Full Nonassociative Lambek Calculus with Modalities and Its Applications in Type Grammars

Wydział Matematyki i InformatykiRozprawa jest poświęcona pełnemu niełącznemu rachunkowi Lambeka wzbogaconemu o różne modalności. Te systemy tworzą pewną rodzinę logik substrukturalnych. W rozprawie badamy rachunki NL (niełączny rachunek Lambeka), DFNL (pełny niełączny rachunek Lambeka z prawami dystrybutywności dla operacji kratowych) i BFNL (DFNL z negacją spełniającą prawa algebr Boole’a) oraz ich rozszerzenia o operatory modalne, tworzące parę rezyduacji i spełniające standardowe aksjomaty logik modalnych (T), (4) i (5). Rozważamy też gramatyki typów oparte na tych rachunkach. Główne wyniki: twierdzenie o eliminacji cięć dla modalnych rozszerzeń NL z założeniami, wielomianowa złożoność relacji konsekwencji dla tych systemów, lemat interpolacyjny dla modalnych rozszerzeń DFNL i BFNL z założeniami, silna własność skończonego modelu dla tych systemów, rozstrzygalność relacji konsekwencji dla tyc systemów, PSPACE-zupełność rachunku BFNL, bezkontekstowość języków generowanych przez gramatyki typów oparte na tych rachunkach. Rozprawa kontynuuje wcześniejsze badania W. Buszkowskiego, M. Farulewskiego, M. Moortgata, A.. Plummera, N. Kurtoniny i innych.The thesis is devoted to full nonassociative Lambek calculus enriched with different modalities. These systems form a family of substrutural logics. In this thesis we study systems NL (nonassociative Lambek calculus), DFNL (full nonassociative Lambek calculus with the distributive laws for lattice operations) and BFNL (DFNL with negation satisfying the laws of Boolean algebras) and their extensions by modal operators, being a residuation pair and fulfilling standard axioms of modal logics (T), (4), (5). We also consider the type grammars based on these calculi. Main results: the cut-elimination theorem for modal extensions of NL with assumptions, the polynomial-time complexity of the consequence relations for these systems, an interpolation lemma for modal extensions of DFNL and BFNL with assumptions, the strong finite model property of the latter systems, the decidability of the consequence relations for the latter systems, the PSPACE-completeness of BFNL, the context-freeness of the languages generated by the type grammars based on these systems. The thesis continues some research of W. Buszkowski, M. Farulewski, M. Moortgat, A. Plummer,, N. Kurtonina and others

Reasoning with Polarity in Categorial Type Logic

The research presented in this thesis follows the parsing as deduction approach to lin- guistics. We use the tools of Categorial Type Logic (CTL) to study the interface of natural language syntax and semantics. Our aim is to investigate the mathematical structure of CTL and explore the possibilities it offers for analyzing natural language structures and their interpretation. The thesis is divided into three parts. Each of them has an introductory chapter. In Chapter 1, we introduce the background assumptions of the categorial approach in linguistics, and we sketch the developments that have led to the introduction of CTL. We discuss the motivation for using logical methods in linguistic analysis. In Chapter 3, we propose our view on the use of unary modalities as `logical features'. In Chapter 5, we set up a general notion of grammatical composition taking into account the form and the meaning dimensions of linguistic expressions. We develop a logical theory of licensing and antilicensing relations that cross-cuts the form and meaning dimensions. Throughout the thesis we focus attention on polarity. This term refers both to the polarity of the logical operators of CTL and to the polarity items one finds in natural language, which, furthermore, are closely connected to natural reasoning. Therefore, the title of this thesis Reasoning with Polarity in Categorial Type Logic is intended to express three meanings. Firstly, we reason with the polarity of the logical operators of CTL and study their derivability patterns. In Chapter 2, we explore the algebraic principles that govern the behavior of the type-forming operations of the Lambek calculus. We extend the categorial vocabulary with downward entailing unary operations obtaining the full tool- kit that we use in the rest of the thesis. We employ unary operators to encode and compute monotonicity information (Chapter 4), to account for the different ways of scope taking of generalized quantifiers (Chapter 6), and to model licensing and antilicensing relations (Chapter 7). Secondly, in Chapter 4, we model natural reasoning inferences drawn from structures suitable for negative polarity item occurrences. In particular, we describe a system of inference based on CTL. By decorating functional types with unary operators we encode the semantic distinction between upward and downward monotone functions. Moreover, we study the advantages of this encoding by exploring the contribution of v monotone functions to the study of natural reasoning and to the analysis of the syntactic distribution of negative polarity items. Thirdly, in Chapter 7, we study the distribution of polarity-sensitive expressions. We show how our theory of licensing and antilicensing relations successfully differentiates between negative polarity items, which are `attracted' by their triggers, and positive polarity items, which are `repelled' by them. We investigate these compatibility and incompatibility relations from a cross-linguistic perspective, and show how we reduce distributional differences between polarity-sensitive items in Dutch, Greek and Italian to differences in the lexical type assignments of these languages

Parsing/theorem-proving for logical grammar CatLog3

CatLog3 is a 7000 line Prolog parser/theorem-prover for logical categorial grammar. In such logical categorial grammar syntax is universal and grammar is reduced to logic: an expression is grammatical if and only if an associated logical statement is a theorem of a fixed calculus. Since the syntactic component is invariant, being the logic of the calculus, logical categorial grammar is purely lexicalist and a particular language model is defined by just a lexical dictionary. The foundational logic of continuity was established by Lambek (Am Math Mon 65:154–170, 1958) (the Lambek calculus) while a corresponding extension including also logic of discontinuity was established by Morrill and Valentín (Linguist Anal 36(1–4):167–192, 2010) (the displacement calculus). CatLog3 implements a logic including as primitive connectives the continuous (concatenation) and discontinuous (intercalation) connectives of the displacement calculus, additives, 1st order quantifiers, normal modalities, bracket modalities, and universal and existential subexponentials. In this paper we review the rules of inference for these primitive connectives and their linguistic applications, and we survey the principles of Andreoli’s focusing, and of a generalisation of van Benthem’s count-invariance, on the basis of which CatLog3 is implemented.Peer ReviewedPostprint (author's final draft

Parsing logical grammar: CatLog3

CatLog3 is a Prolog parser/theorem-prover for (type) logical (categorial) grammar. In such logical grammar, grammar is reduced to logic: a string of words is grammatical if and only if an associated logical statement is a theorem. CalLog3 implements a logic extending displacement calculus, a sublinear fragment including as primitive connectives the continuous (Lambek) and discontinuous wrapping connectives of the displacement calculus, additives, 1st order quantifiers, normal modalities, bracket modalities and subexponentials. In this paper we survey how CatLog3 is implemented on the principles of Andreoli’s focusing and a generalisation of van Benthem’s count-invariance.Peer ReviewedPostprint (published version

Da linguística gerativa à gramática categorial : sujeitos lexicais em infinitivos controlados

Orientadores: Marcelo Esteban Coniglio, Sonia Maria Lazzarini CyrinoTese (doutorado) - Universidade Estadual de Campinas, Instituto de Filosofia e Ciências HumanasResumo: A presente tese situa-se na interface da lógica e da linguística; o seu objeto de estudo são os pronomes lexicais em sentenças de controle em três línguas Românicas: Português, Italiano e Espanhol. Esse assunto tem recebido mais atenção na linguística gerativa, especialmente nos anos recentes, do que na gramática de cunho lógico. Talvez como consequência disso, há ainda muito a ser entendido sobre essas estruturas linguísticas e as suas propriedades lógicas. Essa tese tenta preencher as lacunas na literatura \--- ou, pelo menos, avançar nessa direção \--- colocando questões que não foram suficientemente exploradas até agora. Para tal efeito avançamos duas perguntas-chaves, uma linguística e a outra lógica. Elas são, respectivamente: Qual é o estatuto sintático dos pronomes lexicais em estruturas de controle? E: Quais são os mecanismos disponíveis, em uma gramática lógica livre de contração, para se reusar recursos semânticos? A tese divide-se, consequentemente, em duas partes: linguística gerativa e gramática categorial. Na Parte I revisamos algumas das principais teorias de controle gerativistas e a recente discussão acerca das cláusulas infinitivas com sujeito lexical. Na Parte II revisamos a literatura categorial, atendendo principalmente às propostas acerca das estruturas de controle e dos pronomes anafóricos. Em última instância, mostraremos que as propostas linguísticas e lógicas prévias precisam ser modificadas para se explicar o fenômeno linguístico em questão. Com efeito, nos capítulos finais de cada uma das partes avançamos propostas alternativas que, a nosso ver, resultam mais adequadas que as suas rivais. Mais específicamente, na Parte I avançamos uma proposta linguística na linha do cálculo de controle T/Agr de Landau. Na Parte II apresentamos duas propostas categoriais, uma na linha do cálculo categorial combinatório e a outra, na gramática lógica de tipos. Finalmente mostramos a implementação da última proposta em um analisador sintático e de demonstração categorialAbstract: The present thesis lies at the interface of logic and linguistics; its object of study are control sentences with overt pronouns in Romance languages (European and Brazilian Portuguese, Italian and Spanish). This is a topic that has received considerably more attention on the part of linguists, especially in recent years, than from logicians. Perhaps for this reason, much remains to be understood about these linguistic structures and their underlying logical properties. This thesis seeks to fill the lacunas in the literature \--- or at least take steps in this direction \--- by way of addressing a number of issues that have so far been under-explored. To this end we put forward two key questions, one linguistic and the other logical. These are, respectively: What is the syntactic status of the surface pronoun? And: What are the available mechanisms to reuse semantic resources in a contraction-free logical grammar? Accordingly, the thesis is divided into two parts: generative linguistics and categorial grammar. Part I starts by reviewing the recent discussion within the generative literature on infinitive clauses with overt subjects, paying detailed attention to the main accounts in the field. Part II does the same on the logical grammar front, addressing in particular the issues of control and of anaphoric pronouns. Ultimately, the leading accounts from both camps will be found wanting. The closing chapter of each of Part I and Part II will thus put forward alternative candidates, that we contend are more successful than their predecessors. More specifically, in Part I we offer a linguistic account along the lines of Landau's T/Agr theory of control. In Part II we present two alternative categorial accounts: one based on Combinatory Categorial Grammar, the other on Type-Logical Grammar. Each of these accounts offers an improved, more fine-grained perspective on control infinitives featuring overt pronominal subjects. Finally, we include an Appendix in which our type-logical proposal is implemented in a categorial parser/theorem-prover (categorial parser/theorem-prover)DoutoradoFilosofiaDoutora em Filosofia2013/08115-1, 2015/09699-2FAPESPCAPE

Head-Driven Phrase Structure Grammar

Head-Driven Phrase Structure Grammar (HPSG) is a constraint-based or declarative approach to linguistic knowledge, which analyses all descriptive levels (phonology, morphology, syntax, semantics, pragmatics) with feature value pairs, structure sharing, and relational constraints. In syntax it assumes that expressions have a single relatively simple constituent structure. This volume provides a state-of-the-art introduction to the framework. Various chapters discuss basic assumptions and formal foundations, describe the evolution of the framework, and go into the details of the main syntactic phenomena. Further chapters are devoted to non-syntactic levels of description. The book also considers related fields and research areas (gesture, sign languages, computational linguistics) and includes chapters comparing HPSG with other frameworks (Lexical Functional Grammar, Categorial Grammar, Construction Grammar, Dependency Grammar, and Minimalism)

Students´ language in computer-assisted tutoring of mathematical proofs

Truth and proof are central to mathematics. Proving (or disproving) seemingly simple statements often turns out to be one of the hardest mathematical tasks. Yet, doing proofs is rarely taught in the classroom. Studies on cognitive difficulties in learning to do proofs have shown that pupils and students not only often do not understand or cannot apply basic formal reasoning techniques and do not know how to use formal mathematical language, but, at a far more fundamental level, they also do not understand what it means to prove a statement or even do not see the purpose of proof at all. Since insight into the importance of proof and doing proofs as such cannot be learnt other than by practice, learning support through individualised tutoring is in demand. This volume presents a part of an interdisciplinary project, set at the intersection of pedagogical science, artificial intelligence, and (computational) linguistics, which investigated issues involved in provisioning computer-based tutoring of mathematical proofs through dialogue in natural language. The ultimate goal in this context, addressing the above-mentioned need for learning support, is to build intelligent automated tutoring systems for mathematical proofs. The research presented here has been focused on the language that students use while interacting with such a system: its linguistic propeties and computational modelling. Contribution is made at three levels: first, an analysis of language phenomena found in students´ input to a (simulated) proof tutoring system is conducted and the variety of students´ verbalisations is quantitatively assessed, second, a general computational processing strategy for informal mathematical language and methods of modelling prominent language phenomena are proposed, and third, the prospects for natural language as an input modality for proof tutoring systems is evaluated based on collected corpora

Head-Driven Phrase Structure Grammar

Head-Driven Phrase Structure Grammar (HPSG) is a constraint-based or declarative approach to linguistic knowledge, which analyses all descriptive levels (phonology, morphology, syntax, semantics, pragmatics) with feature value pairs, structure sharing, and relational constraints. In syntax it assumes that expressions have a single relatively simple constituent structure. This volume provides a state-of-the-art introduction to the framework. Various chapters discuss basic assumptions and formal foundations, describe the evolution of the framework, and go into the details of the main syntactic phenomena. Further chapters are devoted to non-syntactic levels of description. The book also considers related fields and research areas (gesture, sign languages, computational linguistics) and includes chapters comparing HPSG with other frameworks (Lexical Functional Grammar, Categorial Grammar, Construction Grammar, Dependency Grammar, and Minimalism)