27 research outputs found
Argument filterings and usable rules in higher-order rewrite systems
The static dependency pair method is a method for proving the termination of
higher-order rewrite systems a la Nipkow. It combines the dependency pair
method introduced for first-order rewrite systems with the notion of strong
computability introduced for typed lambda-calculi. Argument filterings and
usable rules are two important methods of the dependency pair framework used by
current state-of-the-art first-order automated termination provers. In this
paper, we extend the class of higher-order systems on which the static
dependency pair method can be applied. Then, we extend argument filterings and
usable rules to higher-order rewriting, hence providing the basis for a
powerful automated termination prover for higher-order rewrite systems
SAT Solving for Argument Filterings
This paper introduces a propositional encoding for lexicographic path orders
in connection with dependency pairs. This facilitates the application of SAT
solvers for termination analysis of term rewrite systems based on the
dependency pair method. We address two main inter-related issues and encode
them as satisfiability problems of propositional formulas that can be
efficiently handled by SAT solving: (1) the combined search for a lexicographic
path order together with an \emph{argument filtering} to orient a set of
inequalities; and (2) how the choice of the argument filtering influences the
set of inequalities that have to be oriented. We have implemented our
contributions in the termination prover AProVE. Extensive experiments show that
by our encoding and the application of SAT solvers one obtains speedups in
orders of magnitude as well as increased termination proving power
Dynamic Dependency Pairs for Algebraic Functional Systems
We extend the higher-order termination method of dynamic dependency pairs to
Algebraic Functional Systems (AFSs). In this setting, simply typed lambda-terms
with algebraic reduction and separate {\beta}-steps are considered. For
left-linear AFSs, the method is shown to be complete. For so-called local AFSs
we define a variation of usable rules and an extension of argument filterings.
All these techniques have been implemented in the higher-order termination tool
WANDA
First-order formative rules
This paper discusses the method of formative rules for first-order term rewriting, which was previously defined for a higher-order setting. Dual to the well-known usable rules, formative rules allow dropping some of the term constraints that need to be solved during a termination proof. Compared to the higher-order definition, the first-order setting allows for significant improvements of the technique
Proof Theory at Work: Complexity Analysis of Term Rewrite Systems
This thesis is concerned with investigations into the "complexity of term
rewriting systems". Moreover the majority of the presented work deals with the
"automation" of such a complexity analysis. The aim of this introduction is to
present the main ideas in an easily accessible fashion to make the result
presented accessible to the general public. Necessarily some technical points
are stated in an over-simplified way.Comment: Cumulative Habilitation Thesis, submitted to the University of
Innsbruc