1,748 research outputs found
Definable Zero-Sum Stochastic Games
International audienceDefinable zero-sum stochastic games involve a finite number of states and action sets, reward and transition functions that are definable in an o-minimal structure. Prominent examples of such games are finite, semi-algebraic or globally subanalytic stochastic games. We prove that the Shapley operator of any definable stochastic game with separable transition and reward functions is definable in the same structure. Definability in the same structure does not hold systematically: we provide a counterexample of a stochastic game with semi-algebraic data yielding a non semi-algebraic but globally subanalytic Shapley operator. %Showing the definability of the Shapley operator in full generality appears thus as a complex and challenging issue. } Our definability results on Shapley operators are used to prove that any separable definable game has a uniform value; in the case of polynomially bounded structures we also provide convergence rates. Using an approximation procedure, we actually establish that general zero-sum games with separable definable transition functions have a uniform value. These results highlight the key role played by the tame structure of transition functions. As particular cases of our main results, we obtain that stochastic games with polynomial transitions, definable games with finite actions on one side, definable games with perfect information or switching controls have a uniform value. Applications to nonlinear maps arising in risk sensitive control and Perron-Frobenius theory are also given
Connection Matrices and the Definability of Graph Parameters
In this paper we extend and prove in detail the Finite Rank Theorem for
connection matrices of graph parameters definable in Monadic Second Order Logic
with counting (CMSOL) from B. Godlin, T. Kotek and J.A. Makowsky (2008) and
J.A. Makowsky (2009). We demonstrate its vast applicability in simplifying
known and new non-definability results of graph properties and finding new
non-definability results for graph parameters. We also prove a Feferman-Vaught
Theorem for the logic CFOL, First Order Logic with the modular counting
quantifiers
The Church Synthesis Problem with Parameters
For a two-variable formula ψ(X,Y) of Monadic Logic of Order (MLO) the
Church Synthesis Problem concerns the existence and construction of an operator
Y=F(X) such that ψ(X,F(X)) is universally valid over Nat.
B\"{u}chi and Landweber proved that the Church synthesis problem is
decidable; moreover, they showed that if there is an operator F that solves the
Church Synthesis Problem, then it can also be solved by an operator defined by
a finite state automaton or equivalently by an MLO formula. We investigate a
parameterized version of the Church synthesis problem. In this version ψ
might contain as a parameter a unary predicate P. We show that the Church
synthesis problem for P is computable if and only if the monadic theory of
is decidable. We prove that the B\"{u}chi-Landweber theorem can be
extended only to ultimately periodic parameters. However, the MLO-definability
part of the B\"{u}chi-Landweber theorem holds for the parameterized version of
the Church synthesis problem
The Church Problem for Countable Ordinals
A fundamental theorem of Buchi and Landweber shows that the Church synthesis
problem is computable. Buchi and Landweber reduced the Church Problem to
problems about ω-games and used the determinacy of such games as one of
the main tools to show its computability. We consider a natural generalization
of the Church problem to countable ordinals and investigate games of arbitrary
countable length. We prove that determinacy and decidability parts of the
Bu}chi and Landweber theorem hold for all countable ordinals and that its full
extension holds for all ordinals < \omega\^\omega
Deciding definability in FO2(<h,<v) on trees
We provide a decidable characterization of regular forest languages definable
in FO2(<h,<v). By FO2(<h,<v) we refer to the two variable fragment of first
order logic built from the descendant relation and the following sibling
relation. In terms of expressive power it corresponds to a fragment of the
navigational core of XPath that contains modalities for going up to some
ancestor, down to some descendant, left to some preceding sibling, and right to
some following sibling. We also show that our techniques can be applied to
other two variable first-order logics having exactly the same vertical
modalities as FO2(<h,<v) but having different horizontal modalities
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