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3.4: Phrase Markers and Fractals

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    21217
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    Consider a finite set of elements (e.g., words, phonemes, morphemes) that can, by applying certain rules, be combined to create a sentence or expression that is finite in length. A language can be defined as the set of all of the possible expressions that can generated in this way from the same set of building blocks and the same set of rules (Chomsky, 1957). From this perspective, one can define a grammar as a device that can distinguish the set of grammatical expressions from all other expressions, including those that are generated from the same elements but which violate the rules that define the language. In modern linguistics, a basic issue to investigate is the nature of the grammar that defines a natural human language.

    Chomsky (1957) noted that one characteristic of a natural language such as English is that a sentence can be lengthened by inserting a clause into its midst. As we see in the following section, this means that the grammar of natural languages is complicated enough that simple machines, such as finite state automata, are not powerful enough to serve as grammars for them.

    The complex, clausal structure of a natural language is instead captured by a more powerful device—a Turing machine—that can accommodate the regularities of a context-free grammar (e.g., Chomsky, 1957, 1965). A context-free grammar can be described as a set of rewrite rules that convert one symbol into one or more other symbols. The application of these rewrite rules produces a hierarchically organized symbolic structure called a phrase marker (Radford, 1981). A phrase marker is a set of points or labelled nodes that are connected by branches. Nonterminal nodes represent lexical categories; at the bottom of a phrase marker are the terminal nodes that represent lexical categories (e.g., words). A phrase marker for the simple sentence Dogs bark is illustrated in Figure 3-6.

    clipboard_e92c8b00811cff45efad9f2702830327a.png

    Figure 3-6. A phrase marker for the sentence Dogs bark.

    The phrase marker for a sentence can be illustrated as an upside-down tree whose structure is grown from the root node S (for sentence). The application of the rewrite rule S → NP VP produces the first layer of the Figure 3-6 phrase marker, showing how the nodes NP (noun phrase) and VP (verb phrase) are grown from S. Other rewrite rules that are invoked to create that particular phrase marker are NP → , → N, N → dogs, VP → , → V, and V → bark. When any of these rewrite rules are applied, the symbol to the left of the → is rewritten as the symbol or symbols to the right. In the phrase marker, this means the symbols on the right of the → are written as nodes below the original symbol, and are connected to the originating node above, as is shown in Figure 3-6.

    In a modern grammar called x-bar syntax (Jackendoff, 1977), nodes like NP and VP in Figure 3-6 are symbols that represent phrasal categories, nodes like and are symbols that represent lexical categories, and nodes like “and” are symbols that represent categories that are intermediates between lexical categories and phrasal categories. Such intermediate categories are required to capture some regularities in the syntax of natural human languages.

    In some instances, the same symbol can be found on both sides of the → in a rewrite rule. For instance, one valid rewrite rule for the intermediate node of a noun NP N S N dogs bark VP V V Elements of Classical Cognitive Science 67 phrase is → AP , where AP represents an adjective phrase. Because the same symbol occurs on each side of the equation, the context-free grammar is recursive. One can apply this rule repeatedly to insert clauses of the same type into a phrase. This is shown in Figure 3-7, which illustrates phrase markers for noun phrases that might apply to my dog Rufus. The basic noun phrase is the dog. If this recursive rule is applied once, it permits a more elaborate noun phrase to be created, as in the cute dog. Recursive application of this rule permits the noun phrase to be elaborated indefinitely, (e.g., the cute brown scruffy dog).

    clipboard_eca5701654ea317430f9aa7da4312f088.png

    Figure 3-7. Phrase markers for three noun phrases: (A) the dog, (B) the cute dog, and (C) the cute brown scruffy dog. Note the recursive nature of (C).

    The recursive nature of a context-free grammar is revealed in a visual inspection of a phrase marker like the one illustrated in Figure 3-7C. As one inspects the figure, one sees the same pattern recurring again and again, as was the case with the Sierpinski triangle. The recursive nature of a context-free grammar produces self-similarity within a phrase marker. The recursion of such a grammar is also responsible for its ability to use finite resources (a finite number of building blocks and a finite number of rewrite rules) to produce a potentially infinite variety of expressions, as in the sentences of a language, each of which is represented by its own phrase marker.


    This page titled 3.4: Phrase Markers and Fractals is shared under a CC BY-NC-ND license and was authored, remixed, and/or curated by Michael R. W. Dawson (Athabasca University Press) .

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