TRS Innermost Runtime Complexity

The TRS Innermost Runtime Complexity category is concerned with the question "What is the time complexity of algorithms represented by a given TRS with relation to the size of the initial term if we only consider innermost rewriting".

Category Motivation

Analyzing the time complexity of a rewrite system is interesting because it reveals how its performance scales as input size grows.

While derivational complexity considers arbitrary initial terms, runtime complexity restricts the initial terms to model an algorithm that is given fixed data as arguments, which relates to the intuitive definition of time complexity.

Moreover, since innermost rewriting (call-by-value) is one of the most important evaluation strategies, it is an interesting restriction to find bounds on the innermost time complexity.

Syntax & Semantic

The syntax and the semantics of term rewrite systems are described here and here.

Formally, a term rewrite system R = {l₁ → r₁,...,l_n → r_n} is a finite set of rewrite rules.

Let |t| be the size of the term t, i.e., the number of its positions. Then the innermost derivation height dh(t) of a term t is the supremum over all →ⁱ_R-rewrite sequences starting with t.

We split the function symbols into defined symbols (those that occur at the root of a left-hand side of a rule, the algorithms) and constructor symbols (the remaining function symbols, the data). A basic term t has a defined symbol at the root and otherwise only constructors and variables, i.e., it represents an algorithm that is given fixed input data. The set of all basic terms is denoted T_B.

The innermost runtime complexity of an ARS is a function that maps every natural number n to the greatest derivation height of all basic terms of size at most n, i.e., dc_→R (n) = sup{m | t ∈ T_B, |t| <= n, t →ⁱ_Rⁿ s}.

The goal is to find an asymptotic upper bound and lower bound on the innermost runtime complexity of a given TRS R. Possible complexity classes are O(1), O(n), O(n²), ..., O(EXP), O(2-EXP). The '?' indicates that no bound was found.

Problem

Input: A term rewrite system R.

Question: What is the asymptotic innermost runtime complexity of R?

Possible Outputs:

• "WORST_CASE(f(n),g(n))" where f(n) and g(n) are lower and upper complexity bounds, respectively, or '?', followed by a proof of these bounds, e.g., a ranking function proving termination of R.
• "MAYBE" (indicating that the solver cannot prove any complexity bound).