The computational complexity of sentence derivation in functional unification grammar

Functional unification (FU) grammar is a general linguistic formalism based on the merging of feature-sets. An informal outline is given of how the definition of derivation within FU grammar can be used to represent the satisfiability of an arbitrary logical formula in conjunctive normal form. This suggests that the generation of a structure from an arbitrary FU g~ammar is NP-hard, which is an undesirably high level of computational complexity. I. Functional Unification Grammar There is not space here to give a full definition of FU grammar (see Kay (1979, 1984, 1985), Ritchie(1984)); the aim is rather to outline how the problem of satisfiability of a propositional logic expression in conjunctive normal form (CNF) can be expressed as a derivation in FU grammar, thereby suggesting that the derivation question in FU grammar is "NP-hard" (Garey and Johnson (1979)). 0nly those aspects of FU grammar which are relevant to the sketch of the proof will be outlined. The argument here is wholly independent of the generative power discussion in Ritchie(1984). Functional unification (FU) grammar is a grammatical formalism which allows descriptions of linguistic structures to be expressed as functional descriptions (FDs), which are sets of "features" [attribute-value pairs), and grammatical derivation is expressed in terms of these structures. Within a level of an FD, each feature-name can appear only once; i.e. no feature can appear with two different values. Constituent structure within FDs is indicated as follows. In an FD E, any feature F whose feature-name is listed in the value of the PATTERN feature at the same level of nesting within E is a constituent. Feature-values written in anglebrackets (e.g. I are not simple datavalues, but are pointers to other positions within the structure. These "paths" indicate a structural position that can be found by starting at the outermost level of nesting and tracing feature-names inward along the path. An FD El is said to be an extension of another FD E2 if there is a sub-structure of El which is isomorphic to EY, including identity of feature-n~nes and all feature-values. In determining if El is an extension of E2, the comparison process must start at the outermost level. An FU grammar can be thought of as a set of FDs, each one describing a possible shape for a constituent in the language. A FD F is well-formed with respect to the grammar G if there is an FD E in G such that F is an extension of E, and every constituent of F (see above) is well-formed with respect to G. An arbitrary FD can be used as the initial structure in deriving a fuller FD. Suppose G is a FU grammar, FI and F2 are PDs. Then FI derives F2 using grammar G if F2 is well-formed with respect to G, and F2 is an extension of FI. In the textual representation of an FU grammar, it is normal to represent several similar FDs by writing just one FD containing disjunctive lists of the possible variations between braces (curly brackets). This is an abbreviation for the full set of basic FDs, each corresponding to choosing one item from each disjunctive list. 2. Representing CNF expressions In representing CNF-satisfiability as FU grammar derivation, we will divide the information contained in the CNF expression between two structures an FD (which will act as the initial functional description for the derivation) and an FU grammar (with respect to which the derivation is defined). The former encodes, in a very direct way, the structure of the CNF expression, whereas the latter is of a very general form which varies only in size laccording to the number of propositional symbols and number of conjuncts in the CNF expression). Suppose the CNF expression has n propositional symbols PI,..Pn, and k conj uncts. The FU representation will involve the feature-nm~es "CAT", "PATTERN", "PI ", .... "Pn" , "NOT-PI ", ..... ,"NOT-Pn" , "CI", "CY",..."Ck", with the feature-values CNFEXPRESSION, CONJUNCT,TRUE, FALSE, NONE and the ktuple [CI .... Ck). A conjunct of the CNF expression which mentions the literals All], AI2 ] .... Aim 1 explicitly but omits A(m+1] ..... A[Yn) teach A[ ]i being either an atomic proposition or a negated atomic proposition) will be represented by an FD of the general form given in (I].