Tractable Shape Grammars

Abstract

This paper explores the theoretical basis for a concept of "computation-friendly" shape grammars, through a formal examination of tractability of the grammar formalism.Although a variety of shape grammar definitions have evolved over time, it is possible to unify these to be backwards compatible. Under this unified definition, a shape grammar can be constructed to simulate any Turing machine from which it follows that: a shape grammar may not halt; its language space can be exponentially large; and in general, its membership problem is unsolvable. Moreover, parametric subshape recognition is shown to be NP. This implies that it is unlikely in general to find a polynomial time algorithm to interpret parametric shape grammars, and that more pragmatic approaches need to be sought. Factors that influence the tractability of shape grammars are identified and discussed.

Algorithmic complexity of shape grammar implementation

“…The parametric shape grammar implementation of Grasl and Economou (2013) achieves subshape detection through non-planar subgraph matching, which is NP-complete. Yue and Krishnamurti (2013) argue that subshape detection for parametric shape grammars is NP-complete in general. Their proof rests on the claim that – by converting a graph into a shape – one could use a hypothetical, polynomial-time subshape detection algorithm for subgraph matching.…”

Typological Descriptions as Generative Guides for Historical Architecture

“…We use the term shape grammar in reference to its definition by Stiny (1992) (initially Stiny and Gips 1971;see also Yue and Krishnamurti 2013), except that we consider the grammar to apply over an algebra that is not only composed of shape algebras (extended with labels and/or weights) but also (a) description algebra(s). To some extent, this can be considered a small variation on the definition of shape grammars and mostly in line with earlier redefinitions of shape grammars by Stiny (see Yue and Krishnamurti 2013). However, to avoid further ambiguity, we will talk about rules Fig.…”

Developing a tractable shape grammar

“…Tractable shape grammars have polynomial time and language space complexity (Yue and Krishnamurti, 2013). Grammars exhibit a variety of characteristics thereby rendering impossible a single uniform shape grammar interpreter.…”

“…Previously, we examined tractable parametric shape grammars (Yue and Krishnamurti, 2013), and developed a general paradigm for implementing classes of such grammars (Yue and Krishnamurti, 2014). A tractable shape grammar has polynomial computing complexity, and is specified in a way that is readily transformable to a computer program.…”

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AbstractPreviously, we examined tractable parametric shape grammars (Yue and Krishnamurti, 2013), and developed a general paradigm for implementing classes of such grammars (Yue and Krishnamurti, 2014). A tractable shape grammar has polynomial computing complexity, and is specified in a way that is readily transformable to a computer program.

By contrast, traditionally, shape grammars have been typically developed without a computer implementation in mind, either requiring ambiguity to be clarified, or it is hardly possible for the grammar to be implemented by a polynomial algorithm.

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