Black Box White Arrow

The present paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box, we consider categories of black boxes and their morphisms. This makes new classes of black box problems accessible. For example, we can enrich black box groups by actions of outer automorphisms. As an example of application of this technique, we construct Frobenius maps on black box groups of untwisted Lie type in odd characteristic (Section 6) and inverse-transpose automorphisms on black box groups encrypting ${\rm (P)SL}_n(\mathbb{F}_q)$. One of the advantages of our approach is that it allows us to work in black box groups over finite fields of big characteristic. Another advantage is explanatory power of our methods; as an example, we explain Kantor's and Kassabov's construction of an involution in black box groups encrypting ${\rm SL}_2(2^n)$. Due to the nature of our work we also have to discuss a few methodological issues of the black box group theory. The paper is further development of our text "Fifty shades of black" [arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248

Iterated multiplication in VTC0VTC^0

We show that $VTC^0$, the basic theory of bounded arithmetic corresponding to the complexity class $\mathrm{TC}^0$, proves the $IMUL$ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the $\mathrm{TC}^0$ iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, $VTC^0$ can also prove the integer division axiom, and (by results of Je\v{r}\'abek) the RSUV-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories $\Delta^b_1$-$CR$ and $C^0_2$. As a side result, we also prove that there is a well-behaved $\Delta_0$ definition of modular powering in $I\Delta_0+WPHP(\Delta_0)$.Comment: 57 page