Elaboration in Dependent Type Theory

To be usable in practice, interactive theorem provers need to provide convenient and efficient means of writing expressions, definitions, and proofs. This involves inferring information that is often left implicit in an ordinary mathematical text, and resolving ambiguities in mathematical expressions. We refer to the process of passing from a quasi-formal and partially-specified expression to a completely precise formal one as elaboration. We describe an elaboration algorithm for dependent type theory that has been implemented in the Lean theorem prover. Lean's elaborator supports higher-order unification, type class inference, ad hoc overloading, insertion of coercions, the use of tactics, and the computational reduction of terms. The interactions between these components are subtle and complex, and the elaboration algorithm has been carefully designed to balance efficiency and usability. We describe the central design goals, and the means by which they are achieved

Really Natural Linear Indexed Type Checking

Recent works have shown the power of linear indexed type systems for enforcing complex program properties. These systems combine linear types with a language of type-level indices, allowing more fine-grained analyses. Such systems have been fruitfully applied in diverse domains, including implicit complexity and differential privacy. A natural way to enhance the expressiveness of this approach is by allowing the indices to depend on runtime information, in the spirit of dependent types. This approach is used in DFuzz, a language for differential privacy. The DFuzz type system relies on an index language supporting real and natural number arithmetic over constants and variables. Moreover, DFuzz uses a subtyping mechanism to make types more flexible. By themselves, linearity, dependency, and subtyping each require delicate handling when performing type checking or type inference; their combination increases this challenge substantially, as the features can interact in non-trivial ways. In this paper, we study the type-checking problem for DFuzz. We show how we can reduce type checking for (a simple extension of) DFuzz to constraint solving over a first-order theory of naturals and real numbers which, although undecidable, can often be handled in practice by standard numeric solvers

More On The Connection Between Planar Field Theory And String Theory

We continue work on the connection between world sheet representation of the planar phi^3 theory and string formation. The present article, like the earlier work, is based on the existence of a solitonic solution on the world sheet, and on the zero mode fluctuations around this solution. The main advance made in this paper is the removal of the cutoff and the transition to the continuum limit on the world sheet. The result is an action for the modes whose energies remain finite in this limit (light modes). The expansion of this action about a dense background of graphs on the world sheet leads to the formation of a string.Comment: 27 pages, 3 figure

Cosmology with orthogonal nilpotent superfields

We study the application of a supersymmetric model with two constrained supermultiplets to inflationary cosmology. The first superfield S is a stabilizer chiral superfield satisfying a nilpotency condition of degree 2, S^2=0. The second superfield Phi is the inflaton chiral superfield, which can be combined into a real superfield B=(Phi-Phi*)/2i. The real superfield B is orthogonal to S, S B=0, and satisfies a nilpotency condition of degree 3, B^3=0. We show that these constraints remove from the spectrum the complex scalar sgoldstino, the real scalar inflaton partner (i.e. the "sinflaton"), and the fermionic inflatino. The corresponding supergravity model with de Sitter vacua describes a graviton, a massive gravitino, and one real scalar inflaton, with both the goldstino and inflatino being absent in unitary gauge. We also discuss relaxed superfield constraints where S^2=0 and S Phi* is chiral, which removes the sgoldstino and inflatino, but leaves the sinflaton in the spectrum. The cosmological model building in both of these inflatino-less models offers some advantages over existing constructions.Comment: 20+9 pages; v2: version to appear in PR

Gauge Freedom in Orbital Mechanics

In orbital and attitude dynamics the coordinates and the Euler angles are expressed as functions of the time and six constants called elements. Under disturbance, the constants are endowed with time dependence. The Lagrange constraint is then imposed to guarantee that the functional dependence of the perturbed velocity on the time and constants stays the same as in the undisturbed case. Constants obeying this condition are called osculating elements. The constants chosen to be canonical are called Delaunay elements, in the orbital case, or Andoyer elements, in the spin case. (As some Andoyer elements are time dependent even in the free-spin case, the role of constants is played by their initial values.) The Andoyer and Delaunay sets of elements share a feature not readily apparent: in certain cases the standard equations render them non-osculating. In orbital mechanics, elements furnished by the standard planetary equations are non-osculating when perturbations depend on velocities. To preserve osculation, the equations must be amended with extra terms that are not parts of the disturbing function. In the case of Delaunay parameterisation, these terms destroy canonicity. So under velocity-dependent disturbances, osculation and canonicity are incompatible. (Efroimsky and Goldreich 2003, 2004) Similarly, the Andoyer elements turn out to be non-osculating under angular-velocity-dependent perturbation. Amendment of only the Hamiltonian makes the equations render nonosculating elements. To make them osculating, more terms must enter the equations (and the equations will no longer be canonical). In practical calculations, is often convenient to deliberately deviate from osculation by substituting the Lagrange constraint with a condition that gives birth to a family of nonosculating elements.Comment: Talk at the annual Princeton conference ``New Trends in Astrodynamics" 2005 http://www.math.princeton.edu/astrocon

Polymonadic Programming

Monads are a popular tool for the working functional programmer to structure effectful computations. This paper presents polymonads, a generalization of monads. Polymonads give the familiar monadic bind the more general type forall a,b. L a -> (a -> M b) -> N b, to compose computations with three different kinds of effects, rather than just one. Polymonads subsume monads and parameterized monads, and can express other constructions, including precise type-and-effect systems and information flow tracking; more generally, polymonads correspond to Tate's productoid semantic model. We show how to equip a core language (called lambda-PM) with syntactic support for programming with polymonads. Type inference and elaboration in lambda-PM allows programmers to write polymonadic code directly in an ML-like syntax--our algorithms compute principal types and produce elaborated programs wherein the binds appear explicitly. Furthermore, we prove that the elaboration is coherent: no matter which (type-correct) binds are chosen, the elaborated program's semantics will be the same. Pleasingly, the inferred types are easy to read: the polymonad laws justify (sometimes dramatic) simplifications, but with no effect on a type's generality.Comment: In Proceedings MSFP 2014, arXiv:1406.153