3,974 research outputs found
Computing with Classical Real Numbers
There are two incompatible Coq libraries that have a theory of the real
numbers; the Coq standard library gives an axiomatic treatment of classical
real numbers, while the CoRN library from Nijmegen defines constructively valid
real numbers. Unfortunately, this means results about one structure cannot
easily be used in the other structure. We present a way interfacing these two
libraries by showing that their real number structures are isomorphic assuming
the classical axioms already present in the standard library reals. This allows
us to use O'Connor's decision procedure for solving ground inequalities present
in CoRN to solve inequalities about the reals from the Coq standard library,
and it allows theorems from the Coq standard library to apply to problem about
the CoRN reals
An Optimal Decision Procedure for MPNL over the Integers
Interval temporal logics provide a natural framework for qualitative and
quantitative temporal reason- ing over interval structures, where the truth of
formulae is defined over intervals rather than points. In this paper, we study
the complexity of the satisfiability problem for Metric Propositional Neigh-
borhood Logic (MPNL). MPNL features two modalities to access intervals "to the
left" and "to the right" of the current one, respectively, plus an infinite set
of length constraints. MPNL, interpreted over the naturals, has been recently
shown to be decidable by a doubly exponential procedure. We improve such a
result by proving that MPNL is actually EXPSPACE-complete (even when length
constraints are encoded in binary), when interpreted over finite structures,
the naturals, and the in- tegers, by developing an EXPSPACE decision procedure
for MPNL over the integers, which can be easily tailored to finite linear
orders and the naturals (EXPSPACE-hardness was already known).Comment: In Proceedings GandALF 2011, arXiv:1106.081
Begin, After, and Later: a Maximal Decidable Interval Temporal Logic
Interval temporal logics (ITLs) are logics for reasoning about temporal
statements expressed over intervals, i.e., periods of time. The most famous ITL
studied so far is Halpern and Shoham's HS, which is the logic of the thirteen
Allen's interval relations. Unfortunately, HS and most of its fragments have an
undecidable satisfiability problem. This discouraged the research in this area
until recently, when a number non-trivial decidable ITLs have been discovered.
This paper is a contribution towards the complete classification of all
different fragments of HS. We consider different combinations of the interval
relations Begins, After, Later and their inverses Abar, Bbar, and Lbar. We know
from previous works that the combination ABBbarAbar is decidable only when
finite domains are considered (and undecidable elsewhere), and that ABBbar is
decidable over the natural numbers. We extend these results by showing that
decidability of ABBar can be further extended to capture the language
ABBbarLbar, which lays in between ABBar and ABBbarAbar, and that turns out to
be maximal w.r.t decidability over strongly discrete linear orders (e.g. finite
orders, the naturals, the integers). We also prove that the proposed decision
procedure is optimal with respect to the complexity class
Some new results on decidability for elementary algebra and geometry
We carry out a systematic study of decidability for theories of (a) real
vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces,
Banach spaces and metric spaces, all formalised using a 2-sorted first-order
language. The theories for list (a) turn out to be decidable while the theories
for list (b) are not even arithmetical: the theory of 2-dimensional Banach
spaces, for example, has the same many-one degree as the set of truths of
second-order arithmetic.
We find that the purely universal and purely existential fragments of the
theory of normed spaces are decidable, as is the AE fragment of the theory of
metric spaces. These results are sharp of their type: reductions of Hilbert's
10th problem show that the EA fragments for metric and normed spaces and the AE
fragment for normed spaces are all undecidable.Comment: 79 pages, 9 figures. v2: Numerous minor improvements; neater proofs
of Theorems 8 and 29; v3: fixed subscripts in proof of Lemma 3
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