4,175 research outputs found
A functional interpretation for nonstandard arithmetic
We introduce constructive and classical systems for nonstandard arithmetic
and show how variants of the functional interpretations due to Goedel and
Shoenfield can be used to rewrite proofs performed in these systems into
standard ones. These functional interpretations show in particular that our
nonstandard systems are conservative extensions of extensional Heyting and
Peano arithmetic in all finite types, strengthening earlier results by
Moerdijk, Palmgren, Avigad and Helzner. We will also indicate how our rewriting
algorithm can be used for term extraction purposes. To conclude the paper, we
will point out some open problems and directions for future research and
mention some initial results on saturation principles
On Elementary Theories of Ordinal Notation Systems based on Reflection Principles
We consider the constructive ordinal notation system for the ordinal
that were introduced by L.D. Beklemishev. There are fragments of
this system that are ordinal notation systems for the smaller ordinals
(towers of -exponentiations of the height ). This
systems are based on Japaridze's provability logic . They are
closely related with the technique of ordinal analysis of and
fragments of based on iterated reflection principles. We consider
this notation system and it's fragments as structures with the signatures
selected in a natural way. We prove that the full notation system and it's
fragments, for ordinals , have undecidable elementary theories.
We also prove that the fragments of the full system, for ordinals
, have decidable elementary theories. We obtain some results
about decidability of elementary theory, for the ordinal notation systems with
weaker signatures.Comment: 23 page
Linear Temporal Logic and Propositional Schemata, Back and Forth (extended version)
This paper relates the well-known Linear Temporal Logic with the logic of
propositional schemata introduced by the authors. We prove that LTL is
equivalent to a class of schemata in the sense that polynomial-time reductions
exist from one logic to the other. Some consequences about complexity are
given. We report about first experiments and the consequences about possible
improvements in existing implementations are analyzed.Comment: Extended version of a paper submitted at TIME 2011: contains proofs,
additional examples & figures, additional comparison between classical
LTL/schemata algorithms up to the provided translations, and an example of
how to do model checking with schemata; 36 pages, 8 figure
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