MoL-2011-07: Tightening the Compression Hierarchies

MoL-2011-07: Talebanfard, Navid (2011) Tightening the Compression Hierarchies. [Report]

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Fix t, k and n such that k < n <= t and let EASY^n_{t,k} be the set of all strings of length n that are generated by programs of size k in at most t steps. It is not hard to see that for sufficiently large t (in fact for t' > t log t for technical reasons) we have EASY^n_{t,k} ⊆ EASY^n_{t',k'}. But can we get a strict inclusion, or equivalently, is there an string x that is generated by some k-bit program in t' steps but cannot be generated in t steps from k-bit programs? If so what is the smallest t for which a strict inclusion holds? Consider "the first (in lexicographic order) x \in {0,1}^n that is not generated in t steps by any program of size k". The statement in quotations is already a description of that x and it takes 2^k*t*log(t) steps to output x. This shows that EASY^n_{t,k} ⊂ EASY^n_{2^k*t*log(t), k}. But is this exponential gap really needed? This question that still remains open will be the central topic of this thesis. We will examine different variants of it and will demonstrate its connection with deterministic simulations of randomized computation.

Item Type: Report
Report Nr: MoL-2011-07
Series Name: Master of Logic Thesis (MoL) Series
Year: 2011
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38

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