Armin B. Cremers

The structure of context-free grammatical families*

Let ℒ CI be the family of context-free languages. Two characterization theorems are given for (context-free) grammatical families. The first says that the class of grammatical families not ℒ CF is the smallest collection of sets of languages which contains the trivial ones and is closed under a union operator ∨, a concatenation operator ⊙, and ℐ, where ℐ is a ternary operator involving substitution into linear languages. The second asserts that the collection of all nontrivial grammatical families not ℒ CF is the smallest collection of sets of languages which contains the family of regular sets, and is closed under ∨, ⊙, ℐ, and the full AFL operator ℱ ^ .

Normal forms for context-sensitive grammars

SummaryIn this paper, we mainly study the relation between scattered context grammars, which are an example for regulated context-free rewriting devices, and context-sensitive grammars. Emphasis is laid upon both normal form characterizations of context-sensitive grammars and an argument in how far scattered context grammars are stronger, with respect to generative capacity than unordered scattered context grammars.

Mutual exclusion of N processors using an O(N)-valued message variable

The concurrent programming control problem of mutual exclusion can be precisely defined in the framework of data spaces. The size of the shared message variable is introduced as a machine-independent complexity measure for the effort of implementing scheduling schemes for mutual exclusion. Along the lines of a minimal solution for two-processor mutual exclusion using a three-valued message variable, a general n-processor solution is developed using uninterruptable test-and-set instructions on a (2n-1)-valued message variable.

Characterization of context-free grammatical families

Two characterization theorems for the class of context-free grammatical families are given. The characterizations are expressed in terms of the family of regular sets, the family of linear sets, and the following language-family operators: union, concatenation, the full semi-AFL operator, the full AFL operator, and a new ternary substitution operator.

The Semantical Definition of Programming Languages in Terms of Their Data Spaces

In this paper we extend the notion of data type with a control structure to define “data spaces” that are useful in describing semantical aspects of programming languages as well as properties of software systems. The latter, in fact, are included in the former, since a programming language usually serves as a vehicle for expressing something that a computer system does.

Axioms for Concurrent Processes

The theoretical construct data space is intended for modeling computing processes in general 1, 2, 3] and proved of considerable utility in modelling algorithms designed for VLSI 4, 5]. It is here applied, along with a sharpened deenition of subspace, to the formalization of two notions of concurrent processing: the functional equivalence of message passing and shared memory, and the mutual exclusion problem. A method for constructing big mutual exclusion schemes from small ones is presented formally. Fundamental Deenitions. A data space D consists of a set X(D) of states, a relation p(D), called the moves, on the states such that no (x; x) is in p, a subset of X called the start states, and a set F(D) of functions on the states which is complete: if f(x) = f(y) for all f 2 F then x = y, and orthogonal: for any function on F such that (f) is in the range of f for each f, there is a state x such that f(x) = (f) for each f in F, The functions F are often referred to as the memory cells. The range of one of them corresponds to a data type. The orthogonality property says in eeect that the memory cells are independent: that for any set of values of the memory cells there is a state in which all those values are found in the memory cells. We use the notation x ! y, or x ! y in D when D needs to be speciied, to say that x; y is in p. We also call x ! y a move of D, which is technically not compatible with the rst use but will always be clear in the context. A history of D, or D-history, is a ((nite or innnite) sequence x 1 ; x 2 ; : : : of states such that x i ! x i+1 for each i. A state x is reachable if there is a nite history beginning with a start state and ending with x. We use the notation x(f 1 = v 1) to denote the state y such that f 1 (y) = v 1 and f(y) = f(x) for all f 6 = f 1. Continuing by induction, x(f 1 = v 1 ; : : : ; f n+1 = v n+1) means x(f 1 = v 1 ; : : …

Context-Free Grammar Forms

In [3] an attempt was made to formalize a notion of “family of grammars” which would yield, as special instances, many of the well-known families of phrase structure grammars. The model presented here (1) is a variant of that in [3], but is much simpler in nature, It involves two concepts, a “grammar form” and an “interpretation.” A grammar form, the analogue to a grammar schema of [3], is a pharase structure grammar (v, l, θ, σ) together with two sets V and Σ, where Σ and V-Σ are infinite, Σ ⊆ V, (v−l) ⊆ (V-Σ), and l ⊆ Σ. Informally, the productions in θ are the “skeleton productions” and determine the form of the productions in the grammars to be defined, while Σ and V-Σ consist of the terminal letters and variables, respectively, of the grammars to be defined. An interpretation I, the analogue to an interpretation in [3], consists of (i) a substitution μ on v* such that μ(a) is a finite subset of Σ* for each element a in l, μ(ξ) is a finite subset of V-Σ for each element ξ in v−l, and μ(ξ) ⋂ μ(η) = o for all ξ and η, ξ ≠ η, in v−l, and (ii) a phrase structure grammar G1= (V1, E1, P1, σ1), where P1 is a subset of , μ(α → β) being the set {u → v/u in μ(α), v in μ(β)}, V1, and Σ1 are the sets of all symbols in V and Σ, respectively, occurring in productions of P1, and σ1 is in μ(σ). Thus an interpretation consists essentially of a finite substitution μ and a phrase structure grammar whose set of productions is a subset of the productions in θ under the substitution μ and whose start variable is in μ(σ).

Observations about bounded languages and developmental systems

We prove that every strictly bounded language with a semilinear Parikh mapping is an EDTOL language, and use the main proofargument to establish that all finite intersections of bounded context-free languages, and also the complements of these intersections, are EDTOL languages.