This paper characterizes alternation trading-based proofs that the Boolean satisfiability problem is not in the time- and space-bounded class DTISP(nc,nϵ)\documentclass[12pt]{minimal}
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\begin{document}$${(n^c,n^\epsilon)}$$\end{document}, for various values c < 2 and ϵ<1\documentclass[12pt]{minimal}
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\begin{document}$${\epsilon < 1}$$\end{document}. We characterize exactly what can be proved in the ϵ=0\documentclass[12pt]{minimal}
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\begin{document}$${\epsilon = 0}$$\end{document} case with currently known methods and prove the conjecture of Williams that c=2cos(π/7)\documentclass[12pt]{minimal}
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\begin{document}$${c=2\cos(\pi/7)}$$\end{document} is optimal for this. For time–space trade-offs and lower bounds on satisfiability, we give a theoretical and computational analysis of the alternation trading proofs for 0<ϵ<1\documentclass[12pt]{minimal}
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\begin{document}$${0 < \epsilon < 1}$$\end{document}.
