Grammatical structures and logical deductions

The three essays presented here concern natural connections between grammatical derivations and structures provided by certain standard grammar formalisms, on the one hand, and deductions in logical systems, on the other hand. In the first essay we analyse the adequacy of Polish notation for higher-order languages. The Ajdukiewicz algorithm (Ajdukiewicz 1935) is discussed in terms of generalized MP-deductions. We exhibit a failure in Ajdukiewicz’s original version of the algorithm and give a correct one; we prove that generalized MP-deductions have the frontier property, which is essential for the plausibility of Polish notation. The second essay deals with logical systems corresponding to different grammar formalisms, as e.g. Finite State Acceptors, Context-Free Grammars, Categorial Grammars, and others. We show how can logical methods be used to establish certain linguistically significant properties of formal grammars. The third essay discusses the interplay between Natural Deduction proofs in grammar oriented logics and semantic structures expressible by typed lambda terms and combinators

The Category of the Conjuction in Categorial Grammar

En aquest treball es proposa un tipus categorial per a les conjuncions (i, o, etc.) dins del formalisme de la Gràmatica Categorial. En primer lloc s'exposen tres característiques fonamentals que qualsevol tractament de la conjunció ha de poder explicar. Després es contemplen les diferents aportacions que s'han fet dins aquest formalisme per a delinear una categoria de la conjunció que permeti donar compte dels fenòmens del llenguatge natural. Totes aquestes aportacions es comenten respecte de la seva adequació amb les tres característiques de la conjunció exposades al principi. Seguidament, es proposa una categoria per a les conjuncions que pot donar compte de les característiques esmentades. Aquesta categoria introdueix un nou operador n-tuple que resulta també molt útil per a l'anàlisi d'altres fenòmens del llenguatge natural.In this work a categorial type for conjunctions (and, or, etc) is proposed within the Categorial Grammar formalism. First of all, I present three main characteristics that have to be accounted for in any analysis of conjunction. Secondly, I explain the different contributions that have been made within this formalism to fmd a category for conjunction that allows us to account for natural language phenomena. All those proposals are commented on with regard to the three properties to be explained. Next, a categorial type for conjunctions is proposed which can account for those characteristics. This category introduces a new n-tuple operator which is also useful for analysing other natural language phenomena

Gapping as Constituent Coordination

A number of coordinate constructions in natural languages conjoin sequences which do not appear to correspond to syntactic constituents in the traditional sense. One striking instance of the phenomenon is afforded by the gapping construction of English, of which the following sentence is a simple example: (1) Harry eats beans, and Fred, potatoes Since all theories agree that coordination must in fact be an operation upon constituents, most of them have dealt with the apparent paradox presented by such constructions by supposing that such sequences as the right conjunct in the above example, Fred, potatoes, should be treated in the grammar as traditional constituents, of type S, but with pieces missing or deleted

Extraction and Coordination in Phrase Structure Grammar and Categorial Grammar

A large proportion of computationally-oriented theories of grammar operate within the confines of monostratality (i.e. there is only one level of syntactic analysis), compositionality (i.e. the meaning of an expression is determined by the meanings of its syntactic parts, plus their manner of combination), and adjacency (i.e. the only operation on terminal strings is concatenation). This thesis looks at two major approaches falling within these bounds: that based on phrase structure grammar (e.g. Gazdar), and that based on categorial grammar (e.g. Steedman). The theories are examined with reference to extraction and coordination constructions; crucially a range of 'compound' extraction and coordination phenomena are brought to bear. It is argued that the early phrase structure grammar metarules can characterise operations generating compound phenomena, but in so doing require a categorial-like category system. It is also argued that while categorial grammar contains an adequate category apparatus, Steedman's primitives such as composition do not extend to cover the full range of data. A theory is therefore presented integrating the approaches of Gazdar and Steedman. The central issue as regards processing is derivational equivalence: the grammars under consideration typically generate many semantically equivalent derivations of an expression. This problem is addressed by showing how to axiomatise derivational equivalence, and a parser is presented which employs the axiomatisation to avoid following equivalent paths

Mathematical linguistics

but in fact this is still an early draft, version 0.56, August 1 2001. Please d