X-bar theory
In linguistics, X-bar theory is a model of phrase-structure grammar and a theory of syntactic category formation that was first proposed by Noam Chomsky in 1970 and further developed by Jackendoff (1974, 1977a, 1977b), along the lines of the theory of generative grammar put forth in the 1950s by Chomsky. It attempts to capture the structure of phrasal categories with a single uniform structure called the X-bar schema, basing itself on the assumption that any phrase in natural language is an XP (X phrase) that is headed by a given syntactic category X. It  played a significant role in resolving issues that phrase structure rules had, representative of which is the proliferation of grammatical rules, which is against the thesis of generative grammar. X-bar theory was incorporated into both transformational and nontransformational theories of syntax, including Government and binding theory (GB), Generalized phrase structure grammar (GPSG), Lexical-functional grammar (LFG), and Head-driven phrase structure grammar (HPSG). Although recent work in the Minimalist Program has largely abandoned X-bar schemata in favor of bare phrase structure approaches, the theory's central assumptions are still valid in different forms and terms in many theories  of minimalist syntax. Background. The X-bar theory was developed to resolve the issues that phrase structure rules (PSR) under the Standard Theory had. The PSR approach has the following four main issues. The X-bar theory is a theory that attempts to resolve these issues by assuming the mold or template phrasal structure of "XP". X-bar Schema. Basic Principles. The "X" in the X-bar theory is equivalent to a variable in mathematics: It can be substituted by syntactic categories such as N, V, A, and P. These categories are lexemes and not phrases: The "X-bar" is  a grammatical unit larger than X, thus than a lexeme, and the X-double-bar (=XP) outsizes the X(-single)-bar. X-double-bar categories are equal to phrasal categories such as NP, VP, AP, and PP. The X-bar theory assumes that all phrasal categories have the structure in Figure 1. This structure is called the X-bar schema. As in Figure 1, the phrasal category XP is notated by an X with a double overbar. For typewriting reasons, the bar symbol is often substituted by the prime ('), as in X'. The X-bar theory embodies two central principles. The  headedness principle resolves the issues 1 and 3 above simultaneously. The binarity principle is important to projection and ambiguity, which will be explained below. The X-bar schema consists of a head and its circumstantial components, in accordance with the headedness principle. The relevant components are as follows: The specifier, head, and complement are obligatory; hence, a phrasal category XP must contain one specifier, one head, and one complement. On the other hand, the adjunct is optional; hence, a phrasal category contains zero or more adjuncts. Accordingly, when a phrasal category XP does not have an adjunct, it forms the  structure in Figure 2. For example, the NP "linguistics" in the sentence "John studies linguistics" has the structure in Figure 3. It is important that even if there are no candidates that can fit into the specifier and complement positions, these positions are syntactically present, and thus they are merely empty and unoccupied. (This is a natural consequence of the binarity principle.) This means that all phrasal categories have fundamentally uniform structures under the X-bar schema, which makes it unnecessary to assume that different phrases have different structures, unlike when we adopt the PSR. (This resolves the second issue  above.) In the meanwhile, one needs to be wary of when such empty positions are representationally omitted as in Figure 4. In illustrating syntactic structures this way, at least one X'-level node is present in any circumstance because the complement is obligatory. Next, the X<nowiki>"</nowiki> and X' inherit the characteristics of the head X. This trait inheritance is referred to as projection. Figure 5 suggests that syntactic structures are derived in a bottom-up fashion under the X-bar theory. More specifically, the structures are derived via the following processes. It is important that all the processes  except for the third are obligatory. This means that one phrasal category necessarily includes X, X, and XP (=X<nowiki>"</nowiki>). Moreover, nodes bigger than X (thus, X and XP nodes) are called constituents. Directionality of Branching. Figures 1-5 are based on the word order of English, but the X-bar schema does not specify the directionality of branching because the binarity principle does not have a rule on it. For example, "John read a long book of linguistics with a red cover", which involves two adjuncts, may have either of the structures in Figure 6 or  Figure 7. (The figures follow the convention of omitting the inner structures of certain phrasal categories with triangles.) The structure in Figure 6 yields the meaning "the book of linguistics with a red cover is long", and the one in Figure 7 "the long book of linguistics is with a red cover" (see also #Hierarchical Structure). What is important is the directionality of the nodes N' and N': One is left-branching, while the other is right-branching. Accordingly, the X-bar theory, more specifically the binarity principle, does not impose a restriction on how a node branches. When it  comes to the head and the complement, their relative order is determined based on the principles-and-parameters model of language, more specifically by the head parameter (not by the X-bar schema itself). A principle is a shared, invariable rule of grammar across languages, whereas a parameter is a typologically variable aspect of the grammars. One can either set their parameter with the values of "+" or "-": In the case of the head parameter, one configures the parameter of <nowiki>[±head first]</nowiki>, depending on what language they primarily speak. If this parameter is configured to be <nowiki>[+head  first]</nowiki>, what results is head-initial languages such as English, and if it is configured to be <nowiki>[-head first]</nowiki>, what results is head-final languages such as Japanese. For example, the English sentence "John ate an apple" and its corresponding Japanese sentence "John-ga ringo-o tabe-ta" (ジョンがリンゴを食べた; John-NOM apple-ACC eat-PAST) have the structures in Figure 8 and Figure 9, respectively. Note finally that the directionality of the specifier node is in essence unspecified as well, although this is subject to debate: Some argue that the relevant node is necessarily left-branching across  languages, the idea of which is (partially) motivated by the fact that both English and Japanese have subjects on the left of a VP, whereas others such as Saito and Fukui (1998) argue that the directionality of the node is not fixed and needs to be externally determined, for example by the head parameter. Structure of Sentence. Structure of S. Under the PSR, the structure of S(entence) is illustrated as follows. However, this structure violates the headedness principle because it has an exocentric, headless structure, and would also violate the binarity principle if an Aux(iliary) occurs, because the S node  will then be ternary-branching. Given these, Chomsky (1981) proposed that S is an InflP headed by the functional category Infl(ection), and later in Chomsky (1986a), this category was relabelled as I (hence constitutes an IP), following the notational convention that phrasal categories are represented in the form of XP, with two letters. The category I includes auxiliary verbs such as "will" and "can", clitics such as "-s" of the third person singular present and "-ed" of the past tense. This is consistent with the headedness principle, which requires that a phrase have a head, because a sentence (or a  clause) necessarily involves an element that determines the inflection of a verb. Assuming that S constitutes an IP, the structure of the sentence "John studies linguistics at the university", for example, can be illustrated as in Figure 10. As is obvious, the IP hypothesis makes it possible to regard the grammatical unit of sentence as a phrasal category. It is also important that the configuration in Figure 10 is fully compatible with the central assumptions of the X-bar theory, namely the headedness principle and the binarity principle. Structure of S'. Words that introduce subordinate or complement clauses are called  complementizers, and representative of them are "that", "if", and "for." Under the PSR, complement clauses were assumed to constitute the category S'. Chomsky (1986a) proposed that this category is in fact a CP headed by the functional category C. The sentence "I think that John is honest", for example, then has the following structure. Moreover, Chomsky (1986a) assumes that the landing site of wh-movement is the specifier position of CP (Spec-CP). Accordingly, the wh-question "What did John eat?", for example, is derived as in Figure 12. In this derivation, the I-to-C movement is an instance  of subject-auxiliary inversion (SAI), or more generally, head movement. Hierarchical Structure. The PSR has the shortcoming of being incapable of capturing sentence ambiguities. This sentence is ambiguous between the reading "I saw a man, using binoculars", in which "with binoculars" modifies the VP, and the reading "I saw a man who had binoculars", in which the PP modifies the NP. Under the PSR model, the sentence above is subject to the following two parsing rules. The sentence's structure under these PSRs would be as in Figure 13. It is obvious that this structure fails to capture the NP  modification reading because [ with binoculars] modifies the VP no matter how one tries to illustrate the structure. The X-bar theory, however, successfully captures the ambiguity as demonstrated in the configurations in Figure 14 and 15 below, because it assumes hierarchical structures in accordance with the binarity principle. Thus, the X-bar theory resolves the fourth issue mentioned in #Background as well. Note that there is always a unilateral relation from syntax to semantics (never from semantics to syntax) in any version of generative grammar because syntactic computation starts from the lexicon, then continues into the syntax, then into Logical  Form (LF) at which meanings are computed. This is so under any of Standard Theory (Chomsky 1965), Extended Standard Theory (Chomsky 1972), and Revised Extended Standard Theory (Chomsky 1981).