Stergios Chatzikyriakidis

Natural Language Inference in Coq

In this paper we propose a way to deal with natural language inference (NLI) by implementing Modern Type Theoretical Semantics in the proof assistant Coq. The paper is a first attempt to deal with NLI and natural language reasoning in general by using the proof assistant technology. Valid NLIs are treated as theorems and as such the adequacy of our account is tested by trying to prove them. We use Luo’s Modern Type Theory (MTT) with coercive subtyping as the formal language into which we translate natural language semantics, and we further implement these semantics in the Coq proof assistant. It is shown that the use of a MTT with an adequate subtyping mechanism can give us a number of promising results as regards NLI. Specifically, it is shown that a number of inference cases, i.e. quantifiers, adjectives, conjoined noun phrases and temporal reference among other things can be successfully dealt with. It is then shown, that even though Coq is an interactive and not an automated theorem prover, automation of all of the test examples is possible by introducing user-defined automated tactics. Lastly, the paper offers a number of innovative approaches to NL phenomena like adjectives, collective predication, comparatives and factive verbs among other things, contributing in this respect to the theoretical study of formal semantics using MTTs.

Adjectives in a Modern Type-Theoretical Setting

In this paper we discuss the semantics of adjectives from the perspective of a Modern Type Theory (MTT) with an adequate subtyping mechanism. In an MTT, common nouns (CNs) can be interpreted as types and, in particular, CNs modified by intersective and subsective adjectives can be given semantics by means of Σ-types. However, an interpretation of CNs as types would not be viable without a proper notion of subtyping which, as we explain, is given by coercive subtyping, an adequate notion of subtyping for MTTs. It is also shown that suitable uses of universes are one of the key ingredients that have made such an analysis adequate. Privative and non-committal adjectives require different treatments than the use of Σ-types. We propose to deal with privative adjectives using the disjoint union type while non-committal adjectives by making use of the type-theoretical notion of context, as used by Ranta [27] to approximate the model-theoretic notion of a possible world. Our approach to adjectives has a number of advantages over those proposed within the Montagovian setting, one of which is that the inferences related with the adjectives arise via typing and not by some kind of extra semantic meaning in the form of a meaning postulate.

An Account of Natural Language Coordination in Type Theory with Coercive Subtyping

We discuss the semantics of NL coordination in modern type theories MTTs with coercive subtyping. The issue of conjoinable types is handled by means of a type universe of linguistic types. We discuss quantifier coordination, arguing that they should be allowed in principle and that the semantic infelicity of some cases of quantifier coordination is due to the incompatible semantics of the relevant quantifiers. Non-Boolean collective readings of conjunction are also discussed and, in particular, treated as involving the vectors of type VecA,n, an inductive family of types in an MTT. Lastly, the interaction between coordination and copredication is briefly discussed, showing that the proposed account of coordination and that of copredication by means of dot-types combine consistently as expected.

Natural Language Reasoning Using Proof-Assistant Technology: Rich Typing and Beyond

In this paper, we study natural language inference based on the formal semantics in modern type theories (MTTs) and their implementations in proof-assistants such as Coq. To this end, the type theory UTT with coercive subtyping is used as the logical language in which natural language semantics is translated to, followed by the implementation of these semantics in the Coq proof-assistant. Valid inferences are treated as theorems to be proven via Coq’s proof machinery. We shall emphasise that the rich typing mechanisms in MTTs (much richer than those in the simple type theory as used in the Montagovian setting) provide very useful tools in many respects in formal semantics. This is exemplified via the formalisation of various linguistic examples, including conjoined NPs, comparatives, adjectives as well as various linguistic coercions. The aim of the paper is thus twofold: a) to show that the use of proof-assistant technology has indeed the potential to be developed into a new way of dealing with inference, and b) to exemplify the advantages of having a rich typing system to the study of formal semantics in general and natural language inference in particular.

Adverbs in a Modern Type Theory

This paper is the first attempt to deal with aspects of the semantics of adverbs within a modern type theoretical setting. A number of issues pertaining to the semantics of different classes of adverbs like verididality and intensionality will be discussed and further shown to be captured straightforwardly within a modern type theoretical setting. In particular, I look at the issue of veridicality and show that the inferences associated with veridical adverbs can be dealt with via typing alone, i.e. without the aid of meaning postulates. In case of intensional adverbs like intentionally or allegedly, I show that these can be captured by making use of the type theoretical notion of context, i.e. without the use of possible worlds.

Individuation Criteria, Dot-types and Copredication: A View from Modern Type Theories

In this paper we revisit the issue of copredication from the perspective of modern type theories. Specifically, we look at: a) the counting properties of dot-types, and b) the case of a complex dot-type that has remained unsolved in the literature, i.e. that of newspaper. As regards a), we show that the account proposed in (Luo, 2010) for dot-types makes the correct predictions as regards counting. In order to verify this, we implement the account in the Coq proof-assistant and check that the desired inferences follow. Then, we look at the case of b), the case of a dot-type which is both resource and context sensitive. We propose a further resource sensitive version of the dottype, in effect a linear dot-type. This along with local coercions can account for the behaviour attested.

Using Signatures in Type Theory to Represent Situations

Signatures have been introduced to represent situations in formal semantics based on modern type theories. In this paper, we study the notion of signature in more details, presenting it formally and discussing its use in representations of situations. In particular, the new forms of signature entries, the subtyping entries and the manifest entries, are formally presented and studied. Besides being signature entries, these two forms of entries may be introduced to form contextual entries as well and this may have interesting implications in applications of the notion of context to, for example, belief contexts.

Modern Perspectives in Type-Theoretical Semantics

This book is a collective volume that reports the state of the art in the applications of type theory to linguistic semantics. The volume fills a 20 year gap from the last published book on the issue and aspires to bring researchers closer to cutting edge alternatives in formal semantics research. It consists of unpublished work by some key researchers on various issues related to the type theoretical study of formal semantics and further exemplifies the advantages of using modern type theoretical approaches to linguistic semantics. Themes that are covered include modern developments of type theories in formal semantics, foundational issues in linguistic semantics like anaphora, modality and plurals, innovational interdisciplinary research like the introduction of probability theory to type theories as well as computational implementations of type theoretical approaches. This volume will be of great interest to formal semanticists that are looking for alternative ways to study linguistic semantics, but will also be of interest to theoretical computer scientists and mathematicians that are interested in the applications of type theory.

Proof Assistants for Natural Language Semantics

In this paper we discuss the use of interactive theorem provers also called proof assistants in the study of natural language semantics. It is shown that these provide useful platforms for NL semantics and reasoning on the one hand, and allow experiments to be performed on various frameworks and new theories, on the other. In particular, we show how to use Coq, a prominent type theory based proof assistant, to encode type theoretical semantics of various NL phenomena. In this respect, we can encode the NL semantics based on type theory for quantifiers, adjectives, common nouns, and tense, among others, and it is shown that Coq is a powerful engine for checking the formal validity of these accounts as well as a powerful reasoner about the implemented semantics. We further show some toy semantic grammars for formal semantic systems, like the Montagovian Generative Lexicon, Type Theory with Records and neo-Davidsonian semantics. It is also explained that experiments on new theories can be done as well, testing their validity and usefulness. Our aim is to show the importance of using proof assistants as useful tools in natural language reasoning and verification and argue for their wider application in the field.