Formal grammar

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In formal semantics, computer science and linguistics, a formal grammar (also called formation rules) is a precise description of a formal language – that is, of a set of strings over some alphabet. In other words, a grammar describes which of the possible sequences of symbols (strings) in a language constitute valid words or statements in that language, but it does not describe their semantics (i.e. what they mean). The branch of mathematics that is concerned with the properties of formal grammars and languages is called formal language theory.

A grammar is usually regarded as a means to generate all the valid strings of a language; it can also be used as the basis for a recognizer that determines for any given string whether it is grammatical (i.e. belongs to the language). To describe such recognizers, formal language theory uses separate formalisms, known as automata.

A grammar can also be used to analyze the strings of a language – i.e. to describe their internal structure. In computer science, this process is known as parsing. Most languages have very compositional semantics, i.e. the meaning of their utterances is structured according to their syntax; therefore, the first step to describing the meaning of an utterance in language is to analyze it and look at its analyzed form (known as its parse tree in computer science, and as its deep structure in generative grammar).

1 Background
2 Formal grammars
3 Analytic grammars
4 References
5 See also
6 External links

[edit] Background

[edit] Formal language

Main article: Formal language

A formal language is an organized set of symbols the essential feature of which is that it can be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without any reference to any meanings of any of its expressions; it can exist before any formal interpretation is assigned to it -- that is, before it has any meaning. First order logic is expressed in some formal language. A formal grammar determines which symbols and sets of symbols are formulas in a formal language.

[edit] Formal systems

Main article: Formal system

A formal system (also called a logical calculus, or a logical system) consists of a formal language together with a deductive apparatus (also called a deductive system). The deductive apparatus may consist of a set of transformation rules (also called inference rules) or a set of axioms, or have both. A formal system is used to derive one expression from one or more other expressions.

[edit] Formal proofs

Main article: Formal proof

A formal proof is a sequences of well-formed formulas of a formal language, the last one of which is a theorem of a formal system. The theorem is a syntactic consequence of all the wffs preceding it in the proof. For a wff to qualify as part of a proof, it must be the result of applying a rule of the deductive apparatus of some formal system to the previous wffs in the proof sequence.

[edit] Formal interpretations

Main articles: Formal semantics and Formal interpretation

An interpretation of a formal system is the assignment of meanings to the symbols, and truth-values to the sentences of a formal system. The study of formal interpretations is called formal semantics. Giving an interpretation is synonymous with constructing a model.

[edit] Formal grammars

Main articles: formal language and generative grammar

A grammar consists of a set of rules for transforming strings. To generate a string in the language, one begins with a string consisting of only a single start symbol, and then successively applies the rules (any number of times, in any order) to rewrite this string. The language consists of all the strings that can be generated in this manner. Any particular sequence of legal choices taken during this rewriting process yields one particular string in the language. If there are multiple ways of generating the same single string, then the grammar is said to be ambiguous.

For example, assume the alphabet consists of $a$ and $b$ , the start symbol is $S$ and we have the following rules:

1. $S \rightarrow aSb$

2. $S \rightarrow ba$

then we start with $S$ , and can choose a rule to apply to it. If we choose rule 1, we obtain the string $a S b$ . If we choose rule 1 again, we replace $S$ with $a S b$ and obtain the string $a a S b b$ . This process can be repeated at will until all occurrences of S are removed, and only symbols from the alphabet remain (i.e., $a$ and $b$ ). For example, if we now choose rule 2, we replace $S$ with $b a$ and obtain the string $a a b a b b$ , and are done. We can write this series of choices more briefly, using symbols: $S \Rightarrow aSb \Rightarrow aaSbb \Rightarrow aababb$ . The language of the grammar is the set of all the strings that can be generated using this process: $\left \{ba, abab, aababb, aaababbb, ...\right \}$ .

[edit] Formal definition

In the classic formalization of generative grammars first proposed by Noam Chomsky in the 1950s,^[1]^[2] a grammar G consists of the following components:

A finite set $N$ of nonterminal symbols.
A finite set $Σ$ of terminal symbols that is disjoint from $N$ .
A finite set $P$ of production rules, each of the form

$(\Sigma \cup N)^{*} N (\Sigma \cup N)^{*} \rightarrow (\Sigma \cup N)^{*}$

where

*

is the Kleene star operator and $\cup$ denotes set union. That is, each production rule maps from one string of symbols to another, where the first string contains at least one nonterminal symbol. In the case that the second string is the empty string – that is, that it contains no symbols at all – a lambda (

λ

) is typically written in its place to avoid ambiguity.

A distinguished symbol $S \in N$ that is the start symbol.

A grammar is formally defined as the ordered quad-tuple $(N,Σ, P, S)$ . Such a formal grammar is often called a rewriting system or a phrase structure grammar in the literature.^[3]^[4]

The language of a formal grammar $G = (N,Σ, P, S)$ , denoted as $\boldsymbol{L}(G)$ , is defined as all those strings over $Σ$ that can be generated by starting with the start symbol $S$ and then applying the production rules in $P$ until no more nonterminal symbols are present.

[edit] Example

For these examples, formal languages are specified using set-builder notation.

Consider the grammar $G$ where $N = \left \{S, B\right \}$ , $\Sigma = \left \{a, b, c\right \}$ , $S$ is the start symbol, and $P$ consists of the following production rules:

1. $S \rightarrow aBSc$

2. $S \rightarrow abc$

3. $Ba \rightarrow aB$

4. $Bb \rightarrow bb$

Some examples of the derivation of strings in $\boldsymbol{L}(G)$ are:

$\boldsymbol{S} \Rightarrow_2 \boldsymbol{abc}$
$\boldsymbol{S} \Rightarrow_1 \boldsymbol{aBSc} \Rightarrow_2 aB\boldsymbol{abc}c \Rightarrow_3 a\boldsymbol{aB}bcc \Rightarrow_4 aa\boldsymbol{bb}cc$
$\boldsymbol{S} \Rightarrow_1 \boldsymbol{aBSc} \Rightarrow_1 aB\boldsymbol{aBSc}c \Rightarrow_2 aBaB\boldsymbol{abc}cc \Rightarrow_3 a\boldsymbol{aB}Babccc \Rightarrow_3 aaB\boldsymbol{aB}bccc$ $\Rightarrow_3 aa\boldsymbol{aB}Bbccc \Rightarrow_4 aaaB\boldsymbol{bb}ccc \Rightarrow_4 aaa\boldsymbol{bb}bccc$

(Note on notation: $L \Rightarrow_i R$ reads "L generates R by means of production i" and the generated part is each time indicated in bold.)

This grammar defines the language $L = \left \{ a^{n}b^{n}c^{n} | n \ge 1 \right \}$ where $a n$ denotes a string of n consecutive $a$ 's. Thus, the language is the set of strings that consist of 1 or more $a$ 's, followed by the same number of $b$ 's, followed by the same number of $c$ 's.

[edit] The Chomsky hierarchy

Main article: Chomsky hierarchy

When Noam Chomsky first formalized generative grammars in 1956,^[1] he classified them into types now known as the Chomsky hierarchy. The difference between these types is that they have increasingly strict production rules and can express fewer formal languages. Two important types are context-free grammars (Type 2) and regular grammars (Type 3). The languages that can be described with such a grammar are called context-free languages and regular languages, respectively. Although much less powerful than unrestricted grammars (Type 0), which can in fact express any language that can be accepted by a Turing machine, these two restricted types of grammars are most often used because parsers for them can be efficiently implemented.^[5] For example, all regular languages can be recognized by a finite state machine, and for useful subsets of context-free grammars there are well-known algorithms to generate efficient LL parsers and LR parsers to recognize the corresponding languages those grammars generate.

[edit] Context-free grammars

A context-free grammar is a grammar in which the left-hand side of each production rule consists of only a single nonterminal symbol. This restriction is non-trivial; not all languages can be generated by context-free grammars. Those that can are called context-free languages.

The language defined above is not a context-free language, and this can be strictly proven using the pumping lemma for context-free languages, but for example the language $\left \{ a^{n}b^{n} | n \ge 1 \right \}$ (at least 1 $a$ followed by the same number of $b$ 's) is context-free, as it can be defined by the grammar $G 2$ with $N=\left \{S\right \}$ , $\Sigma=\left \{a,b\right \}$ , $S$ the start symbol, and the following production rules:

1. $S \rightarrow aSb$

2. $S \rightarrow ab$

A context-free language can be recognized in $O (n 3)$ time (see Big O notation) by an algorithm such as Earley's algorithm. That is, for every context-free language, a machine can be built that takes a string as input and determines in $O (n 3)$ time whether the string is a member of the language, where $n$ is the length of the string.^[6] Further, some important subsets of the context-free languages can be recognized in linear time using other algorithms.

[edit] Regular grammars

In regular grammars, the left hand side is again only a single nonterminal symbol, but now the right-hand side is also restricted: It may be the empty string, or a single terminal symbol, or a single terminal symbol followed by a nonterminal symbol, but nothing else. (Sometimes a broader definition is used: one can allow longer strings of terminals or single nonterminals without anything else, making languages easier to denote while still defining the same class of languages.)

The language defined above is not regular, but the language $\left \{ a^{n}b^{m} \,| \, m,n \ge 1 \right \}$ (at least 1 $a$ followed by at least 1 $b$ , where the numbers may be different) is, as it can be defined by the grammar $G 3$ with $N=\left \{S, A,B\right \}$ , $\Sigma=\left \{a,b\right \}$ , $S$ the start symbol, and the following production rules:

$S \rightarrow aA$
$A \rightarrow aA$
$A \rightarrow bB$
$B \rightarrow bB$
$B \rightarrow \epsilon$

All languages generated by a regular grammar can be recognized in linear time by a finite state machine. Although, in practice, regular grammars are commonly expressed using regular expressions, some forms of regular expression used in practice do not strictly generate the regular languages and do not show linear recognitional performance due to those deviations.

[edit] Other forms of generative grammars

Many extensions and variations on Chomsky's original hierarchy of formal grammars have been developed more recently, both by linguists and by computer scientists, usually either in order to increase their expressive power or in order to make them easier to analyze or parse. Some forms of grammars developed include:

Tree-adjoining grammars increase the expressiveness of conventional generative grammars by allowing rewrite rules to operate on parse trees instead of just strings.^[7]
Affix grammars^[8] and attribute grammars^[9]^[10] allow rewrite rules to be augmented with semantic attributes and operations, useful both for increasing grammar expressiveness and for constructing practical language translation tools.

[edit] Analytic grammars

Though there is a tremendous body of literature on parsing algorithms, most of these algorithms assume that the language to be parsed is initially described by means of a generative formal grammar, and that the goal is to transform this generative grammar into a working parser. Strictly speaking, a generative grammar does not in any way correspond to the algorithm used to parse a language, and various algorithms have different restrictions on the form of production rules that are considered well-formed.

An alternative approach is to formalize the language in terms of an analytic grammar in the first place, which more directly corresponds to the structure and semantics of a parser for the language. Examples of analytic grammar formalisms include the following:

The Language Machine ^[11] directly implements unrestricted analytic grammars. Substitution rules are used to transform an input to produce outputs and behaviour. The system can also produce the lm-diagram which shows what happens when the rules of an unrestricted analytic grammar are being applied.
Top-down parsing language (TDPL): a highly minimalist analytic grammar formalism developed in the early 1970s to study the behavior of top-down parsers.^[12]
Link grammars: a form of analytic grammar designed for linguistics, which derives syntactic structure by examining the positional relationships between pairs of words.^[13]^[14]
Parsing expression grammars (PEGs): a more recent generalization of TDPL designed around the practical expressiveness needs of programming language and compiler writers.^[15]

[edit] References

^ ^a ^b Chomsky, Noam (1956). "Three Models for the Description of Language". IRE Transactions on Information Theory 2 (2): 113–123.
^ Chomsky, Noam (1957). Syntactic Structures. The Hague: Mouton.
^ Ginsburg, Seymour (1975). Algebraic and automata theoretic properties of formal languages. North-Holland, 8-9. ISBN 0720425069.
^ Harrison, Michael A. (1978). Introduction to Formal Language Theory, 13. ISBN 0201029553.
^ Grune, Dick & Jacobs, Ceriel H., Parsing Techniques – A Practical Guide, Ellis Horwood, England, 1990.
^ Earley, Jay, "An Efficient Context-Free Parsing Algorithm," Communications of the ACM, Vol. 13 No. 2, pp. 94-102, February 1970.
^ Joshi, Aravind K., et al., "Tree Adjunct Grammars," Journal of Computer Systems Science, Vol. 10 No. 1, pp. 136-163, 1975.
^ Koster , Cornelis H. A., "Affix Grammars," in ALGOL 68 Implementation, North Holland Publishing Company, Amsterdam, p. 95-109, 1971.
^ Knuth, Donald E., "Semantics of Context-Free Languages," Mathematical Systems Theory, Vol. 2 No. 2, pp. 127-145, 1968.
^ Knuth, Donald E., "Semantics of Context-Free Languages (correction)," Mathematical Systems Theory, Vol. 5 No. 1, pp 95-96, 1971.
^ the language machine
^ Birman, Alexander, The TMG Recognition Schema, Doctoral thesis, Princeton University, Dept. of Electrical Engineering, February 1970.
^ Sleator, Daniel D. & Temperly, Davy, "Parsing English with a Link Grammar," Technical Report CMU-CS-91-196, Carnegie Mellon University Computer Science, 1991.
^ Sleator, Daniel D. & Temperly, Davy, "Parsing English with a Link Grammar," Third International Workshop on Parsing Technologies, 1993. (Revised version of above report.)
^ Ford, Bryan, Packrat Parsing: a Practical Linear-Time Algorithm with Backtracking, Master’s thesis, Massachusetts Institute of Technology, Sept. 2002.

[edit] See also

[edit] External links

Yearly Formal Grammar conference

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Automata theory: formal languages and **formal grammars**
Chomsky hierarchy	Grammars	Languages	Minimal automaton
Type-0	Unrestricted	Recursively enumerable	Turing machine
n/a	(no common name)	Recursive	Decider
Type-1	Context-sensitive	Context-sensitive	Linear-bounded
n/a	Indexed	Indexed	Nested stack
n/a	Tree-adjoining etc.	(Mildly context-sensitive)	Embedded pushdown
Type-2	Context-free	Context-free	Nondeterministic pushdown
n/a	Deterministic context-free	Deterministic context-free	Deterministic pushdown
Type-3	Regular	Regular	Finite
n/a		Star-free	Counter-Free
Each category of languages or grammars is a proper subset of the category directly above it, and any automaton in each category has an equivalent automaton in the category directly above it.

Categories: Formal languages

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