3,557 research outputs found
Admissible Bases Via Stable Canonical Rules
We establish the dichotomy property for stable canonical multi-conclusion rules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics
Admissible Bases Via Stable Canonical Rules
We establish the dichotomy property for stable canonical multi-conclusionrules for IPC, K4, and S4. This yields an alternative proof of existence of explicit bases of admissible rules for these logics
Canonical formulas for k-potent commutative, integral, residuated lattices
Canonical formulas are a powerful tool for studying intuitionistic and modal
logics. Actually, they provide a uniform and semantic way to axiomatise all
extensions of intuitionistic logic and all modal logics above K4. Although the
method originally hinged on the relational semantics of those logics, recently
it has been completely recast in algebraic terms. In this new perspective
canonical formulas are built from a finite subdirectly irreducible algebra by
describing completely the behaviour of some operations and only partially the
behaviour of some others. In this paper we export the machinery of canonical
formulas to substructural logics by introducing canonical formulas for
-potent, commutative, integral, residuated lattices (-).
We show that any subvariety of - is axiomatised by canonical
formulas. The paper ends with some applications and examples.Comment: Some typo corrected and additional comments adde
The topological classification of one-dimensional symmetric quantum walks
We give a topological classification of quantum walks on an infinite 1D
lattice, which obey one of the discrete symmetry groups of the tenfold way,
have a gap around some eigenvalues at symmetry protected points, and satisfy a
mild locality condition. No translation invariance is assumed. The
classification is parameterized by three indices, taking values in a group,
which is either trivial, the group of integers, or the group of integers modulo
2, depending on the type of symmetry. The classification is complete in the
sense that two walks have the same indices if and only if they can be connected
by a norm continuous path along which all the mentioned properties remain
valid. Of the three indices, two are related to the asymptotic behaviour far to
the right and far to the left, respectively. These are also stable under
compact perturbations. The third index is sensitive to those compact
perturbations which cannot be contracted to a trivial one. The results apply to
the Hamiltonian case as well. In this case all compact perturbations can be
contracted, so the third index is not defined. Our classification extends the
one known in the translation invariant case, where the asymptotic right and
left indices add up to zero, and the third one vanishes, leaving effectively
only one independent index. When two translationally invariant bulks with
distinct indices are joined, the left and right asymptotic indices of the
joined walk are thereby fixed, and there must be eigenvalues at or
(bulk-boundary correspondence). Their location is governed by the third index.
We also discuss how the theory applies to finite lattices, with suitable
homogeneity assumptions.Comment: 36 pages, 7 figure
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