6,343 research outputs found
The prospects for mathematical logic in the twenty-first century
The four authors present their speculations about the future developments of
mathematical logic in the twenty-first century. The areas of recursion theory,
proof theory and logic for computer science, model theory, and set theory are
discussed independently.Comment: Association for Symbolic Logi
Encoding TLA+ set theory into many-sorted first-order logic
We present an encoding of Zermelo-Fraenkel set theory into many-sorted
first-order logic, the input language of state-of-the-art SMT solvers. This
translation is the main component of a back-end prover based on SMT solvers in
the TLA+ Proof System
On the expressive power of planar perfect matching and permanents of bounded treewidth matrices
Valiant introduced some 25 years ago an algebraic model of computation along
with the complexity classes VP and VNP, which can be viewed as analogues of the
classical classes P and NP. They are defined using non-uniform sequences of
arithmetic circuits and provides a framework to study the complexity for
sequences of polynomials. Prominent examples of difficult (that is,
VNP-complete) problems in this model includes the permanent and hamiltonian
polynomials. While the permanent and hamiltonian polynomials in general are
difficult to evaluate, there have been research on which special cases of these
polynomials admits efficient evaluation. For instance, Barvinok has shown that
if the underlying matrix has bounded rank, both the permanent and the
hamiltonian polynomials can be evaluated in polynomial time, and thus are in
VP. Courcelle, Makowsky and Rotics have shown that for matrices of bounded
treewidth several difficult problems (including evaluating the permanent and
hamiltonian polynomials) can be solved efficiently. An earlier result of this
flavour is Kasteleyn's theorem which states that the sum of weights of perfect
matchings of a planar graph can be computed in polynomial time, and thus is in
VP also. For general graphs this problem is VNP-complete. In this paper we
investigate the expressive power of the above results. We show that the
permanent and hamiltonian polynomials for matrices of bounded treewidth both
are equivalent to arithmetic formulas. Also, arithmetic weakly skew circuits
are shown to be equivalent to the sum of weights of perfect matchings of planar
graphs.Comment: 14 page
Torsion homology and regulators of isospectral manifolds
Given a finite group G, a G-covering of closed Riemannian manifolds, and a
so-called G-relation, a construction of Sunada produces a pair of manifolds M_1
and M_2 that are strongly isospectral. Such manifolds have the same dimension
and the same volume, and their rational homology groups are isomorphic. We
investigate the relationship between their integral homology. The
Cheeger-Mueller Theorem implies that a certain product of orders of torsion
homology and of regulators for M_1 agrees with that for M_2. We exhibit a
connection between the torsion in the integral homology of M_1 and M_2 on the
one hand, and the G-module structure of integral homology of the covering
manifold on the other, by interpreting the quotients Reg_i(M_1)/Reg_i(M_2)
representation theoretically. Further, we prove that the p-primary torsion in
the homology of M_1 is isomorphic to that of M_2 for all primes p not dividing
#G. For p <= 71, we give examples of pairs of isospectral hyperbolic
3-manifolds for which the p-torsion homology differs, and we conjecture such
examples to exist for all primes p.Comment: 21 pages; minor changes; included a data file; to appear in J.
Topolog
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