185 research outputs found
Tropical polar cones, hypergraph transversals, and mean payoff games
We discuss the tropical analogues of several basic questions of convex
duality. In particular, the polar of a tropical polyhedral cone represents the
set of linear inequalities that its elements satisfy. We characterize the
extreme rays of the polar in terms of certain minimal set covers which may be
thought of as weighted generalizations of minimal transversals in hypergraphs.
We also give a tropical analogue of Farkas lemma, which allows one to check
whether a linear inequality is implied by a finite family of linear
inequalities. Here, the certificate is a strategy of a mean payoff game. We
discuss examples, showing that the number of extreme rays of the polar of the
tropical cyclic polyhedral cone is polynomially bounded, and that there is no
unique minimal system of inequalities defining a given tropical polyhedral
cone.Comment: 27 pages, 6 figures, revised versio
Vertex adjacencies in the set covering polyhedron
We describe the adjacency of vertices of the (unbounded version of the) set
covering polyhedron, in a similar way to the description given by Chvatal for
the stable set polytope. We find a sufficient condition for adjacency, and
characterize it with similar conditions in the case where the underlying matrix
is row circular. We apply our findings to show a new infinite family of
minimally nonideal matrices.Comment: Minor revision, 22 pages, 3 figure
Proof Automation in the Theory of Finite Sets and Finite Set Relation Algebra
{log} ('setlog') is a satisfiability solver for formulas of the theory of
finite sets and finite set relation algebra (FSTRA). As such, it can be used as
an automated theorem prover (ATP) for this theory. {log} is able to
automatically prove a number of FSTRA theorems, but not all of them.
Nevertheless, we have observed that many theorems that {log} cannot
automatically prove can be divided into a few subgoals automatically
dischargeable by {log}. The purpose of this work is to present a prototype
interactive theorem prover (ITP), called {log}-ITP, providing evidence that a
proper integration of {log} into world-class ITP's can deliver a great deal of
proof automation concerning FSTRA. An empirical evaluation based on 210
theorems from the TPTP and Coq's SSReflect libraries shows a noticeable
reduction in the size and complexity of the proofs with respect to Coq
Duality between invariant spaces for max-plus linear discrete event systems
We extend the notions of conditioned and controlled invariant spaces to
linear dynamical systems over the max-plus or tropical semiring. We establish a
duality theorem relating both notions, which we use to construct dynamic
observers. These are useful in situations in which some of the system
coefficients may vary within certain intervals. The results are illustrated by
an application to a manufacturing system.Comment: 22 pages, 3 figures (6 eps files
Reachability problems for products of matrices in semirings
We consider the following matrix reachability problem: given square
matrices with entries in a semiring, is there a product of these matrices which
attains a prescribed matrix? We define similarly the vector (resp. scalar)
reachability problem, by requiring that the matrix product, acting by right
multiplication on a prescribed row vector, gives another prescribed row vector
(resp. when multiplied at left and right by prescribed row and column vectors,
gives a prescribed scalar). We show that over any semiring, scalar reachability
reduces to vector reachability which is equivalent to matrix reachability, and
that for any of these problems, the specialization to any is
equivalent to the specialization to . As an application of this result and
of a theorem of Krob, we show that when , the vector and matrix
reachability problems are undecidable over the max-plus semiring
. We also show that the matrix, vector, and scalar
reachability problems are decidable over semirings whose elements are
``positive'', like the tropical semiring .Comment: 21 page
Tropical Fourier-Motzkin elimination, with an application to real-time verification
We introduce a generalization of tropical polyhedra able to express both
strict and non-strict inequalities. Such inequalities are handled by means of a
semiring of germs (encoding infinitesimal perturbations). We develop a tropical
analogue of Fourier-Motzkin elimination from which we derive geometrical
properties of these polyhedra. In particular, we show that they coincide with
the tropically convex union of (non-necessarily closed) cells that are convex
both classically and tropically. We also prove that the redundant inequalities
produced when performing successive elimination steps can be dynamically
deleted by reduction to mean payoff game problems. As a complement, we provide
a coarser (polynomial time) deletion procedure which is enough to arrive at a
simply exponential bound for the total execution time. These algorithms are
illustrated by an application to real-time systems (reachability analysis of
timed automata).Comment: 29 pages, 8 figure
The tropical analogue of polar cones
We study the max-plus or tropical analogue of the notion of polar: the polar
of a cone represents the set of linear inequalities satisfied by its elements.
We establish an analogue of the bipolar theorem, which characterizes all the
inequalities satisfied by the elements of a tropical convex cone. We derive
this characterization from a new separation theorem. We also establish variants
of these results concerning systems of linear equalities.Comment: 21 pages, 3 figures, example added, figures improved, notation
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