Second-order Logic - Relation To Computational Complexity

Relation To Computational Complexity

The expressive power of various forms of second-order logic on finite structures is intimately tied to computational complexity theory. The field of descriptive complexity studies which computational complexity classes can be characterized by the power of the logic needed to express languages (sets of finite strings) in them. A string w = w₁···w_n in a finite alphabet A can be represented by a finite structure with domain D = {1,...,n}, unary predicates P_a for each a ∈ A, satisfied by those indices i such that w_i = a, and additional predicates which serve to uniquely identify which index is which (typically, one takes the graph of the successor function on D or the order relation <, possibly with other arithmetic predicates). Conversely, the table of any finite structure can be encoded by a finite string.

With this identification, we have the following characterizations of variants of second-order logic over finite structures:

• REG (the set of regular languages) is definable by monadic, second-order formulas (Büchi's theorem, 1960)
• NP is the set of languages definable by existential, second-order formulas (Fagin's theorem, 1974).
• co-NP is the set of languages definable by universal, second-order formulas.
• PH is the set of languages definable by second-order formulas.
• PSPACE is the set of languages definable by second-order formulas with an added transitive closure operator.
• EXPTIME is the set of languages definable by second-order formulas with an added least fixed point operator.

Relationships among these classes directly impact the relative expressiveness of the logics over finite structures; for example, if PH = PSPACE, then adding a transitive closure operator to second-order logic would not make it any more expressive over finite structures.