On the construction of parallel computers from various bases of Boolean functions

The effects of bases of two-input boolean functions are characterised in terms of their impact on some questions in parallel computation. It is found that a certain set of bases (called the P-complete set) which are not necessarily complete in the classical sense, apparently makes the circuit value problem difficult, and renders extended Turing machines and conglomerates equal to general parallel computers. A class of problems called EP arises naturally from this study, relating to the parity of the number of solutions to a problem, in contrast to previously defined classes concerning the count of the number of solutions (#P) or the existence of solutions to a problem (NP). Tournament isomorphism is a member of EP

The New Generation of Computer Literacy

A tremendous mismatch is developing between two of the most critical components of any computer literacy course: the textbooks and the students. We are encountering a new generation of students (literally as well as figuratively!) who are much better acquainted with computer usage than their earlier counterparts. Yet many textbooks with increasing emphasis in those same computer tools continue to appear. There are signs of a coming change in that a few authors and publishers apparently are becoming aware of the need for innovations in texts for non-scientists. These textbooks open the door for a new orientation to principles in the teaching of computer literacy

Balancing Bounded Treewidth Circuits

Algorithmic tools for graphs of small treewidth are used to address questions in complexity theory. For both arithmetic and Boolean circuits, it is shown that any circuit of size $n^{O(1)}$ and treewidth $O(\log^i n)$ can be simulated by a circuit of width $O(\log^{i+1} n)$ and size $n^c$, where $c = O(1)$, if $i=0$, and $c=O(\log \log n)$ otherwise. For our main construction, we prove that multiplicatively disjoint arithmetic circuits of size $n^{O(1)}$ and treewidth $k$ can be simulated by bounded fan-in arithmetic formulas of depth $O(k^2\log n)$. From this we derive the analogous statement for syntactically multilinear arithmetic circuits, which strengthens a theorem of Mahajan and Rao. As another application, we derive that constant width arithmetic circuits of size $n^{O(1)}$ can be balanced to depth $O(\log n)$, provided certain restrictions are made on the use of iterated multiplication. Also from our main construction, we derive that Boolean bounded fan-in circuits of size $n^{O(1)}$ and treewidth $k$ can be simulated by bounded fan-in formulas of depth $O(k^2\log n)$. This strengthens in the non-uniform setting the known inclusion that $SC^0 \subseteq NC^1$. Finally, we apply our construction to show that {\sc reachability} for directed graphs of bounded treewidth is in $LogDCFL$

Counting Homomorphisms to Square-Free Graphs, Modulo 2

We study the problem ⊕HomsToH of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph H. A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (nonmodular) counting; thus, subtle dichotomy theorems can arise. We show the following dichotomy: for any H that contains no 4-cycles, ⊕HomsToH is either in polynomial time or is ⊕P-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs, including graphs of unbounded tree-width. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example, in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach