19,665 research outputs found
On the power of ordering in linear arithmetic theories
We study the problems of deciding whether a relation definable by a first-order formula in linear rational or linear integer arithmetic with an order relation is definable in absence of the order relation. Over the integers, this problem was shown decidable by Choffrut and Frigeri [Discret. Math. Theor. C., 12(1), pp. 21 - 38, 2010], albeit with non-elementary time complexity. Our contribution is to establish a full geometric characterisation of those sets definable without order which in turn enables us to prove coNP-completeness of this problem over the rationals and to establish an elementary upper bound over the integers. We also provide a complementary ??^P lower bound for the integer case that holds even in a fixed dimension. This lower bound is obtained by showing that universality for ultimately periodic sets, i.e., semilinear sets in dimension one, is ??^P-hard, which resolves an open problem of Huynh [Elektron. Inf.verarb. Kybern., 18(6), pp. 291 - 338, 1982]
On interpretations of bounded arithmetic and bounded set theory
In a recent paper, Kaye and Wong proved the following result, which they
considered to belong to the folklore of mathematical logic.
THEOREM: The first-order theories of Peano arithmetic and ZF with the axiom
of infinity negated are bi-interpretable: that is, they are mutually
interpretable with interpretations that are inverse to each other.
In this note, I describe a theory of sets that stands in the same relation to
the bounded arithmetic IDelta0 + exp. Because of the weakness of this theory of
sets, I cannot straightforwardly adapt Kaye and Wong's interpretation of
arithmetic in set theory. Instead, I am forced to produce a different
interpretation.Comment: 12 pages; section on omega-models removed due to error; references
added and typos correcte
Interpretations of Presburger Arithmetic in Itself
Presburger arithmetic PrA is the true theory of natural numbers with
addition. We study interpretations of PrA in itself. We prove that all
one-dimensional self-interpretations are definably isomorphic to the identity
self-interpretation. In order to prove the results we show that all linear
orders that are interpretable in (N,+) are scattered orders with the finite
Hausdorff rank and that the ranks are bounded in terms of the dimension of the
respective interpretations. From our result about self-interpretations of PrA
it follows that PrA isn't one-dimensionally interpretable in any of its finite
subtheories. We note that the latter was conjectured by A. Visser.Comment: Published in proceedings of LFCS 201
Categorical characterizations of the natural numbers require primitive recursion
Simpson and the second author asked whether there exists a characterization
of the natural numbers by a second-order sentence which is provably categorical
in the theory RCA. We answer in the negative, showing that for any
characterization of the natural numbers which is provably true in WKL,
the categoricity theorem implies induction. On the other hand, we
show that RCA does make it possible to characterize the natural numbers
categorically by means of a set of second-order sentences. We also show that a
certain -conservative extension of RCA admits a provably
categorical single-sentence characterization of the naturals, but each such
characterization has to be inconsistent with WKL+superexp.Comment: 17 page
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