Valiant's 2013 book is Probably Approximately Correct: Nature's Algorithms for Learning and Prospering in a Complex World (Basic Books, ISBN 9780465032716). In it he argues, among other things, that evolutionary biology cannot explain the rate at which evolution occurs, writing, for example, "The evidence for Darwin's general schema for evolution being essentially correct is convincing to the great majority of biologists. This author has been to enough natural history museums to be convinced himself. All this, however, does not mean the current theory of evolution is adequately explanatory. At present the theory of evolution can offer no account of the rate at which evolution progresses to develop complex mechanisms or to maintain them in changing environments."
Valiant has contributed in a decisive way to the growth of almost every branch of theoretical computer science. His work is concerned mainly with quantifying mathematically the resource costs of solving problems on a computer. In early work (1975) he found the asymptotically fastest algorithm known for recognising context-free languages. At the same time, he pioneered the use of communication properties of graphs for analysing computations. In 1977 he defined the notion of #P-completeness ("sharp-P") and established its utility in classifying counting or enumeration problems according to computational tractability. The first application was to counting matchings (the matrix permanent function). In 1984 Valiant introduced a definition of inductive learning that for the first time reconciles computational feasibility with the applicability to non-trivial classes of logical rules to be learned.* More recently he has devised a scheme for efficient routing of communications in a multiprocessor system. He showed that the overheads involved even in a sparse network need not grow with the size of the system. This establishes, from a theoretical viewpoint, the possibility of efficient general purpose parallel computers.