By Bruno Courcelle (auth.), Luca Aceto, Ivan Damgård, Leslie Ann Goldberg, Magnús M. Halldórsson, Anna Ingólfsdóttir, Igor Walukiewicz (eds.)

The two-volume set LNCS 5125 and LNCS 5126 constitutes the refereed complaints of the thirty fifth foreign Colloquium on Automata, Languages and Programming, ICALP 2008, held in Reykjavik, Iceland, in July 2008.

The 126 revised complete papers offered including four invited lectures have been rigorously reviewed and chosen from a complete of 407 submissions. The papers are grouped in 3 significant tracks on algorithms, automata, complexity and video games, on common sense, semantics, and thought of programming, and on protection and cryptography foundations. LNCS 5125 comprises 70 contributions of song a particular from 269 submissions in addition to 2 invited lectures. The papers are prepared in topical sections on complexity: boolean capabilities and circuits, facts buildings, random walks and random constructions, layout and research of algorithms, scheduling, codes and coding, coloring, randomness in computation, on-line and dynamic algorithms, approximation algorithms, estate trying out, parameterized algorithms and complexity, graph algorithms, computational complexity, video games and automata, crew trying out, streaming, and quantum, algorithmic video game thought, and quantum computing.

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We will derive from F a minimum depth-d formula for F that is of the form pictured in Figure 1. We prove this in the next three subsections. Note that if F has size larger than 4u + 2k + 2n + 1, then we are done; therefore we will assume the contrary in what follows. The z variable. First, we show that there is some i for which zi occurs exactly once in F and that it does not occur negated. 1. If a formula for F has size at most 4u + 2k + 2n + 1, then there is some i such that zi occurs exactly once.

Vm , x1 , . . , xn and an integer k, is there a subset I ⊆ {1, 2, . . n} with |I| ≤ k and D ∨ i∈I xi ≡ 1? This can be seen as a succinct version of Set Cover, in which the sets are implicitly speciﬁed by the ones of the formulas D, x1 , x2 , . . , because it covers some point not covered by any of the other sets). Throughout this paper, we assume that the formula D accepts the all-true assignment, as it only requires polynomial time to check this and the SSC instance is trivially false otherwise.

For each xi , let 1 ≤ (j ) ji ≤ w(xi ) be the integer for which xi i occurs least among the xi -leaves of F . (j) Consider the restriction ρ that for each i sets xi to true for j = ji . By our choice of ji , |F | is at least the w-weighted size of Fρ . But the formula Fρ clearly is equivalent to F , so its w-weighted size is an upper bound on the minimum w-weighted size of a formula equivalent to F . 1 The Problems As mentioned in the introduction, we will reduce from the Σ2P -complete problem succinct set cover.