Processing Succinct Matrices and Vectors

We study the complexity of algorithmic problems for matrices that are represented by multi-terminal decision diagrams (MTDD). These are a variant of ordered decision diagrams, where the terminal nodes are labeled with arbitrary elements of a semiring (instead of 0 and 1). A simple example shows that the product of two MTDD-represented matrices cannot be represented by an MTDD of polynomial size. To overcome this deficiency, we extended MTDDs to MTDD_+ by allowing componentwise symbolic addition of variables (of the same dimension) in rules. It is shown that accessing an entry, equality checking, matrix multiplication, and other basic matrix operations can be solved in polynomial time for MTDD_+-represented matrices. On the other hand, testing whether the determinant of a MTDD-represented matrix vanishes PSPACE$-complete, and the same problem is NP-complete for MTDD_+-represented diagonal matrices. Computing a specific entry in a product of MTDD-represented matrices is #P-complete.Comment: An extended abstract of this paper will appear in the Proceedings of CSR 201

Expander Construction in VNC1

We give a combinatorial analysis (using edge expansion) of a variant of the iterative expander construction due to Reingold, Vadhan, and Wigderson (2002), and show that this analysis can be formalized in the bounded arithmetic system VNC^1 (corresponding to the "NC^1 reasoning"). As a corollary, we prove the assumption made by Jerabek (2011) that a construction of certain bipartite expander graphs can be formalized in VNC^1. This in turn implies that every proof in Gentzen\u27s sequent calculus LK of a monotone sequent can be simulated in the monotone version of LK (MLK) with only polynomial blowup in proof size, strengthening the quasipolynomial simulation result of Atserias, Galesi, and Pudlak (2002)

On Matrix Powering in Low Dimensions

We investigate the Matrix Powering Positivity Problem, PosMatPow: given an m X m square integer matrix M, a linear function f: Z^{m X m} -> Z with integer coefficients, and a positive integer n (encoded in binary), determine whether f(M^n) geq 0. We show that for fixed dimensions m of 2 and 3, this problem is decidable in polynomial time

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

Powering a Biosensor Using Wearable Thermoelectric Technology

Wearable medical devices such as insulin pumps, glucose monitors, hearing aids, and electrocardiograms provide necessary medical aid and monitoring to millions of users worldwide. These battery powered devices require battery replacement and frequent charging that reduces the freedom and peace of mind of users. Additionally, the significant portion of the world without access to electricity is unable to use these medical devices as they have no means to power them constantly. Wearable thermoelectric power generation aims to charge these medical device batteries without a need for grid power. Our team has developing a wristband prototype that uses body heat, ambient air, and heat sinks to create a temperature difference across thermoelectric modules thus generating ultra-low voltage electrical power. A boost converter is implemented to boost this voltage to the level required by medical device batteries. Our goal was to use this generated power to charge medical device batteries off-the-grid, increasing medical device user freedom and allowing medical device access to those without electricity. We successfully constructed a wearable prototype that generates the voltage required by an electrocardiogram battery; however, further thermoelectric module and heat dissipation optimization is necessary to generate sufficient current to charge the battery