Can Chimps Go It Alone?

Consider a smart chimpanzee named M from a tribe afflicted with a form of Alzheimer’s disease. Think of M as a logspace-bounded Turing machine. M can do simple things like integer arithmetic and matrix multiplication, but M turns sullen and calls for help when asked to perform seemingly equally simple tasks, such as simulating deterministic tree and dag automata. Is M acting difficult or is she just not smart enough? Even before the P versus NP question, Cook [Coo71] conjectured that no amount of smarts can compensate for Alzheimer’s disease. We will review some of the attempts at separating L from P inspired by pebbling arguments. Emphasis will be placed on branching programs for the tree evaluation problem, recently studied anew [CMW12]. The problem consists of determining the value that percolates to the root of a (binary) tree when a value from a domain D is prescribed at each tree leaf and an explicit function f : D×D → D is prescribed at each internal node. In a nutshell, lower bounds for restricted branching programs can be proved, but approaches to attack the general model strangely come up against the same barrier that Nec̆iporuk encountered in a two-page note 50 years ago and that still stands today. Tree evaluation naturally extends to tree generation [Cha13], where the functions f : D ×D → D at internal tree nodes are replaced with functions f : D ×D → {S : S ⊆ D}. This is interpreted as allowing to pick, as the D-value of a node labelled f with left child ` and right child r, any value from f(D-value of `, D-value of r). Tree generation can then be turned into a monotone boolean function. Strong lower bounds for this function have been derived from pebbling intuition [CP12,Cha13] and we will further discuss some of these. For a suitable bibliography please consult [CMW12,CP12,Cha13].