JCKassing: Added TRS Derivational Complexity

2026-03-19T10:23:56Z

Added TRS Derivational Complexity

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The TRS Derivational Complexity category is concerned with the question "What is the time complexity of a given TRS with relation to the size of the initial term".

== Category Motivation ==

Analyzing the time complexity of a rewrite system is interesting because it reveals how its performance scales as input size grows.
his helps to detect rewrite systems that seem fine for small inputs but become impractical for large problems.

== Syntax & Semantic ==

The syntax and the semantics of term rewrite systems are described [[Term Rewriting | here]] and [https://project-coco.uibk.ac.at/ARI/trs.php here].

Formally, a term rewrite system R = {l1 → r1,...,ln → rn} 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 derivation height dh(t) of a term t is the supremum over all →R-rewrite sequences starting with t.
The derivational complexity of an ARS is a function that maps every natural number n
to the greatest derivation height of all objects of size at most n, i.e., dc→R (n) = sup{m | |t| <= n, t →Rn s}.

The goal is to find an asymptotic upper bound and lower bound on the derivational 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 derivational 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).

== References ==

[[Category:Categories]]

TRS Derivational Complexity - Revision history

JCKassing: Added TRS Derivational Complexity