Classical System of Martin-Lof's Inductive Definitions is not Equivalent to Cyclic Proofs

A cyclic proof system, called CLKID-omega, gives us another way of representing inductive definitions and efficient proof search. The 2005 paper by Brotherston showed that the provability of CLKID-omega includes the provability of LKID, first order classical logic with inductive definitions in Martin-L\"of's style, and conjectured the equivalence. The equivalence has been left an open question since 2011. This paper shows that CLKID-omega and LKID are indeed not equivalent. This paper considers a statement called 2-Hydra in these two systems with the first-order language formed by 0, the successor, the natural number predicate, and a binary predicate symbol used to express 2-Hydra. This paper shows that the 2-Hydra statement is provable in CLKID-omega, but the statement is not provable in LKID, by constructing some Henkin model where the statement is false

Type theoretic semantics for semantic networks: an application to natural language engineering

Semantic Networks have long been recognised as an important tool for natural language processing. This research has been a formal analysis of a semantic network using constructive type theory. The particular net studied is SemNet, the internal knowledge representation for LOLITA(^1): a large scale natural language engineering system. SemNet has been designed with large scale, efficiency, integration and expressiveness in mind. It supports many different forms of plausible and valid reasoning, including: epistemic reasoning, causal reasoning and inheritance. The unified theory of types (UTT) integrates two well known type theories, Coquand-Huet's (impredicative) calculus of constructions and Martin-Lof's (predicative) type theory. The result is a strong and expressive language which has been used for formalization of mathematics, program specification and natural language. Motivated by the computational and richly expressive nature of UTT, this research has used it for formalization and semantic analysis of SemNet. Moreover, because of applications to software engineering, type checkers/proof assistants have been built. These tools are ideal for organising and managing the analysis of SemNet. The contribution of the work is twofold. First the semantic model built has led to improved and deeper understanding of SemNet. This is important as many researchers that work on different aspects of LOLITA, now have a clear and un- ambigious interpertation of the meaning of SemNet constructs. The model has also been used to show soundess of the valid reasoning and to give a reasonable semantic account of epistemic reasoning. Secondly the research contributes to NLE generally, both because it demonstrates that UTT is a useful formalization tool and that the good aspects of SemNet have been formally presented