60 research outputs found
Modules over Monads and Operational Semantics
This paper is a contribution to the search for efficient and high-level mathematical tools to specify and reason about (abstract) programming languages or calculi. Generalising the reduction monads of Ahrens et al., we introduce transition monads, thus covering new applications such as ???-calculus, ?-calculus, Positive GSOS specifications, differential ?-calculus, and the big-step, simply-typed, call-by-value ?-calculus. Finally, we design a suitable notion of signature for transition monads
Coinduction in Flow: The Later Modality in Fibrations
This paper provides a construction on fibrations that gives access to the so-called later modality, which allows for a controlled form of recursion in coinductive proofs and programs. The construction is essentially a generalisation of the topos of trees from the codomain fibration over sets to arbitrary fibrations. As a result, we obtain a framework that allows the addition of a recursion principle for coinduction to rather arbitrary logics and programming languages. The main interest of using recursion is that it allows one to write proofs and programs in a goal-oriented fashion. This enables easily understandable coinductive proofs and programs, and fosters automatic proof search.
Part of the framework are also various results that enable a wide range of applications: transportation of (co)limits, exponentials, fibred adjunctions and first-order connectives from the initial fibration to the one constructed through the framework. This means that the framework extends any first-order logic with the later modality. Moreover, we obtain soundness and completeness results, and can use up-to techniques as proof rules. Since the construction works for a wide variety of fibrations, we will be able to use the recursion offered by the later modality in various context. For instance, we will show how recursive proofs can be obtained for arbitrary (syntactic) first-order logics, for coinductive set-predicates, and for the probabilistic modal mu-calculus. Finally, we use the same construction to obtain a novel language for probabilistic productive coinductive programming. These examples demonstrate the flexibility of the framework and its accompanying results
Second quantization of the elliptic Calogero-Sutherland model
We use loop group techniques to construct a quantum field theory model of
anyons on a circle and at finite temperature. We find an anyon Hamiltonian
providing a second quantization of the elliptic Calogero-Sutherland model. This
allows us to prove a remarkable identity which is a starting point for an
algorithm to construct eigenfunctions and eigenvalues of the elliptic
Calogero-Sutherland Hamiltonian (this algorithm is elaborated elsewhere).
This paper contains a detailed introduction, technical details and proofs.Comment: 36 page
First-Order Guarded Coinduction in Coq
We introduce two coinduction principles and two proof translations which, under certain conditions, map coinductive proofs that use our principles to guarded Coq proofs. The first principle provides an "operational" description of a proof by coinduction, which is easy to reason with informally. The second principle extends the first one to allow for direct proofs by coinduction of statements with existential quantifiers and multiple coinductive predicates in the conclusion. The principles automatically enforce the correct use of the coinductive hypothesis. We implemented the principles and the proof translations in a Coq plugin
Probabilistic Mu-Calculus: Decidability and Complete Axiomatization
We introduce a version of the probabilistic mu-calculus (PMC) built on top of a probabilistic modal logic that allows encoding n-ary inequational conditions on transition probabilities. PMC extends previously studied calculi and we prove that, despite its expressiveness, it enjoys a series of good meta-properties. Firstly, we prove the decidability of satisfiability checking by establishing the small model property. An algorithm for deciding the satisfiability problem is developed. As a second major result, we provide a complete axiomatization for the alternation-free fragment of PMC. The completeness proof is innovative in many aspects combining various techniques from topology and model theory
Arithmetic theory of q-difference equations. The q-analogue of Grothendieck-Katz's conjecture on p-curvatures
Grothendieck's conjecture on p-curvatures predicts that an arithmetic
differential equation has a full set of algebraic solutions if and only if its
reduction in positive characteristic has a full set of rational solutions for
almost all finite places. It is equivalent to Katz's conjectural description of
the generic Galois group. In this paper we prove an analogous statement for
arithmetic q-difference equation.Comment: 45 pages. Defintive versio
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