Dependency grammar - henceforth: DG - is the approach to the explanation of language and its structures where the accent is on hierarchical relations between words. Nowadays, virtually every grammar theory acknowledges hierarchical relations as a major and central concept. The global distinction between two different theoretical positions is captured by the assumption between which elements hierarchical relations are considered as basic. In constituency grammars (CG), hierarchical relations are assumed to hold between constituents, namely words and phrases, in the form of dominance relations. However, although often misunderstood even by experts, the difference between DGs and CGs is not that DGs assume hierarchical relations between words, and CGs between constituents. The difference between DGs and CGs is that DGs assume that hierarchical relations between words are basic, while CGs do not. This misunderstanding has led many DGians to suggest that hierachical relations between constituents have no place in DGs, and even further that the notion of phrase is irrelevant for DGs.
Even though words are the basic syntactic elements, and even though the hierarchical relations between words constitute the basic syntax, the notion of phrases is a desirable concept in DG. Otherwise various problems such as for instance binding, word order, coordination etc. that are considered as syntactic mainstays cannot be properly described in a DG.
In a DG, a hierarchical relation between at least two words is considered as the basic syntactic relation. In my proposal for a DG, however, even though these kind of relations are considered as basic syntactic relations, they are not basic in terminology. A hierarchical relation between two words - or a government relation - must be verified, and without further assumptions about the words in question, such a verification runs afoul. I think that government relations are constituted by two different instances: classematic control and semanteme selection.
Classematic control is a hierarchical relation between two morphs or morphemes (take your pick!) in the way that morpheme M1 determines the class of which a cooccurring morpheme M2 must be a member. The class of a morpheme M is considered as a property of M and called classeme. So, control means that a certain classeme must or must not cooccur. For instance, German subjunctions demand that the verb in the subordinated clause must have a finite flexeme. In other words, German subjunctions control the classeme "finite flexeme" in verbs.
Semanteme selection is also a hierarchical relation between two morphemes. Here, a morpheme M1 determines one or more semantic properties a cooccurring morpheme M2 must have. These semantic properties are calles semantemes and comprise sememes (the lexical meaning of a morpheme), functions (grammatical functions such as tense etc.), and features. Features are distinguished into semes (semantical features such as [animated] etc.) and grammemes (grammatical features such as [masculine]). For example, psychological verbs such as see demand that the subject refers to an animated object. Therefore a sentence such as the book sees him is wrong because the noun book violates this specific selection relation.
Classematic control and selection constitute proto-dependency relations if the relata of the respective relations are two morphemes of two different words. If then a morpheme M1 of the word W1 is proto-dependent on a morpheme M2 of the word W2, W2 directly governs W1. Since the direct government relation requires its relata to be words, it is the basic syntactic relation.
I have refrained from using the term "dependency" as the opposite to "government", because I wanted to reserve that term as name for a hierarchical relation between a word and one or more phrases. In order to do that the term phrase is necessary, and its definition is somewhat demanding. In my theory, this is done by defining direct successor, indirect government, and indirect successor. A direct successor of a word W2 is a word W1, if W2 governs W1. A word W3 indirectly governs W1, if W3 directly governs another word W2, and W2 directly governs W1. Direct and indirect government are unified as general government. An indirect successor of a word W3 is any word W1, if W3 indirectly governs W1, or if another indirect successor of W3, for example W2, generally governs W1. This is called a recursive definition, because the definiendum appears in the second logical adjunct of the definiens. However, that poses no problems because the last step of the verification whether the term in the definiendum is valid, requires only the first logical adjunct but not the second. Direct and indirect successor relations are unified as general successor relations.
A phrase then contains a word W and all its general successors. W is called the head of the phrase. Once this concept has become terminologized, relations between phrases and a word must be defined. A phrase P is directly dependent on a word W, if W directly governs the head of P. P is indirectly dependent on W, if the head of P is an indirect successor of W. Direct and indirect dependency are unified as general dependency.
In order to visualize the syntactic relations between words, a theory for graphical representation of these relations is desirable (but not necessary). This theory is called stemmatology. Other than all other approaches known to me, which are basically algebraic, I have devised a purely geometric theory. In order to make an algebraic theory of graphical representations work, theorems are required that - in my view - are too restrictive to analyze all proper utterances of natural languages. These theorems include the provisions that every structure has one and only one "root", that every element is connected etc. However, it is easy to come up with acceptable structure that violate these theorems. A geometric approach such as my own, only considers the alignment and hierarchy relations between elements in a structure, and does not condition the wellformedness of that structure. I.e. my theory works in any case.
In a first step, every word in a structure is assigned a linear index designating its ordinal position. Next, every word is assigned a government index that equals the linear index of that word that governs the first one. The head is assigned a government index equal {0}.
Dimensional relations are defined next. The vertical relation under is defined by a hierarchy relation, and the horizontal relations left and right are defined by linear indices. These relations are then fused. For example, in little boys little is one unit under and one unit left of boys. Fused relations are then abstracted to vector relations. Connecting branches are actual vectors with definite values for length and angle. Both values can be computed by using only the previously assigned values of the linear and government indices.
In a further operation, a Cartesian coordinate system is used to locate the words. The resulting graphical structures called stemmata are projective. Discontinuities are characterized by crossing of connecting or projecting branches. A discontinuous element D and the highest located element E located between D and its governor exhibit an unidentity relation concerning their indices. Among these four indices, all index values are different, and no index values of one element are numerally contigue.
In a stemma, the vertical dimension is called the hierarchy dimension, and the horizontal one time dimension. A plane is defined as two dimensions: the hierachy and the time dimension define the dependency plane.
In order to deal with coordination, the theory provides the possibility to create a three-dimensional stemma. Coordination is instituted by the repeated employment of previously used formation rules. The first time these rules are enacted, they create a primary procedure, the second time is called secondary procedure asf. Structures that are created by different procedures are to be placed into different planes by the employment of procedure indices. This third dimension is called procedural time dimension. The procedural time dimension and the time dimension define the time plane, and the procedural time dimension and the hierarchy dimension define the procedural hierarchy plane.
In order to capture and describe word order, I propose a theory called zone theory. Based on proto- and normal dependency relations, zones can be defined as locations for phrasal primary constituents in a sentence. I distinguish remote zones for the location of primary constituents that somehow agree with the finite head, and close zones for all other primary constituents. Zones can be split into loci if there is need to do so; for instance in the case of more than just one complement. The head's internal structure determines the sequence of the zones. Certain alignments - that are in general language-sensitive - act as bases. In a base, every element is located in a source location. In structures that are not bases, at least one element is not located in its source location but in a goal location. This element is then considered as having moved, and the whole structure as having derived from its base. Parameters determine which constituents can take what kind of goal locations. Parameters can be violated, and there is a degree to which violations constitute inacceptable structures. The idea of parameters and their violation has been adopted from Optimality Theory, and seems to be promising because it provides a softer approach to grammaticality since this property is now regarded as a continuum.