The continuity of monadic stream functions

Brouwer’s continuity principle states that all functions from infinite sequences of naturals to naturals are continuous, that is, for every sequence the result depends only on a finite initial segment. It is an intuitionistic axiom that is incompatible with classical mathematics. Recently Mart́ín Escardó proved that it is also inconsistent in type theory. We propose a reformulation of the continuity principle that may be more faithful to the original meaning by Brouwer. It applies to monadic streams, potentially unending sequences of values produced by steps triggered by a monadic action, possibly involving side effects. We consider functions on them that are uniform, in the sense that they operate in the same way independently of the particular monad that provides the specific side effects. Formally this is done by requiring a form of naturality in the monad. Functions on monadic streams have not only a foundational importance, but have also practical applications in signal processing and reactive programming. We give algorithms to determine the modulus of continuity of monadic stream functions and to generate dialogue trees for them (trees whose nodes and branches describe the interaction of the process with the environment)

The continuity of monadic stream functions

Brouwer’s continuity principle states that all functions from infinite sequences of naturals to naturals are continuous, that is, for every sequence the result depends only on a finite initial segment. It is an intuitionistic axiom that is incompatible with classical mathematics. Recently Mart́ín Escardó proved that it is also inconsistent in type theory. We propose a reformulation of the continuity principle that may be more faithful to the original meaning by Brouwer. It applies to monadic streams, potentially unending sequences of values produced by steps triggered by a monadic action, possibly involving side effects. We consider functions on them that are uniform, in the sense that they operate in the same way independently of the particular monad that provides the specific side effects. Formally this is done by requiring a form of naturality in the monad. Functions on monadic streams have not only a foundational importance, but have also practical applications in signal processing and reactive programming. We give algorithms to determine the modulus of continuity of monadic stream functions and to generate dialogue trees for them (trees whose nodes and branches describe the interaction of the process with the environment)

General Recursion via Coinductive Types

A fertile field of research in theoretical computer science investigates the representation of general recursive functions in intensional type theories. Among the most successful approaches are: the use of wellfounded relations, implementation of operational semantics, formalization of domain theory, and inductive definition of domain predicates. Here, a different solution is proposed: exploiting coinductive types to model infinite computations. To every type A we associate a type of partial elements Partial(A), coinductively generated by two constructors: the first, return(a) just returns an element a:A; the second, step(x), adds a computation step to a recursive element x:Partial(A). We show how this simple device is sufficient to formalize all recursive functions between two given types. It allows the definition of fixed points of finitary, that is, continuous, operators. We will compare this approach to different ones from the literature. Finally, we mention that the formalization, with appropriate structural maps, defines a strong monad.Comment: 28 page

Contractive functions on infinite data structures

Coinductive data structures, such as streams or infinite trees, have many applications in functional programming and type theory, and are naturally defined using recursive equations. But how do we ensure that such equations make sense, i.e. that they actually generate a productive infinite object? A standard means to achieve productivity is to use Banach’s fixed-point theorem, which guarantees the unique existence of solutions to recursive equations on metric spaces under certain conditions. Functions satisfying these conditions are called contractions. In this article, we give a new characterization of contractions on streams in the form of a sound and complete representation theorem, and generalize this result to a wide class of non-well-founded structures, first to infinite binary trees, then to final coalgebras of container functors. These results have important potential applications in functional programming, where coinduction and corecursion are successfully deployed to model continuous reactive systems, dynamic interactivity, signal processing, and other tasks that require flexible manipulation of non-well-founded data. Our representation theorems provide a definition paradigm to compactly compute with such data and easily reason about them

An Improved Implementation and Abstract Interface for Hybrid

Hybrid is a formal theory implemented in Isabelle/HOL that provides an interface for representing and reasoning about object languages using higher-order abstract syntax (HOAS). This interface is built around an HOAS variable-binding operator that is constructed definitionally from a de Bruijn index representation. In this paper we make a variety of improvements to Hybrid, culminating in an abstract interface that on one hand makes Hybrid a more mathematically satisfactory theory, and on the other hand has important practical benefits. We start with a modification of Hybrid's type of terms that better hides its implementation in terms of de Bruijn indices, by excluding at the type level terms with dangling indices. We present an improved set of definitions, and a series of new lemmas that provide a complete characterization of Hybrid's primitives in terms of properties stated at the HOAS level. Benefits of this new package include a new proof of adequacy and improvements to reasoning about object logics. Such proofs are carried out at the higher level with no involvement of the lower level de Bruijn syntax.Comment: In Proceedings LFMTP 2011, arXiv:1110.668

Formal proofs in real algebraic geometry: from ordered fields to quantifier elimination

This paper describes a formalization of discrete real closed fields in the Coq proof assistant. This abstract structure captures for instance the theory of real algebraic numbers, a decidable subset of real numbers with good algorithmic properties. The theory of real algebraic numbers and more generally of semi-algebraic varieties is at the core of a number of effective methods in real analysis, including decision procedures for non linear arithmetic or optimization methods for real valued functions. After defining an abstract structure of discrete real closed field and the elementary theory of real roots of polynomials, we describe the formalization of an algebraic proof of quantifier elimination based on pseudo-remainder sequences following the standard computer algebra literature on the topic. This formalization covers a large part of the theory which underlies the efficient algorithms implemented in practice in computer algebra. The success of this work paves the way for formal certification of these efficient methods.Comment: 40 pages, 4 figure

Recursive Definitions of Monadic Functions

Using standard domain-theoretic fixed-points, we present an approach for defining recursive functions that are formulated in monadic style. The method works both in the simple option monad and the state-exception monad of Isabelle/HOL's imperative programming extension, which results in a convenient definition principle for imperative programs, which were previously hard to define. For such monadic functions, the recursion equation can always be derived without preconditions, even if the function is partial. The construction is easy to automate, and convenient induction principles can be derived automatically.Comment: In Proceedings PAR 2010, arXiv:1012.455

Step-Indexed Normalization for a Language with General Recursion

The Trellys project has produced several designs for practical dependently typed languages. These languages are broken into two fragments-a_logical_fragment where every term normalizes and which is consistent when interpreted as a logic, and a_programmatic_fragment with general recursion and other convenient but unsound features. In this paper, we present a small example language in this style. Our design allows the programmer to explicitly mention and pass information between the two fragments. We show that this feature substantially complicates the metatheory and present a new technique, combining the traditional Girard-Tait method with step-indexed logical relations, which we use to show normalization for the logical fragment.Comment: In Proceedings MSFP 2012, arXiv:1202.240

journal homepage: www.elsevier.com/locate/tcs

AbstractThis is a survey article on the use of coalgebras in functional programming and type theory. It presents the basic theory underlying the implementation of coinductive types, families and predicates. It gives an overview of the application of corecursive methods to the study of general recursion, formal power series, tabulations of functions on inductive data. It also sketches some advanced topics in the study of the solutions to non-guarded corecursive equations and the design of non-standard type theory

Universal Algebra in Type Theory

We present a development of Universal Algebra inside Type Theory, formalized using the proof assistant Coq. We define the notion of a signature and of an algebra over a signature. We use setoids, i.e. ..