment: 43 pages; published without appendices in: H. Geuvers et al., eds, Intelligent Computer Mathematics (CICM 2017), Lecture Notes in Computer Science, Vol. 10383, pp. 9-24, Springer, 201

Automatically proving equivalence by type-safe reflection

We are also grateful for the support of the Scottish Informatics and Computer Science Alliance (SICSA) and EPSRC grant EP/N024222/1.One difficulty with reasoning and programming with dependent types is that proof obligations arise naturally once programs become even moderately sized. For example, implementing an adder for binary numbers indexed over their natural number equivalents naturally leads to proof obligations for equalities of expressions over natural numbers. The need for these equality proofs comes, in intensional type theories, from the fact that the propositional equality enables us to prove as equal terms that are not judgementally equal, which means that the typechecker can’t always obtain equalities by reduction. As far as possible, we would like to solve such proof obligations automatically. In this paper, we show one way to automate these proofs by reflection in the dependently typed programming language Idris. We show how defining reflected terms indexed by the original Idris expression allows us to construct and manipulate proofs. We build a hierarchy of tactics for proving equivalences in semi-groups, monoids, commutative monoids, groups, commutative groups, semi-rings and rings. We also show how each tactic reuses those from simpler structures, thus avoiding duplication of code and proofs.Postprin

Adding an Abstraction Barrier to ZF Set Theory

Much mathematical writing exists that is, explicitly or implicitly, based on set theory, often Zermelo-Fraenkel set theory (ZF) or one of its variants. In ZF, the domain of discourse contains only sets, and hence every mathematical object must be a set. Consequently, in ZF, with the usual encoding of an ordered pair ${\langle a, b\rangle}$, formulas like ${\{a\} \in \langle a, b \rangle}$ have truth values, and operations like ${\mathcal P (\langle a, b\rangle)}$ have results that are sets. Such 'accidental theorems' do not match how people think about the mathematics and also cause practical difficulties when using set theory in machine-assisted theorem proving. In contrast, in a number of proof assistants, mathematical objects and concepts can be built of type-theoretic stuff so that many mathematical objects can be, in essence, terms of an extended typed ${\lambda}$-calculus. However, dilemmas and frustration arise when formalizing mathematics in type theory. Motivated by problems of formalizing mathematics with (1) purely set-theoretic and (2) type-theoretic approaches, we explore an option with much of the flexibility of set theory and some of the useful features of type theory. We present ZFP: a modification of ZF that has ordered pairs as primitive, non-set objects. ZFP has a more natural and abstract axiomatic definition of ordered pairs free of any notion of representation. This paper presents axioms for ZFP, and a proof in ZF (machine-checked in Isabelle/ZF) of the existence of a model for ZFP, which implies that ZFP is consistent if ZF is. We discuss the approach used to add this abstraction barrier to ZF

Automating Change of Representation for Proofs in Discrete Mathematics (Extended Version)

Representation determines how we can reason about a specific problem. Sometimes one representation helps us find a proof more easily than others. Most current automated reasoning tools focus on reasoning within one representation. There is, therefore, a need for the development of better tools to mechanise and automate formal and logically sound changes of representation. In this paper we look at examples of representational transformations in discrete mathematics, and show how we have used Isabelle's Transfer tool to automate the use of these transformations in proofs. We give a brief overview of a general theory of transformations that we consider appropriate for thinking about the matter, and we explain how it relates to the Transfer package. We show our progress towards developing a general tactic that incorporates the automatic search for representation within the proving process

Designing an automated clinical decision support system to match clinical practice guidelines for opioid therapy for chronic pain

Abstract Background Opioid prescribing for chronic pain is common and controversial, but recommended clinical practices are followed inconsistently in many clinical settings. Strategies for increasing adherence to clinical practice guideline recommendations are needed to increase effectiveness and reduce negative consequences of opioid prescribing in chronic pain patients. Methods Here we describe the process and outcomes of a project to operationalize the 2003 VA/DOD Clinical Practice Guideline for Opioid Therapy for Chronic Non-Cancer Pain into a computerized decision support system (DSS) to encourage good opioid prescribing practices during primary care visits. We based the DSS on the existing ATHENA-DSS. We used an iterative process of design, testing, and revision of the DSS by a diverse team including guideline authors, medical informatics experts, clinical content experts, and end-users to convert the written clinical practice guideline into a computable algorithm to generate patient-specific recommendations for care based upon existing information in the electronic medical record (EMR), and a set of clinical tools. Results The iterative revision process identified numerous and varied problems with the initially designed system despite diverse expert participation in the design process. The process of operationalizing the guideline identified areas in which the guideline was vague, left decisions to clinical judgment, or required clarification of detail to insure safe clinical implementation. The revisions led to workable solutions to problems, defined the limits of the DSS and its utility in clinical practice, improved integration into clinical workflow, and improved the clarity and accuracy of system recommendations and tools. Conclusions Use of this iterative process led to development of a multifunctional DSS that met the approval of the clinical practice guideline authors, content experts, and clinicians involved in testing. The process and experiences described provide a model for development of other DSSs that translate written guidelines into actionable, real-time clinical recommendations.http://deepblue.lib.umich.edu/bitstream/2027.42/78267/1/1748-5908-5-26.xmlhttp://deepblue.lib.umich.edu/bitstream/2027.42/78267/2/1748-5908-5-26.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/78267/3/1748-5908-5-26-S3.TIFFhttp://deepblue.lib.umich.edu/bitstream/2027.42/78267/4/1748-5908-5-26-S2.TIFFhttp://deepblue.lib.umich.edu/bitstream/2027.42/78267/5/1748-5908-5-26-S1.TIFFPeer Reviewe

Modeling and discretization of flow in porous media with thin, full-tensor permeability inclusions

When modeling fluid flow in fractured reservoirs, it is common to represent the fractures as lower-dimensional inclusions embedded in the host medium. Existing discretizations of flow in porous media with thin inclusions assume that the principal directions of the inclusion permeability tensor are aligned with the inclusion orientation. While this modeling assumption works well with tensile fractures, it may fail in the context of faults, where the damage zone surrounding the main slip surface may introduce anisotropy that is not aligned with the main fault orientation. In this article, we introduce a generalized dimensional reduced model which preserves full-tensor permeability effects also in the out-of-plane direction of the inclusion. The governing equations of flow for the lower-dimensional objects are obtained through vertical averaging. We present a framework for discretization of the resulting mixed-dimensional problem, aimed at easy adaptation of existing simulation tools. We give numerical examples that show the failure of existing formulations when applied to anisotropic faulted porous media, and go on to show the convergence of our method in both two-dimensional and three-dimensional.publishedVersio

Smc5/6 coordinates formation and resolution of joint molecules with chromosome morphology to ensure meiotic divisions

During meiosis, Structural Maintenance of Chromosome (SMC) complexes underpin two fundamental features of meiosis: homologous recombination and chromosome segregation. While meiotic functions of the cohesin and condensin complexes have been delineated, the role of the third SMC complex, Smc5/6, remains enigmatic. Here we identify specific, essential meiotic functions for the Smc5/6 complex in homologous recombination and the regulation of cohesin. We show that Smc5/6 is enriched at centromeres and cohesin-association sites where it regulates sister-chromatid cohesion and the timely removal of cohesin from chromosomal arms, respectively. Smc5/6 also localizes to recombination hotspots, where it promotes normal formation and resolution of a subset of joint-molecule intermediates. In this regard, Smc5/6 functions independently of the major crossover pathway defined by the MutLγ complex. Furthermore, we show that Smc5/6 is required for stable chromosomal localization of the XPF-family endonuclease, Mus81-Mms4Eme1. Our data suggest that the Smc5/6 complex is required for specific recombination and chromosomal processes throughout meiosis and that in its absence, attempts at cell division with unresolved joint molecules and residual cohesin lead to severe recombination-induced meiotic catastroph

The Evolution of Compact Binary Star Systems

We review the formation and evolution of compact binary stars consisting of white dwarfs (WDs), neutron stars (NSs), and black holes (BHs). Binary NSs and BHs are thought to be the primary astrophysical sources of gravitational waves (GWs) within the frequency band of ground-based detectors, while compact binaries of WDs are important sources of GWs at lower frequencies to be covered by space interferometers (LISA). Major uncertainties in the current understanding of properties of NSs and BHs most relevant to the GW studies are discussed, including the treatment of the natal kicks which compact stellar remnants acquire during the core collapse of massive stars and the common envelope phase of binary evolution. We discuss the coalescence rates of binary NSs and BHs and prospects for their detections, the formation and evolution of binary WDs and their observational manifestations. Special attention is given to AM CVn-stars -- compact binaries in which the Roche lobe is filled by another WD or a low-mass partially degenerate helium-star, as these stars are thought to be the best LISA verification binary GW sources.Comment: 105 pages, 18 figure