A Decidable Linear Logic for Speech Translation

The structure of objects employed in the study of Natural Language Semantics has been increasingly being complicated to represent the items of information conveyed by utterances. The complexity becomes a source of troubles when we employ those theories in building linguistic applications such as speech translation system. To understand better how programs operating on semantic representations work, we adopt a logical approach and present a monadic and multiplicative linear logic. In designing the fragment, we refine on the multiplicative conjunction to employ both the commutative and non-commutative connectives. The commutative connective is used to glue up a set of formulae representing a semantic object conjointly. The non-commutative connective is used to glue up a list of formulae representing the argument structure of an object, where the order matters. We also introduce to our fragment a Lambek slash, the directional implication, to concatenate the formula representing the predicative part of the object and the list of formulae representing the argument part. The monadic formulae encode each element of the argument part by representing its sort with the predicate and the element as the place-holder. The fragment enjoys the nice property of being decidable. To encode contextual information involved in utterances, however, we extend the fragment with the exponential operator. The context is regarded as a resource available as many as required, but not infinitely many. We encode the items of context with the exponential operator, but ensure that the operator should appear only in the antecedent. The extention keeps the fragment decidable because the proof search will not fall into an endless search caused by the coupling of unlimited supply and consumption. We show that the fragment is rich enough to encode and transform semantic objects employed in the contemporary linguistic theories. The result guarantees that the theories on natural language semantics can be implemented reasonably and safely on computers. Keyward: multiplicative linear logic, natural language semantics, monadic logic, Lambek calculus