Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userA computer verified, monadic, functional implementation of the integral
151
0
0.0
(
0
)
Added by
Bas Spitters
Publication date
2010
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
We provide a computer verified exact monadic functional implementation of the Riemann integral in type theory. Together with previous work by OConnor, this may be seen as the beginning of the realization of Bishops vision to use constructive mathematics as a programming language for exact analysis.
rate research
Read More
Compact sets in constructive mathematics capture our intuition of what computable subsets of the plane (or any other complete metric space) ought to be. A good representation of compact sets provides an efficient means of creating and displaying images with a computer. In this paper, I build upon existing work about complete metric spaces to define compact sets as the completion of the space of finite sets under the Hausdorff metric. This definition allowed me to quickly develop a computer verified theory of compact sets. I applied this theory to compute provably correct plots of uniformly continuous functions.
In control theory, to solve a finite-horizon sequential decision problem (SDP) commonly means to find a list of decision rules that result in an optimal expected total reward (or cost) when taking a given number of decision steps. SDPs are routinely solved using Bellmans backward induction. Textbook authors (e.g. Bertsekas or Puterman) typically give more or less formal proofs to show that the backward induction algorithm is correct as solution method for deterministic and stochastic SDPs. Botta, Jansson and Ionescu propose a generic framework for finite horizon, monadic SDPs together with a monadic version of backward induction for solving such SDPs. In monadic SDPs, the monad captures a generic notion of uncertainty, while a generic measure function aggregates rewards. In the present paper we define a notion of correctness for monadic SDPs and identify three conditions that allow us to prove a correctness result for monadic backward induction that is comparable to textbook correctness proofs for ordinary backward induction. The conditions that we impose are fairly general and can be cast in category-theoretical terms using the notion of Eilenberg-Moore-algebra. They hold in familiar settings like those of deterministic or stochastic SDPs but we also give examples in which they fail. Our results show that backward induction can safely be employed for a broader class of SDPs than usually treated in textbooks. However, they also rule out certain instances that were considered admissible in the context of Botta et al.s generic framework. Our development is formalised in Idris as an extension of the Botta et al. framework and the sources are available as supplementary material.
We introduce a syntactic translation of Goedels System T parametrized by a weak notion of a monad, and prove a corresponding fundamental theorem of logical relation. Our translation structurally corresponds to Gentzens negative translation of classical logic. By instantiating the monad and the logical relation, we reveal the well-known properties and structures of T-definable functionals including majorizability, continuity and bar recursion. Our development has been formalized in the Agda proof assistant.
We present sqire, a low-level language for quantum computing and verification. sqire uses a global register of quantum bits, allowing easy compilation to and from existing `quantum assembly languages and simplifying the verification process. We demonstrate the power of sqire as an intermediate representation of quantum programs by verifying a number of useful optimizations, and we demonstrate sqires use as a tool for general verification by proving several quantum programs correct.
Hidden Markov Models, HMMs, are mathematical models of Markov processes with state that is hidden, but from which information can leak. They are typically represented as 3-way joint-probability distributions. We use HMMs as denotations of probabilistic hidden-state sequential programs: for that, we recast them as `abstract HMMs, computations in the Giry monad $mathbb{D}$, and we equip them with a partial order of increasing security. However to encode the monadic type with hiding over some state $mathcal{X}$ we use $mathbb{D}mathcal{X}to mathbb{D}^2mathcal{X}$ rather than the conventional $mathcal{X}{to}mathbb{D}mathcal{X}$ that suffices for Markov models whose state is not hidden. We illustrate the $mathbb{D}mathcal{X}to mathbb{D}^2mathcal{X}$ construction with a small Haskell prototype. We then present uncertainty measures as a generalisation of the extant diversity of probabilistic entropies, with characteristic analytic properties for them, and show how the new entropies interact with the order of increasing security. Furthermore, we give a `backwards uncertainty-transformer semantics for HMMs that is dual to the `forwards abstract HMMs - it is an analogue of the duality between forwards, relational semantics and backwards, predicate-transformer semantics for imperative programs with demonic choice. Finally, we argue that, from this new denotational-semantic viewpoint, one can see that the Dalenius desideratum for statistical databases is actually an issue in compositionality. We propose a means for taking it into account.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
