Abstract. We describe a new approach to proving termination with size change graphs. This is the first decision procedure for size change termination (SCT) which makes direct use of global ranking functions. It handles a well-defined and significant subset of SCT instances, designed to be amenable to a SAT-based solution. We have implemented the approach using a state-of-the-art Boolean satisfaction solver. Experimentation indicates that the approach is a viable alternative to the complete SCT decision procedure based on closure computation and local ranking functions. Our approach has the extra benefit of producing an explicit witness to prove termination in the form of a global ranking function. 1

Abstract. We present a simple method to formally prove termination of recursive functions by searching for lexicographic combinations of size measures. Despite its simplicity, the method turns out to be powerful enough to solve a large majority of termination problems encountered in daily theorem proving practice. 1

Abstract. Size-Change Termination (SCT) is a method of proving program termination based on the impossibility of infinite descent. To this end we use a program abstraction in which transitions are described by monotonicity constraints over (abstract) variables. When only constraints of the form x> y ′ and x ≥ y ′ are allowed, we have size-change graphs. In the last decade, both theory and practice have evolved significantly in this restricted framework. The crucial underlying assumption of most of the past work is that the domain of the variables is well-founded. In a recent paper I showed how to extend and adapt some theory from the domain of size-change graphs to general monotonicity constraints, thus complementing previous work, but remaining in the realm of well-founded domains. However, monotonicity constraints are, interestingly, capable of proving termination also in the integer domain, which is not well-founded. The purpose of this paper is to explore the application of monotonicity constraints in this domain. We lay the necessary theoretical foundation, and present precise decision procedures for termination; finally, we provide a procedure to construct explicit global ranking functions from monotonicity constraints in singlyexponential time, and of optimal worst-case size and dimension (ordinal). 1.

Abstract. Size-Change Termination is an increasingly-popular technique for verifying program termination. These termination proofs are deduced from an abstract representation of the program in the form of size-change graphs. We present algorithms that, for certain classes of size-change graphs, deduce a global ranking function: an expression that ranks program states, and decreases on every transition. A ranking function serves as a witness for a termination proof, and is therefore interesting for program certification. The particular form of the ranking expressions that represent SCT termination proofs sheds light on the scope of the proof method. The complexity of the expressions is also interesting, both practicaly and theoretically. While deducing ranking functions from size-change graphs has already been shown possible, the constructions in this paper are simpler and more transparent than previously known. They improve the upper bound on the size of the ranking expression from triply exponential down to singly exponential (for certain classes of instances). We claim that this result is, in some sense, optimal. To this end, we introduce a framework for lower bounds on the complexity of ranking expressions and prove exponential lower bounds. 1.

We develop a Coq formalization of the matrix interpretation method, which is a recently developed, powerful approach to proving termination of term rewriting. Our formalization is a contribution to the CoLoR project and allows to automatically certify matrix interpretation proofs produced by tools for proving termination. Thanks to this development the combination of CoLoR and our

Abstract. We show how to automate termination proofs for recursive functions in (a first-order subset of) Isabelle/HOL by encoding them as term rewrite systems and invoking an external termination prover. Our link to the external prover includes full proof reconstruction, where all necessary properties are derived inside Isabelle/HOL without oracles. Apart from the certification of the imported proof, the main challenge is the formal reduction of the proof obligation produced by Isabelle/HOL to the termination of the corresponding term rewrite system. We automate this reduction via suitable tactics which we added to the IsaFoR library. 1

Abstract. 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. 1

Abstract. The size-change abstraction (SCA) is an important program abstraction for termination analysis, which has been successfully implemented in many tools for functional and logic programs. In this paper, we demonstrate that SCA is also a highly effective abstract domain for the bound analysis of imperative programs. We have implemented a bound analysis tool based on SCA for imperative programs. We abstract programs in a pathwise and context dependent manner, which enables our tool to analyze real-world programs effectively. Our work shows that SCA captures many of the essential ideas of previous termination and bound analysis and goes beyond in a conceptually simpler framework. 1

Abstract. We introduce the All-Termination(T) problem: given a termination solver T and a collection of functions F, find every subset of the formal parameters to F whose consideration is sufficient to show, using T, that F terminates. An important and motivating application is enhancing theorem proving systems by constructing the set of strongest induction schemes for F, modulo T. These schemes can be derived from the set of termination cores, the minimal sets returned by All-Termination(T), without any reference to an explicit measure function. We study the All-Termination(T) problem as applied to the size-change termination analysis (SCT), a PSpace-complete problem that underlies many termination solvers. Surprisingly, we show that All-Termination(SCT) is also PSpace-complete, even though it substantially generalizes SCT. We develop a practical algorithm for All-Termination(SCT), and show experimentally that on the ACL2 regression suite (whose size is over 100MB) our algorithm generates stronger induction schemes on 90 % of multiargument functions. 1

Abstract. We recently introduced the All-Termination(T) problem: given a termination solver T and a function F, find every subset of the formal parameters to F whose consideration is sufficient to show, using T, that F terminates. These subsets can be harnessed by a theorem prover to locate and justify induction schemes, and are also useful for guiding rewriting heuristics and ensuring their termination. In this paper, we study the All-Termination problem for SCP (polynomial size-change analysis), a powerful, cubic-time termination analysis. SCP is the first nonmonotonic termination analysis studied in the context of All-Termination, making its analysis both challenging and informative. We develop an algorithm for solving the All-Termination(SCP) problem, and briefly report on initial experimental results obtained on the ACL2 regression suite. 1