We study five extensions of the polymorphically typed lambda-calculus (system F) by type constructs intended to model fixed-points of monotone operators. Building on work by Geuvers concerning the relation between term rewrite systems for least pre-fixed-points and greatest post-fixed-points of positive type schemes (i.e., non-nested positive inductive and coinductive types) and so-called retract types, we show that there are reduction-preserving embeddings even between systems of monotone (co)inductive types and non-inter leav ing positive fixed-point types (which are essentially those retract types). The reduction relation considered is β- and η-reduction for system FF plus either (full) primitive recursion on the inductive types or (full) primitive corecursion on the coinductive types or an extremely simple rule for the fixed-point types. Monotonicity is not confined to the syntactic restriction on type formation of having only positive occurrences of the type variable α in ρ for the inductive type µαρ or the coinductive type ναρ. Instead of that only a “monotonicity witness” which is a term of type ∀α∀β.(α → β) → ρ → ρ[α:=β] is required. This term may already use (co)recursion such that our monotone (co)inductive types may even be “interleaved” and not only nested.

In this paper we give a simple proof that the alternation-depth hierarchy of the μ-calculus for binary trees is strict. The witnesses for this strictness are the automata that determine whether there is a winning strategy for the parity game played on a tree.

We provide an elementary proof of the fixpoint alternation hierarchy in arithmetic, which in turn allows us to simplify the proof of the modal mu-calculus alternation hierarchy. We further show that the alternation hierarchy on the binary tree is strict, resolving a problem of Niwiński.

The notion of a bisimulation relation is of basic importance in many areas of computation theory and logic. Of late, it has come to take a particular significance in work on the formal analysis and verification of hybrid control systems, where system properties are expressible by formulas of the modal μ-calculus or weaker temporal logics. Our purpose here is to give an analysis of the concept of bisimulation, starting with the observation that the zig-zag conditions are suggestive of some form of continuity. We give a topological characterization of bisimularity for preorders, and then use the topology as a route to examining the algebraic semantics for the µ-calculus, developed in recent work of Kwiatkowska et al., and its relation to the standard set-theoretic semantics. In our setting, μ-calculus sentences evaluate as clopen sets of an Alexandroff topology, rather than as clopens of a (compact, Hausdorff) Stone topology, as arises in the Stone space representation of Boolean algebras (with operators). The paper concludes by applying the topological characterization to obtain the decidability of μ-calculus properties for a class of first-order definable hybrid dynamical systems, slightly extending and considerably simplifying the proof of a recent result of Lafferriere et al.

Proof systems with sequents of the form U ⊢ Φ for proving validity of a propositional modal μ-calculus formula Φ over a set U of states in a given model usually handle fixed-point formulae through unfolding, thus allowing such formulae to reappear in a proof. Tagging is a technique originated by Winskel for annotating fixed-point formulae with information about the proof states at which these are unfolded. This information is used later in the proof to avoid unnecessary unfolding, without having to investigate the history of the proof. Depending on whether tags are used for acceptance or for rejection of a branch in the proof tree, we refer to “positive” or “negative” tagging, respectively. In their simplest form, tags consist of the sets U at which fixed-point formulae are unfolded. In this paper, we generalise results of earlier work by Andersen et al. which, in the case of least fixed-point formulae, are applicable to singleton U sets only.

An illustration of coinduction in terms of a notion of weak bisimilarity is presented. First, an operational semantics $\mbox{${\cal O}$}$ for while programs is defined in terms of a final automaton. It identifies any two programs that are weakly bisimilar, and induces in a canonical manner a compositional model $\mbox{${\cal D}$}$. Next $\mbox{${\cal O}$}= \mbox{${\cal D}$}$ is proved by coinduction.

The study of inductive and coinductive types (like finite lists and streams, respectively) is usually conducted within the framework of category theory, which to all intents and purposes is a theory of sets and functions between sets. Allegory theory, an extension of category theory due to Freyd, is better suited to modelling relations between sets as opposed to functions between sets. The question thus arises of how to extend the standard categorical results on the existence of final objects in categories (for example, coalgebras and products) to their existence in allegories. The motivation is to streamline current work on generic programming, in which the use of a relational theory rather than a functional theory has proved to be desirable. In this paper, we define the notion of a relational final dialgebra and prove, for an important class of dialgebras, that a relational final dialgebra exists in an allegory if and only if a final dialgebra exists in the underlying category of maps. Instances subsumed by the class we consider include coalgebras and products. An important lemma expresses bisimulations in allegorical terms and proves this equivalent to Aczel and Mendler's categorical definition.

For every fixed-point expression e of alternation-depth r, we construct a new fixed-point expression e' of alternation-depth 2 and size ${\cal O}(r\cdot|e|)$. Expression e' is equivalent to e whenever operators are distributive and the underlying complete lattice has a co-continuous least upper bound. We alternation-depth but also w.r.t. the increase in size of the resulting expression.

An alternative (tree-based) semantics for a class of regular expressions is proposed that assigns a central rôle to the + operator and thus to nondeterminism and nondeterministic choice. For the new semantics a consistent and complete axiomatization is obtained from the original axiomatization of regular expressions by Salomaa and by Kozen by dropping the idempotence law for + and the distribution law of • over +.

We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the well-accepted operational definition proposed by Barendregt et al. – but for the case of circular redexes , for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms.