Downward Closures of Indexed Languages

ions. A fruitful idea in the analysis of complex systems is that of abstractions: Instead of working with the original model, one considers a model that has a simpler structure but preserves pertinent properties. A prominent example of such an abstraction is the Parikh image, which is available whenever the semantics of a model is given as a formal language. If L is a language over an alphabet X = {x1, . . . , xn}, then its Parikh image Ψ(L) consists of all vectors (a1, . . . , an) ∈ N such that there is a w ∈ L in which xi occurs ai times for i ∈ {1, . . . , n}. The Parikh image is a useful abstraction, provided that for the languages L at hand, Ψ(L) can be described using simpler means than L itself. This means, most applications of Parikh images are confined to situations where Ψ(L) is semilinear (or, equivalently, definable in Presburger arithmetic). For example, this is the case for context-free languages [10], languages accepted by blind (or revesal-bounded) counter automata [7], and combinations thereof [1, 4, 6]. Of course, there are important types of languages that do not guarantee semilinearity of their Parikh image. For example, Parikh images of languages accepted by higher-order pushdown automata are not semilinear in general: It is easy to construct a second-order pushdown automaton for the language {a2n | n ≥ 0}. Downward closures. However, there is an abstraction of languages that guarantees a simple description for every language whatsoever and still reflects important properties of the abstracted language—the downward closure. For words u, v ∈ X∗, we write u v if u = u1 · · ·un and v = v0u1v1 · · ·unvn for some u1, . . . , un, v0, . . . , vn ∈ X∗. Then, the downward closure of L is defined as L↓ = {u ∈ X∗ | ∃v ∈ L : u v}. In other words, the downward closure consists of the set of all (not necessarily contiguous) subwords of members of L. It has the remarkable property that it is regular for every set of words L ⊆ X∗, which is due to the fact that the subword ordering is a well-quasi-ordering [5]. Moreover, downward closures preserve useul information about the abstracted language. Suppose L describes the behavior of a system that is observed through a lossy channel, meaning that on the way to the observer, arbitrary actions can get lost. Then, L↓ is the set of words received by the observer [3]. Hence, given the downward closure as a finite automaton, we can decide whether two systems are equivalent under such observations, and even whether one system includes the behavior of another. However, while we know that for every L, there exists a finite automaton for L↓, it is not clear in general how to compute them. In fact, there are examples of languages classes for which downward closures are known not to be computable [2, 9]. For an overview of the language classes for which computability is known, see [11].