Chomsky hierarchy

About: Chomsky hierarchy is a research topic. Over the lifetime, 601 publications have been published within this topic receiving 31067 citations. The topic is also known as: Chomsky–Schützenberger hierarchy.

On the Computational Capabilities of Physical Systems

Abstract: In this first of two papers, strong limits on the accuracy of physical computation are established. First it is proven that there cannot be a physical computer C to which one can pose any and all computational tasks concerning the physical universe. Next it is proven that no physical computer C can correctly carry out any computational task in the subset of such tasks that can be posed to C. This result holds whether the computational tasks concern a system that is physically isolated from C, or instead concern a system that is coupled to C. As a particular example, this result means that there cannot be a physical computer that can, for any physical system external to that computer, take the specification of that external system's state as input and then correctly predict its future state before that future state actually occurs; one cannot build a physical computer that can be assured of correctly 'processing information faster than the universe does'. The results also mean that there cannot exist an infallible, general-purpose observation apparatus, and that there cannot be an infallible, general-purpose control apparatus. These results do not rely on systems that are infinite, and/or non-classical, and/or obey chaotic dynamics. They also hold even if one uses an infinitely fast, infinitely dense computer, with computational powers greater than that of a Turing Machine. This generality is a direct consequence of the fact that a novel definition of computation - a definition of 'physical computation' - is needed to address the issues considered in these papers. While this definition does not fit into the traditional Chomsky hierarchy, the mathematical structure and impossibility results associated with it have parallels in the mathematics of the Chomsky hierarchy. The second in this pair of papers presents a preliminary exploration of some of this mathematical structure, including in particular that of prediction complexity, which is a 'physical computation analogue' of algorithmic information complexity. It is proven in that second paper that either the Hamiltonian of our universe proscribes a certain type of computation, or prediction complexity is unique (unlike algorithmic information complexity), in that there is one and only version of it that can be applicable throughout our universe.

A Generalisation of Parikh's Theorem in Formal Language Theory

Abstract: We show that when a family of languages F has a few appropriate closure-properties, all languages algebraic over F are still equivalent to languages in F when occurrences of symbols are permuted At the same time, the methods used imply a new and simple algebraic proof of Parikh’s original theorem, directly transforming an arbitrary context-free grammar into a letter-equivalent regular grammar Further applications are discussed

Efficient context-free grammar constraints

Abstract: With the introduction of constraints based on finite automata a new line of research has opened where constraints are based on formal languages. Recently, constraints based on grammars higher up in the Chomsky hierarchy were introduced. We devise a time- and space-efficient incremental arc-consistency algorithm for context-free grammars. Particularly, we show how to filter a sequence of monotonically tightening problems in cubic time and quadratic space. Experiments on a scheduling problem show orders of magnitude improvements in time and space consumption.

Operators and Nucleus: A Contribution to the Theory of Grammar

Abstract: 1. Preliminaries 2. Chomsky's model of grammatical description 3. Criticisms of Chomsky's model 4. Deep structure and operators 5. A specimen of description Bibliography Index.

Formal Languages: Origins and Directions

Abstract: Origins of the theory of formal languages and automata are surveyed starting from 1936 with the work of Turing and Post. Special attention is given to the machine translation projects of the 1950s and early 1960s and associated work in mathematical linguistics. The development of the Chomsky hierarchy of grammars, machines, and languages from 1956 to 1964 is traced. It is observed that the same important ideas emerged independently for the automatic analysis and translation of both natural and artificial languages. Since 1964, formal language theory is part of theoretical computer science. A few of the directions since 1964 are considered: restrictions and extensions of context-free grammars and pushdown store automata, unifying frameworks, and complexity questions.