Presenting Distributive Laws

Distributive laws of a monad T over a functor F are categorical tools for specifying algebra-coalgebra interaction. They proved to be important for solving systems of corecursive equations, for the specification of well-behaved structural operational semantics and, more recently, also for enhancements of the bisimulation proof method. If T is a free monad, then such distributive laws correspond to simple natural transformations. However, when T is not free it can be rather difficult to prove the defining axioms of a distributive law. In this paper we describe how to obtain a distributive law for a monad with an equational presentation from a distributive law for the underlying free monad. We apply this result to show the equivalence between two different representations of context-free languages

Beating the Productivity Checker Using Embedded Languages

Some total languages, like Agda and Coq, allow the use of guarded corecursion to construct infinite values and proofs. Guarded corecursion is a form of recursion in which arbitrary recursive calls are allowed, as long as they are guarded by a coinductive constructor. Guardedness ensures that programs are productive, i.e. that every finite prefix of an infinite value can be computed in finite time. However, many productive programs are not guarded, and it can be nontrivial to put them in guarded form. This paper gives a method for turning a productive program into a guarded program. The method amounts to defining a problem-specific language as a data type, writing the program in the problem-specific language, and writing a guarded interpreter for this language.Comment: In Proceedings PAR 2010, arXiv:1012.455

Generalised coinduction

Final coalgebras of a functor F are suited for an abstract description of infinite datatypes and dynamical systems. Functions into such a domain are specified by coinductive definitions. The format these specifications take when their justification is directly based on finality is called the coiteration schema here. In applications it often turns out to be too rigid to allow for a convenient description of the functions under consideration. Thus, generalisations or variations are desired. We introduce a generic ?-coiteration schema that can be instantiated by a distributive law ? of some functor T over F and show that - under mild assumptions on the underlying category - one obtains principles which uniquely characterise arrows into the carrier of a final F-coalgebra as well. Certain instances of ?-coiteration can be shown to specify arrows that fail to be coiterative. Examples are the duals of primitive recursion and course-of-value iteration, which are known extensions of coiteration. One can furthermore obtain schemata justifying recursive specifications that involve operators such as arithmetic operations on power series, regular operators for languages, or parallel and sequential composition of processes. Next, the same type of distributive law ? is used to generalise coinductive proof techniques. To this end, we introduce the notion of a ?-bisimulation relation, many instances of which are weaker than the conventional definition of a bisimulation. It specialises e.g. to what could be called bisimulation up-to-equality or bisimulation up-to-context for contexts built from operators of the type mentioned above. We give a proof showing that every ?-bisimulation only contains pairs of bisimilar states. This principle leads to simpler proofs through the use of less complex relations

Proper Functors and Fixed Points for Finite Behaviour

The rational fixed point of a set functor is well-known to capture the behaviour of finite coalgebras. In this paper we consider functors on algebraic categories. For them the rational fixed point may no longer be fully abstract, i.e. a subcoalgebra of the final coalgebra. Inspired by \'Esik and Maletti's notion of a proper semiring, we introduce the notion of a proper functor. We show that for proper functors the rational fixed point is determined as the colimit of all coalgebras with a free finitely generated algebra as carrier and it is a subcoalgebra of the final coalgebra. Moreover, we prove that a functor is proper if and only if that colimit is a subcoalgebra of the final coalgebra. These results serve as technical tools for soundness and completeness proofs for coalgebraic regular expression calculi, e.g. for weighted automata

Witnessing (co)datatypes

Datatypes and codatatypes are useful for specifying and reasoning about (possibly infinite) computational processes. The Isabelle/HOL proof assistant has recently been extended with a definitional package that supports both. We describe a complete procedure for deriving nonemptiness witnesses in the general mutually recursive, nested case—nonemptiness being a proviso for introducing types in higher-order logic

Witnessing (co)datatypes

Datatypes and codatatypes are useful for specifying and reasoning about (possibly infinite) computational processes. The Isabelle/HOL proof assistant has recently been extended with a definitional package that supports both. We describe a complete procedure for deriving nonemptiness witnesses in the general mutually recursive, nested case—nonemptiness being a proviso for introducing types in higher-order logic

The productivity of polymorphic stream equations and the composition of circular traversals

This thesis has two independent parts concerned with different aspects of laziness in functional programs. The first part is a theoretical study of productivity for very restricted stream programs. In the second part we define a programming abstraction over a recursive pattern for defining circular traversals modularly. Productivity is in general undecidable. By restricting ourselves to mutually recursive polymorphic stream equations having only three basic operations, namely "head", "tail", and "cons", we aim to prove interesting properties about productivity. Still undecidable for this restricted class of programs, productivity of polymorphic stream functions is equivalent to the totality of their indexing function, which characterise their behaviour in terms of operations on indices. We prove that our equations generate all possible polymorphic stream functions, and therefore their indexing functions are all the computable functions, whose totality problem is indeed undecidable. We then further restrict our language by reducing the numbers of equations and parameters, but despite those constraints the equations retain their expressiveness. In the end we establish that even two non-mutually recursive equations on unary stream functions are undecidable with complexity $Π_2^0$. However, the productivity of a single unary equation is decidable. Circular traversals have been used in the eighties as an optimisation to combine multiple traversals in a single traversal. In particular they provide more opportunities for applying deforestation techniques since it is the case that an intermediate datastructure can only be eliminated if it is consumed only once. Another use of circular programs is in the implementation of attribute grammars in lazy functional languages. There is a systematic transformation to define a circular traversal equivalent to multiple traversals. Programming with this technique is not modular since the individual traversals are merged together. Some tools exist to transform programs automatically and attribute grammars have been suggested as a way to describe the circular traversals modularly. Going to the root of the problem, we identify a recursive pattern that allows us to define circular programs modularly in a functional style. We give two successive implementations, the first one is based on algebras and has limited scope: not all circular traversals can be defined this way. We show that the recursive scheme underlying attribute grammars computation rules is essential to combine circular programs. We implement a generic recursive operation on a novel attribute grammar abstraction, using containers as a parametric generic representation of recursive datatypes. The abstraction makes attribute grammars first-class objects. Such a strongly typed implementation is novel and make it possible to implement a high level embedded language for defining attribute grammars, with many interesting new features promoting modularity

Interaction Trees: Representing Recursive and Impure Programs in Coq

"Interaction trees" (ITrees) are a general-purpose data structure for representing the behaviors of recursive programs that interact with their environments. A coinductive variant of "free monads," ITrees are built out of uninterpreted events and their continuations. They support compositional construction of interpreters from "event handlers", which give meaning to events by defining their semantics as monadic actions. ITrees are expressive enough to represent impure and potentially nonterminating, mutually recursive computations, while admitting a rich equational theory of equivalence up to weak bisimulation. In contrast to other approaches such as relationally specified operational semantics, ITrees are executable via code extraction, making them suitable for debugging, testing, and implementing software artifacts that are amenable to formal verification. We have implemented ITrees and their associated theory as a Coq library, mechanizing classic domain- and category-theoretic results about program semantics, iteration, monadic structures, and equational reasoning. Although the internals of the library rely heavily on coinductive proofs, the interface hides these details so that clients can use and reason about ITrees without explicit use of Coq's coinduction tactics. To showcase the utility of our theory, we prove the termination-sensitive correctness of a compiler from a simple imperative source language to an assembly-like target whose meanings are given in an ITree-based denotational semantics. Unlike previous results using operational techniques, our bisimulation proof follows straightforwardly by structural induction and elementary rewriting via an equational theory of combinators for control-flow graphs.Comment: 28 pages, 4 pages references, published at POPL 202

The productivity of polymorphic stream equations and the composition of circular traversals

This thesis has two independent parts concerned with different aspects of laziness in functional programs. The first part is a theoretical study of productivity for very restricted stream programs. In the second part we define a programming abstraction over a recursive pattern for defining circular traversals modularly. Productivity is in general undecidable. By restricting ourselves to mutually recursive polymorphic stream equations having only three basic operations, namely "head", "tail", and "cons", we aim to prove interesting properties about productivity. Still undecidable for this restricted class of programs, productivity of polymorphic stream functions is equivalent to the totality of their indexing function, which characterise their behaviour in terms of operations on indices. We prove that our equations generate all possible polymorphic stream functions, and therefore their indexing functions are all the computable functions, whose totality problem is indeed undecidable. We then further restrict our language by reducing the numbers of equations and parameters, but despite those constraints the equations retain their expressiveness. In the end we establish that even two non-mutually recursive equations on unary stream functions are undecidable with complexity $Π_2^0$. However, the productivity of a single unary equation is decidable. Circular traversals have been used in the eighties as an optimisation to combine multiple traversals in a single traversal. In particular they provide more opportunities for applying deforestation techniques since it is the case that an intermediate datastructure can only be eliminated if it is consumed only once. Another use of circular programs is in the implementation of attribute grammars in lazy functional languages. There is a systematic transformation to define a circular traversal equivalent to multiple traversals. Programming with this technique is not modular since the individual traversals are merged together. Some tools exist to transform programs automatically and attribute grammars have been suggested as a way to describe the circular traversals modularly. Going to the root of the problem, we identify a recursive pattern that allows us to define circular programs modularly in a functional style. We give two successive implementations, the first one is based on algebras and has limited scope: not all circular traversals can be defined this way. We show that the recursive scheme underlying attribute grammars computation rules is essential to combine circular programs. We implement a generic recursive operation on a novel attribute grammar abstraction, using containers as a parametric generic representation of recursive datatypes. The abstraction makes attribute grammars first-class objects. Such a strongly typed implementation is novel and make it possible to implement a high level embedded language for defining attribute grammars, with many interesting new features promoting modularity

Formal Frameworks for Circular Phenomena. Possibilities of Modeling Pathological Expressions in Formal and Natural Languages

This dissertation has four parts. The first one is a general introduction into the topic of the work separated in a chapter that explains the used notation, a chapter that discusses typical examples, and a chapter that gives an overview of the three main parts. An important aspect of this first part is the attempt of a conceptual clarification of circularity, in particular in relation to the non-well-foundedness of a phenomenon. This clarification represents the philosophical core of the primarily formal dissertation. In the second part, Kripke's fixed point approach concerning partially defined truth predicates is examined: the algebraic foundations are introduced and problems of the construction are discussed. The main results of this second part are three characterization theorems of subclasses of interlaced bilattices and their applications. In the third part, revision theories are introduced. Their adequacy for the representation of circularity is discussed. Additionally, the complexity of these theories, the relation of revision theories to a wider thematic context, and their empirical properties are examined. In the last part of this dissertation, circularity is introduced on the level of set theory. The crucial idea is the concept of a coalgebraic modeling. In particular, the modeling of truth and the representation of the difference between private and common knowledge is emphasized. A comparison of the different accounts is provided in the last chapter.This dissertation has four parts. The first one is a general introduction into the topic of the work separated in a chapter that explains the used notation, a chapter that discusses typical examples, and a chapter that gives an overview of the three main parts. An important aspect of this first part is the attempt of a conceptual clarification of circularity, in particular in relation to the non-well-foundedness of a phenomenon. This clarification represents the philosophical core of the primarily formal dissertation. In the second part, Kripke's fixed point approach concerning partially defined truth predicates is examined: the algebraic foundations are introduced and problems of the construction are discussed. The main results of this second part are three characterization theorems of subclasses of interlaced bilattices and their applications. In the third part, revision theories are introduced. Their adequacy for the representation of circularity is discussed. Additionally, the complexity of these theories, the relation of revision theories to a wider thematic context, and their empirical properties are examined. In the last part of this dissertation, circularity is introduced on the level of set theory. The crucial idea is the concept of a coalgebraic modeling. In particular, the modeling of truth and the representation of the difference between private and common knowledge is emphasized. A comparison of the different accounts is provided in the last chapter