Ground Confluence Prover based on Rewriting Induction

Ground confluence of term rewriting systems guarantees that all ground terms are confluent. Recently, interests in proving confluence of term rewriting systems automatically has grown, and confluence provers have been developed. But they mainly focus on confluence and not ground confluence. In fact, little interest has been paid to developing tools for proving ground confluence automatically. We report an implementation of a ground confluence prover based on rewriting induction, which is a method originally developed for proving inductive theorems

Improving Rewriting Induction Approach for Proving Ground Confluence

In (Aoto&Toyama, FSCD 2016), a method to prove ground confluence of many-sorted term rewriting systems based on rewriting induction is given. In this paper, we give several methods that add wider flexibility to the rewriting induction approach for proving ground confluence. Firstly, we give a method to deal with the case in which suitable rules are not presented in the input system. Our idea is to construct additional rewrite rules that supplement or replace existing rules in order to obtain a set of rules that is adequate for applying rewriting induction. Secondly, we give a method to deal with non-orientable constructor rules. This is accomplished by extending the inference system of rewriting induction and giving a sufficient criterion for the correctness of the system. Thirdly, we give a method to deal with disproving ground confluence. The presented methods are implemented in our ground confluence prover AGCP and experiments are reported. Our experiments reveal the presented methods are effective to deal with problems for which state-of-the-art ground confluence provers can not handle

Deciding Confluence and Normal Form Properties of Ground Term Rewrite Systems Efficiently

It is known that the first-order theory of rewriting is decidable for ground term rewrite systems, but the general technique uses tree automata and often takes exponential time. For many properties, including confluence (CR), uniqueness of normal forms with respect to reductions (UNR) and with respect to conversions (UNC), polynomial time decision procedures are known for ground term rewrite systems. However, this is not the case for the normal form property (NFP). In this work, we present a cubic time algorithm for NFP, an almost cubic time algorithm for UNR, and an almost linear time algorithm for UNC, improving previous bounds. We also present a cubic time algorithm for CR

Proving Ground Confluence of Equational Specifications Modulo Axioms

Terminating functional programs should be deterministic, i.e., should evaluate to a unique result, regardless of the evaluation order. For equational functional programs such determinism is exactly captured by the ground confluence property. For terminating equations this is equivalent to ground local confluence, which follows from local confluence. Checking local confluence by computing critical pairs is the standard way to check ground confluence. The problem is that some perfectly reasonable equational programs are not locally confluent and it can be very hard or even impossible to make them so by adding more equations. We propose a three-step strategy to prove that an equational program as is is ground confluent: First: apply the strategy proposed in [8] to use non-joinable critical pairs as completion hints to either achieve local confluence or reduce the number of critical pairs. Second: use the inductive inference system proposed in this paper to prove the remaining critical pairs ground joinable. Third: to show ground confluence of the original specification, prove also ground joinable the equations added. These methods apply to order-sorted and possibly conditional equational programs modulo axioms such as, e.g., Maude functional modules.This work has been partially supported by NRL under contract number N00173-17-1-G002.Ope

Interactions between groundwater and surface water at river banks and the confluence of rivers

Riparian vegetation depends on hydrological resources and has to adapt to changes in water levels and soil moisture conditions. The origin and mixing of water in the streamside corridor were studied in detail. The development of riparian woodland often reflects the evolution of hydrological events. River water levels and topography are certainly the main causes of the exchange between groundwater and river water through the riverbank. Stable isotopes, such as 18O, are useful tools that allow water movement to be traced. Two main water sources are typically present: (i) river water, depleted of heavy isotopes, originating upstream, and (ii) groundwater, which comes mainly from the local rainfall. On the Garonne River bank field site downstream of Toulouse, the mixing of these two waters is variable, and depends mainly on the river level and the geographical position. The output of the groundwater into the river water is not diffuse on a large scale, but localised at few places. At the confluence of two rivers, the water-mixing area is more complex because of the presence of a third source of water. In this situation, groundwater supports the hydrologic pressure of both rivers until they merge, this pressure could influence its outflow. Two cases will be presented. The first is the confluence of the Garonne and the Ariège Rivers in the south-west of France, both rivers coming from the slopes of the Pyrénées mountains. Localised groundwater outputs have been detected about 200 m before the confluence. The second case presented is the confluence of the Ganges and the Yamuna Rivers in the north of India, downstream of the city of Allahabad. These rivers are the two main tributaries of the Ganges, and both originate in the Himalayas. A strong stream of groundwater output was measured at the point of confluence