The Grail theorem prover: Type theory for syntax and semantics

As the name suggests, type-logical grammars are a grammar formalism based on logic and type theory. From the prespective of grammar design, type-logical grammars develop the syntactic and semantic aspects of linguistic phenomena hand-in-hand, letting the desired semantics of an expression inform the syntactic type and vice versa. Prototypical examples of the successful application of type-logical grammars to the syntax-semantics interface include coordination, quantifier scope and extraction.This chapter describes the Grail theorem prover, a series of tools for designing and testing grammars in various modern type-logical grammars which functions as a tool . All tools described in this chapter are freely available

Category Theoretic Analysis of Hierarchical Protein Materials and Social Networks

Materials in biology span all the scales from Angstroms to meters and typically consist of complex hierarchical assemblies of simple building blocks. Here we describe an application of category theory to describe structural and resulting functional properties of biological protein materials by developing so-called ologs. An olog is like a “concept web” or “semantic network” except that it follows a rigorous mathematical formulation based on category theory. This key difference ensures that an olog is unambiguous, highly adaptable to evolution and change, and suitable for sharing concepts with other olog. We consider simple cases of beta-helical and amyloid-like protein filaments subjected to axial extension and develop an olog representation of their structural and resulting mechanical properties. We also construct a representation of a social network in which people send text-messages to their nearest neighbors and act as a team to perform a task. We show that the olog for the protein and the olog for the social network feature identical category-theoretic representations, and we proceed to precisely explicate the analogy or isomorphism between them. The examples presented here demonstrate that the intrinsic nature of a complex system, which in particular includes a precise relationship between structure and function at different hierarchical levels, can be effectively represented by an olog. This, in turn, allows for comparative studies between disparate materials or fields of application, and results in novel approaches to derive functionality in the design of de novo hierarchical systems. We discuss opportunities and challenges associated with the description of complex biological materials by using ologs as a powerful tool for analysis and design in the context of materiomics, and we present the potential impact of this approach for engineering, life sciences, and medicine.Presidential Early Career Award for Scientists and Engineers (N000141010562)United States. Army Research Office. Multidisciplinary University Research Initiative (W911NF0910541)United States. Office of Naval Research (grant N000141010841)Massachusetts Institute of Technology. Dept. of MathematicsStudienstiftung des deutschen VolkesClark BarwickJacob Luri

Modal auxiliaries and negation : A type-logical account

National Institute for Japanese Language and LinguisticsOhio State UniversityThis paper proposes an analysis of modal auxiliaries in English in Type-Logical Grammar. The proposed analysis captures the scopal interactions between different types of modal auxiliaries and negation by incorporating the key analytic idea of Iatridou and Zeijlstra [6], who classify English modal auxiliaries into PPI and NPI types. In order to technically implement this analysis, we build on Kubota and Levine’s [8, 10] treatment of modal auxiliaries as higher-order operators that take scope at the clausal level. The proposed extension of the Kubota/Levine analysis is shown to have several interesting consequences, including a formal derivability relation from the higher-order entry for auxiliaries to a lower-order VP/VP entry traditionally recognized in categorial grammar (CG) research. The systematic analysis of the scopal properties of auxiliaries and the somewhat more abstract meta-comparison between ‘transformational’ and ‘non-transformational’ analytic ideas that become possible in a type-logical setup highlight the value of taking a logical perspective on the syntax of natural language embodied in Type-Logical Grammar research