A Nearly Optimal Lower Bound on the Approximate Degree of AC0^0

The approximate degree of a Boolean function $f \colon \{-1, 1\}^n \rightarrow \{-1, 1\}$ is the least degree of a real polynomial that approximates $f$ pointwise to error at most $1/3$. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function $f$ with approximate degree $d$ into a function $F$ on $O(n \cdot \operatorname{polylog}(n))$ variables with approximate degree at least $D = \Omega(n^{1/3} \cdot d^{2/3})$. In particular, if $d= n^{1-\Omega(1)}$, then $D$ is polynomially larger than $d$. Moreover, if $f$ is computed by a polynomial-size Boolean circuit of constant depth, then so is $F$. By recursively applying our transformation, for any constant $\delta > 0$ we exhibit an AC$^0$ function of approximate degree $\Omega(n^{1-\delta})$. This improves over the best previous lower bound of $\Omega(n^{2/3})$ due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of $n$ that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant $\delta > 0$, an $\Omega(n^{1-\delta})$ lower bound on the quantum communication complexity of a function in AC$^0$. * A Boolean function $f$ with approximate degree at least $C(f)^{2-o(1)}$, where $C(f)$ is the certificate complexity of $f$. This separation is optimal up to the $o(1)$ term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC$^0$.Comment: 40 pages, 1 figur

Subset Semantics for Justifications

Justification logic is a variant of modal logic where the modal operators are replaced be justification terms. So we deal with formulas like t:A where t is a term denoting some justification that justifies the formula A. There are many justification logics among which the Logic of Proof established by Artemov was the first. However, since a long time the framework of justification logic is also used in a wide range of epistemic logics. In this field justification terms represent reasons to belief or know something. A standard interpretation of a justification term t is then the set of formulas that are supported by the reason t. This thesis establishes in the first part another way to interpret terms, namely as sets of worlds. We use so-called subset models in which t:A is true in a normal world, when the interpretation of t in this world is a subset of the truthset of A. These models are shown to be sound and complete towards a whole family of justification logics, including the Logic of Proof. As is shown in the second part of this thesis, subset models can easily be adapted to model new kinds of justification terms and operations: finer distinctions between several variants of combining justifications, justifications with presumptions, probabilistic evidence. Furthermore, it is shown, how subset models can be used to model dynamic reasoning and forgetting