Bodlaender's Theorem states that for every k there is a linear-time algorithm that decides whether an input graph has tree width k and, if so, computes a width-k tree composition. Courcelle's Theorem builds on Bodlaender's Theorem and states that for every monadic second-order formula φ and for every k there is a linear-time algorithm that decides whether a given logical structure A of tree width at most k satisfies φ . We prove that both theorems still hold when "linear time" is replaced by "logarithmic space." The transfer of the powerful theoretical framework of monadic second-order logic and bounded tree width to logarithmic space allows us to settle a number of both old and recent open problems in the logspace world.

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We study the role of connectivity of communication networks in private computations under information theoretic settings. It will be shown that some functions can be computed by private protocols even if the underlying network is 1-connected but not 2-connected. Then we give a complete characterisation of non-degenerate functions that can be computed on non-2-connected networks. Furthermore, a general technique for simulating private protocols on arbitrary networks will be presented. Using this technique every private protocol can be simulated on arbitrary k-connected networks using only a small number of additional random bits. Finally, we give matching lower and upper bounds for the number of random bits needed to compute the parity function on k-connected networks.On leave from Instytut Informatyki, Uniwersytet Wroc lawski, Poland.