Editing EXPTIME (section)

=== Succinct circuits ===
Another set of important EXPTIME-complete problems relates to succinct circuits. The idea is that if we can exponentially compress the description of a problem that requires polynomial time, then that compressed problem would require exponential time.

As one example, some graphs can be succinctly described by a small Boolean circuit. The circuit has <math>2n</math> inputs, 1 output and <math>\mathsf{poly}(n)</math> gates, thus requiring <math>\mathsf{poly}(n)</math> bits to describe. The circuit represents a graph with <math>2^n</math> vertices. For each pair of vertices, if the binary code for the two vertices are put into the circuit, then the output of the circuit states whether the two vertices are connected by an edge.

For many naturally occurring [[P-complete]] decision problems about graphs, where the graph is expressed in a natural representation such as an [[adjacency matrix]], solving the same problem on a succinct circuit representation is EXPTIME-complete, because the input is exponentially smaller; but this requires nontrivial proof, since succinct circuits can only describe a subclass of graphs.<ref>{{harvtxt|Papadimitriou|1994|p=495|loc=Section 20.1}}</ref>

Generically, a Boolean circuit with <math>n</math> inputs and a single output is a succinct representation of a string of <math>2^n</math> bits, which can be used to describe some other object, such as a graph, a 3-[[Conjunctive normal form|CNF formula]], etc. For essentially all known NP-complete problems, the succinct version of it is NEXP-complete. In particular, SUCCINCT 3-SAT is NEXP-complete under polynomial-time reductions.<ref>{{Cite journal |last=Papadimitriou |first=Christos H. |last2=Yannakakis |first2=Mihalis |date=1986-12-01 |title=A note on succinct representations of graphs |url=https://www.sciencedirect.com/science/article/pii/S0019995886800092 |journal=Information and Control |volume=71 |issue=3 |pages=181–185 |doi=10.1016/S0019-9958(86)80009-2 |issn=0019-9958}}</ref><ref>{{Cite journal |last=Williams |first=Ryan |date=2011-10-14 |title=Guest column: a casual tour around a circuit complexity bound |url=https://dl.acm.org/doi/10.1145/2034575.2034591 |journal=ACM SIGACT News |language=en |volume=42 |issue=3 |pages=54–76 |doi=10.1145/2034575.2034591 |issn=0163-5700}}</ref>