Fagin's theorem

Fagin's theorem is the oldest result of descriptive complexity theory, a branch of computational complexity theory that characterizes complexity classes in terms of logic-based descriptions of their problems rather than by the behavior of algorithms for solving those problems. The theorem states that the set of all properties expressible in existential second-order logic is precisely the complexity class NP. It was proven by Ronald Fagin in 1973 in his doctoral thesis, and appears in his 1974 paper. The arity required by the second-order formula was improved (in one direction) in , and several results of Grandjean have provided tighter bounds on nondeterministic random-access machines. In addition to Fagin's 1974 paper, provides a detailed proof of the theorem. It is straightforward to show that every existential second-order formula can be recognized in NP, by nondeterministically choosing the value of all existentially-qualified variables, so the main part of the proof is to show that every language in NP can be described by an existential second-order formula. To do so, one can use second-order existential quantifiers to arbitrarily choose a computation tableau. In more detail, for every timestep of an execution trace of a non-deterministic Turing machine, this tableau encodes the state of the Turing machine, its position in the tape, the contents of every tape cell, and which nondeterministic choice the machine makes at that step. A first-order formula can constrain this encoded information so that it describes a valid execution trace, one in which the tape contents and Turing machine state and position at each timestep follow from the previous timestep. A key lemma used in the proof is that it is possible to encode a linear order of length (such as the linear orders of timesteps and tape contents at any timestep) as a -ary relation on a universe of size . One way to achieve this is to choose a linear ordering of and then define to be the lexicographical ordering of -tuples from with respect to .

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related concepts (2)

Descriptive complexity theory

Descriptive complexity is a branch of computational complexity theory and of finite model theory that characterizes complexity classes by the type of logic needed to express the languages in them. For example, PH, the union of all complexity classes in the polynomial hierarchy, is precisely the class of languages expressible by statements of second-order logic.

Second-order logic

In logic and mathematics, second-order logic is an extension of first-order logic, which itself is an extension of propositional logic. Second-order logic is in turn extended by higher-order logic and type theory. First-order logic quantifies only variables that range over individuals (elements of the domain of discourse); second-order logic, in addition, also quantifies over relations. For example, the second-order sentence says that for every formula P, and every individual x, either Px is true or not(Px) is true (this is the law of excluded middle).