Type classes for efficient exact real arithmetic in Coq

Floating point operations are fast, but require continuous effort on the part of the user in order to ensure that the results are correct. This burden can be shifted away from the user by providing a library of exact analysis in which the computer handles the error estimates. Previously, we [Krebbers/Spitters 2011] provided a fast implementation of the exact real numbers in the Coq proof assistant. Our implementation improved on an earlier implementation by O'Connor by using type classes to describe an abstract specification of the underlying dense set from which the real numbers are built. In particular, we used dyadic rationals built from Coq's machine integers to obtain a 100 times speed up of the basic operations already. This article is a substantially expanded version of [Krebbers/Spitters 2011] in which the implementation is extended in the various ways. First, we implement and verify the sine and cosine function. Secondly, we create an additional implementation of the dense set based on Coq's fast rational numbers. Thirdly, we extend the hierarchy to capture order on undecidable structures, while it was limited to decidable structures before. This hierarchy, based on type classes, allows us to share theory on the naturals, integers, rationals, dyadics, and reals in a convenient way. Finally, we obtain another dramatic speed-up by avoiding evaluation of termination proofs at runtime.Comment: arXiv admin note: text overlap with arXiv:1105.275

Certified Context-Free Parsing: A formalisation of Valiant's Algorithm in Agda

Valiant (1975) has developed an algorithm for recognition of context free languages. As of today, it remains the algorithm with the best asymptotic complexity for this purpose. In this paper, we present an algebraic specification, implementation, and proof of correctness of a generalisation of Valiant's algorithm. The generalisation can be used for recognition, parsing or generic calculation of the transitive closure of upper triangular matrices. The proof is certified by the Agda proof assistant. The certification is representative of state-of-the-art methods for specification and proofs in proof assistants based on type-theory. As such, this paper can be read as a tutorial for the Agda system

Beables/Observables in Classical and Quantum Gravity

Observables 'are observed' whereas beables just 'are'. This gives beables more scope in the cosmological and quantum domains. Both observables and beables are entities that form 'brackets' with 'the constraints' that are 'equal to' zero. We explain how depending on circumstances, these could be, e.g., Poisson, Dirac, commutator, histories, Schouten-Nijenhuis, double or Nambu brackets, first-class, gauge, linear or effective constraints, and strong, weak or weak-effective equalities. The Dirac-Bergmann distinction in notions of gauge leads to further notions of observables or beables, and is tied to a number of diffeomorphism-specific subtleties. Thus we cover a wide range of notions of observables or beables that occur in classical and quantum gravitational theories: Dirac, Kucha\v{r}, effective, Bergmann, histories, multisymplectic, master, Nambu and bi-. Indeed this review covers a representatively wide range of such theories: general relativity, loop quantum gravity, histories theory, supergravity and M-theory

Formalization of Universal Algebra in Agda

In this work we present a novel formalization of universal algebra in Agda. We show that heterogeneous signatures can be elegantly modelled in type-theory using sets indexed by arities to represent operations. We prove elementary results of heterogeneous algebras, including the proof that the term algebra is initial and the proofs of the three isomorphism theorems. We further formalize equational theory and prove soundness and completeness. At the end, we define (derived) signature morphisms, from which we get the contravariant functor between algebras; moreover, we also proved that, under some restrictions, the translation of a theory induces a contra-variant functor between models.Fil: Gunther, Emmanuel. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Gadea, Alejandro Emilio. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Pagano, Miguel Maria. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentin

Functorial Semantics for Petri Nets under the Individual Token Philosophy

Although the algebraic semantics of place/transition Petri nets under the collective token philosophy has been fully explained in terms of (strictly) symmetric (strict) monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory because it lacks universality and also functoriality. We introduce the notion of pre-net to recover these aspects, obtaining a fully satisfactory categorical treatment centered on the notion of adjunction. This allows us to present a purely logical description of net behaviours under the individual token philosophy in terms of theories and theory morphisms in partial membership equational logic, yielding a complete match with the theory developed by the authors for the collective token view of net

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