Systemic grammar, in distinction to value-attribute grammars, distinguishes type logic (the classes of units) and role logic (the constituency and dependency relations between units). The type logic is expressed in a network, called a system network. The role logic is expressed as a set of constraints on the types of the grammar.
Systemic grammar (e.g., Halliday, 1985, Hudson, 1971, Matthiessen and Mann, 1985) uses an inheritance network to organise grammatical types (or `feature' in Systemics), and their structural consequences. A Systemic inheritance network is called a system network.
A system network is used to organise the co-occurrence potential of grammatical types, showing which types are mutually compatible, and which are incompatible. It consists of a set of systems, which are sets of mutually exclusive types. There is also a covering relation between the types of a system, meaning that if the entry condition of the system is satisfied, then one of the types in the cover must be selected.
Figure 1 shows a system network for a simple grammar of English. It includes 11 systems, representing various grammatical distinctions, for instance, between clause and word, between transitive and intransitive clauses, or between nominative and accusative pronouns.
Figure 1: A partial Systemic network
Each type inherits the properties of types to its left in the network. Note that the system network may be logically complex, since entry conditions (the logical condition on a system) may consist of conjunctions and disjunctions of types.
Types of the system network are associated with structural realisations -- the structural consequence of the type. Figure 2 shows the realisations of the types in Figure 1.
Figure 2: Realisation Rules
This grammar deals mainly with some systems involving the Subject and Object, what types of units fill these roles, and how these roles conflate with two other roles: Actor and Actee. The grammar assumes that both roles are filled by pronouns, which are either [nominative] or [accusative], [singular] or [plural], and [human] (e.g., ``I'', ``you'', ``he'') or [nonhuman] (e.g., ``it'', ``that''). Only [human] pronouns can fill the Actor role of a clause.
The realisation operators used in the formalism are as follows:
Insert e.g., Finite = [ ]: indicates that the function Finite must be present in the structure.
Conflate e.g., Modal/Finite: indicates that the two functions Modal and Finite are filled by the same grammatical unit.
Order e.g., Subject ^/ Finite: indicates the sequencing of functions in the surface structure. In this example, the Subject is sequenced directly before the Finite. Any number of elements can be sequenced in a single rule.
Partition e.g., Thing Event End: Another sequence operator, specifies that the appear in this order, but not necessarily immediately adjacent (linear precedence).
Preselect e.g., Subject: nominal-group: indicates that the Subject element must be filled by a unit of type nominal-group.
Lexify e.g., Deict = ``the'': used to assign lexical items directly to elements of structure. Note that lexify overrides any preselect which may apply to the same element of structure.
For the purposes of the expansion of this grammar, we re-express it in a logical formalism. Figure 3 shows Logical Form I of this grammar, including the structural constraints embedded in the form. Note that :xor indicates exclusive disjunction.
Figure 3: Logical Form I of the Grammar