Proving Quadratic Derivational Complexities Using Context Dependent Interpretations

Abstract

Abstract. In this paper we study context dependent interpretations, a semantic termination method extending interpretations over the natural numbers, introduced by Hofbauer. We present two subclasses of context dependent interpretations and establish tight upper bounds on the induced derivational complexities. In particular we delineate a class of interpretations that induces quadratic derivational complexity. Furthermore, we present an algorithm for mechanically proving termination of rewrite systems with context dependent interpretations. This algorithm has been implemented and we present ample numerical data for the assessment of the viability of the method.

Cdiprover3: A Tool for Proving Derivational Complexities of Term Rewriting Systems

“…The encoding is an adaptation of the procedure in (Contejean et al 2005) to context-dependent interpretations. It is described in detail in (Schnabl 2007, Moser & Schnabl 2008). Once we have built the constraints, we continue using the same techniques as for searching polynomial interpretations: we encode the constraints in a propositional satisfiability problem, apply the SAT solver, and use a satisfying assignment to construct a context-dependent interpretation.…”

“…Theorem 6 ( (Moser & Schnabl 2008)). Let R be a TRS and suppose that there exists a compatible ∆-linear interpretation.…”

Polynomial Path Orders

“…Example 3.15. Consider the TRS R nc given by the rules 43 : f(n; ) → h(; gs(n; )) 44 : gs(0; ) → 0 45 : h(; g(; n)) → c(; h(; n), h(; n)) 46 : gs(s(; n); ) → g(; gs(n; )) 47 :…”

“…Polynomial complexity analysis is an active research area in rewriting. Starting from [46] interest in automated polynomial complexity analysis greatly increased over the last years, see for example [30,31,45,48,57]. This is partly due to the incorporation of a dedicated category for complexity into the annual termination competition (TERMCOMP).…”

Synthesis of sup-interpretations: A survey

“…Many studies have demonstrated that termination techniques can be used to study the runtime complexity of a given TRS. See [11,12,22,36], among others. In this subsection, we show that, as expected, bounding the runtime complexity of a TRS, allows us to recover a sup-interpretation.…”