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

Seesaw mechanism in magnetic compactifications

In this paper, we explore a new avenue to a natural explanation of the observed tiny neutrino masses with a dynamical realization of the three-generation structure in the neutrino sector. Under the magnetized background based on $T^2/Z_2$, matter consists of multiply-degenerated zero modes and the whole intergenerational structure is dynamically determined. In this sense, we can conclude that our scenario is favored by minimality, where no degree of freedom remains to deform the intergenerational structure by hand freely. Under the consideration of brane-localized Majorana-type mass terms for an $SU(2)_L$ singlet neutrino, it is sufficient to introduce one Higgs doublet for reproducing the observed neutrino data. In all reasonable flux configurations with three right-handed neutrinos, phenomenologically acceptable parameter configurations are found.Comment: 20 pages, 3 figures, 4 tables; published version from JHEP (v2

Internal models of system F for decompilation

AbstractThis paper considers Girard’s internal coding of each term of System F by some term of a code type. This coding is the type-erasing coding definable already in the simply typed lambda-calculus using only abstraction on term variables. It is shown that there does not exist any decompiler for System F in System F, where the decompiler maps a term of System F to its code. An internal model of F is given by interpreting each type of F by some type equipped with maps between the type and the code type. This paper gives a decompiler–normalizer for this internal model in F, where the decompiler–normalizer maps any term of the internal model to the code of its normal form. It is also shown that for any model of F the composition of this internal model and the model produces another model of F whose equational theory is below untyped beta–eta-equality

Type Inference for Bimorphic Recursion

This paper proposes bimorphic recursion, which is restricted polymorphic recursion such that every recursive call in the body of a function definition has the same type. Bimorphic recursion allows us to assign two different types to a recursively defined function: one is for its recursive calls and the other is for its calls outside its definition. Bimorphic recursion in this paper can be nested. This paper shows bimorphic recursion has principal types and decidable type inference. Hence bimorphic recursion gives us flexible typing for recursion with decidable type inference. This paper also shows that its typability becomes undecidable because of nesting of recursions when one removes the instantiation property from the bimorphic recursion.Comment: In Proceedings GandALF 2011, arXiv:1106.081