A Transformational Approach to Prove Outermost Termination Automatically (original) (raw)

2D Dependency Pairs for Proving Operational Termination of CTRSs

Lecture Notes in Computer Science, 2014

The notion of operational termination captures nonterminating computations due to subsidiary processes that are necessary to issue a single 'main' step but which often remain 'hidden' when the main computation sequence is observed. This highlights two dimensions of nontermination: one for the infinite sequencing of computation steps, and the other that concerns the proof of some single steps. For conditional term rewriting systems (CTRSs), we introduce a new dependency pair framework which exploits the bidimensional nature of conditional rewriting (rewriting steps + satisfaction of the conditions as reachability problems) to obtain a powerful and more expressive framework for proving operational termination of CTRSs.

Reducing Right-Hand Sides for Termination

Lecture Notes in Computer Science, 2005

We propose two transformations on term rewrite systems (TRSs) based on reducing right hand sides: one related to the transformation order and a variant of dummy elimination. Under mild conditions we prove that the transformed system is terminating if and only if the original one is terminating. Both transformations are very easy to implement, and make it much easier to prove termination of some TRSs automatically.

Termination Analysis by Dependency Pairs and Inductive Theorem Proving

Lecture Notes in Computer Science, 2009

Current techniques and tools for automated termination analysis of term rewrite systems (TRSs) are already very powerful. However, they fail for algorithms whose termination is essentially due to an inductive argument. Therefore, we show how to couple the dependency pair method for TRS termination with inductive theorem proving. As confirmed by the implementation of our new approach in the tool AProVE, now TRS termination techniques are also successful on this important class of algorithms.

Termination analysis with compositional transition invariants

Computer Aided …, 2010

Modern termination provers rely on a safety checker to construct disjunctively well-founded transition invariants. This safety check is known to be the bottleneck of the procedure. We present an alternative algorithm that uses a light-weight check based on transitivity of ranking relations to prove program termination. We provide an experimental evaluation over a set of 87 Windows drivers, and demonstrate that our algorithm is often able to conclude termination by examining only a small fraction of the program. As a consequence, our algorithm is able to outperform known approaches by multiple orders of magnitude.

The Dependency Pair Framework: Combining Techniques for Automated Termination Proofs

Lecture Notes in Computer Science, 2005

The dependency pair approach is one of the most powerful techniques for automated termination proofs of term rewrite systems. Up to now, it was regarded as one of several possible methods to prove termination. In this paper, we show that dependency pairs can instead be used as a general concept to integrate arbitrary techniques for termination analysis. In this way, the benefits of different techniques can be combined and their modularity and power are increased significantly. We refer to this new concept as the "dependency pair framework " to distinguish it from the old "dependency pair approach". Moreover, this framework facilitates the development of new methods for termination analysis. To demonstrate this, we present several new techniques within the dependency pair framework which simplify termination problems considerably. We implemented the dependency pair framework in our termination prover AProVE and evaluated it on large collections of examples.

A path ordering for proving termination of term rewriting systems

Mathematical Foundations of …, 1985

A new partial ordering scheme for proving uniform termination of term rewriting systems is presented. The basic idea is that two terms are compared by comparing the paths through them. It is shown that the ordering is a well-founded simplification ordering and also a ...

Improved Modular Termination Proofs Using Dependency Pairs

Lecture Notes in Computer Science, 2004

The dependency pair approach is one of the most powerful techniques for automated (innermost) termination proofs of term rewrite systems (TRSs). For any TRS, it generates inequality constraints that have to be satisfied by well-founded orders. However, proving innermost termination is considerably easier than termination, since the constraints for innermost termination are a subset of those for termination. We show that surprisingly, the dependency pair approach for termination can be improved by only generating the same constraints as for innermost termination. In other words, proving full termination becomes virtually as easy as proving innermost termination. Our results are based on splitting the termination proof into several modular independent subproofs. We implemented our contributions in the automated termination prover AProVE and evaluated them on large collections of examples. These experiments show that our improvements increase the power and efficiency of automated termination proving substantially. ⋆

Decidability of Innermost Termination for Semi-Constructor Term Rewriting Systems

Yi and Sakai [7] showed that the termination problem is decidable for a daae of semiconstructor term rewriting systems, which is a superclepsilonssupercl\epsilon\ ssuperclepsilons of the class of right ground term rewriting systems. The decidability w&8 8hown8hown8hown by the fact that every non-terminating TRS in the cl&ae has aloop. In this paper we modify the prinftyfpr\infty fprinftyf of [7] to show that innermost termination is decidable for the class of semi-constnictor TRSs.

Automated termination proofs with AProVE

2004

We describe the system AProVE, an automated prover to verify (innermost) termination of term rewrite systems (TRSs). For this system, we have developed and implemented efficient algorithms based on classical simplification orders, dependency pairs, and the size-change principle. In particular, it contains many new improvements of the dependency pair approach that make automated termination proving more powerful and efficient. In AProVE, termination proofs can be performed with a user-friendly graphical interface and the system is currently among the most powerful termination provers available.

Some undecidable termination problems for semi-Thue systems

Lecture Notes in Computer Science, 1993

We show that the uniform termination problem is undecidable for length-preserving semi-Thue systems having 9 rules. We then give an explicit uniformly terminating semi-Thue system J having 9 rules which is "universal with respect to termination problems" in some sense. It follows that there exists a fixed rule (Uo,Vo) such that 3-w ((Uo, Vo) } has 10 rules and undecidable termination problem.