Tilburg University A Natural Proof System for Natural Language Abzianidze, Lasha Publication date: 2017 Document Version Publisher's PDF, also known as Version of record Link to publication in Tilburg University Research Portal Citation for published version (APA): Abzianidze, L. (2017). A Natural Proof System for Natural Language. Ridderprint. General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal Take down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Download date: 04. okt. 2021 A In the �tle of this thesis, a proof system refers to a system that carries out formal proofs (e.g., of theorems). Logicians would interpret a system as a symbolic calculus while com- Natural puter scien�sts might understand it as a computer program. Both interpreta�ons are fine for the current purpose as the thesis develops a proof calculi and a computer program A NATURAL PROOF SYSTEM based on it. What makes the proof system natural is that it operates on formulas with a natural appearance, i.e. resembling natural language phrases. This is contrasted to the Proof artificial formulas logicians usually use. Put differently, the natural proof system is FOR NATURAL LANGUAGE designed for a natural logic, a logic that has formulas similar to natural language sentenc- es. The natural proof system represents a further development of an analy�c tableau system for natural logic (Muskens, 2010). The implementa�on of the system acts as a System pp theorem prover for wide-coverage natural language sentences. For instance, it can prove x n that not all PhD theses are interesting entails some dissertations are not interesting. On z certain textual entailment datasets, the prover achieves results compe��ve to the y s state-of-the-art. t γT pp for zγ αF s Natural vp x t β e s vp t T F e np t s n e Language y z λ e s T t T z γ α z λα t n x e F e s T npα λ γ t x β y pp T pp y np x t γ e T n λ T s e x γ t T z F λ z y γ t T s e z e λ αn s β t T np α λ λ e F t α γ Lasha Abzianidze vp x s e s t vp pp F λ s y t z z T λ β n x αpp γ Lasha Abzianidze γ ISBN 978-94-6299-494-2 A Natural Proof System for Natural Language ISBN 978-94-6299-494-2 cb 2016 by Lasha Abzianidze Cover & inner design by Lasha Abzianidze Used item: leaf1 font by Rika Kawamoto Printed and bound by Ridderprint BV, Ridderkerk A Natural Proof System for Natural Language Proefschrift ter verkrijging van de graad van doctor aan Tilburg University op gezag van de rector magnificus, prof.dr. E.H.L. Aarts, in het openbaar te verdedigen ten overstaan van een door het college voor promoties aangewezen commissie in de aula van de Universiteit op vrijdag 20 januari 2017 om 14.00 uur door Lasha Abzianidze geboren op 24 januari 1987 te Tbilisi, Georgie¨ Promotiecommissie Promotor Prof. dr. J.M. Sprenger Copromotor Dr. R.A. Muskens Overige leden Prof. dr. Ph. de Groote Dr. R. Moot Prof. dr. L.S. Moss Prof. dr. Y. Winter Acknowledgments During working on this dissertation, there were many people who helped and supported me, both inside and outside the academic life. Here I would like to thank them. First, my sincere thanks go to my supervisors Jan and Reinhard. I learned a lot from them during this period. Their joint feedback and guidance were vital for my research. This dissertation would not have been possible without their input. Many thanks go to Reinhard who believed that I was a suitable candidate for his innovative research program. He continuously provided me with crucial advice and suggestions during the research. I thank him for believing in me and giving me a freedom to choose a more exciting and demanding research direction towards wide-coverage semantics. Jan gave me valuable and needed support in the later stage of the project. His comments and suggestions have significantly improved the presentation in the thesis. I also benefited from his advice about job applications. I am grateful to him. I feel very lucky to have so much expertise and leading researchers of my field on the committee. I am thankful to Philippe de Groote, Richard Moot, Larry Moss and Yoad Winter for that. Additionally, I am indebted to Philippe for his support during my LCT master study, and I want express my gratitude to Larry whose comments and opinion on my work has been inspiring. Special thanks go to my office mate Janine and my close friend Irakli who agreed to be my paranymphs. I also thank Janine for preparing delicious cakes and sweets for me. Irakli has been a friend I can rely on any time. I appreciate it. I acknowledge the valuable benefit and feedback I got from participating in several ESSLLI schools, attending and presenting at several workshops and conferences, among them LENLS, WoLLIC, EMNLP, TbiLLC, Amsterdam Colloquium, ACL and *SEM. The local TiLPS seminars were very helpful, where I presented early results of my re- search and widened my knowledge of logic, language and philosophy. My working environment and stay in Tilburg were enjoyable due to my colleagues at TiLPS: Alessandra, Alfred, Barbara, Bart, Chiara, Colin, Dominik, Georgi, Jan, Janine, Jasmina, Jun, Silvia, Machteld, Matteo, Michal, Naftali, Reinhard, and Thomas. Thank you guys; with you I enjoyed playing footy in the corridor, having drinks, dinners or parties, watching football, going on TiLPS excursions, or simply spending time together. I also want to thank my close friend Sandro, who was also doing his PhD meanwhile. The (skype) conversations with him were helpful, encouraging and entertaining for me. Many thanks go to my family on the other side of the continent. Especially I want to thank my mother, who always did her best to support and create conditions for me to feel comfortable and to concentrate on my studies. I am deeply grateful to her for everything. Finally and mostly, I would like to thank mu kallis Eleri for her continuous and endless support, patience and unconditional love during this period. She bared me when I was obsessed with research and writing. Her help and care have been indispensable for me. I cannot thank her enough. v vi Abstract We humans easily understand semantics of natural language text and make inferences based on it. But how can we make machines reason and carry out inferences on the text? The question represents a crucial problem for several research fields and is at the heart of the current thesis. One of the most intuitive approaches to model reasoning in natural language is to translate semantics of linguistic expressions into some formal logic that comes with an automated inference procedure. Unfortunately, designing such automatized translation turns out to be almost as hard as the initial problem of reasoning. In this thesis, following Muskens(2010), we develop a model for natural reasoning that takes a proof theoretic stand. Instead of a heavy translation procedure, we opt for a light-weight one. Logical forms obtained after the translation are natural in the sense that they come close to the original linguistic expressions. But at the same time, they represent terms of higher-order logic. In order to reason over such logical forms, we extend a tableau proof system for type theory by Muskens. An obtained natural tableau system employs inference rules specially tailored for various linguistic constructions. Since we employ the logical forms close to linguistic expressions, our approach con- tributes to the project of natural logic. Put differently, we employ the higher-order logic disguised as natural logic. Due to this hybrid nature, the proof system can carry out both shallow and deep reasoning over multiple-premised arguments. While doing so, a main vehicle for shallow reasoning is monotonicity calculus. In order to evaluate the natural proof system, we automatize it as a theorem prover for wide-coverage natural language text. In particular, first the logical forms are obtained from syntactic trees of linguistic expressions, and then they are processed by the prover. Despite its simple architecture, on certain textual entailment benchmarks the prover ob- tains high competitive results for both shallow and deep reasoning. Notice that this is done while it employs only WordNet as a knowledge base. The theorem prover also rep- resents the first wide-coverage system for natural logic which can reason over multiple propositions at the same time. The thesis makes three major contributions. First, it significantly extends the natu- ral tableau system for wide-coverage natural reasoning. After extending the type system and the format of tableau entries (Chapter 2), we collect a plethora of inference rules for a wide range of constructions (Chapter 4). Second, the thesis proposes a procedure to obtain the logical forms from syntactic derivations of linguistic expressions (Chapter 3).