Parallel computation thesis

In computational complexity theory, the parallel computation thesis is a hypothesis which states that the time used by a (reasonable) parallel machine is polynomially related to the space used by a sequential machine. The parallel computation thesis was set forth by Chandra and Stockmeyer in 1976.^[1]

In other words, for a computational model which allows computations to branch and run in parallel without bound, a formal language which is decidable under the model using no more than ${\displaystyle t(n)}$ steps for inputs of length n is decidable by a non-branching machine using no more than ${\displaystyle t(n)^{k}}$ units of storage for some constant k. Similarly, if a machine in the unbranching model decides a language using no more than ${\displaystyle s(n)}$ storage, a machine in the parallel model can decide the language in no more than ${\displaystyle s(n)^{k}}$ steps for some constant k.

The parallel computation thesis is not a rigorous formal statement, as it does not clearly define what constitutes an acceptable parallel model. A parallel machine must be sufficiently powerful to emulate the sequential machine in time polynomially related to the sequential space; compare Turing machine, non-deterministic Turing machine, and alternating Turing machine. N. Blum (1983) introduced a model for which the thesis does not hold.^[2] However, the model allows ${\displaystyle 2^{2^{O(T(n))}}}$ parallel threads of computation after ${\displaystyle T(n)}$ steps. (See Big O notation.) Parberry (1986) suggested a more "reasonable" bound would be ${\displaystyle 2^{O(T(n))}}$ or ${\displaystyle 2^{T(n)^{O(1)}}}$, in defense of the thesis.^[3] Goldschlager (1982) proposed a model which is sufficiently universal to emulate all "reasonable" parallel models, which adheres to the thesis.^[4] Chandra and Stockmeyer originally formalized and proved results related to the thesis for deterministic and alternating Turing machines, which is where the thesis originated.^[5]