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In Descriptive Complexity, Immerman has Corollary 7.23. The following conditions are equivalent: 1. P = NP. 2. Over finite, ordered structures, FO(LFP) = SO. This can be thought of as "ampli …

Ladner's Theorem states that if P ≠ NP, then there is an infinite hierarchy of complexity classes strictly containing P and strictly contained in NP. The proof uses the completeness of SAT under many …

The Immerman-Vardi theorem states that PTIME (or P) is precisely the class of languages that can be described by a sentence of First-Order Logic together with a fixed-point operator, over the class of …

The Complexity Zoo points out in the entry on EXP that if L = P then PSPACE = EXP. Since NPSPACE = PSPACE by Savitch, as far as I can tell the underlying padding argument extends to show that $$(\tex …

Bounded nondeterminism associates a function $g(n)$ with a class $C$ of languages accepted by resource-bounded deterministic Turing machines, to form a new class $g$-$C$. This class consists of those …

RP is the class of problems decidable by a nondeterministic Turing machine that terminates in polynomial time, but that is also allowed one-sided error. P is the usual class of problems decidable by …

In The computational complexity of linear optics (ECCC TR10-170), Scott Aaronson and Alex Arkhipov argue that if quantum computers can be efficiently simulated by classical computers then the polynomi …

Dinur's paper (linked in the answer by Daniel Apon) is well written and worth reading. An extended discussion was also published about this paper and the proof, which is useful when reading the paper …

There is no Zoo-style reference yet, but a recent automata-theoretic survey of Giovanni Pighizzini has been useful to me, especially the slides from his talk. Giovanni Pighizzini, Investigations on …

Following on from a previous question, what are the best current space lower bounds for SAT? With a space lower bound I here mean the number of worktape cells used by a Turing machine which uses …

The original Nondeterministic Time Hierarchy Theorem is due to Cook (the link is to S. Cook, A hierarchy for nondeterministic time complexity, JCSS 7 343–353, 1973). The theorem states that for any r …

USTCONN is the problem that requires deciding whether there is a path from the source vertex $s$ to the target vertex $t$ in a graph $G$, where these are all given as part of the input. Omer Reingold …

INDEPENDENT SET is NP-complete for (cross,triangle)-free graphs, but can be solved in linear time for (chair,triangle)-free graphs. (The X-free graphs are those that contain no graph from X as an ind …

As Walter says, clauses of 2-SAT have a special form. This can be exploited to find solutions quickly. There are actually several classes of SAT instances that can be decided in polynomial time, and …

Denote the $k$-variable fragment of logic $L$ by $L^{(k)}$. The model-checking problem for a logic $L$ with respect to a class of structures $C$, denoted $MC(L,C)$, is the decision problem $MC(L, …