the monadic second order logic of graphs ix machines and their behaviours abstract we establish that every monadic second order property of the behaviour of a machine transition systems and tree automata are typical examples of machines is a monadic second order property of the machine itself in this way we clarify the distinction between dynamic properties of machines i e properties of their behaviours and their static properties i e properties of the graphs or relational structures representing them it is important for program verification that the dynamic properties that one wants to verify can be formulated statically in the simplest possible way as a corollary of our main result we also obtain that the monadic theory of an algebraic tree is decidable

fundamental properties of infinite trees infinite trees naturally arise in the formalization and the study of the semantics of programming languages this paper investigates some of their combinatorial and algebraic properties that are especially relevant to semantics

the role of terminal cost reward in finite horizon discrete time lq optimal control the optimal control problem for time invariant linear systems with quadratic cost is considered for arbitrary i e non necessarily positive semidefinite terminal cost matrices a classification of such matrices is proposed based on the maximum horizon for which there is a finite minimum cost for all initial states when such an horizon is infinite the classification is further refined based on the asymptotic behavior of the optimal control law a number of characterizations and other properties of the proposed classification are derived in the study of the asymptotic behavior a characterization is given of those matrices a such that the image of ats0 converges in the gap metric for any subspace s0

a note on finite horizon lq problems with indefinite cost this paper addresses the discrete time finite horizon linear quadratic lq optimal control problem in the case in which the quadratic forms in the performance index are not assumed to be positive semidefinite but only symmetric we provide a necessary and sufficient condition for the existence of an optimal control

an approximate kam renormalization group scheme for hamiltonian systems we construct an approximate renormalization scheme for hamiltonian systems with two degrees of freedom this scheme is a combination of kolmogorov arnold moser kam theory and renormalization group techniques it makes the connection between the approximate renormalization procedure derived by escande and doveil and a systematic expansion of the transformation in particular we show that the two main approximations consisting in keeping only the quadratic terms in the actions and the two main resonances keep the essential information on the threshold of the breakup of invariant tori

determination of the threshold of the break up of invariant tori in a class of three frequency hamiltonian systems we consider a class of hamiltonians with three degrees of freedom that can be mapped into quasi periodically driven pendulums the purpose of this paper is to determine the threshold of the break up of invariant tori with a specific frequency vector we apply two techniques the frequency map analysis and renormalization group methods the renormalization transformation acting on a hamiltonian is a canonical change of coordinates which is a combination of a partial elimination of the irrelevant modes of the hamiltonian and a rescaling of phase space around the considered torus we give numerical evidence that the critical coupling at which the renormalization transformation starts to diverge is the same as the value given by the frequency map analysis for the break up of invariant tori furthermore we obtain by these methods numerical values of the threshold of the break up of the last invariant torus

regular sets over extended tree structures we investigate notions of decidability and definability for the monadic second order logic over labeled tree structures and its relations with finite automata using oracles to test input prefixes a general framework is defined allowing to transfer some mso properties from a graph structure to a labeled tree structure transferred properties are the decidability of sentences and the existence of a definable model for every satisfiable formula a class of finite automata with prefix oracles is also defined recognizing exactly languages defined by mso formulas in any labeled tree structure applying these results the well known equivalence between languages recognized by finite automata sets of vertices mso definable in a tree structure and sets of pushdown contexts generated by pushdown automata is extended to k iterated pushdown automata

the evaluation of first order substitution is monadic second order compatible we denote first order substitutions of finite and infinite terms by function symbols indexed by the sequences of first order variables to which substitutions are made we consider the evaluation mapping from infinite terms to infinite terms that evaluates these substitution operations this mapping may perform infinitely many nested substitutions so that a term which has the structure of an infinite string can be transformed into one isomorphic to an infinite binary tree we prove that this mapping is monadic second order compatible which means that a monadic second order formula expressing a property of the output term produced by the evaluation mapping can be translated into a monadic second order formula expressing this property over the input term this implies that deciding the monadic second order theory of the output term reduces to deciding that of the input term as an application we obtain another proof that the monadic second order properties of the algebraic trees which represent the behaviours of recursive applicative program schemes are decidable this proof extends to hyperalgebraic trees these infinite trees correspond to certain recursive program schemes with functional parameters of arbitrary high type

stabilization control for ito stochastic system with indefinite state and control weight costs in standard linear quadratic lq control the first step in investigating infinite horizon optimal control is to derive the stabilisation condition with the optimal lq controller this paper focus

dpda s in atomic normal form and applications to equivalence problems abstract in this paper we introduce a new normal form for dpda s the atomic normal form as an application using also the concept of address language due to gorn 19 20 we give alternate and more direct proofs of results of courcelle 3 4 relating recursion schemes and dpda s address languages enable us to encode trees even when they are not locally finite as a consequence the decidability of a new class of schemes corresponding to the stateless dpda s of 26 is obtained

the monadic second order logic of graphs x linear orderings graphs are finite and handled as relational structures we give some answers to the following general questions 1 1 for which classes of graphs c is it possible to specify a linear ordering of the set of vertices of each graph of c by fixed monadic second order formulas 2 2 for which classes of graphs c does there exist an extension l of monadic second order logic such that a subclass l of c is recognizable if and only if it is the class of graphs in c that satisfy a formula of l in this paper recognizability is understood in an algebraic sense relative to a finite set of graph operations and basic graphs that generate all graphs of c 3 3 for which classes of graphs c is it possible to construct in every graph of the class and by fixed formulas of a suitable extension of monadic second order logic its hierarchichal structure i e a finite term written with the operations and basic graphs of 2 that defines the considered graph applications concern dependency graphs of partially commutative words partial k paths cographs and graphs the modular decomposition of which uses prime graphs of bounded size

equivalences and transformations of regular systems applications to recursive program schemes and grammars abstract this work presents a unified theory of recursive program schemes context free grammars grammars on arbitrary algebraic structures and in fact recursive definitions of all kind by means of regular systems the equivalences of regular systems associated with either all their solutions or their least solutions in all domains of appropriate type satisfying a set of algebraic laws expressed by equations are systematically investigated and characterized in some cases in terms of system transformations by folding unfolding and rewriting according to the equational algebraic laws grammars are better characterized in terms of polynomial systems which are regular systems involving the operation of set union and the same questions are raised for them we also examine conditions insuring the uniqueness of the solution of a regular or of a polynomial system this theory applies to grammars of many kinds which generate trees graphs etc we formulate some classical transformations of context free grammars in terms of correct transformations which only use folding unfolding and algebraic laws and we immediately obtain their correctness

the dual algebraic riccati equations and the set of all solutions of the discrete time riccati equation abstractin this paper two new pairs of dual continuous time algebraic riccati equations cares and dual discrete time algebraic riccati equations dares are proposed the dual dares are first studied with some nonsingularity assumptions on the system matrix and the parameter matrix then in the case of singular matrices a generalised inverse is introduced to deal with the dual dare problem these dual ares can easily lead us to an iterative procedure for finding the anti stabilising solutions especially to dare by means of that for the stabilising solutions furthermore we provide the counterpart results on the set of all solutions to dare inspired by the results for care two examples are presented to illustrate the theoretical results

computer science and the fine structure of borel sets i wadge defined a natural refinement of the borel hierarchy now called the wadge hierarchy wh the fundamental properties of wh follow from the results of kuratowski martin wadge and louveau we give a transparent restatement and proof of wadge s main theorem our method is new for it yields a wide and unexpected extension from borel sets of reals to a class of natural but non borel sets of infinite sequences wadge s theorem is quite ineffective and our generalization clearly worsens in this respect yet paradoxically our method is appropriate to effectivize this whole theory in the context discussed below ii wagner defined on buchi automata accepting words of length a hierarchy and proved for it an effective analog of wadge s results we extend wagner s results to more general kinds of automata counters push down automata and buchi automata reading transfinite words the notions and methods developed in i are quite useful for this extension and we start to use them in order to look for extensions of the fundamental effective determinacy results of buchi landweber rabin and of courcelle walukiewicz

on the structure of the solutions of discrete time algebraic riccati equation with singular closed loop matrix the classical discrete time algebraic riccati equation dare is considered in the case when the corresponding closed loop matrix is singular it is shown that in this case all the symmetric solutions of the dare coincide along some directions a parametrization of the set of solutions in terms of a reduced order dare is then obtained this parametrization provides an algorithm that appears to be computationally very attractive when the multiplicity of the eigenvalue spl lambda 0 of the closed loop matrix is large for the computation of the solutions of the dare the same issue for the generalized dare is also addressed

kolmogorov arnold moser renormalization group approach to the breakup of invariant tori in hamiltonian systems we analyze the breakup of invariant tori in hamiltonian systems with two degrees of freedom using a combination of kolmogorov arnold moser kam theory and renormalization group techniques we consider a class of hamiltonians quadratic in the action variables that is invariant under the chosen kam transformations following the approach of thirring the numerical implementation of the transformation shows that the kam iteration converges up to the critical coupling at which the torus breaks up by combining this iteration with a renormalization consisting of a shift of resonances and rescalings of momentum and energy we obtain a more efficient method that allows one to determine the critical coupling with high accuracy this transformation is based on the physical mechanism of the breakup of invariant tori we show that the critical surface of the transformation is the stable manifold of codimension one of a nontrivial fixed point and we discuss its universality properties

on a hardy space approach to the analysis of spectral factors the paper presents a study of several problems related to spectral factorizations we assume only a very weak form of coercivity for the p x p spectral function phi and look at the set wm of all rectangular p x m spectral factors the main object is the arithmetization of the geometry of the set of minimal stable spectral factors by employing hardy space techniques and the arithmetic of inner functions particular attention is paid to the study of various partial orders associated with the set of spectral factors

the monadic second order logic of graphs i recognizable sets of finite graphs abstract the notion of a recognizable set of finite graphs is introduced every set of finite graphs that is definable in monadic second order logic is recognizable but not vice versa the monadic second order theory of a context free set of graphs is decidable

semiclassical spectrum using secular perturbation theory the quartic oscillator we analyse the quartic oscillator using the technique of removal of resonances in a region of parameter space where the system exhibits soft chaos we obtain analytic expressions for all primary resonances and thus obtain a semiclassical spectrum a criterion relating to overlapping resonances gives an estimate about the break down of the removal of resonance method

a version of thirring s approach to the kolmogorov arnold moser theorem for quadratic hamiltonians with degenerate twist we give a proof of the kolmogorov arnold moser kam theorem on the existence of invariant tori for weakly perturbed hamiltonian systems based on thirring s approach for hamiltonians that are quadratic in the action variables the main point of this approach is that the iteration of canonical transformations on which the proof is based stays within the space of quadratic hamiltonians we show that thirring s proof for nondegenerate hamiltonians can be adapted to hamiltonians with degenerate twist this case in fact drastically simplifies thirring s proof

transformation properties of perturbation expansions around elliptic fixed points transformation properties of perturbation expansions of vibrational quasi periodic orbits upon the modular transformations of frequencies are studied using simple but nontrivial examples the siegel complex quadratic map and special solutions of the real area preserving quadratic map it is shown that the transformation properties are similar to those in the previously studied case of the rotational invariant tori except for some special features of the real map related to the atypical nature of 1 3 resonance in this case

quasiperiodic hamiltonian motions scale invariance harmonic oscillators the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point the work of kolmogorov arnold and moser appeared just before the renormalization group approach to statistical mechanics was proposed by wilson phys rev b 4 3174 3183 1971 it can be classified as a multiscale approach which also appeared in works on the convergence of fourier s series l carleson acta math 116 135 157 1966 and c fefferman ann math 98 551 571 1973 or construction of euclidean quantum fields e nelson j math phys 5 1190 1964 or the scaling analysis of the short scale behavior of navier stokes fluids l caffarelli r kohn and l nirenberg commun pure appl math 35 771 831 1982 to name a few which resulted in numerous problems in this review the kam theorem proof will be presented as a classical renormalization problem with the harmonic oscillator as a trivial fixed point

hermitian solutions of the discrete time algebraic riccati equation this paper deals with the existence and uniqueness of unmixed solutions of the discrete time algebraic riccati equation dare sets of solutions of the dare are parametrized by invariant subspaces of the state transition matrix

monadic second order logic on tree like structures an operation m which constructs from a given structure m a tree like structure whose domain consists of the finite sequences of elements of m is considered a notion of automata running on such tree like structures is defined it is shown that automata of this kind characterise expressive power of monadic second order logic msol over tree like structures using this characterisation it is proved that msol theory of a tree like structure is effectively reducible to that of the original structure as another application of the characterisation it is shown that msol on trees of arbitrary degree is equivalent to first order logic extended with unary least fixpoint operator