Representations of stream processors using nested fixed points

We define representations of continuous functions on infinite streams of discrete values, both in the case of discrete-valued functions, and in the case of stream-valued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite. In the case of discrete-valued functions, the representatives are well-founded (finite-path) trees of a certain kind. The underlying idea can be traced back to Brouwer's justification of bar-induction, or to Kreisel and Troelstra's elimination of choice-sequences. In the case of stream-valued functions, the representatives are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an alternating fixpoint construction of some ubiquity

Terminal semantics for codata types in intensional Martin-L\"of type theory

In this work, we study the notions of relative comonad and comodule over a relative comonad, and use these notions to give a terminal coalgebra semantics for the coinductive type families of streams and of infinite triangular matrices, respectively, in intensional Martin-L\"of type theory. Our results are mechanized in the proof assistant Coq.Comment: 14 pages, ancillary files contain formalized proof in the proof assistant Coq; v2: 20 pages, title and abstract changed, give a terminal semantics for streams as well as for matrices, Coq proof files updated accordingl

Representations of first order function types as terminal coalgebras

Cosmic rays provide an important source for free electrons in Earth's atmosphere and also in dense interstellar regions where they produce a prevailing background ionization. We utilize a Monte Carlo cosmic ray transport model for particle energies of 10(6) eV <E <10(9) eV, and an analytic cosmic ray transport model for particle energies of 10(9) eV <E <10(12) eV in order to investigate the cosmic ray enhancement of free electrons in substellar atmospheres of free-floating objects. The cosmic ray calculations are applied to Drift-Phoenix model atmospheres of an example brown dwarf with effective temperature T-eff = 1500 K, and two example giant gas planets (T-eff = 1000 K, 1500 K). For the model brown dwarf atmosphere, the electron fraction is enhanced significantly by cosmic rays when the pressure p(gas) <10(-2) bar. Our example giant gas planet atmosphere suggests that the cosmic ray enhancement extends to 10(-4)-10(-2) bar, depending on the effective temperature. For the model atmosphere of the example giant gas planet considered here (T-eff = 1000 K), cosmic rays bring the degree of ionization to f(e) greater than or similar to 10(-8) when p(gas) <10(-8) bar, suggesting that this part of the atmosphere may behave as a weakly ionized plasma. Although cosmic rays enhance the degree of ionization by over three orders of magnitude in the upper atmosphere, the effect is not likely to be significant enough for sustained coupling of the magnetic field to the gas.Publisher PDFPeer reviewe

Structured general corecursion and coinductive graphs [extended abstract]

Bove and Capretta's popular method for justifying function definitions by general recursive equations is based on the observation that any structured general recursion equation defines an inductive subset of the intended domain (the "domain of definedness") for which the equation has a unique solution. To accept the definition, it is hence enough to prove that this subset contains the whole intended domain. This approach works very well for "terminating" definitions. But it fails to account for "productive" definitions, such as typical definitions of stream-valued functions. We argue that such definitions can be treated in a similar spirit, proceeding from a different unique solvability criterion. Any structured recursive equation defines a coinductive relation between the intended domain and intended codomain (the "coinductive graph"). This relation in turn determines a subset of the intended domain and a quotient of the intended codomain with the property that the equation is uniquely solved for the subset and quotient. The equation is therefore guaranteed to have a unique solution for the intended domain and intended codomain whenever the subset is the full set and the quotient is by equality.Comment: In Proceedings FICS 2012, arXiv:1202.317

Non-wellfounded trees in Homotopy Type Theory

We prove a conjecture about the constructibility of coinductive types - in the principled form of indexed M-types - in Homotopy Type Theory. The conjecture says that in the presence of inductive types, coinductive types are derivable. Indeed, in this work, we construct coinductive types in a subsystem of Homotopy Type Theory; this subsystem is given by Intensional Martin-L\"of type theory with natural numbers and Voevodsky's Univalence Axiom. Our results are mechanized in the computer proof assistant Agda.Comment: 14 pages, to be published in proceedings of TLCA 2015; ancillary files contain Agda files with formalized proof