274 research outputs found
Reducibility and TT-lifting for computation types
Abstract. We propose â€â€-lifting as a technique for extending operational predicates to Moggiâs monadic computation types, independent of the choice of monad. We demonstrate the method with an application to Girard-Tait reducibility, using this to prove strong normalisation for the computational metalanguage λml. The particular challenge with reducibility is to apply this semantic notion at computation types when the exact meaning of âcomputation â (stateful, side-effecting, nondeterministic, etc.) is left unspecified. Our solution is to define reducibility for continuations and use that to support the jump from value types to computation types. The method appears robust: we apply it to show strong normalisation for the computational metalanguage extended with sums, and with exceptions. Based on these results, as well as previous work with local state, we suggest that this âleap-frog â approach offers a general method for raising concepts defined at value types up to observable properties of computations.
Proving termination of evaluation for System F with control operators
We present new proofs of termination of evaluation in reduction semantics
(i.e., a small-step operational semantics with explicit representation of
evaluation contexts) for System F with control operators. We introduce a
modified version of Girard's proof method based on reducibility candidates,
where the reducibility predicates are defined on values and on evaluation
contexts as prescribed by the reduction semantics format. We address both
abortive control operators (callcc) and delimited-control operators (shift and
reset) for which we introduce novel polymorphic type systems, and we consider
both the call-by-value and call-by-name evaluation strategies.Comment: In Proceedings COS 2013, arXiv:1309.092
Stable Flags and the Riemann-Hilbert Problem
We tackle the Riemann-Hilbert problem on the Riemann sphere as stalk-wise
logarithmic modifications of the classical R\"ohrl-Deligne vector bundle. We
show that the solutions of the Riemann-Hilbert problem are in bijection with
some families of local filtrations which are stable under the prescribed
monodromy maps. We introduce the notion of Birkhoff-Grothendieck
trivialisation, and show that its computation corresponds to geodesic paths in
some local affine Bruhat-Tits building. We use this to compute how the type of
a bundle changes under stalk modifications, and give several corresponding
algorithmic procedures.Comment: 39 page
Strongly Normalizing Audited Computation
Auditing is an increasingly important operation for computer programming, for example in security (e.g. to enable history-based access control) and to enable reproducibility and accountability (e.g. provenance in scientific programming). Most proposed auditing techniques are ad hoc or treat auditing as a second-class, extralinguistic operation; logical or semantic foundations for auditing are not yet well-established. Justification Logic (JL) offers one such foundation; Bavera and Bonelli introduced a computational interpretation of JL called lambda^h that supports auditing. However, lambda^h is technically complex and strong normalization was only established for special cases. In addition, we show that the equational theory of lambda^h is inconsistent. We introduce a new calculus lambda^hc that is simpler than lambda^hc, consistent, and strongly normalizing. Our proof of strong normalization is formalized in Nominal Isabelle
On the Values of Reducibility Candidates
The straightforward elimination of union types is known to break subject reduction, and for some extensions of the lambda-calculus, to break strong normalization as well. Similarly, the straightforward elim- ination of implicit existential types breaks subject reduction. We propose elimination rules for union types and implicit existential quantification which use a form call-by-value issued from Girard's re- ducibility candidates. We show that these rules remedy the above men- tioned difficulties, for strong normalization and, for the existential quan- tification, for subject reduction as well. Moreover, for extensions of the lambda-calculus based on intuitionistic logic, we show that the obtained existential quantification is equivalent to its usual impredicative encoding w.r.t. provability in realizability models built from reducibility candidates and biorthogonals
Jordan blocks of cuspidal representations of symplectic groups
Let G be a symplectic group over a nonarchimedean local field of characteristic zero and odd residual characteristic. Given an irreducible cuspidal representation of G, we determine its Langlands parameter (equivalently, its Jordan blocks in the language of MĆglin) in terms of the local data from which the representation is explicitly constructed, up to a possible unramified twist in each block of the parameter. We deduce a ramification theorem for G, giving a bijection between the set of endoparameters for G and the set of restrictions to wild inertia of discrete Langlands parameters for G, compatible with the local Langlands correspondence. The main tool consists in analyzing the Hecke algebra of a good cover, in the sense of BushnellâKutzko, for parabolic induction from a cuspidal representation of G Ă GL n , seen as a maximal Levi subgroup of a bigger symplectic group, in order to determine reducibility points; a criterion of MĆglin then relates this to Langlands parameters
Computability in constructive type theory
We give a formalised and machine-checked account of computability theory in the Calculus of Inductive Constructions (CIC), the constructive type theory underlying the Coq proof assistant. We first develop synthetic computability theory, pioneered by Richman, Bridges, and Bauer, where one treats all functions as computable, eliminating the need for a model of computation. We assume a novel parametric axiom for synthetic computability and give proofs of results like Riceâs theorem, the Myhill isomorphism theorem, and the existence of Postâs simple and hypersimple predicates relying on no other axioms such as Markovâs principle or choice axioms. As a second step, we introduce models of computation. We give a concise overview of definitions of various standard models and contribute machine-checked simulation proofs, posing a non-trivial engineering effort. We identify a notion of synthetic undecidability relative to a fixed halting problem, allowing axiom-free machine-checked proofs of undecidability. We contribute such undecidability proofs for the historical foundational problems of computability theory which require the identification of invariants left out in the literature and now form the basis of the Coq Library of Undecidability Proofs. We then identify the weak call-by-value λ-calculus L as sweet spot for programming in a model of computation. We introduce a certifying extraction framework and analyse an axiom stating that every function of type â â â is L-computable.Wir behandeln eine formalisierte und maschinengeprĂŒfte Betrachtung von Berechenbarkeitstheorie im Calculus of Inductive Constructions (CIC), der konstruktiven Typtheorie die dem Beweisassistenten Coq zugrunde liegt. Wir entwickeln erst synthetische Berechenbarkeitstheorie, vorbereitet durch die Arbeit von Richman, Bridges und Bauer, wobei alle Funktionen als berechenbar behandelt werden, ohne Notwendigkeit eines Berechnungsmodells. Wir nehmen ein neues, parametrisches Axiom fĂŒr synthetische Berechenbarkeit an und beweisen Resultate wie das Theorem von Rice, das Isomorphismus Theorem von Myhill und die Existenz von Postâs simplen und hypersimplen PrĂ€dikaten ohne Annahme von anderen Axiomen wie Markovâs Prinzip oder Auswahlaxiomen. Als zweiten Schritt fĂŒhren wir Berechnungsmodelle ein. Wir geben einen kompakten Ăberblick ĂŒber die Definition von verschiedenen Berechnungsmodellen und erklĂ€ren maschinengeprĂŒfte Simulationsbeweise zwischen diesen Modellen, welche einen hohen Konstruktionsaufwand beinhalten. Wir identifizieren einen Begriff von synthetischer Unentscheidbarkeit relativ zu einem fixierten Halteproblem welcher axiomenfreie maschinengeprĂŒfte Unentscheidbarkeitsbeweise erlaubt. Wir erklĂ€ren solche Beweise fĂŒr die historisch grundlegenden Probleme der Berechenbarkeitstheorie, die das Identifizieren von Invarianten die normalerweise in der Literatur ausgelassen werden benötigen und nun die Basis der Coq Library of Undecidability Proofs bilden. Wir identifizieren dann den call-by-value λ-KalkĂŒl L als sweet spot fĂŒr die Programmierung in einem Berechnungsmodell. Wir fĂŒhren ein zertifizierendes Extraktionsframework ein und analysieren ein Axiom welches postuliert dass jede Funktion vom Typ NâN L-berechenbar ist
Multiple Permitting and Bounded Turing Reducibilities
We look at various properties of the computably enumerable (c.e.) not totally Ï-c.e. Turing degrees.
In particular, we are interested in the variant of multiple permitting given by those degrees. We
define a property of left-c.e. sets called universal similarity property which can be viewed as a
universal or uniform version of the property of array noncomputable c.e. sets of agreeing with any
c.e. set on some component of a very strong array. Using a multiple permitting argument, we
prove that the Turing degrees of the left-c.e. sets with the universal similarity property coincide
with the c.e. not totally Ï-c.e. degrees. We further introduce and look at various notions of socalled
universal array noncomputability and show that c.e. sets with those properties can be found
exactly in the c.e. not totally Ï-c.e. Turing degrees and that they guarantee a special type of
multiple permitting called uniform multiple permitting. We apply these properties of the c.e. not
totally Ï-c.e. degrees to give alternative proofs of well-known results on those degrees as well as
to prove new results. E.g., we show that a c.e. Turing degree contains a left-c.e. set which is not
cl-reducible to any complex left-c.e. set if and only if it is not totally Ï-c.e. Furthermore, we prove
that the nondistributive finite lattice S7 can be embedded into the c.e. Turing degrees precisely
below any c.e. not totally Ï-c.e. degree.
We further look at the question of join preservation for bounded Turing reducibilities r and râČ
such that r is stronger than râČ. We say that join preservation holds for two reducibilities r and
râČ if every join in the c.e. r-degrees is also a join in the c.e. râČ-degrees. We consider the class of
monotone admissible (uniformly) bounded Turing reducibilities, i.e., the reflexive and transitive
Turing reducibilities with use bounded by a function that is contained in a (uniformly computable)
family of strictly increasing computable functions. This class contains for example identity bounded
Turing (ibT-) and computable Lipschitz (cl-) reducibility. Our main result of Chapter 3 is that join
preservation fails for cl and any strictly weaker monotone admissible uniformly bounded Turing
reducibility. We also look at the dual question of meet preservation and show that for all monotone
admissible bounded Turing reducibilities r and râČ such that r is stronger than râČ, meet preservation
holds. Finally, we completely solve the question of join and meet preservation in the classical
reducibilities 1, m, tt, wtt and T
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