# Confluence Competition 2019

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## Abstract

We report on the 2019 edition of the Confluence Competition, a competition of software tools that aim to prove or disprove confluence and related (undecidable) properties of rewrite systems automatically.

## Keywords

Confluence Term rewriting Automation## 1 Introduction

The Confluence Competition (CoCo)^{1} is an annual competition of software tools that aim to prove or disprove confluence and related (undecidable) properties of a variety of rewrite formalisms automatically. Initiated in 2012, CoCo runs live in a single slot at a conference or workshop and is executed on the cross-community competition platform StarExec [1]. For each category, 100 suitable problems are randomly selected from the online database of confluence problems (COPS). Participating tools must answer YES or NO within 60 s, followed by a justification that is understandable by a human expert; any other output signals that the tool could not determine the status of the problem. CoCo 2019 features new categories on commutation, infeasibility problems, and confluence of string rewrite systems.

Confluence provides a general notion of determinism and has been conceived as one of the central properties of rewriting. A rewrite system \(\mathcal {R}\) is a set of directed equations, so called rewrite rules, which induces a rewrite relation \(\rightarrow _\mathcal {R}\) on terms. We provide a simple example.

### Example 1

*s*matches the left-hand side of the rule \(\mathsf {s}(x) + y \rightarrow \mathsf {s}(x + y)\), with matching substitution \(\{ x \mapsto \mathsf {0}, y \mapsto \mathsf {s}(\mathsf {s}(\mathsf {0})) \}\). Hence the subterm can be replaced by \(\mathsf {s}(\mathsf {0} + \mathsf {s}(\mathsf {s}(\mathsf {0})))\). It follows that

*s*rewrites (in a single step) to the term \(t = \mathsf {s}(\mathsf {s}(\mathsf {0})) \times \mathsf {s}(\mathsf {0} + \mathsf {s}(\mathsf {s}(\mathsf {0})))\). Continuing this process from

*t*eventually results in the term \(\mathsf {s}(\mathsf {s}(\mathsf {s}(\mathsf {s}(\mathsf {s}(\mathsf {s}(\mathsf {0}))))))\). This term cannot be simplified further. Such terms are called normal forms.

*s*. The choice does not matter since all maximal rewrite sequences terminate in the same normal form, which is readily checked. This property not only holds for the term

*s*, but for all terms that can be constructed from the symbols in the rules.

*Confluence*is the property that guarantees this. A rewrite system \(\mathcal {R}\) is confluent if the inclusion Open image in new window holds. Here \(\rightarrow _\mathcal {R}^*\) denotes the transitive reflexive closure of the one-step rewrite relation \(\rightarrow _\mathcal {R}^{ {*}}\), Open image in new window denotes the inverse of \(\rightarrow _\mathcal {R}^*\), and \(\cdot \) denotes relational composition. A more graphical definition of confluence is presented in Fig. 1. The precise notions of rewrite rules, associated rewrite steps, and terms to be rewritten vary from formalism to formalism.

## 2 Categories

**TRS/CPF-TRS**The two original categories are about confluence of*first-order term rewriting*. CPF-TRS is a category for*certified*confluence proofs, where participating tools must generate certificates that are checked by an independent certifier.**CTRS/CPF-CTRS**These two categories, introduced respectively in 2014 and 2015, are concerned (certified) confluence of*conditional term rewriting*, a formalism in which rewrite rules come equipped with conditions that are evaluated recursively using the rewrite relation.**HRS**This category, introduced in 2015, deals with confluence of*higher-order rewriting*, i.e., rewriting with binders and functional variables.**GCR**This category is about*ground*confluence of*many-sorted*term rewrite systems and was also introduced in 2015.**NFP/UNC/UNR**These three categories, introduced in 2016, are about properties of first-order term rewrite systems related to unique normal forms, namely,*the normal form property*(NFP),*unique normal forms with respect to conversion*(UNC), and*unique normal forms with respect to reduction*(UNR).**COM**This new category is about*commutation*of first-order rewrite systems.**INF**This new category is about*infeasibility*problems.**SRS**This new category is concerned with confluence of*string rewriting*.

The new categories are described in detail in Sect. 5. Descriptions of the other categories can be found in the CoCo 2015 [2] and 2018 [3] reports, and on the CoCo website (see Footnote 1). The underlying problem format is the topic of the next section.

## 3 Confluence Problems

Tools participating in CoCo are given problems from the database of *confluence problems* (COPS)^{2} in a format suitable for the category in which the tools participate. Besides commutation and infeasibility problems, which are described in Sect. 5, four different formats are supported: TRS, CTRS, MSTRS, and HRS. As these formats were simplified recently, we present the official syntax below in four subsections.

In addition to the format, *tags* are used to determine suitable problems for CoCo categories. For instance, for the CTRS category, selected problems must have the 3_ctrs and oriented tags. Such tags are automatically computed when problems are submitted to COPS. Detailed information on COPS, including a description of the tagging mechanism, can be found in [4].

### 3.1 TRS Format

The format for first-order rewrite systems comes in two versions: a basic version and an extended version. The latter contains an additional signature declaration which is used to define function symbols that do not appear in the rewrite rules.

^{3}according to the following grammar: Here Open image in new window is any sequence of characters and Open image in new window is any nonempty sequence of characters not containing whitespace, the characters Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , Open image in new window , and the sequences Open image in new window , Open image in new window , Open image in new window , Open image in new window , and Open image in new window . In Open image in new window the variables of the TRS are declared. If this is missing, the TRS is ground. Symbols ( Open image in new window ) appearing in the Open image in new window declaration that were not declared as variables are function symbols. If they appear multiple times, they must be used with the same number (arity) of arguments. Here is an example of the basic format, COPS #1: In the extended format, a signature declaration specifying the set of function symbols and their arities is added. In this format, every symbol appearing in the rules must be declared as a function symbol or a variable. Formally, the Open image in new window declaration in the basic format is replaced by where Open image in new window is a nonempty sequence of digits. An example of the extended format is provided by COPS #557:

### 3.2 CTRS Format

### 3.3 MSTRS Format

### 3.4 HRS Format

## 4 Competition

Since 2012 a total of 17 tools participated in CoCo. Many of the tools participated in multiple categories. The proceedings of the International Workshop on Confluence^{4} contain (short) descriptions of the contenders. For each category, 100 problems are randomly selected from COPS. Problem selection for CoCo 2019 is subject to the following constraints. For the TRS, CPF-TRS, NFP, UNC, and UNR categories, problems in TRS format are selected. The problems for the SRS category are further restricted to those having the srs tag. For the CTRS and CPF-CTRS categories, problems must be in CTRS format and have the tags 3_ctrs and oriented, since participating tools handle only *oriented* CTRSs of *type 3*. In an oriented CTRS the conditions in the rules are interpreted as reachability and type 3 is a syntactic restriction on the distribution of variables in rewrite rules which ensure that rewriting does not introduce fresh variables [6]. For the GCR category, eligible problems must be in TRS or MSTRS format. Being in HRS format is a prerequisite for problems to be selected for the HRS category. For the new COM and INF categories, problems must have the commutation and infeasibility tags, respectively. The respective formats are described in the next section. New in 2019 is the possibility for tool authors to submit *secret* problems just before the competition. These will be included in the selected problems.

Earlier editions of CoCo only considered problems stemming from the *literature*. This restriction was put in place to avoid bias towards one particular tool or technique. Since both COPS and CoCo have grown and diversified considerably since their inception, this restriction has become hard to maintain in a meaningful way, while at the same time losing its importance. Consequently it has been dropped for CoCo 2019. Further selection details are available from the CoCo website.

*LiveView*[4] tool which continuously polls new results from StarExec while the competition is running. A screenshot of the LiveView of CoCo 2018 is shown in Fig. 2. New is the realtime display of YES/NO conflicts. Since all categories deal with undecidable problems, and developing software tools is error-prone, conflicts appear once a while. In the past they were identified after the live competition finished, now action by the SC can be taken before winners are announced. As can be seen from the screenshot, in last year’s competition there was a YES/NO conflict in the HRS category, which led to lively discussion about the semantics of the HRS format. After each competition, the results are made available from the results page.

^{5}

The certification categories (CPF-TRS and CPF-CTRS) are there to ensure that tools produce correct answers. In these categories tools have to produce certified (non-)confluence proofs with their answers. The predominant approach to achieve this uses a combination of confluence prover and independent certifier. First the confluence prover analyses confluence as usual, restricting itself to criteria supported by the certifier. If it is successful the prover prints its proof in the certification problem format (CPF),^{6} which is then checked by the certifier. To ensure correctness of this check, soundness of the certifier is mechanized in a proof assistant like Isabelle/HOL. So far only one certifier has participated in CoCo:
Open image in new window
.^{7}

## 5 New Categories in 2019

### 5.1 Commutation

TRSs \(\mathcal {R}\) and \(\mathcal {S}\) *commute* if the inclusion Open image in new window holds. Commutation is an important generalization of confluence: Apart from direct applications in rewriting, e.g. for confluence,^{8} standardization, normalization, and relative termination, commutation is the basis of many results in computer science, like correctness of program transformations [7], and bisimulation up-to [8].

Currently, commutation is supported by the tools CoLL [9] and FORT [10]. The former supports commutation versions of three established confluence techniques: development closedness [11], rule labeling [12], and an adaption of a confluence modulo result by Jouannaud and Kirchner [13]. The latter is a decision tool for the first-order theory of rewriting based on tree automata techniques, but restricted to left-linear right-ground TRSs.

Commutation problems consist of two TRSs \(\mathcal {R}\) and \(\mathcal {S}\). The question to be answered is whether these commute. To ensure compatibility of the signatures of the involved TRSs, we rename function symbols and variables in \(\mathcal {S}\) on demand. Before we describe this precisely, we give an example of a commutation problem that illustrates the problem.

### 5.2 Infeasibility Problems

Infeasibility problems originate from different sources. Critical pairs in a conditional rewrite system are equipped with conditions. If no satisfying substitution for the variables in the conditions exists, the critical pair is harmless and can be ignored when analyzing confluence of the rewrite system in question. In this case the critical pair is said to be *infeasible* [14, Definition 7.1.8]. Sufficient conditions for infeasibility of conditional critical pairs are reported in [15, 16].

Another source of infeasibility problems is the dependency graph in termination analysis of rewrite systems [17]. An edge from dependency pair \(\ell _1 \rightarrow r_1\) to dependency pair \(\ell _2 \rightarrow r_2\) exists in the dependency graph if two substitutions \(\sigma \) and \(\tau \) can be found such that \(r_1\sigma \) rewrites to \(\ell _2\tau \). (By renaming the variables in the dependency pairs apart, a single substitution suffices.) If no substitutions exists, there is no edge, which may ease the task of proving termination of the underlying rewrite system [18, 19].

### 5.3 String Rewriting

*x*is a variable. A concrete example (COPS #442) is given below: The correct answer of this problem is Open image in new window since the addition of the redundant rules [20] Open image in new window and Open image in new window makes the critical pairs of the SRS development closed [11].

The SRS category has been established to stimulate further research on confluence of string rewriting. In the Termination Competition^{9} there is an active community developing powerful techniques for (relative) termination of SRSs. We anticipate that these are beneficial when applied to confluence analysis.

## 6 Outlook

In the near future we plan to merge CoCo with COPS and CoCoWeb,^{10} a convenient web interface to execute the tools that participate in CoCo without local installation, to achieve a single entry point for confluence problems, tools, and competitions. Moreover, the submission interface of COPS will be extended with functionality to support submitters of new problems as well as the CoCo SC. We anticipate that in the years ahead new categories will be added to CoCo. Natural candidates are rewriting modulo AC and nominal rewriting.

## Footnotes

## Notes

### Acknowledgments

We are grateful to Nao Hirokawa for continuous support for the infrastructure of CoCo. Fabian Mitterwallner contributed to the inlining and renaming tools for the new commutation and infeasibility categories. Raúl Gutiérrez, Naoki Nishida, and Salvador Lucas contributed the initial set of infeasibility problems (COPS #818 – #936). Johannes Waldmann contributed challenging SRS problems (COPS #987 – #1036). Finally, we acknowledge the TOOLympics 2019 initiators for giving us the opportunity to present CoCo 2019.

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