Coinductive Proof Principles Research Articles

Fixpoint Theory -- Upside Down

Knaster-Tarski's theorem, characterising the greatest fixpoint of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form $\mathbb{M}^Y$, where $Y$ is a finite set and $\mathbb{M}$ an MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, metric transition systems, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games.

Well-definedness and observational equivalence for inductive–coinductive programs

Abstract We define notions of well-definedness and observational equivalence for programs of mixed inductive and coinductive types. These notions are defined by means of tests formulas which combine structural congruence for inductive types and modal logic for coinductive types. Tests also correspond to certain evaluation contexts. We define a program to be well-defined if it is strongly normalizing under all tests, and two programs are observationally equivalent if they satisfy the same tests. We show that observational equivalence is sufficiently coarse to ensure that least and greatest fixed point types are initial algebras and final coalgebras, respectively. This yields inductive and coinductive proof principles for reasoning about program behaviour. On the other hand, we argue that observational equivalence does not identify too many terms, by showing that tests induce a topology that, on streams, coincides with usual topology induced by the prefix metric. As one would expect, observational equivalence is, in general, undecidable, but in order to develop some practically useful heuristics we provide coinductive techniques for establishing observational normalization and observational equivalence, along with up-to techniques for enhancing these methods.

Modular Control of Discrete-Event Systems With Coalgebra

Modular supervisory control of discrete-event systems, where the overall system is a synchronous (parallel) product of subsystems, is considered. The main results of this paper are formulations of sufficient conditions for the compatibility between the synchronous product and various operations stemming from supervisory control as supervised product and supremal controllable sublanguages. These results are generalized to the case of modules with partial observations: e.g., modular computation of supremal normal sublanguages is studied. Coalgebraic techniques based on the coinduction proof principle are used in our main results. Sufficient conditions are derived for modular to equal global control synthesis. An algorithmic procedure for checking the new conditions is proposed and the computational benefit of the modular approach is discussed and illustrated by comparing the time complexity of modular and monolithic computation.

Coinductive Proof Principles for Stochastic Processes

We give an explicit coinduction principle for recursively-defined stochastic processes. The principle applies to any closed property, not just equality, and works even when solutions are not unique. The rule encapsulates low-level analytic arguments, allowing reasoning about such processes at a higher algebraic level. We illustrate the use of the rule in deriving properties of a simple coin-flip process.

Normal Form Simulation for McCarthy's Amb

This paper presents co-inductive operational theories of program refinement and equivalence, called whnf similarity and whnf simulation equivalence, for the λ-calculus extended with McCarthy's ambiguous choice operator amb. The associated whnf simulation co-induction proof principle is useful for establishing non-trivial equivalences and refinement relationships between programs. Whnf similarity is a pre-congruence and whnf simulation equivalence is a congruence and a conservative extension of the Lévy-Longo tree theory for the pure λ-calculus.

A Coalgebraic Approach to Process Equivalence and a Coinduction Principle for Traces

An abstract coalgebraic approach to well-structured relations on processes is presented, based on notions of tests and test suites. Preorders and equivalences on processes are modelled as coalgebras for behaviour endofunctors lifted to a category of test suites. The general framework is specialized to the case of finitely branching labelled transition systems. It turns out that most equivalences from the so-called van Glabbeek spectrum can be described by well-structured test suites. As an immediate application, coinductive proof principles are described for these equivalences, in particular for trace equivalence.

On Equivalences for a Class of Timed Regular Expressions

Timed regular expressions are an extension of regular expressions that capture a notion of time. Roughly speaking, timed regular expressions can be used to represent timed sequences of events, with new operators to control the duration of those sequences. These timed regular expressions correspond to a form of timed automaton equipped with clocks, of the kind introduced by Alur and Dill. We develop a coalgebraic treatment of such timed regular expressions, along the lines of the coalgebraic treatment of regular expressions based on deterministic automata. This yields a coinductive proof principle, that can be used to establish equivalence of a class of timed regular expressions.

Bisimilarity for the Region Calculus

A region calculus is a programming language calculus with explicit instrumentation for memory management. Every value is annotated with a region in which it is stored and regions are allocated and deallocated in a stack-like fashion. The annotations can be statically inferred by a type and effect system, making a region calculus suitable as an intermediate language for a compiler of statically typed programming languages. Although a lot of attention has been paid to type soundness properties of different flavors of region calculi, it seems that little effort has been made to develop a semantic framework. In this paper, we present a theory based on bisimulation, which serves as a coinductive proof principle for showing equivalences of polymorphically region-annotated terms. Our notion of bisimilarity is reminiscent of open bisimilarity for the ?-calculus and we prove it sound and complete with respect to Morris-style contextual equivalence. As an application, we formulate a syntactic equational theory, which is used elsewhere to prove the soundness of a specializer based on region inference. We use our bisimulation framework to show that the equational theory is sound with respect to contextual equivalence.

Supremal normal sublanguages of large distributed discrete-event systems

Coalgebraic methods provide new results and insights for modular supervisory control of discrete-event systems (DES), where the overall system is composed of subsystems that are themselves partially observed DES. It is well known that the computation of supremal normal sublanguages is computationally very difficult. The attention of this paper is focused on complex distributed systems that are composed of a large number of small subsystems that are combined in a modular fashion. Conditions are derived under which supremal normal sublanguages commute with synchronous product, i.e. the computation of supremal normal sublanguages can be done locally. The coinduction proof principle is used to obtain our main result.

A coalgebraic approach to Kleene algebra with tests

Kleene algebra with tests is an extension of Kleene algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene algebra with Tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. Since the known automata-theoretic presentation of Kleene algebra with tests does not lend itself to a coalgebraic theory, we define a new interpretation of Kleene algebra with tests expressions and a corresponding automata-theoretic presentation. One outcome of the theory is a coinductive proof principle, that can be used to establish equivalence of our Kleene algebra with tests expressions.

A Coalgebraic Approach to Kleene Algebra with Tests

Kleene Algebra with Tests is an extension of Kleene Algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene Algebra with Tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. Since the known automata-theoretic presentation of Kleene Algebra with Tests does not lend itself to a coalgebraic theory, we define a new interpretation of Kleene Algebra with Tests expressions and a corresponding automata-theoretic presentation. One outcome of the theory is a coinductive proof principle, that can be used to establish equivalence of our Kleene Algebra with Tests expressions.

Towards Weak Bisimulation For Coalgebras

This paper contains a novel approach to observational equivalence for coalgebras. We describe how to define weak homomorphisms, weak bisimulation, and investigate the connection between them as well as the relation to the known theory of bisimulation for coalgebras. The ultimate result of the paper is the correctness-proof for a weak coinduction proof principle.

From Set-theoretic Coinduction to Coalgebraic Coinduction: some results, some problems

We investigate the relation between the set-theoretical description of coinduction based on Tarski Fixpoint Theorem, and the categorical description of coinduction based on coalgebras. In particular, we introduce set-theoretic generalizations of the coinduction proof principle, in the spirit of Milner's bisimulation “up-to”, and we discuss categorical counterparts for these. Moreover, we investigate the connection between these and the equivalences induced by T-coiterative functions. These are morphisms into final coalgebras, satisfying the T-coiteration scheme, which is a generalization of the corecursion scheme. We show how to describe coalgebraic F-bisimulations as set-theoretical ones. A list of examples of set-theoretic coinductions which appear not to be easily amenable to coalgebraic terms are discussed.