Controlled languages are regimented fragments of natural languagedesigned to make the processing of natural language more efficient andreliable. This paper defines a controlled language, E2V, whose principalgrammatical resources include determiners, relative clauses, reflexivesand pronouns. We provide a formal syntax and semantics for E2V, in whichanaphoric ambiguities are resolved in a linguistically natural way. Weshow that the expressive power of E2V is equal to that of thetwo-variable fragment of first-order logic. It follows that the problemof determining the satisfiability of a set (...) of E2V sentences is NEXPTIMEcomplete. We show that E2V can be extended in various ways withoutcompromising these complexity results; however, relaxing our policy onanaphora resolution renders the satisfiability problem for E2Vundecidable. Finally, we argue that our results have a bearing on thebroader philosophical issue of the relationship between natural andformal languages. (shrink)

Fluted Logic is essentially first-order predicate logic deprived of variables. The lack of variables results in reduced expressiveness. Nevertheless, many logical problems that can be stated in natural language, such as the famous Schubert's Steamroller, can be rendered in fluted logic. Further evidence of the expressiveness of fluted logic is its close relation to description logics. Already it has been shown that fluted logic is decidable and has the finite-model property. This paper shows that fluted logic has the exponential-model property (...) and that deciding satisfiability is NEXPTIME-complete. It is shown further that fluted logic is 'nice’, that is, it shares with first-order predicate logic the interpolation property and model preservation properties. (shrink)

What precisely are fragments of classical first-order logic showing “modal” behaviour? Perhaps the most influential answer is that of Gabbay 1981, which identifies them with so-called “finite-variable fragments”, using only some fixed finite number of variables (free or bound). This view-point has been endorsed by many authors (cf. van Benthem 1991). We will investigate these fragments, and find that, illuminating and interesting though they are, they lack the required nice behaviour in our sense. (Several new negative results support this claim.) (...) As a counterproposal, then, we define a large fragment of predicate logic characterized by its use of only bounded quantification. This so-called guarded fragment enjoys the above nice properties, including decidability, through an effectively bounded finite model property. (These are new results, obtained by generalizing notions and techniques from modal logic.) Moreover, its own internal finite variable hierarchy turns out to work well. Finally, we shall make another move. The above analogy works both ways. Modal operators are like quantifiers, but quantifiers are also like modal operators. This observation inspires a generalized semantics for first-order predicate logic with accessibility constraints on available assignments (cf. N´emeti 1986, 1992) which moves the earlier quantifier restrictions into the semantics. This provides a fresh look at the landscape of possible predicate logics, including candidates sharing various desirable features with basic modal logic – in particular, its decidability. (shrink)

By a fragment of a natural language we mean a subset of thatlanguage equipped with semantics which translate its sentences intosome formal system such as first-order logic. The familiar conceptsof satisfiability and entailment can be defined for anysuch fragment in a natural way. The question therefore arises, for anygiven fragment of a natural language, as to the computational complexityof determining satisfiability and entailment within that fragment. Wepresent a series of fragments of English for which the satisfiabilityproblem is polynomial, NP-complete, EXPTIME-complete,NEXPTIME-complete (...) and undecidable. Thus, this paper represents a casestudy in how to approach the problem of determining the logicalcomplexity of various natural language constructions. In addition, wedraw some general conclusions about the relationship between naturallanguage and formal logic. (shrink)

In the predicate calculus, variables provide a flexible indexing service which selects the actual arguments to a predicate letter from among possible arguments that precede the predicate letter (in the parse of the formula). In the process of selection, the possible arguments can be permuted, repeated (used more than once), and skipped. If this service is withheld, so that arguments must be the immediately preceding ones, taken in the order in which they occur, the formula is said to be fluted. (...) Quine showed that if a fluted formula contains only homogeneous conjunction (conjoins only subformulas of equal arity), then the satisfiability of the formula is decidable. It remained an open question whether the satisfiability of a fluted formula without this restriction is decidable. This paper answers that question. (shrink)

This paper develops a version of Natural Logic – an inference system that works directly on natural language syntactic representations, with no intermediate translation to logical formulae. Following work by Sánchez, we develop a small fragment that computes semantic order relations between derivation trees in Categorial Grammar. The proposed system has the following new characteristics: It uses orderings between derivation trees as purely syntactic units, derivable by a formal calculus. The system is extended for conjunctive phenomena like coordination and relative (...) clauses. This allows a simple account of non-monotonic expressions that are reducible to conjunctions of monotonic ones. A decision procedure for provability is developed for a fragment of Natural Logic. (shrink)

Fluted Logic is essentially first-order predicate logic deprived of variables. The lack of variables results in reduced expressiveness. Nevertheless, many logical problems that can be stated in natural language, such as the famous Schubert's Steamroller, can be rendered in fluted logic. Further evidence of the expressiveness of fluted logic is its close relation to description logics. Already it has been shown that fluted logic is decidable and has the finite-model property. This paper shows that fluted logic has the exponential-model property (...) and that deciding satisfiability is NEXPTIME-complete. It is shown further that fluted logic is 'nice, that is, it shares with first-order predicate logic the interpolation property and model preservation properties. (shrink)