Time-space lower bounds for satisfiability

We establish the first polynomial time-space lower bounds for satisfiability on general models of computation. We show that for any constant <i>c</i> less than the golden ratio there exists a positive constant <i>d</i> such that no deterministic random-access Turing machine can solve satisfiability in time <i>n</i><sup><i>c</i></sup> and space <i>n</i><sup><i>d</i></sup>, where <i>d</i> approaches 1 when <i>c</i> does. On conondeterministic instead of deterministic machines, we prove the same for any constant <i>c</i> less than &2radic;.Our lower bounds apply to nondeterministic linear time and almost all natural NP-complete problems known. In fact, they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space <i>n</i><sup><i>1/c</i></sup>.Our proofs follow the paradigm of indirect diagonalization. We also use that paradigm to prove time-space lower bounds for languages higher up in the polynomial-time hierarchy.

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