Sergey Goncharov

A coinductive calculus for asynchronous side-effecting processes

We present an abstract framework for concurrent processes in which atomic steps have generic side effects, handled according to the principle of monadic encapsulation of effects. Processes in this framework are potentially infinite resumptions, modelled using final coalgebras over the monadic base. As a calculus for such processes, we introduce a concurrent extension of Moggi@?s monadic meta-language of effects. We establish soundness and completeness of a natural equational axiomatization of this calculus. Our main result is a corecursion scheme that is explicitly definable over the base language and provides flexible expressive means for the definition of new operators on processes, such as parallel composition. Moreover, we present initial results on verification methods for generic side-effecting processes.

Kleene monads: handling iteration in a framework of generic effects

Monads are a well-established tool for modelling various computational effects. They form the semantic basis of Moggis computational metalanguage, the metalanguage of effects for short, which made its way into modern functional programming in the shape of Haskells do-notation. Standard computational idioms call for specific classes of monads that support additional control operations. Here, we introduce Kleene monads, which additionally feature nondeterministic choice and Kleene star, i.e. nondeterministic iteration, and we provide a metalanguage and a sound calculus for Kleene monads, the metalanguage of control and effects, which is the natural joint extension of Kleene algebra and the metalanguage of effects. This provides a framework for studying abstract program equality focussing on iteration and effects. These aspects are known to have decidable equational theories when studied in isolation. However, it is well known that decidability breaks easily; e.g. the Horn theory of continuous Kleene algebras fails to be recursively enumerable. Here, we prove several negative results for the metalanguage of control and effects; in particular, already the equational theory of the unrestricted metalanguage of control and effects over continuous Kleene monads fails to be recursively enumerable. We proceed to identify a fragment of this language which still contains both Kleene algebra and the metalanguage of effects and for which the natural axiomatisation is complete, and indeed the equational theory is decidable.

A Relatively Complete Generic Hoare Logic for Order-Enriched Effects

Monads are the basis of a well-established method of encapsulating side-effects in semantics and programming. There have been a number of proposals for monadic program logics in the setting of plain monads, while much of the recent work on monadic semantics is concerned with monads on enriched categories, in particular in domain-theoretic settings, which allow for recursive monadic programs. Here, we lay out a definition of order-enriched monad which imposes cpo structure on the monad itself rather than on base category. Starting from the observation that order-enrichment of a monad induces a weak truth-value object, we develop a generic Hoare calculus for monadic side-effecting programs. For this calculus, we prove relative completeness via a calculus of weakest preconditions, which we also relate to strongest postconditions.

Unguarded Recursion on Coinductive Resumptions

We study a model of side-effecting processes obtained by starting from a monad modelling base effects and adjoining free operations using a cofree coalgebra construction; one thus arrives at what one may think of as types of non-wellfounded side-effecting trees, generalizing the infinite resumption monad. Types of this kind have received some attention in the recent literature; in particular, it has been shown that they admit guarded iteration. Here, we show that they also admit unguarded iteration, i.e. form complete Elgot monads, provided that the underlying base effect supports unguarded iteration.

A generic complete dynamic logic for reasoning about purity and effects

For a number of programming languages, among them Eiffel, C, Java, and Ruby, Hoare-style logics and dynamic logics have been developed. In these logics, pre- and postconditions are typically formulated using potentially effectful programs. In order to ensure that these pre- and postconditions behave like logical formulae (that is, enjoy some kind of referential transparency), a notion of purity is needed. Here, we introduce a generic framework for reasoning about purity and effects. Effects are modelled abstractly and axiomatically, using Moggi’s idea of encapsulation of effects as monads. We introduce a dynamic logic (from which, as usual, a Hoare logic can be derived) whose logical formulae are pure programs in a strong sense. We formulate a set of proof rules for this logic, and prove it to be complete with respect to a categorical semantics. Using dynamic logic, we then develop a relaxed notion of purity which allows for observationally neutral effects such writing on newly allocated memory.

Towards a Coalgebraic Chomsky Hierarchy

The Chomsky hierarchy plays a prominent role in the foundations of theoretical computer science relating classes of formal languages of primary importance. In this paper we use recent developments on coalgebraic and monad-based semantics to obtain a generic notion of a \(\mathbb{T}\)-automaton, where \(\mathbb{T}\) is a monad, which allows the uniform study of various notions of machines (e.g. finite state machines, multi-stack machines, Turing machines, weighted automata). We use the generalized powerset construction to define a generic (trace) semantics for \(\mathbb{T}\)-automata, and we show by numerous examples that it correctly instantiates for some known classes of machines/languages captured by the Chomsky hierarchy. Moreover, our approach provides new generic techniques for studying expressivity power of various machine-based models.

Coalgebraic Weak Bisimulation from Recursive Equations over Monads

Strong bisimulation for labelled transition systems is one of the most fundamental equivalences in process algebra, and has been generalised to numerous classes of systems that exhibit richer transition behaviour. Nearly all of the ensuing notions are instances of the more general notion of coalgebraic bisimulation. Weak bisimulation, however, has so far been much less amenable to a coalgebraic treatment. Here we attempt to close this gap by giving a coalgebraic treatment of (parametrized) weak equivalences, including weak bisimulation. Our analysis requires that the functor defining the transition type of the system is based on a suitable order-enriched monad, which allows us to capture weak equivalences by least fixpoints of recursive equations. Our notion is in agreement with existing notions of weak bisimulations for labelled transition systems, probabilistic and weighted systems, and simple Segala systems.

Powermonads and Tensors of Unranked Effects

In semantics and in programming practice, algebraic concepts such as monads or, essentially equivalently, (large) Lawvere theories are a well-established tool for modelling generic side-effects. An important issue in this context are combination mechanisms for such algebraic effects, which allow for the modular design of programming languages and verification logics. The most basic combination operators are sum and tensor: while the sum of effects is just their non-interacting union, the tensor imposes commutation of effects. However, for effects with unbounded arities, these combinations need not in general exist. Here, we introduce the class of uniform effects, which includes unbounded nondeterminism and continuations, and prove that the tensor does always exist if one of the component effects is uniform, thus in particular improving on previous results on tensoring with continuations. We then treat the case of nondeterminism in more detail, and give an order-theoretic characterization of effects for which tensoring with nondeterminism is conservative, thus enabling nondeterministic arguments such as a generic version of the Fischer-Ladner encoding of control operators.

Completeness of global evaluation logic

Monads serve the abstract encapsulation of side effects in semantics and functional programming. Various monad-based specification languages have been introduced in order to express requirements on generic side-effecting programs. A basic role is played here by global evaluation logic, concerned with formulae which may be thought of as being universally quantified over the state space; this formalism is the fundament of more advanced logics such as monad-based Hoare logic or dynamic logic. We prove completeness of global evaluation logic for models in cartesian categories with a distinguished Heyting algebra object.

Unifying Guarded and Unguarded Iteration

Models of iterated computation, such as completely iterative monads, often depend on a notion of guardedness, which guarantees unique solvability of recursive equations and requires roughly that recursive calls happen only under certain guarding operations. On the other hand, many models of iteration do admit unguarded iteration. Solutions are then no longer unique, and in general not even determined as least or greatest fixpoints, being instead governed by quasi-equational axioms. Monads that support unguarded iteration in this sense are called complete Elgot monads. Here, we propose to equip monads with an abstract notion of guardedness and then require solvability of abstractly guarded recursive equations; examples of such abstractly guarded pre-iterative monads include both iterative monads and Elgot monads, the latter by deeming any recursive definition to be abstractly guarded. Our main result is then that Elgot monads are precisely the iteration-congruent retracts of abstractly guarded iterative monads, the latter being defined as admitting unique solutions of abstractly guarded recursive equations; in other words, models of unguarded iteration come about by quotienting models of guarded iteration.