Nicolas Robin
email: robin.nicolas@hotmail.fr

Laurent Dominique
email: laurent-dom@hotmail.fr

Mental Imagery and Tennis: A Review, Applied Recommendations and New Research Directions

Keywords: Motor imagery, performance, review, tennis, training

Mental Imagery (MI), which can be defined as the brain's ability to recreate motor experiences in the absence of actual actions, is a mental strategy commonly used by tennis players and coaches in the context of learning and performance. The purpose of this review was to examine the MI use, interventions and effects on the performance of tennis players.

Preferred reporting items for reviews and meta-analyses guidelines were employed and research studies were collected via SCOPUS, PubMed, PsycINFO, Science Direct and Google Scholar, in English and French, from the earliest record up to August 2021. Forty-one studies met the inclusion criteria. This review (number 2020-05-87) resulted in a wide variety of MI use in tennis and its beneficial effects, whatever the level or age of the players. MI interventions generally improve motor performance, motivational or affective outcomes. Moreover, MI is frequently combined with other mental skills and integrated in preperformance routines including breathing, positive self-talk, and/or concentration. This study also highlighted the weak representation of female tennis players that's why future research is needed to better explore the gender effect in MI interventions in tennis. Overall, MI appears to be beneficial for players, whatever their age or level, and should be incorporated in training programs. Practical implications are discussed.

Imagerie mentale et tennis : Revue de littérature, recommandations appliquées et nouvelles orientations de recherche Résumé L'imagerie mentale (IM), définie comme la capacité du cerveau à recréer des expériences motrices en l'absence d'actions réelles, est une stratégie mentale communément utilisée par les joueurs de tennis et les entraîneurs dans le cadre de l'apprentissage et de la performance. Le but de cette revue était d'examiner l'utilisation de l'IM et ses effets sur la performance des joueurs de tennis. La méthode PRISMA a été utilisée et les travaux de recherche ont été collectés via SCOPUS, PubMed, PsycINFO, Science Direct et Google Scholar, en anglais et en français, jusqu'en août 2021. Quarante-et-une études ont respecté les critères d'inclusion.

Cette revue a révélé une grande variété d'utilisation de l'IM au tennis et des effets bénéfiques, quel que soit le niveau ou l'âge des joueurs sur les performances motrices, les affects et la motivation. De plus, l'IM est fréquemment associée à d'autres habiletés mentales et intégrée dans les routines de pré-performance. Cette étude a mis en évidence la faible représentation des joueuses de tennis, des recherches futures sont donc nécessaires pour explorer s'il existe un effet du genre en lien avec l'IM. Dans l'ensemble, l'IM semble être bénéfique pour les joueurs, indépendamment de leur âge ou de leur niveau et devrait être intégrée dans les programmes d'entraînement. Des recommandations pratiques sont proposées.

Mots clés: Entrainement ; imagerie motrice ; performance ; revue de littérature ; tennis.

Mental Imagery and Tennis: A Review, Applied Recommendations and New Research

Directions

Introduction

Tennis, which is one of the racket sports that involves hitting a mobile like a ball [START_REF] Lees | Science and the major racket sports: A review[END_REF], requires the development and use of technical, physiological, tactical, mental skills and cognitive processes [START_REF] Cece | Mental training program in racket sports: A systematic review[END_REF]. For example, researchers highlighted the importance of attention control and focus in tennis (e.g., [START_REF] Guillot | Motor imagery and tennis serve performance: The external focus efficacy[END_REF][START_REF] Schmid | Strategies for training concentration[END_REF]. [START_REF] Akpinar | Coincidence-anticipation timing requirements are different in racket sports[END_REF] added that this sport requires coincidence-anticipation, accuracy in every stroke and requires a significant amount of practice, learning sessions and repetitions, generally spread over several years and of which the achievement and quality can be facilitated by the use of mental techniques such as mental imagery [START_REF] Cece | Mental training program in racket sports: A systematic review[END_REF]. The aim of the current study was to review the studies that concerned Mental Imagery (MI) specifically in tennis in order to provide consistency from data and to identify the main findings concerning the MI effects, uses, outcomes and characteristics in order to give information, precisions or applied recommendations to coaches and practitioners and also to identify limitations and new research directions.

Tennis and Psychological Components

Tennis players need to invest psychological resources (e.g., motivation), by notably using MI [START_REF] Dana | Internal and external imagery effects on tennis skills among novices[END_REF], in order to improve learning, to progress, to continue participation and to obtain long-term performance [START_REF] Crespo | Motivation in tennis[END_REF]. In addition, because of dual competition and the need to continue playing despite of under-performance, errors or defeats, [START_REF] Covassin | The relationship between self-confidence, mood state, and anxiety among collegiate tennis players[END_REF] identified self-confidence as a salient factor in tennis. Moreover, anxiety control [START_REF] Bolgar | Trait anger, appraisal, and coping differences among adolescent tennis players[END_REF] and the emotional regulation of control [START_REF] Kosslyn | The role of trait emotional intelligence in emotion regulation and performance under pressure[END_REF] have also been considered as really important mental skills to take into account, especially before and during tennis competitions. Indeed [START_REF] Jackson | Flow in sports[END_REF] argued that athletes competing at their best characterized their ideal state of performance as feeling confident and in control; and conceptualised these optimal experiences as flow state including pleasure, concentration or experiencing a balance between the challenges and one's ability levels. Authors found that awareness and flow was related to tennis performance and that a MI intervention increased flow state and performance in tennis competition [START_REF] Koehn | Correlates of dispositional and state flow in tennis competition[END_REF], and that the use of this mental technique was particularly useful in tennis [START_REF] Cece | Mental training program in racket sports: A systematic review[END_REF]. For example, during a match, tennis players perform an intermittent activity, consisting of playing time interspersed with frequent rest times between points and games during which they can use MI to notably improve their accuracy, strategy, concentration or emotional control [START_REF] Dohme | Development, implementation, and evaluation of an athlete-informed mental skills training program for elite youth tennis players[END_REF][START_REF] Mathers | Professional tennis on the ATP tour: A case study of mental skills support[END_REF]. That's why we reviewed the use and the potential beneficial effect of MI to improve learning, performance and motivational or affective outcomes, and to give practical suggestions, to tennis players and coaches, in order to incorporate this mental technique in training programs as suggested by Di [START_REF] Di Corrado | Mental Imagery Skills in Competitive Young Athletes and Non-athletes[END_REF].

Imagery Definition

According to [START_REF] Kosslyn | The role of trait emotional intelligence in emotion regulation and performance under pressure[END_REF], imagining places, sounds and actions in the absence of appropriate stimuli for the relevant perception is done through MI. [START_REF] White | An in-depth analysis of the uses of imagery by high-level slalom canoeists and artistic gymnasts[END_REF] evoked that athletes use MI in various ways including analysing past performance, reducing anxiety, improving self confidence, focus, quality of training or mental rehearsing of match strategies or motor actions specifically called motor imagery. Motor imagery can be defined as a dynamic state during which the representation of a specific motor action is reactivated in the brain without any over movements [START_REF] Decety | Mentally simulated movements in virtual reality: does Fitts's law hold in motor imagery?[END_REF].

Finally, MI is considered a mental technique that uses all senses to re-create or create an experience in the brain and programs the mind and body to respond optimally (Williams et al., 2012).

Imagery Outcomes

MI is commonly used by tennis players and coaches [START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF][START_REF] Lindsay | The effect of mental imagery on skill performance in sport: A systematic review[END_REF], on its own and usually implicitly or as a part of a mental training program combined with other techniques or mental skills such as relaxation, breathing, self-talk, arousal regulation or concentration [START_REF] Cece | Mental training program in racket sports: A systematic review[END_REF]. MI is generally associated with a wide range of positives outcomes in the context of tennis such as motor learning (e.g., [START_REF] Cherappurath | Combined effects of SAQ and PETTLEP imagery training: A study on the learning process of new skills among novice tennis players[END_REF][START_REF] Guillot | Motor imagery and 'placebo-racket effects' in tennis serve performance[END_REF], motor performance (e.g., Cherappurath et al., 2020a;[START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF]Fekih et al., 2020b), regulation andemotional control (e.g., Dohm et al., 2019), anxiety (e.g., [START_REF] Mousavi | The effect of mental imagery upon the reduction of athletes anxiety during sport performance[END_REF] or self-confidence (e.g., [START_REF] Mamassis | The effects of a mental training program on juniors precompetitive anxiety, self-confidence, and tennis performance[END_REF].

Imagery Function

Authors evoked that MI can affect motor behaviour through both cognitive and motivational mechanisms operating at a general or specific level and identified five subcategories of MI practice also called function [START_REF] Hall | Imagery use by athletes: development of the Sport Imagery Questionnaire[END_REF][START_REF] Paivio | Cognitive and motivational functions of imagery in human performance[END_REF]. Cognitive Specific (CS) imagery involves the rehearsal of specific sport skills (e.g., imagining yourself performing a flat serve down the middle) whereas the Cognitive General (CG) imagery, focuses on playing strategies, routines and tactical points that players can use during matches (e.g., imagining performing a serve-volley and winning with a drop shot). Motivational Specific (MS) imagery (e.g., seeing your self winning a match) entails imaging specific goal and the ways to achieve it, influencing intrinsic motivation; Motivational General-Arousal (MG-A) imagery (e.g., imagining the anxiety associated with significant competition) is associated with stress, anxiety and arousal regulation, and Motivational General-Mastery (MG-M) imagery is associated with self-confidence, control and beliefs over an individual's abilities (e.g., seing yourself coping under challenging situations like after losing a set).

Imagery Modality

Regardless of the type (i.e., subcategories) of MI practice, many authors pointed out that MI modalities and perspectives are very important factors to consider [START_REF] Cumming | The cognitive and motivational effects of imagery training: A matter of perspective[END_REF]Robin et al., 2020). Indeed, each MI modality can have differentiated impacts on motor performance [START_REF] Hardy | Efficacy of external and internal visual imagery perspectives for the enhan-cement of performance on tasks in which form is important[END_REF]. The kinesthetic imagery (KI) modality is defined as what one feels during an action and includes sensations such as force, relaxation and effort involved in movement [START_REF] Callow | The effect of kinaesthetic imagery on the sport confidence of flat-race jockeys[END_REF]. Visual imagery modality consists to see an object, an action or a scene. It can be done from an external perspective (i.e., the athlete "views him or herself from another person's perspective") or an internal perspective (i.e., an individual "views the image through their own eyes") (Williams et al., 2012, p. 631).

Given the existence of different imagery modalities and perspectives, clear and precise instructions must be given to participants before MI intervention [START_REF] Callow | Performance improvements from imagery: Evidence that internal visual imagery is superior to external visual imagery for slalom performance[END_REF][START_REF] Hall | Imagery and the acquisition of motor skills[END_REF]Robin et al., 2020). Finally, it is important to note that the effectiveness of the MI intervention may depend on the athlete's capacity to generate mental images [START_REF] Seiler | Biological evidence of imagery abilities: Intraindividual differences[END_REF].

Imagery Ability

MI ability, which can be defined as an individual's capability to form vivid, controllable images and retain them for sufficient time to effect the desired rehearsal appears as a variable moderating the effects of imagery on sport outcomes [START_REF] Martin | Imagery use in sport: A literature review and applied model[END_REF].

Indeed, studies showed that individuals with higher imagery abilities (i.e., good imagers) achieved higher levels of performance than participants with lower imagery abilities (i.e., poor imagers) after MI intervention [START_REF] Goss | Imagery ability and the acquisition and retention of movements[END_REF][START_REF] Robin | Effects of motor imagery training on service return accuracy in tennis: The role of imagery ability[END_REF]. Therefore, [START_REF] Cumming | Imagery interventions in sport[END_REF] suggested that it seems important to screen athletes for MI abilities before and during MI practice.

Aim and Objectives

According to [START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF], the mental aspect of performance in tennis represents a central preoccupation for players and coaches. [START_REF] Cece | Mental training program in racket sports: A systematic review[END_REF], who reviewed the use of mental training program in racket sports, recently showed that the most frequently used mental skill was MI. However, the use of this mental technic requires certain knowledge, know-how and precautions on the part of the users and to our knowledge no studies has so far synthetized the literature to make an inventory of the characteristics, effects and uses in the field of MI, in particular in tennis. Therefore the main aim of the current study was to provide a synthesis of the researches focused on both tennis and MI. The first objective was to list and collect the researches that concerned MI uses and interventions in healthy non-expert (i.e., novice and intermediate levels) and expert tennis players, regardless of their age (i.e., child, teenager and adult), in particular to verify if this mental technique can be beneficial whatever the age or the level of the players, and to provide an organized synthesis of these studies in order to better understand the effects of MI interventions on outcomes such as learning, performances or affective and motivational psychological factors (e.g., motivation, anxiety, self-confidence) and to give arguments to use it. The second objective was to propose practical recommendations to the players, coaches, researchers or even parents who wish to use imagery in the context of tennis. Finally, the third objective was to suggest potential further investigations concerning MI and tennis.

Materials and Methods

The present review was performed in accordance with Preferred Reporting Items for Systematic Reviews and Meta-analyses (PRISMA) guidelines [START_REF] Moher | Preferred reporting items for systematic reviews and meta-analyses: the PRISMA statement[END_REF]. The protocol was written before conducting the review. The literature search procedure, including the systematic approach of identification, screening, eligibility, and inclusion, is detailed by using a PRISMA flow diagram (Figure 1). The present review was obtained through electronic journal and manual searches, from the earliest record up to August 2021, in SCOPUS, PubMed, PsycINFO, Science Direct and Google Scholar databases. The searches, restricted to English and French language publications, were carried out using the following Boolean phrases: 'mental imagery' OR 'motor imagery' OR 'mental rehearsal' AND 'tennis'.

The authors (who are both Ph.D in sport science university, doctor in psychology, tennis coaches and mental trainers) screened all the studies, based on the information available in the title and the abstract, meeting the inclusion and exclusion criteria. This study assessed quality and included: (1) original studies, (2) published in scientific peer-reviewed journal, (3) including healthy participants, (4) using mental or motor imagery interventions, (5) in relation to tennis activity. The exclusion criteria were: Publication language other than English or French, MI realized during hypnosis or MI not related to movements or tennis environments. This review was approved by the local ethics committee of the university (registration number 2020-05-87) and was conducted in accordance with the Declaration of Helsinki. The authors independently realized the search and identified records (N = 870) through database searching (N = 847) and from other sources (N = 23). Articles duplicated in two or more databases were identified and removed from this review. Then, the authors screened the records (N = 329) and excluded articles that do not meet the criteria (e.g., irrelevant language or subjects, unhealthy participants, not tennis related). In order to determine the interrater reliability concerning the eligibility of the articles, Cohen's Kappa [START_REF] Mchugh | Interrater reliability: The kappa statistic[END_REF] was calculated and showed a strong level of agreement (Kappa = 0.893) between the reviewers.

Then during meetings, the authors compared and discussed their findings, and a third external reviewer was consulted to resolve any disagreements and avoid bias. The articles assessed for eligibility (N = 115) were independently screened (e.g., title, abstract and key-words) to determine if they met the criteria and could be included in the review. The authors screened the full-text articles assessed for eligibility (N = 58), for bias such as lack of blinding, absence of randomization, lack of study design or disclosure of conflicts of interest. The selected studies (N = 41), found identically by the two authors, were extracted according to references, type of study (e.g., laboratory, ecological condition), MI modalities (e.g., internal visual, external visual, kinesthetic imageries) and types (i.e., cognitive specific, cognitive general, motivational general-arousal, motivational general-mastery, motivational specific) characteristics of the sample (i.e., sample size, gender, mean age, athletic level), experimental groups and duration, outcome measures and main results. The latters were analysed using descriptive statistics with the software package Statistica 12, and the characteristics of the selected studies were summarised (see Table 1).

The evaluation of the quality was realized by using two rating lists because studies with different quantitative designs were included in this review. Randomised controlled trials and non randomized trials were evaluated on the basis on the Physiotherapy Evidence Database guidelines [START_REF] Maher | Reliability of the PEDro scale for rating quality of randomized controlled trials[END_REF] with a maximum of eleven points or eight points respectively. Single cases experimental designs were evaluated by using the Single Case Experimental Design guideline [START_REF] Tate | Rating the methodological quality of single-subject designs and n-of-1 trials: Introducing the Single-Case Experimental Design (SCED) Scale[END_REF] with a maximum of eleven points [START_REF] Schuster | Best practice for motor imagery: A systematic literature review on motor imagery training elements in five different disciplines[END_REF]. Therefore, the studies (N = 41), which were included in the final review and were independently scrutinized by the experimenters, received one point for fulfilled methodological criterion such as random allocation of the participants to the groups, similar pre-intervention levels, blinding of all the participants and assessors, statistical comparisons reported, replication and generalisation on the respective rating list (see Table 1) with the higher the score obtained, the better the quality of the study. The evaluation of the quality, which may help the reader to interpret and to use the results of a study with caution especially for a low score, was performed by the authors (for similar procedure see [START_REF] Schuster | Best practice for motor imagery: A systematic literature review on motor imagery training elements in five different disciplines[END_REF]. A third external reviewer was consulted to resolve any disagreements.

Results

Type of Participants and Sample Size

This review included a total of 1341 tennis players, from 41 studies, including children (12.1%), teenagers (38,7%), and adults (49,2%). An over-representation of males (70.5%) compared to females (29.5%) was highlighted. The participants were novice (N = 227, 16.9%), or had intermediate (N = 402, 29.9%), regional to national (N = 571, 42.6%) or international (N = 71, 5.3%) levels. Only 3 articles were considered as case studies (7.3%) with less than 5 participants (and only 1 for two articles), the majority of the studies had sample size from 10 to 49 participants (N = 32, 78.0%), and 6 studies had 50 or more participants (14.6%).

Outcomes and Tasks

Most of the studies reviewed (N = 31, 75.6%) found beneficial effects of MI interventions, primarily on performance improvements (i.e., accuracy, speed, efficacy, technique of the strokes, global performance, rank and match wins, see Table 2). Among the studies that showed improved performance, this improvement was observed during tests (N = 22, 53.65%), practice matches (N = 11, 26.82%), or real competitions (N = 8, 19.51%). More specifically, during matches, MI was performed only prior to execution (N = 6, 14.63%), only during execution (N = 5, 12.19%) or during both (N = 7, 17.07%). The outcome measures of the studies reviewed mainly focused on quantitative motor performance gains (e.g., accuracy, speed, percentage of success) in service (N = 18, 43.9%), forehand, backhand and volley (N = 17, 41.15%), service return (N = 2, 4.9%) or duration of movement (i.e., isochrony between real and imagined movement; N = 4, 9.8%). In addition, other dependant variables concerned indicated (14.6%) the use of internal visual imagery perspectives.

Imagery Ability

Several studies have used questionnaires or Likert-type scales to measure: MI ability (N = 19, 46.3%), MI use (N = 7, 17.1%), anxiety (N = 5, 12.2%), flow (N = 1, 2.4%), or perceived stress (N = 1, 2.4%). Most studies controlled the MI ability of the participants (N = 18, 43.91%). One study revealed that national level tennis players exhibit better vividness visual MI ability scores than participants of team sports and another study revealed that good imagers had greater performance than poor imagers. Concerning the experimental procedures, twenty-five studies (61.0%) used a pre-post measurement design. The duration was: 1 day for 14 studies (34.1%), more than 4 weeks and 12 sessions for 23 studies (56.1%), and up to 1 year (N = 1, 2.4%) or more (N = 2, 4.8%). The intervention times varied between 10 and 120 minutes per session and between 1 and 3 sessions per week. While only 7 studies (17.1%) were carried out in laboratory, the others were performed in ecological (N = 35, 85.4%) conditions (i.e., on the court).

Imagery and Practice

Concerning

[Table 1 near here]

Discussion

Apart from a recent study that reviewed the effects of mental training programs in racket sports, to our knowledge, no review has specifically investigated the effects of MI in tennis context. Based on 27 studies, including 12 studies that concerned tennis players, [START_REF] Cece | Mental training program in racket sports: A systematic review[END_REF] found that the most frequently used technique in mental training programs was MI, which confirms previous researches (e.g., [START_REF] Daw | Evaluation of a comprehensive psychological skills training program for collegiate tennis players[END_REF][START_REF] Defrancesco | Performance enhancement strategies used in a professional tennis tournament[END_REF]. The current review revealed that MI is effective in tennis by improving learning and performance in different tennis tasks, regardless of participant type, as well as positive influence on motivational and affective factors. However, it is important to specify that the current review revealed that MI is generally combined with other techniques such as: Goal setting (e.g., Mamassis & Doganis, 2014), arousal regulation (e.g., [START_REF] Daw | Evaluation of a comprehensive psychological skills training program for collegiate tennis players[END_REF], positive statement (e.g., [START_REF] Mathers | Professional tennis on the ATP tour: A case study of mental skills support[END_REF], self-talk (e.g., Dohm et al., 2019), relaxation (e.g., [START_REF] Koehn | Correlates of dispositional and state flow in tennis competition[END_REF], breathing (e.g., [START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF][START_REF] Mamassis | Improving serving speed in young tennis players[END_REF], or video (e.g., [START_REF] Atienza | Video modeling and imaging training on performance of tennis service of 9-to 12-year-old children[END_REF][START_REF] De Sousa Fortes | Effect of motor imagery training on tennis service performance in young tennis athletes[END_REF]. The latter combination can be used whatever the age or level of the participants [START_REF] Di Corrado | Mental Imagery Skills in Competitive Young Athletes and Non-athletes[END_REF]. For example, the use of the video of a model can make it easier to activate the mental representation of the action that will be used in MI [START_REF] Mcneill | Kinaesthetic imagery ability moderates the effect of an AO+MI intervention on golf putt performance: A pilot study[END_REF]. This is why it seems recommended to coaches and players to use video of a model before MI, especially for tennis players who have difficulty in imagining actions, and to develop the use of psychological skills whatever the level or age of the participants.

Imagery and Participants

The present review revealed that MI can be used with tennis players of all skill levels. Indeed, while beneficial effects have been observed in novice players (e.g., [START_REF] Dana | Internal and external imagery effects on tennis skills among novices[END_REF][START_REF] Féry | Kinesthetic and visual images in modelling closed motor skills: the example of the tennis serve[END_REF][START_REF] Jung | Effect of imagery tennis training on cerebral activity[END_REF], other studies have also shown improvements in performance in intermediate level (e.g., [START_REF] Cherappurath | PETTLEP imagery and tennis service performance: an applied investigation[END_REF][START_REF] De Sousa Fortes | Effect of motor imagery training on tennis service performance in young tennis athletes[END_REF][START_REF] Morais | Pre-service routines, mental toughness and performance enhancement of young tennis athletes[END_REF]) or experts (e.g., Di Corrado et al., 2019;[START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF][START_REF] Hegazy | The effect of mental training on motor performance of tennis and field hockey strokes in novice players[END_REF] tennis players. These results confirm those of previous studies which have shown that MI interventions can be beneficial regardless of the level of practice of the participants [START_REF] Toth | Does mental practice still enhance performance? A 24 year follow-up and meta-analytic replication and extension[END_REF] and can be used by children and pre-teens [START_REF] Cherappurath | Combined effects of SAQ and PETTLEP imagery training: A study on the learning process of new skills among novice tennis players[END_REF][START_REF] Dereceli | Investigating the imagination levels of students playing tennis[END_REF][START_REF] Di Corrado | Mental Imagery Skills in Competitive Young Athletes and Non-athletes[END_REF]Dohm et al., 2019;[START_REF] Guillot | Motor imagery and tennis serve performance: The external focus efficacy[END_REF] in motor learning context. Therefore, it is suggested that tennis coaches start using MI with beginners and school tennis players, and encourage them to use it as they progress. It is possible that the use of a particular imagery modality (e.g., kinesthetic imagery), which may be difficult to use in young players [START_REF] Martini | Development and validation of the Movement Imagery Questionnaire for Children (MIQ-C)[END_REF] could induce a lack of (short-term) performance improvement, especially among non-experts. That's why authors suggested the use of general MI training, in the targeted imagery type, to make the athletes more aware of the stimulus and response information in their MI [START_REF] Smith | The effect of imagery modality on golf putting performance[END_REF], and to progressively increase the amount of detail and complexity of the MI instructions [START_REF] Calmels | Effects of an imagerytraining programme on selective attention of national softball players[END_REF]. Apart from studies on deliberate imagery use (e.g., [START_REF] Cumming | Deliberate imagery practice: the development of imagery skills in competitive athletes[END_REF], few tennis studies investigated the effects of MI intervention according to skill level. Further research is needed to compare, the potential difference in the effects of imagery modalities (e.g., kinesthetic, internal visual, external visual), in tennis players of different skill levels (i.e., novice, intermediate and experts), performing the same tennis tasks such as service or service return for examples.

Imagery and Outcomes

A recent meta-analysis revealed that MI interventions can enhance motor performance, motivational and affective outcomes (Simonsmeir et al., 2020). These results

were corroborated in the current review, in which it was observed that MI was used to increase intrinsic motivation (e.g., [START_REF] Gabr | Mental imagery and self-talk as approach to cope with pressure among individual sports[END_REF]Whitbread & Newell, 2013), self-confidence (e.g., [START_REF] Coelho | Use of multimodal imagery with precompetitive anxiety and stress of elite tennis players[END_REF][START_REF] Mamassis | The effects of a mental training program on juniors precompetitive anxiety, self-confidence, and tennis performance[END_REF], or emotional control [START_REF] Di Corrado | Imagery ability of elite level athletes from individual vs. team and contact vs. no-contact sports[END_REF]Dohm et al., 2019), and to lower anxiety (e.g., [START_REF] Mamassis | The effects of a mental training program on juniors precompetitive anxiety, self-confidence, and tennis performance[END_REF][START_REF] Mousavi | The effect of mental imagery upon the reduction of athletes anxiety during sport performance[END_REF] or perceived stress [START_REF] Coelho | Use of multimodal imagery with precompetitive anxiety and stress of elite tennis players[END_REF] in tennis players. As suggested by [START_REF] Cumming | Deliberate imagery practice: the development of imagery skills in competitive athletes[END_REF], it seems important for tennis players to realize that MI should not only be used as a strategy to improve motor performance but can have other functions such as cognitive general (e.g., use of different tactical solutions during a match) or motivational specific (e.g., seeing your self winning a medal) for examples, that can be beneficial to progress and perform in match.

Imagery Function

Although participants reported using MI for all five functions: cognitive specific, cognitive general, motivational general-arousal, motivational general-mastery, motivational specific imageries [START_REF] Cumming | Deliberate imagery practice: the development of imagery skills in competitive athletes[END_REF], motivational general-mastery and cognitive specific imageries were the most frequently used by the tennis players as generally observed in the literature [START_REF] Slimani | Effects of mental imagery on muscular strength in healthy and patient participants: A systematic review[END_REF]. Indeed, studies revealed that MI facilitated the learning and increased performance in service (e.g., [START_REF] Blankert | Imagining success: Multiple achievement goals and the effectiveness of imagery[END_REF][START_REF] Cherappurath | PETTLEP imagery and tennis service performance: an applied investigation[END_REF][START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF][START_REF] Fekih | Effects of motor mental imagery training on tennis service performance during the ramadan fasting: a randomized, controlled trial[END_REF][START_REF] Noel | The effect of visuo-motor behavior rehearsal on tennis performance[END_REF], service return [START_REF] Coelho | Imagery intervention in open and closed tennis motor skill performance[END_REF][START_REF] Robin | Effects of motor imagery training on service return accuracy in tennis: The role of imagery ability[END_REF], volley [START_REF] Cherappurath | Combined effects of SAQ and PETTLEP imagery training: A study on the learning process of new skills among novice tennis players[END_REF][START_REF] Türk | Technical parameters and mental toughness in tennis players[END_REF], forehand and backhand stroke [START_REF] Dana | Internal and external imagery effects on tennis skills among novices[END_REF][START_REF] Hegazy | The effect of mental training on motor performance of tennis and field hockey strokes in novice players[END_REF][START_REF] Koehn | Imagery intervention to increase flow state and performance in competition[END_REF], or general [START_REF] Mathers | Professional tennis on the ATP tour: A case study of mental skills support[END_REF][START_REF] Morais | Pre-service routines, mental toughness and performance enhancement of young tennis athletes[END_REF] performances. One applied implication is the importance of educating tennis players, especially beginners and non-experts, on the value of using different function of MI [START_REF] Cumming | Deliberate imagery practice: the development of imagery skills in competitive athletes[END_REF], including the motivational specific function which can influence intrinsic motivation and the motivational general-arousal function which is associated with stress, anxiety and arousal regulation which will find its use especially during competitions. The different functions of MI can be performed by using different imagery modalities that we will now discuss.

Imagery Modality

Each imagery modality can have differentiated effects on performance depending on the constraints of the actions to be performed [START_REF] Callow | Visual imagery perspectives: A commentary on Morris and Spittle[END_REF][START_REF] Munroe | The four Ws of imagery use: where, when, why and what[END_REF].

Indeed, while proprioceptive imagery (i.e., rediscovering sensations of movement, force, stretching) seems to be beneficial for motor actions involving inter-segmental coordination [START_REF] Féry | Differentiating visual and kinesthetic imagery in mental practice[END_REF] or execution speed [START_REF] Féry | Kinesthetic and visual images in modelling closed motor skills: the example of the tennis serve[END_REF], other authors have put in evidence the superiority of internal visual imagery (i.e., to the first person as if one were looking with his own eyes) for tasks carried out in which environmental changes induce planning constraints as for the service return in tennis [START_REF] Robin | Effects of motor imagery training on service return accuracy in tennis: The role of imagery ability[END_REF]. In addition, it has been shown a superior effect of external visual imagery when it comes to reproducing with precision of forms [START_REF] Hardy | Efficacy of external and internal visual imagery perspectives for the enhan-cement of performance on tasks in which form is important[END_REF] or to pay attention to the trajectory of the ball as is the case for the service on the tennis court (e.g., [START_REF] Atienza | Video modeling and imaging training on performance of tennis service of 9-to 12-year-old children[END_REF][START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF][START_REF] Guillot | Motor imagery and tennis serve performance: The external focus efficacy[END_REF][START_REF] Mamassis | Improving serving speed in young tennis players[END_REF]. Finally, authors using a combination of kinesthetic and visual MI showed beneficial effects in forehand and backhand [START_REF] Guillot | Implementation of motor imagery during specific aerobic training session in young tennis players[END_REF] or service [START_REF] Desliens | Motor imagery and serving precision: A case study[END_REF] performances in experts, but lesser positive effects were obtained in beginners (e.g., Cherappurath & Elayara, 2017), or non-experts players [START_REF] Cherappurath | PETTLEP imagery and tennis service performance: an applied investigation[END_REF]. Indeed, [START_REF] Hardy | Efficacy of external and internal visual imagery perspectives for the enhan-cement of performance on tasks in which form is important[END_REF] have suggested that the use of kinesthetic imagery is more beneficial when participants have certain degree of expertise at a task, non-experts usually have more difficulties in feeling motor actions that they have to imagine [START_REF] Guillot | Duration of mentally simulated movement: A review[END_REF]. It will therefore be recommended to favour visual imagery with novice tennis players and to gradually use kinesthetic imagery with the players especially those who have difficulty in performing MI.

Imagery Ability

It is important to remember that while all tennis players are able to perform MI [START_REF] Goss | Imagery ability and the acquisition and retention of movements[END_REF], and that it is not necessary to be a good imager to begin MI training, its degree of effectiveness depends upon the athlete's ability to generate mental images. Indeed, [START_REF] Robin | Effects of motor imagery training on service return accuracy in tennis: The role of imagery ability[END_REF] showed that good imager tennis players had greater service return performances than poor imagers, who still performed better than participants in a control group who did not performed MI. That's why many authors evoked the necessity to control the tennis player's imagery ability when using MI interventions (e.g., [START_REF] Cherappurath | PETTLEP imagery and tennis service performance: an applied investigation[END_REF][START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF]Fekih et al., 2020a,b;[START_REF] Guillot | Motor imagery and 'placebo-racket effects' in tennis serve performance[END_REF][START_REF] Mizuguchi | The effect of somatosensory input on motor imagery depends upon motor imagery capability[END_REF]Spittle et al., 2007). Indeed, [START_REF] Cumming | Imagery: The role of imagery in performance[END_REF] suggested that if an athlete is not able to accurately form images of particular content or from a certain modality, the potential for using MI to improve the movement or performance can be considerably reduced and it will need more practice time. For example, [START_REF] Goss | Imagery ability and the acquisition and retention of movements[END_REF] showed that good imagers acquired simple movements in fewer trials than poor imagers and that goodimagers were able to reproduce movement patterns more accurately. However, it is important to note that, even if they are poor imagers, all tennis players can benefit from MI interventions [START_REF] Robin | Effects of motor imagery training on service return accuracy in tennis: The role of imagery ability[END_REF] and can improve their MI ability after training [START_REF] Cumming | The influence of an imagery workshop on athletes' use of imagery[END_REF][START_REF] Rodgers | The effect of an imagery training program on imagery ability, imagery use, and figure skating performance[END_REF] by first imagining simple actions or parts of movements and then gradually increasing the complexity of MI scripts. As previously evoked, the use of video (e.g., [START_REF] Atienza | Video modeling and imaging training on performance of tennis service of 9-to 12-year-old children[END_REF] could help poor imagers to access to the motor representation of the action to imagine during MI practice.

Imagery and Practice

It is very important to note that in most of the studies reviewed, MI was used in addition to physical practice, which is likely due to the facts that imagery practice alone is known to have smaller effect than that following physical practice alone [START_REF] Toth | Does mental practice still enhance performance? A 24 year follow-up and meta-analytic replication and extension[END_REF].

However, MI combined with physical practice is generally more effective than physical practice alone [START_REF] Lindsay | The effect of mental imagery on skill performance in sport: A systematic review[END_REF]Simonsmeir et al, 2020), that's why coaches should frequently include MI in training sessions. For example, the positive effects of the combination of MI and physical performance were demonstrated in a recent study in which [START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF] showed the beneficial effects of a pre-performance routine consisting of breathing, ball bounces and MI, performed before serving. The use of routines is very common in skilled and professional tennis players (e.g., [START_REF] Mamassis | The effects of a mental training program on juniors precompetitive anxiety, self-confidence, and tennis performance[END_REF][START_REF] Mathers | Professional tennis on the ATP tour: A case study of mental skills support[END_REF]. For example, [START_REF] Morais | Pre-service routines, mental toughness and performance enhancement of young tennis athletes[END_REF] developed, in tennis athletes, a systematic behavioural and cognitive preparation prior to serve that notably included concentration and MI (i.e., the players imagined hitting theirs shots exactly where they wanted). More importantly, the routine was individualized and automatized during training as recommended by [START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF]. The latter authors, also suggested to specify to the participants the imagery modality and/or perspective they must use during MI intervention and/or routine.

More Applied Recommendations

In order to facilitate the construction of mental representation in racket sports, [START_REF] Guillot | Implementation of motor imagery during specific aerobic training session in young tennis players[END_REF] suggested performing MI on the court, wearing sport clothing, hearing the ball bounces and handling the racket. Indeed, [START_REF] Mizuguchi | The effect of somatosensory input on motor imagery depends upon motor imagery capability[END_REF] suggested that tactile input associated with holding the tennis racket could improve imagery vividness especially in poor imagers or beginners. In addition, [START_REF] Bisio | The tool as the last piece of the athlete's gesture imagery puzzle[END_REF] found, in expert tennis players, that the duration of an imagined tennis gesture was similar to that of the same gesture physically performed when holding a tennis racket while an absence of isochrony was observed when they held a tennis-like racket (i.e., an object resembling a racket, of the same mass: 300 grams but different size) or an umbrella. It is conceivable that the prolonged use of a tennis racket, in skilled tennis players, could induce neuroplastic modification of the body representation in the brain incorporating the racket in their own body representation [START_REF] Maravita | Tools for the body (schema)[END_REF] and considering it as an extension of the hand [START_REF] Fourkas | Kinesthetic imagery and tool-specific modulation of corticospinal representations in expert tennis players[END_REF]. Moreover, the fact that MI can be performed on the court indicates that tennis players can use it during competition before serving, between points and when changing sides for examples.

During tennis competition, pressure is an important problem facing most of the players that can lead to loss of flow experience and performance [START_REF] Gabr | Mental imagery and self-talk as approach to cope with pressure among individual sports[END_REF]. [START_REF] Koehn | Imagery intervention to increase flow state and performance in competition[END_REF], examining the effectiveness of an MI intervention on improving performance and the experience of flow state, showed an increase in flow experiences in three of four junior tennis players. Although these results need to be confirmed by further research, the idea that MI interventions might facilitate optimal flow experiences in competition (e.g., being focused on the task, totally absorbed, feeling in control and confident) is particularly interesting because cognitive and motivational imagery can be easily incorporated into training and competition preparation [START_REF] Koehn | Correlates of dispositional and state flow in tennis competition[END_REF]. Indeed, [START_REF] Koehn | Imagery intervention to increase flow state and performance in competition[END_REF] mentioned that linking flow dimensions of sense of control and challenge-skills balance with motivational general mastery imagery appears to have a beneficial effect on the positive experience, especially on tennis players' confidence. In addition, the use of cognitive specific imagery (e.g., before performing a service or a service return) could allow the tennis players to be more focused and know what to do during a rally. Moreover, cognitive general imagery, focussed on tactical points and strategies, could cognitively better prepare tennis players to react to different match situations. That's why we recommend athletes to use MI both before and during matches. In addition, it is also important to note that tennis mental trainers also recommend athletes to use MI after matches in order to "keep in memory what worked well" and on the contrary "erase" or "replace" with positive imagery of actions [START_REF] Coelho | Imagery intervention in open and closed tennis motor skill performance[END_REF] or strategies the negative outcomes.

Moreover, it is important to note that MI intervention do not consistently provide a beneficial effect on learning and performance. For example, studies showed a lake of service improvement (e.g., [START_REF] Noel | The effect of visuo-motor behavior rehearsal on tennis performance[END_REF]Withbread & Newell, 2013) and Gozalzadeh (2017) who showed similar backhand strike performance between the Control and MI groups, also revealed greater service accuracy when using internal MI and forehand accuracy when using external MI. It seems that clear MI instructions and a sufficient amount of practice are recommended in order to benefit from positive effects of MI whatever the level or type of participant.

Indeed, it is finally important to note that the female tennis players improved their performance (e.g., [START_REF] Atienza | Video modeling and imaging training on performance of tennis service of 9-to 12-year-old children[END_REF][START_REF] Blankert | Imagining success: Multiple achievement goals and the effectiveness of imagery[END_REF] when benefited from motor imagery and had similar MI uses (e.g., Derecli, 2019;Guillot et al., 2004) than males, that's why it is recommended to use this mental technic regardless of gender. But the current study highlighted the weak representation of female tennis players within the studies' participants highlighting a male tennis player predominance in researches that used MI interventions in tennis or racket sports [START_REF] Cece | Mental training program in racket sports: A systematic review[END_REF]. Although many studies have not found gender difference in MI ability and vividness (Robin et al., 2020;[START_REF] Subirats | Age and gender differences in motor imagery[END_REF], more research is needed to better explore the gender influence in MI interventions in tennis.

New Research Directions

Whereas it was previously mentioned that the beneficial effects of MI could be influenced by the imagery ability of the tennis players (e.g., [START_REF] Robin | Effects of motor imagery training on service return accuracy in tennis: The role of imagery ability[END_REF], we may envisage that the combination of MI with other mental technics such as instructional self-talk could help the participants to imagine all the parts of complex motor actions to imagine such as the service. A study will soon be carried out in our laboratory to test this hypothesis in novice tennis players; and further research will also be conducted to assess the potential beneficial effect of combining model video, coach feedback and MI as previously shown by Robin et al. (2020) in a precision task.

Likewise, although several studies showed that MI was effective to improve performance, most of them used acute (e.g., a single day; Deslins et al., 2011;[START_REF] Di Corrado | Imagery ability of elite level athletes from individual vs. team and contact vs. no-contact sports[END_REF] or short duration interventions (i.e., a few sessions or weeks ;Fekih et al., 2020a,b;[START_REF] Turan | The impact of cognitive-based learning and imagery training on tennis skills[END_REF] and very few studies (i.e., less than 10%) used long (several months; e.g., [START_REF] Atienza | Video modeling and imaging training on performance of tennis service of 9-to 12-year-old children[END_REF][START_REF] Dominique | Effet d'une routine centrée sur l'imagerie mentale et sur l'efficacité du service chez des joueurs de tennis experts[END_REF] or chronic (i.e., one year and more; e.g., Dohm et al., 2019;[START_REF] Mamassis | The effects of a mental training program on juniors precompetitive anxiety, self-confidence, and tennis performance[END_REF][START_REF] Mathers | Professional tennis on the ATP tour: A case study of mental skills support[END_REF] interventions. Therefore, further research is needed to determine, using post-performance measures long after the MI intervention, whether the effect of motor imagery lasts over time to stabilize a good performance or could be considered an acute intervention that could be beneficial before a competition for example.

Whereas the use of pre-competition positive MI seems beneficial in closed tennis skill performance [START_REF] Malouff | Effects of pre-competition positive imagery and self-instructions on accuracy of serving in tennis[END_REF], more research is needed to better explore the effect of positive MI during and after competition in open tennis skill (e.g., service return) and match performances.

Finally, due to many different outcomes (i.e., learning, performance improvement, preferences, uses, mental chronometry or neurophysiological measures), participant's characteristics (age, level, gender) and sample sizes, MI interventions durations (only one, weeks, months, years), types of studies, experimental groups and procedures and the differences in study's quality, there are risks of bias and the use of a meta-analysis did not seem relevant which can be considered as a limitation of this study. The increase in number and quality research in the field of tennis, especially in the motor learning or performance improvement following MI intervention context, should make it possible to have more similar studies allowing the realization of meta-analysis in the future.

Conclusion

This study, based on the review of 41 selected published studies, examined the effect of MI on novice to expert tennis players and revealed that most studies used the cognitive specific (e.g., rehearsal of specific sport skills such as service) and motivational general mastery (e.g., self confidence or control of emotions during challenging tennis situations) function of MI. More precisely, this study revealed that MI interventions significantly enhanced motor performance (i.e., improvement of forehand, backhand or volley quality, and service efficacy) as well as motivational and affective outcomes for tennis players, regardless of their level and age. In addition, MI was frequently combined with other mental skills and integrated in pre-performance individualized routines including breathing, positive self-talk, ball bounces and/or concentration. In addition, MI can be used between point and during side changes and it is also recommended to employ it after competition. However, the results of the review highlighted the unequal distribution of the population with over-representation of males in the studies reviewed. To conclude, this review proposed potential implications for tennis players, coaches, parents or researchers, and suggested that the regular practice of motor imagery should be incorporated in tennis player training programs and that could also concerns other dual and racket sports such a table tennis, badminton, paddle or beach-tennis.

Figure 1 .

 1 Figure 1. Flow diagram.

Note.

  Cognitive Specific (CS), Cognitive General (CG), Motivational Specific (MS), Motivational General-Arousal (MG-A), Motivational General-Mastery (MG-M), Mental Imagery (MI).

  the practice, 25 studies (61.0%) used MI technique alone, while other studies (N = 11, 26.8%) included MI in mental training programs or routines (i.e., preperformance or pre-competitive). For example, MI was combined with techniques such as relaxation (N = 2, 4.8%), breathing (N = 1, 2.4%), video observation (N = 5, 11.2%), goal setting (N = 1, 2.4%) or positive self-talk (N = 2, 4.8). Finally, 29 studies (70.7%) used MI intervention in combination with physical practice.

Table 1 .

 1

	253
	254	Summary of article characteristics
		Authors	Titles	Study type	Mental imagery	Participants	Intervention	Outcomes	Main findings or results	Qua-
		and years			modalities and	(N, Gender,	group(s) and	measures		lity
					types	mean age)	duration			rat-
						and athletic				ings
						level
		Atienza	Video modelling and	Experimental	Externally guided	(22 females,	Physical,	Tennis service	Mental practice (i.e., MI	6
		et al.	imaging training on		motor imagery.	10 yr.),	Physical + video,		and/or video) groups had
		(1998)	performance of	Ecological		school tennis	Physical + video	Pre-post	greater post intervention
			tennis service of 9-to	condition	CS	players	+ MI.		performances than the
			12-year-old children				24 weeks		control group.
		Bisio et	The tool as the last	Experimental	Kinesthetic	(11 males, 9	Expert and,	Movement	MI and real movement	7
		al. (2014)	piece of the athlete's		imagery	females, 25	novices groups.	Imagery	durations were similar
			gesture imagery	Laboratory		yr.), 10	Real movement,	Questionnaire-R	with the rackets. When
			puzzle		CS	expert tennis	60 MI trials with	Real and MI	not handled specific tool,
						players, 10	racket, tennis-like	durations,	MI duration increased but
						novices	racket or umbrella	Movement	not in the control
							1 session	kinematic	condition.
		Blankert	Imagining success:	Experimental	Internal visual and	(49 males, 16	Mastery goal,	Service	Post imagery intervention	6
		&	Multiple		kinesthetic	females, 27	performance goal,		service performance was
		Hamstra	achievement goals	Ecological	imagery,	yr.), tennis	mastery plus	Pre-post	greater in the mastery plus
		(2017)	and the effectiveness	condition		players level	performance goal		performance goal group
			of imagery		CS	(4-7 US	groups		than in the other groups.
						rating)	1 session
		Cherap-	Combined effects of	Experimental	Internal visual and	(40 males,	Speed, Agility	Forehand,	Experimental groups had
		purath &	SAQ and PETTLEP		kinesthetic	10-12 yr.),	and Quickness	backhand,	higher post intervention
		Elayaraja	imagery training: A	Ecological	imagery,	novice tennis	training group,	volley	performances than control
		(2017)	study on the learning	condition		players	MI group, SAQ	Movement	group, and the SAQ + MI
			process of new skills		CS		and MI group,	Imagery	group had greater
			among novice tennis				control group	Questionnaire-R	performance than the
			players				12 weeks	Pre-post	other groups.

psychological factors such as anxiety, perceived stress, flow or self-talk (N = 16, 39.0%) mainly evaluated by means of questionnaires, scales and interviews, or physiological activities such as skin resistance, electroencephalography or electromyography activities (N = 3, 7.3%) measured by specific tools.

Table 2.

Summary of the main results concerning the effects of motor imagery

Outcomes

Positive results

Neutral results

Negative results

Service performance the Motivational Specific (MS; 9.8%) functions of MI.

Imagery Modality

Concerning the modalities used during MI interventions, 16 articles (39.0%) mentioned the use of kinesthetic imagery, most of the time combined with visual imagery modality (N = 14). Twenty-five studies (61.0%) reported the use of visual imagery modality only. Specifically, 10 studies (24.4%) reported the use of external visual imagery, and 6

Declaration of Interest Statement

The authors declare that they have no competing interests.


Eugene Asarin
email: asarin@ippi.ac.msk.su

Marius Bozga

Alain Kerbrat

Oded Maler
email: oded.maler@imag.fr

Amir Pnueli

Anne Rasse

Data-Structures for the Veri cation of Timed Automata ?

In this paper we suggest numerical decision diagrams, a bddbased data-structure for representing certain subsets of the Euclidean space, namely those encountered in veri cation of timed automata. Unlike other representation schemes, ndd's are canonical and provide for all the necessary operations needed in the veri cation and synthesis of timed automata. We report some preliminary experimental results.

? This research was supported in part

1 Introduction Consider a transition system A = (Q; ), where Q is the set of states and : Q 7 ! 2 Q is a transition function, mapping each state q 2 Q into the set of q-successors (q) Q.

The problem of calculating or characterizing all the states reachable from a subset F Q of the state-space is one of the central problems in veri cation.

The basic algorithm to calculate this set of states is the following: F 0 := F for i = 0; 1; : : :; repeat F i+1 := F i (F i ) until F i+1 = F i where (F i ) = S q2Fi (q). Symbolic methods BCM + 93], McM93] have proved to be a very useful tool in the analysis of large discrete transition systems composed of many interacting components. Instead of transforming the description of the system into an enormous \ at" transition table over 4 IB m , on which reachability analysis is practically impossible, these methods represent the transition relation as a formula over the state variables. Given such a formula T and a formula P describing the subset F of the state-space one can calculate a new formula P 0 characterizing the set (F ) of immediate successors of F. Iterating the procedure until a xedpoint is reached yields a formula P characterizing the set of all states reachable from F. When is expressed by a formula T (X; X 0 ), and F by a formula P(X).

the above algorithm can be reformulated as: P 0 (X) := P(X) for i = 0; 1; : : : ; repeat P i+1 (X) := P i (X) _ 9Y (P i (Y ) ^T (Y; X)) until P i (X) = P i+1 (X)

The essence of any symbolic method is a data-structure for representing sets (equivalently, the formulas characterizing them) on which the above operations can be performed, in particular the forward (or backward) projection (line 3), boolean operations and equivalence testing. Binary decision diagrams (bdd's) Bry86] are such a data-structure for boolean domains. The calculation of the forward projection is relatively-easy on large practical problems and the space requirements for the representations are reasonable. Given an ordering of the variables, bdd's also have the canonicity property: all equivalent formulas lead to the same bdd and equivalence testing is thus trivial.

The veri cation of timed automata introduces an additional ingredient, that is, a set of continuous variables (clocks) ranging over non-countable domains. The dynamics of the passage of time cannot be captured by a \next-state" transition relation, and symbolic methods are unavoidable as states and trajectories cannot be enumerated. The sets encountered in reachability analysis of timed automata are thus certain subsets of IB m IR d . While the discrete part is standard, the subsets of IR d that need to be represented and manipulated are what we call k-polyhedral sets, namely sets de nable by a boolean combination of basic inequalities of the form x i < c, x i c, x i ? x j < c and x i ? x j c, for i; j 2 f1; : : :; dg and c 2 f0; : : : ; kg. Such polyhedral sets have been called regions in AD94].

As long as these polyhedra are convex (i.e., de nable by conjunctions of basic inequalities and their negations), there exists a canonical representation, the difference bounds matrix (dbm, see for example Dil89]). This is a (d+1) (d+1) matrix with entries taken from f0; : : : ; kg denoting the constants in a non-redundant set of inequalities whose intersection forms the region. For this representation, the intersection is done very easily via min and max operations. The forward and backward projections via elimination of the time quanti er are also done very e ciently on dbm's. Things however get complicated when we have arbitrary unions of convex polyhedra. In this case there is no unique representation and most tools represent such sets as a list of dbm's. The more \non-convex" the set becomes, more matrices are required in order to represent it and this makes equivalence testing and redundancy elimination di cult. Moreover, it is not clear how this representation is to be combined with a symbolic representation of the discrete part.

In this paper, we suggest an alternative bdd-based data-structure, Numerical Decision Diagram (ndd) that has a caonicity property: given an ordering of the clock variables, every k-polyhedral set has a unique minimal representation. For this data-structure we have boolean set-theoretic operations and equivalence testing for free. 5 We present an algorithm to calculate forward and backward projection in time for this data-structure and thus have all the ingredients needed in order to do reachability analysis for timed automata. Since this representation is bdd-based it can be combined naturally with symbolic methods for the discrete part of the system.

The rest of the paper is organized as follows. In section 2 we present timed automata and de ne the components of their reachability analysis algorithms. In section 3 we de ne ndd's and their forward projection algorithm for the discretetime interpretation of timed automata. In section 4 we show how a discretization scheme, rst reported in GPV94], can be used to extend the scope of ndd's to the dense-time interpretation. Finally we present some experimental results.

Timed Automata

First, some notations. We use bold-face letters to denote points in IR d . Thus, v stands for (v 1 ; : : : ; v d ), where v i 2 IR, for every i = 1; : : : ; d. For points u; v 2 IR d , we write u v to denote that u i v i , for every i = 1; : : : ; d. A subset S IR d is called monotonic if v 2 S implies u 2 S, for every u 2 IR d satisfying u v.

For the sake of (the few) readers not familiar with timed automata we start with an informal illustration of the behavior of these creatures. Consider the timed automaton of gure 1. It has two states and two clocks z 1 and z 2 . Suppose it starts operating in the con guration (q 1 ; 0; 0) (the two last coordinates denote the values of the clocks). Then it can stay at q 1 as long as the staying condition for q 1 is true, namely z 1 2. Meanwhile the values of the clocks grow and the set of all con gurations reachable from (q 1 ; 0; 0) without leaving q 1 is f(q 1 ; t; t) : 0 t 2g. However, after one second, the condition z 1 1 (the guard of the transition from q 1 to q 2 ) is satis ed and the automaton can move to q 2 while setting z 2 to 0. Hence the additional reachable con gurations are f(q 2 ; t; 0) : 1 t 2g. Having entered q 2 in one of these con gurations, the automaton can either stay there as long as z 1 5 ^z2 3 or can unconditionally move to (q 1 ; 0; 0), etc. z 1 1=z 2 := 0 =z 1 := z 2 := 0 q 1 z 1 2 q 2 z 1 5ẑ 2 3 Fig. 1. A timed automaton.

Since the state-space of timed automata contains real values, we have an in nite-state automaton and an enumerative approach, where all states and transitions are enumerated, is impossible. We will use notation such as G qq 0 to denote the set of values in the clock space that satisfy the condition (\guard") for the transition from q to q 0 6 = q. Similarly, G qq denotes the set of clock values allowing the automaton to stay in q (\staying conditions"). In timed automata such sets are restricted to be k-polyhedral subsets of IR d , that is, the class of sets obtainable by applying set-theoretic operations to half-spaces of the form fv : v i cg, fv : v i < cg, fv : v i ? v j cg or fv : v i ? v j < cg for some integer c 2 f0; : : : ; kg, where k is some positive integer. 6 These sets constitute the nite region graph AD94] whose properties underlie all analysis methods for timed automata.

A function from IR d to itself is a reset function if it sets some of its arguments to 0 and leaves the others intact. We will use R qq 0 to denote the reset function associated with every pair of states (we take R qq to be the identity function).

We will make the following simplifying assumptions concerning the timed automata that we consider: 1) There is only one transition associated with every pair of states. 2) The values of the clocks are bounded by k. Hence the clock space is 0; k) d . 3) G qq 0 is convex for every q; q 0 2 Q, and 4) G qq is monotonic for every q 2 Q. The readers can convince themselves that it costs few states to convert any timed automaton into one satisfying these properties.

We let K denote the interval 0; k) in the dense-time interpretation or the set f0; : : : ; k ? 1g in the discrete-time interpretation. For every z 2 IR d we use z + t to denote z + t 1 where 1 = (1; 1; : : : ; 1) is a d-dimensional unit vector. De nition 1 (Timed Automaton). A timed automaton is A = (Q; Z; ) such that, Q is a discrete set, Z = K d is the clock space (Q Z is the con guration space) and : Q Z 7 ! 2 Q Z is the transition relation. It is required that admits the following decomposition: For every q; q 0 2 Q, let G qq 0 Z be a k- polyhedral monotonic set and let R qq 0 : Z 7 ! Z be a reset function. Then, for every (q; z) 2 Q Z (q; z) = (q 0 ; z 0 ) : 9t 2 K z + t 2 G qq \ G qq 0 ẑ0 = R qq 0 (z + t)

(1)

The meaning of (q; z) is the set of Q Z con gurations the automaton can reach starting at (q; z) by waiting t time (possibly zero), and then taking at most one transition. 7 Every subset of Q Z encountered in the analysis of timed automata can be decomposed into a nite union of sets of the form fqg P where P is kpolyhedral. We will write such sets as (q; P). We will extend functions on ele-6 In fact, we can use c 2 f0; r; 2r : : : ; krg for some positive rational r. 7 In the treatment here, we assume that all sets of the form Gqq are de nable by a positive boolean combination of inequalities of the form xi ci and xi ? xj cij.

All the techniques presented here can be generalized to apply to the more general case that some of the inequalities de ning Gqq are strict.

ments to functions on sets in the natural way, e.g. (q; P) = S z2P (q; z) and R q;q 0 (P ) = S z2P R qq 0 (z).

Next, we de ne a function : 2 Z 7 ! 2 Z (time forward projection) as:

(P ) = fz + t : z 2 P; t 2 Kg \ Z:

It is not hard to see that the immediate successors of a set of con guration (q; P) can be written as ((q; P)) = (q; b P) q 0 6 =q (q 0 ; P q 0 ) where b P = (P ) \ G qq and for every q 0 , P q 0 = R qq 0 ( b P \ G qq 0 ): This concludes the motivation for the paper as we see that the additional machinery needed to analyze timed automata consists of calculations of boolean operations, R(P), and (P ) on k-polyhedral sets.

3 Numerical Decision Diagrams: Discrete Time

Representation

The idea of ndd's is elementary. When we consider K d under the discrete interpretation, we have nothing but subsets of a nite set. Obviously, every element of K can be coded in binary using b = dlog ke bits, where dlog ke is the smallest integer not smaller than log k. Consequently, we can represent every subset of K d as a boolean function of d b boolean variables. This function can be represented by a bdd in the usual way. We will use standard positional encoding, i.e., every number n 2 K is represented by a set of values x 0 ; : : : x b?1 such that n = P b?1 i=0 x i 2 i .

The rst question concerning the implementation is the ordering of the bits of every number. Although, especially for sets of the form x c, putting the most signi cant bit rst might lead to smaller bdd's, we prefer to put the least signi cant bit rst, because it facilitates the calculation of . Examples of sets and their ndd representation for d = 1 and k = 8 appear in gure 2. When there are more than one clock variables, there are various ways to order their bits, for example, x 0 ; x 1 ; : : : ; y 0 ; y 1 ; : : :.

In order to represent decision trees and bdd's textually we will use the expression bdd(x i ; L; R) to denote a tree that tests x i , branches to the subtree L on zero and to the subtree R on one. For example, the tree obtained from the bdd for x < 5 in gure 2 is written as: bdd(x 0 ; bdd(x 1 ;

1; bdd(x 2 ; 1; 0)); bdd(x 1 ; bdd(x 2 ; 1; 0); 0

)) 0 0 0 7 7 7

x > 5

x < 5

x < 3 0 7 (x > 5) _ (x < 5)

x 2

x 1

x 0 0

1 0 0 1 1 1 0 0 7 (x > 5) _ (x < 3) x 0 x 1 x 1 0 1 0 1 1 x 2 x 2 0 0 0 0 1 1 1 x 0 x 1 x 1 1 0 1 1 0 x 2 0 1 0 0,1 x 0 x 1 x 1 x 2 1 0 1 1 0 0 1 1 0 0 x 0 x 1 x 2 0 1 0,1 0 1 0 1
Fig. 2. Some 8-polyhedral sets in one-dimension and their corresponding ndd's.

Operations

Beside set-theoretic operations that we have for free, the reset operation is also elementary: in order to calculate R(P) for a reset function R that resets, say, the variable x, you build a bdd for the set x = 0 and intersect it with the bdd for 9x P. What remains to show is how to calculate (P ), which we will rst demonstrate on the semantic level.

Given P 2 K d , (P ) can be written as fz : 9t 2 K s:t: z ? t 2 Pg. Before applying the existential quanti er we have a set P 0 2 K d+1 representing all the tuples (t; z) such that z ? t 2 P. We will present a procedure that converts a b d-variable ndd for P into a b (d + 1)-variable ndd for P 0 (with t as an additional K-variable, encoded using the boolean variables t 0 ; t 1 ; : : : ; t b?1 ).

Eliminating the existential quanti er for t from P 0 , we obtain the bdd for (P ).

The procedure will initially create the ndd for the set P 1 = f(t; z) : z t 2 Pg where stands for subtraction modulo k. Then, by intersecting P 1 with the set P 2 = f(t; z) : z t 1g we get P 0 (see gure 3). the time. We rst make the subtraction modulo k (creating P1) and then intersect with P2 to get rid from over ow.

To illustrate the construction of P 1 , we consider rst the case that d = 1, i.e. only one clock. The recursive function sub(B; borrow) presented in table 1 takes an ndd B for P K and produces an ndd for P 1 K 2 as described above. The parameter borrow represents the \borrow" bit which is propagated from right to left on performing binary subtraction. The external invocation of this function is done with borrow = 0. For simplicity of presentation, we assume that B has nodes for all variables, with entries of the form bdd(x i ; L; L) in case the function is independent of x i . An optimized version can be derived for the more general case of skipped variable tests.8

The e ect of applying the function to an arbitrary decision tree over f0; : : : ; 3g is depicted in gure 4. The extension to d > 1 is rather straightforward.

Dense Time

The above construction is su cient for analyzing timed automata under the discrete-time interpretation. It is however known that some timed automata can produce behaviors (state-sequences) under a dense semantics which are not possible under any discrete-time semantics. In this section we introduce a discretization scheme GPV94] 9 having the two following important properties: 1) It preserves the qualitative behavior of the automaton, that is, for every sequence of discrete transitions in the semantics of a timed automaton A, there is a similar sequence in the semantics of its discretization e

A and vice versa.

(I i = I 0 i ) ^(f i > 0 $ f 0 i > 0)
^î;j2f1;:::;dg

(f i > f j $ f 0 i > f 0 j ):
We consider automata with Z = K d . We will use a discretization step = 1=(2d)

and let e K = fn : 0 n < 2kdg. In other words, we cut every unit interval into 2d equal segments and pick the endpoints. The discretized clock space (that is, the domain over which discretized clocks range) is e Z = e K d \ f(z 1 ; : : : ; z d ) : 8i; j jz i ? z j j = 2m g:

Note that we take from e K d only points such that the di erence between any pair of clock valuations is an even multiple of (see gure 5). For any polyhedral

1 0 0 1 1 A B C D x1 x1 x0 A A B B C C D D B D D B C A A 0 0 0 C 0 1 1 1 1 0 1 0 1 0 0 1
Fig. 4. Applying sub to an arbitrary decision tree over f0; 1; 2; 3g. The circled leaves will become zero when we intersect with f(x; t) : x tg.

set P, we let its discretization be e P = P \ e Z. It is not hard to see that, for every k-polyhedral set P, we have P 6 = ; i e P 6 = ;. Another important property of this scheme is the following:

Claim 1. Let z = e z + " for some e z 2 e Z, j"j < . Then z 2 P ) (e z 2 P _ e z + 2 P)

(and hence at least one of them belongs also to e P).

Proof: If e z = (z 1 ; : : : ; z d ) 2 P we are done, otherwise there is one or more inequalities of the form z i > c i satis ed by e z + " but not by e z (which implies that z i = 2dn for some integer n). These inequalities must be satis ed by e z+ as well. On the other hand, if there is an inequality of the form z j < c j satis ed by e z + " but not by e z + , we have z j = (2dn ? 1) , which contradicts the assumption e z 2 e Z. In addition, e z + " and e z + satisfy together every diagonal inequality (of the form z i ? z j < c) and we can conclude that e z + 2 P. Note that this fails to be true for points outside e Z. Consider z = (0; 3=4) and P = (0 < x < 1) ^(0 < y < 1) ^(y > x). Here z + " 2 P but neither z nor z + = (1=4; 1) belong to P. e ( e P) = fe z 0 2 e Z : 9e z 2 e P 9 e t 2 e K s:t: e z 0 = e z + e tg:

Claim 2 Discretization Preserves Forward Projection.

For every k-polyhedral set P and P 0 such that P 0 = (P )

e ( e P) = f P 0 :

Proof: One direction, e ( e P) f P 0 is obvious because e P P and e K K.

For the other direction, suppose some e z 0 2 (P ) \ e Z, implying that e z 0 can be written as (n 1 ; : : : ; n d ), and that for some z 2 Z, t 2 K, z + t = e z 0 hence z = (n 1 ? t; : : : ; n d ? t). Let t = m + t 0 for some t 0 < . Then z = ((n 1 ?m) + t 0 ; : : : ; (n d ?m) + t 0 ). According to the previous claim either e z or e z + is in e P and their temporal successor e z 0 is in e ( e P). Having shown that forward projection (as well as boolean operations) on Z and K can be imitated by discretized operations on e

Z and e K, the only remaining problem is concerned with the reset operator. The problem is that e Z is not closed under reset functions { for example, resetting the rst coordinate of ( ; ) 2 e Z we obtain (0; ) 2 e K d ? e Z (because the di erence between the points is not an even multiple of ). This is important because claim 2 does not hold on e K d but only on e Z. In order to calculate successors on the discretization we need an \adjustment" operator, which, after applying a reset, will delete points that went out of e Z and replace each of them by one or more region-equivalent points in e

Z. This extra operator can be viewed as the price we pay for dense reasoning.

For each m 2 f0; : : : ; d?1g, let us de ne a function m : e K d ! 2 e Z as follows:

m (z) = fz 0 : d î=1 (I i = I 0 i ) ^0 B B B B @ f i = 0 ^f0 i = 0 _ f i = (2l + 1) ^l < m ^f0 i = (2l + 2) _ f i = (2l + 1) ^l > m ^f0 i = (2l) 1 C C C C A g
The function m returns a non-empty set if and only if all the non-zero fractional parts of z are odd multiples of , and none of them falls in the interval 2m ; (2m + 2) ]. Its e ect is to add to all fractional parts f i satisfying 0 < f i < 2m and subtract from all fractional parts satisfying (2m+2) < f i .

Zero fractional parts are left unchanged. One can see that if z satis es these conditions then m (z) = fz 0 g and z 0 z. This operator is illustrated in gure 6. we obtain z2 = (1; 0; 5; 7) 6 2 e Z. (c) Applying 1 we push the non-zero clocks toward the \hole" around 3 and obtain z3 = (2; 0; 4; 6) which is region-equivalent to z2. Note that in this graphical representation the passage of time is via clock-wise rotation.

It is not hard to see that the application of to any P e K d yields a subset P 0 e Z such that, for every z 2 P, there is at least one z 0 2 P 0 satisfying z z 0 . Based on this function, we can de ne for every reset function R : Z ! Z a discretized reset function e R : e Z ! 2 e Z as e R(e z) = (R(e z)). It follows that, for every P, e R( e P) has elements of every region equivalence class which are represented in R(P). This is all we need: we just add to the ndd solution for integer time is d log(2d) bits to represent the ner grid and to replace every reset function R qq 0 by its adjusted version g R qq 0 . The same arithmetical calculation of time successors described in section 3 will work, when we add the fractional bits to the clock and time variables. The adjustment operation seems to be the hard part of the calculation (we have only implemented the discrete-time representation so far) but at least this operation is performed only on the fractional bits (and hence is sensitive to the number of clocks but not to their ranges). In fact there is a tradeo between two discretization schemes (see GPV94]), one with = 1=(d + 1), where resets behave normally but the evolution of time is distorted and loses some of its arithmetical content, and the other one we describe here, were time evolution remains arithmetic while resetting is more involved.

Concluding Remarks

Related Work

Various tools for the analysis of timed and hybrid automata have been developed recently, e.g. kronos DOY94], uppal BGK + ] and Hy-Tech AHH93]. The rst two represent polyhedral sets by dbms. An alternative approaches is to transform the timed automaton into a huge discrete automaton (the region graph) and than encode its using boolean variables and bdd's. 10 The idea of extending bdd's for the purpose of solving arithmetical constraints has been proposed by Rauzy Rau95]. The structures he proposes are, however, not canonical. Our method can be applied as well outside the analysis of timed automata, e.g., as a decision procedure for some decidable theories in bounded arithmetics (see also WB95]). In fact, the forward projection calculation can be easily adapted to clocks having non-uniform rates in f0; ?1; +1g and can be applied to the analysis of larger classes of hybrid systems and to programs with bounded integer variables.

Experimental Results

For experimentation we have used a system developed at Verimag for representing and manipulating communicating automata augmeneted with bounded variables BFK96]. This system takes such automata and translates them into bdds using one of several publicly-available bdd packages { we have used the CUDD package S95] of Colorado University. We have incorporated a discretetime version of the ndd representation into that system and tested its performance on various timed automata corresponding to digital circuits with delays (the exact de nitions and the translation procedure from circuits to timed automata are described in MP95]). We will report here the results obtained with two generic families of automata, for which we tried to calculate all the reachable con gurations starting from an initial state.

The rst family consists of one-state automaton having n clocks and n transitions. The automaton (see gure 7-a) can let time progress as long as none of the clocks has reached its upper bound u i . Whenever a clock C i reaches the lower bound l i < u i , a self-looping transition which resets C i can be taken. These automata allow us to isolate the complexity of representing and manipulating polyhedral sets from that of treating the discrete state-space. The second family consists of a product of n two-state automata of the type appearing in ( gure 7-b). Such a product is the natural way to model n independent non-deterministic input oscillators and it is a necessary ingredient in any attempt to do exhaustive timing analysis of asynchronous circuits.

The main di erence between the two examples is that in the second we have 2 n discrete states. While the set of reachable con guration for this example will be of the form f(q; P 1 )g q2f0;1g n , the set of reachable clock con gurations of the rst automaton will be S q P q , which seems in general to be simpler to represent. In fact, for the constants we have chosen all the clock space is eventually reachable, but some very \hard" sets are encountered in the intermediate stages of the xed-point calculations.

We have taken clock values in the range 0::15, let l i = 9, u i = 12 and compared the results with those of kronos which uses dbms. For this set of examples the ndd results were much better. It should be noted, however, that dbms implement the richer dense semantics and are not sensitive to the range of clock values as long as they do not cross the maximal integer value. In contrast, the performance of ndds depends critically on the number of bits used to encode clock values. Moreover, the examples were chosen so that they generate intermediate polyhedral sets which are very \non-convex", which makes life hard for the dbm implementation (but also for ndds). Finally the current forward simulation algorithm of kronos keeps all reachable regions in a form of a simulation graph11 and this turns out to be ine cient for these examples { we believe that changing this implementation detail will allow kronos to treat larger examples. The results are summarized in table 2. They were obtained on a SUN Ultra-Sparc 1 with 256MB of memory.

  Fig. 3. Calculating (P ) via moving to d+1 dimensions (P 0 ) and then projecting away

  2) It is amenable to representation by ndd's.For each clock value z i , i = 1; : : : ; d, let I i and f i denote the integer and fractional parts of z i , respectively. Two clock valuations z = (I 1 +f 1 ; : : : ; I d +f d ) and z de ned to be region equivalent, written z z 0 , if d î=1

Fig. 5 .

 5 Fig. 5. Left: discretizing 0; 1) 2 : the circled points are the elements of e Z while the

Fig. 6 .

 6 Fig. 6. The e ect of the adjustment operator 1 for K = 0; 1) 4 and = 1=8 (all points are multiples of ): (a) A point z1 = (1; 3; 5; 7) 2 e Z. (b) After restting the second clock

Fig. 7 .

 7 Fig. 7. The automata used for experiments: (a) A one-state automaton with n clocks and n transitions. (b) A basic two-state automaton with one clock.

Table 1 .

 1 The function sub.

	function sub(B; borrow) begin if B is a leaf then return(B); else let B = bdd(xi; L; R) if borrow = 0 then return(bdd(ti;bdd(xi;(sub(L; 0); sub(R; 0))), bdd(xi; (sub(R; 1); sub(L; 0))))); else return(bdd(ti;bdd(xi;(sub(R;1);sub(L; 0))) bdd(xi; (sub(L; 1); sub(R; 1))))); end

That is, for the same price as for bdd's in general.

As usual in bdd applications, all calls are hashed so that repeated calls with the same arguments will not repeatedly traverse the complete subtrees.

In CC95] this approach has been applied to the degenerate case of one-clock automata.

This is the number appearing in the \regions" column in the table.

Acknowledgment: We thank S. Yovine, K. Daws and S. Tripakis for useful discussions concerning veri cation of timed automata in general and the kronos implementation in particular, and P. Raymond for some help in bdds.

by the European Community projects HYBRID EC-US-043 and INTAS-94-697. Verimag is a joint laboratory of cnrs and ujf. 4 We use IB for f0; 1g and IR for the non-negative reals.

Our intermediate conclusion is that the analysis of timed automata with many clocks is not yet feasible. We have managed to handle additional non-trivial examples (such as an interconnected chain of XOR gates) with 10-13 clocks, but a closer investigation of polyhedral sets and their various representation schemes is needed in order to push performance limitations forward.


T h e C u r s e d T r u m p e t s o f T u t a n k h a m u n

The Myth crystallization

From this point onwards, the story repeated itself at various occasions during the major troubles that Egypt faced aftermath, from the Six Days War up to the 2011 revolution: any subsequent incident could be linked to a probable Tutankhamun trumpets blowing.

These allegations are unfortunately quite elusive; their origination is never formaly referenced. They nonetheless evokes the powerful picture of a peculiar facet of the "Curse of the Pharaohs," deemed to appeal to popular culture.

Association of trumpet with dramatic events is indeed not uncommun (though unknown in ancient Egypt itself): it is notably present in the biblic tradition of the battle of Jericho or the famous Apocalypse trumpets, and also in coranic lore, where the angel Isrāfīl shall blow a trumpet to announce the Day of Resurection. This curious turn of events is now part of the narrative of the Tutankhamun Treasure. The belief of an antic curse focused around those two instruments -activated by the performance to unleash their terrible powers-is a striking illustration of the symbolic impact of these museum artefacts on collective imagination.

Blowing a pharaonic trumpet in 1939

On 16 April 1939, less than twenty years after the famous discovery of Tutankhamun's tomb and the spread of the myth of the "Curse of the Pharaohs," BBC broadcast the live performance of both trumpets, to record what sounds could have graced the Pharaonic ears.

If this world premiere attracted tremendous interest, the recording was not without incidents: a rehearsal provoked the shattering of the silver trumpet due to vibrations; after restorations, the very evening of the live broadcast, all lights in Cairo were shutt off because of an electric failure and the BBC team had to operate by candlelight.

When the World War II stroke some months later, people had already begun to think that the prophetic blowing of the trumpet had been instrumental in triggering the disaster…

The Cairo Egptian Museum, where the trumpets are held since 1925.

The 1939 performance by Jame Tappern.

Correlations & coincidences

Unverified rumors of the trumpets blowing.

Discovered in 1925 in the tomb of Tutankhamun, the only two trumpets conveyed to us from ancient Egypt were experimentaly performed and recorded by the BBC in 1939, months before the outbreak of the Secon World War. They are suspected to bring chaos and devastation when blown since then.

  P e r f o r m a n c e H a z a r d s & U r b a n M y t h Thomas Lebée lebeethomas@laposte.net https://hef.hypotheses.org/ The two trumpets of Tutankhamun New Kingdom, 18 th dynasty, reign of Tutankhamun (c. 1330 B.C.) Kings Valley, tomb of Tutankhamun (KV 62Catalogue général de antiquité égptienne du musée du Caire. Instruments de musique (n os 69201-69852), Cairo, Institut français d'archéologie orientale, 1949, p. 143-145 and pl. LXXXVII-XC. JAMES T. G. H., Howard Carter. The Path to Tutankhamun, London -New York, Tauris Parke, 2 nd ed., 2000. KEATING Rex, The Trumpets of Tutankhamun. Adventure of a radio pioneer in the Middle East, Basingstoke, Fisher Miller, 1999.