180,190 research outputs found
The Variable Hierarchy for the Games mu-Calculus
Parity games are combinatorial representations of closed Boolean mu-terms. By
adding to them draw positions, they have been organized by Arnold and one of
the authors into a mu-calculus. As done by Berwanger et al. for the
propositional modal mu-calculus, it is possible to classify parity games into
levels of a hierarchy according to the number of fixed-point variables. We ask
whether this hierarchy collapses w.r.t. the standard interpretation of the
games mu-calculus into the class of all complete lattices. We answer this
question negatively by providing, for each n >= 1, a parity game Gn with these
properties: it unravels to a mu-term built up with n fixed-point variables, it
is semantically equivalent to no game with strictly less than n-2 fixed-point
variables
Fixed-point elimination in the intuitionistic propositional calculus
It is a consequence of existing literature that least and greatest
fixed-points of monotone polynomials on Heyting algebras-that is, the algebraic
models of the Intuitionistic Propositional Calculus-always exist, even when
these algebras are not complete as lattices. The reason is that these extremal
fixed-points are definable by formulas of the IPC. Consequently, the
-calculus based on intuitionistic logic is trivial, every -formula
being equivalent to a fixed-point free formula. We give in this paper an
axiomatization of least and greatest fixed-points of formulas, and an algorithm
to compute a fixed-point free formula equivalent to a given -formula. The
axiomatization of the greatest fixed-point is simple. The axiomatization of the
least fixed-point is more complex, in particular every monotone formula
converges to its least fixed-point by Kleene's iteration in a finite number of
steps, but there is no uniform upper bound on the number of iterations. We
extract, out of the algorithm, upper bounds for such n, depending on the size
of the formula. For some formulas, we show that these upper bounds are
polynomial and optimal
Parallel synchronous algorithm for nonlinear fixed point problems
We give in this paper a convergence result concerning parallel synchronous
algorithm for nonlinear fixed point problems with respect to the euclidian norm
in \Rn. We then apply this result to some problems related to convex analysis
like minimization of functionals, calculus of saddle point, convex
programming..
A Finite-Model-Theoretic View on Propositional Proof Complexity
We establish new, and surprisingly tight, connections between propositional
proof complexity and finite model theory. Specifically, we show that the power
of several propositional proof systems, such as Horn resolution, bounded-width
resolution, and the polynomial calculus of bounded degree, can be characterised
in a precise sense by variants of fixed-point logics that are of fundamental
importance in descriptive complexity theory. Our main results are that Horn
resolution has the same expressive power as least fixed-point logic, that
bounded-width resolution captures existential least fixed-point logic, and that
the polynomial calculus with bounded degree over the rationals solves precisely
the problems definable in fixed-point logic with counting. By exploring these
connections further, we establish finite-model-theoretic tools for proving
lower bounds for the polynomial calculus over the rationals and over finite
fields
A Weakest Pre-Expectation Semantics for Mixed-Sign Expectations
We present a weakest-precondition-style calculus for reasoning about the
expected values (pre-expectations) of \emph{mixed-sign unbounded} random
variables after execution of a probabilistic program. The semantics of a
while-loop is well-defined as the limit of iteratively applying a functional to
a zero-element just as in the traditional weakest pre-expectation calculus,
even though a standard least fixed point argument is not applicable in this
context. A striking feature of our semantics is that it is always well-defined,
even if the expected values do not exist. We show that the calculus is sound,
allows for compositional reasoning, and present an invariant-based approach for
reasoning about pre-expectations of loops
A Coinductive Approach to Proof Search
National audienceWe propose to study proof search from a coinductive point of view. In this paper, we consider intuitionistic logic and a focused system based on Herbelin's LJT for the implicational fragment. We introduce a variant of lambda calculus with potentially infinitely deep terms and a means of expressing alternatives for the description of the "solution spaces" (called Böhm forests), which are a representation of all (not necessarily well-founded but still locally well-formed) proofs of a given formula (more generally: of a given sequent). As main result we obtain, for each given formula, the reduction of a coinductive definition of the solution space to a effective coinductive description in a finitary term calculus with a formal greatest fixed-point operator. This reduction works in a quite direct manner for the case of Horn formulas. For the general case, the naive extension would not even be true. We need to study "co-contraction" of contexts (contraction bottom-up) for dealing with the varying contexts needed beyond the Horn fragment, and we point out the appropriate finitary calculus, where fixed-point variables are typed with sequents. Co-contraction enters the interpretation of the formal greatest fixed points - curiously in the semantic interpretation of fixed-point variables and not of the fixed-point operator
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