For a relational approach to modern literary Arabic conditional clauses 1 .

Introduction

The issue of the conditional in classical Arabic is treated in the classical Arabic grammars, be the authors Arab, both traditional such as Awḍaḥ al-Masālik's Ibn Hišām [START_REF] Hišām | Awḍaḥ = ʿAbd Allāh b[END_REF]) (d. 761/1360) and modern as Ǧamīʿ al-Durūs al-ʿArabiyya 's al-Ġalāyīnī (1886-1944) [START_REF] Salīm | Beyrouth, Dār al-kutub al-ʿilmiyya[END_REF], or foreign (Arabist), for instance Blachère and Demombynes [START_REF] Blachère | Grammaire de l'arabe classique (Morphologie et syntaxe)[END_REF] of Fischer [START_REF] Fischer | Grammatik des Klassischen Arabisch, 2[END_REF] or the work dedicated by Peled to this question [START_REF] Peled | Conditional Structures in Classical Arabic, Wiesbaden, Otto Harrassowitz[END_REF]. Using the novel by Ǧamāl al-Ġiṭānī, Al-Zaynī Barakāt as a starting point, we identified many deviations from the rules of the classical Arabic. The question then arose as to how we express the conditional in Modern Arabic.

Assuming that the answer must be found in the Modern Standard Arabic grammars, we intended to compare what we saw in different contemporary literary texts with what these grammars present to us on the subject. Yet, the study of the literary texts shows that these grammars are descriptively inadequate. Our purpose here will be to study only the literary register of Modern Standard Arabic, highlighting at the same time the descriptive inadequacy of the Modern Standard Arabic grammars and the relationship existing between the operator of the conditional clause and the apodosis of the hypothetical system in question.

Literary corpus, methodology and first observations

Regarding descriptive realism, we chose a linguistic approach based on corpus and thus have reviewed hypothetical systems in extenso through various contemporary literary works. In diachrony, our sample covers the period from 1963 to 2005. These then are novels by authors born after the 1930s, i.e. well after the second generation of the Nahḍa and its effects on the Arabic language, and at a time when the influence of European languages on it must have been already widely felt. Geographically speaking, our corpus ranges from Syria to Morocco. The list of works is as follow: Kanafānī, Riǧāl fī l-šams and Al-ʿĀšiq 2 ; Zafzāf, Ḥiwār layl mutaʾaḫḫir 3 ; Ġiṭānī, Al-Zaynī Barakāt 4 ; Tāmir, Al-Numūr fī l-yawm al-ʿāšir 5 ; Misʿidī, Ḥaddat̠ a Abū Hurayra qāla… 6 ; Ibrāhīm, al-Laǧna, Ḏāt and Warda 7 ; Ibn 1. I dedicate this paper, which is a translation from my French article « Pour une approche relationnelle de la conditionnelle en arabe littéraire moderne » published in Arabica, 2010, 57, pp. 68-98, to ʿAdil and Muḥammad, for them to understand a little bit more what I do to their language… All my special thanks to Hannah Scottdeuchar who has gently read and corrected this English version, with patience and accuracy.

2. Kanafānī, 2002 ;[START_REF] Kanafānī | Muaʾassasat Ġassān Kanafānī al-Ṯaqāfiyya[END_REF] 3. [START_REF] Zafzāf | Ḥiwār Layl Mutaʾaḫḫir, Damas, Manšūrāt wizārat al-ṯaqāfa[END_REF] 4.

Ġiṭānī [START_REF] Fourcade | Manšūrāt wizārat al-ṯaqāfa wa-l-iršād al-qawmī[END_REF]: 225-345 = Ghitani, 1985 : 211-316. 5. [START_REF] Tāmir | Al-Numūr fī al-Yawm al-ʿĀšir[END_REF] 6. [START_REF] Misʿidī | Ḥaddaṯa Abū Hurayra qāla…, Tunis, Dār al-ǧanūb li-l-našr[END_REF] 7. [START_REF] Ibrāhīm | Al-Laǧna, Le Caire, Dār al-mustaqbal al-ʿarabī[END_REF]Ibrāhīm, 1998[START_REF] Ibrahim | Les années de Zeth[END_REF]Ibrāhīm, 2000= Ibrahim, 2005. Haddūqa, Al-Ǧāziya wa-l-darāwīš 8 ; al-Kūnī, Malakūt ṭiflat al-Rabb 9 . It is clear that what both Arab and Arabist Classical Arabic grammars teach us inadequately reflects modern uses. We shoud also note that these uses do not sufficiently shock modern translators who, otherwise, would perhaps not have failed to report them. This is a direct result of the fact that this new syntax is familiar to us, since it is more or less ours… For our study then, we identified all of the conditional clause operators present in the novels mentioned, i.e. both the two "classical" particles in and law (and its derivatives, including law-lā), and the time circonstant (ẓarf zamān) iḏā (and its derivatives including iḏā mā "as soon as, hardly"). From this set, we naturally chose to keep only truly hypothetical systems (where the protasis p logically implies the apodosis q), thus excluding the concessive clauses (wa-law, ḥattā law, ḥattā wa-law, ḥattā iḏā, wa-in, etc.). Of the remaining systems, we then retained only the hypothetical systems that are doubly verbal and assertive. "Doubly verbal" excludes 1) systems that have a protasis introduced by law anna or law-lā, 2) systems that are not fully conditional (i.e. cases of protasis without apodosis which are truncation, optation and frozen uses, like law samaḥta or in šāʾa llah), and 3) systems whose apodosis is a nominal or existential sentence (like lā budda an, ʿalay-hi an, bi-wusʿi-hi an). However, this distinction retains the apodoses which are phrases made up of initial NP/report where the report is itself a verbal sentence (like inna-hu faʿala/yaf ʿalu 10 ). "Assertive" excludes the imperative, the negative imperative, and the interrogative. From this first selection, out of the entire identified corpus, we get 402 systems. Of this total, we will only process the 283 relevant if p q sequences (which represent 70.40% of the total while the sequence q if p only represents 29.60%), to assess the possible importance of fain the Potential systems and of lain the Unreal systems.

In Postclassical Arabic, the Arabic hypothetical system can be schematized as follows, showing the possible verbal forms, both in protasis and apodosis, and also the hypothetical statutes: Ben [START_REF] Haddūqa | Al-Ǧāziya wa-l-Darāwīš, Beyrouth, Dār al-adab[END_REF] 9.

Kūnī (Al-), 2005.

10. We will formalize the verbal forms of protases and apodoses as following: māḍī (perfect) = faʿ ala, muḍāriʿ marfūʿ (imperfect indicative) = yafʿalu, muḍāriʿ manṣūb (imperfect subjunctive) = yafʿala and muḍāriʿ mağzūm (imperfect apocopate) = yafʿal. We are reminding here that the affirmative apocopate does not affect systems in iḏā or law (see Alosh, 2005 : 271 and examples p. 195, 218), which confirms the following example: iḏā ǧunna yasʾalūna-hu ʿannī (Ḥaddaṯa : 69): "When he gets mad, they ask him about me". The apocopate seems still to be in use in the in systems. See example below (11).

11. The distinction between Past Eventual and Potential occurs only with recourse to the context.

12. This line reads in pairs of words: law faʿala… (fa-)yafʿalu or law kana yafʿalu… (la-)faʿala as proposed law faʿala/kāna faʿala faʿala Past Unreal

Within the 283 relevant systems, we first observe the near disappearance of in: the latter represents only 16 occurrences, that is to say 5.65% of the total. In so doing, our purpose will be to focus primarily on the two remaining operators, iḏā and law.

We then observe the overwhelming usage of the perfect form of the protasis verb (97.18%). Only 8 systems in law have a imperfect protasis verb.

Regarding now the apodoses, we find, next to the expected faʿala, the yafʿalu verbal form. We also observe structures that tolerate a verb like sa-yafʿalu in the apodosis. This concerns 28.57% of iḏā systems and 13.84% of law systems. We are thus dealing with three possible forms of apodosis both for iḏā and law: yafʿalu, sa-yafʿalu and faʿala. Without denying the possible existence of cases of ambiguity, as in Classical Arabic, the principle of non synonymy requires looking beyond the three forms of apodosis (yafʿalu, sa-yafʿalu and faʿala) to three distinct conditional meanings.

It appears moreover that some systems in iḏā faʿala…faʿala, classically linked to Eventual and Potential, are here indeed linked to both and thus describe a statute complying with that described in Classical Arabic grammars. But many of these iḏā faʿala…faʿala denote instead the Present Unreal! And if we add to this the fact that Badawi et al. (Badawi et al., 2004 : 647) indicate that law can be synonymous of in, then what about the strict classical dichotomy based on the operators: iḏā/in-Potential vs law-Unreal?

From our corpus, we also note that the "segmentator" fa-13 , far from being systematic 14 , represents only 36.70% of the apodoses in (sa-)yafʿalu; apodoses that should impose it, according to the canons of classical grammar. Thus, it seems no longer to indicate a simultaneous syntactic and semantic break, as it did in Classical Arabic 15 but to occur mainly for contrastive reasons, permitting the indication that what follows is indeed the beginning of the apodosis.

As for la-, it also now appears, possibly in imitation of law, to indicate the Unreal in iḏā systems. However it is not systematic in Unreal systems, representing only 52.35% of the apodoses in faʿala. It should nevertheless be noted that while quasi-systematically absent from Unreal systems in iḏā, it is quasi-systematically present in the case of the Past Unreal in law. This can be called regrettable because, as we will see, it would have allowed differentiation between two iḏā faʿala…faʿala systems, which only the context can disambiguate: the Past Eventual and… the Present Unreal. Like fa-, it may, when present, indicate the beginning of the apodosis, since this is not systematic.

In summary, it appears 1) that various forms of apodoses (yafʿalu, sa-yafʿalu and faʿala) appear in the hypothetical systems in iḏā as in law, 2) that the dichotomy iḏā/in-Potential vs law-Unreal is no longer as strict as it was, and 3) that fais not obligatory in cases classically thought to require it, by Pierre Larcher (Larcher, 2003b). It is also noteworthy that Moïnfar quoted by Abi Aad, 2001 : 107, suggests distinguishing between the Present Unreal in law yafʿalu… yafʿalu (law tadrusu tanǧaḥu "If you were studying you would succeed") and the Past Unreal in law faʿala… la-faʿala (law darasta lanaǧaḥta "If you had studied you would have succeeded"). However, it was impossible for us to verify this assertion.

13. About "segmentator" expression and the introduction in arabics studies of the notion of segmentation from Charles Bally, see Larcher, 2006. 14. Which seems to corroborate Taha, 1995 : 180-182 quoted by Ryding, 2005 : 671. 15. On the presence of fain CA, see among others Ibn Hišām, 1989/IV : 113 and what follows;Zamaḫšarī (Al-), 1999 : 417;[START_REF] Larcher | Subordination vs coordination "sémantiques". L'exemple des systèmes hypothétiques de l'arabe classique[END_REF][START_REF] Ayoub | Corrélation et rupture modales. Formes verbales et particules énonciatives dans les hypothétiques en arabe littéraire[END_REF] just as lafor law seems now to be optional. What do the Modern Arabic grammars say about all of this ?

What the Modern Arabic grammars show and what they do not : the grammars' descriptive inadequacy

As we can see, all this contrasts in a very singular way with what is taught (and then learned and reproduced and taught again...). This is "normal" in the case of the presentation of the normalized expression of the conditional put forward by the modern grammars of Classical Arabic, like Haywood and Nahmad (Haywood et Nahmad, 2001 : 290-300). These mention the classic dichotomy in/iḏā-Potential vs law-Unreal, indicate the predominance of in over iḏā and note, in regard to the verbal forms, the exclusive presence of the perfect in the law systems (or very rarely of the imperfect indicative) while noting the classical verbal possibilities in the case of in 16 . It is the same for Moïnfar, as quoted by Abi Aad 17 (see above footnote 12), [START_REF] Kouloughli | Grammaire de l'arabe d'aujourd'hui, Paris, Pocket[END_REF][START_REF] Neyreneuf | Grammaire active de l'arabe[END_REF] or moreover, as expected, the latest French teaching manual for Arabic, Kullo Tamâm [START_REF] Tahhan | Kullo Tamâm[END_REF], which continues with a classical presentation and whose perspective is more prescriptive than descriptive. On the contrary, that these phenomena are more or less unknown in modern grammars of Modern Arabic, grammars that aim to be more descriptive of a modern, concrete state, is what seems strange.

Beeston (Beeston, 2006 : 94-97) does record the replacement of in by iḏā. For him the arabic conditional clauses are not marked by the verb form, but by the particle used. Therefore, he remains with the classical dichotomy. Without mentioning in, he only shows perfects or jussives and reports the systematic nature of fain the case of system "breaks". For law, he once again only records perfects and then notes the ambiguity between the Present and Past Unreal. Finally, the occurrence of lais presented as optional.

Holes (Holes, 2004 : 292-299) seems to note, but without making it systematic, that iḏā accepts in apodosis verbal forms other than the simple and classical faʿala (Holes, 2004 : 296-297). According to him, "The salient features of conditional sentences in MSA are the sequence of verb forms used and the particles used to introduce the conditional clause (the protasis) and, in some types of sentences, the answering clause (the apodosis) (Holes, 2004 : 293). Nevertheless, the author continues: "But, unlike English, the type of conditionreal, possible or unrealis signaled chiefly by the particle used to introduce the conditional clause, rather than verb form per se" (Holes, 2004 : 293) 18 . He then concludes: "Thus the different shades of probability of a conditional clause being fulfilled are signaled in written Arabic by the choice of particle and not, as in English, by the form of the verb" (Holes, 2004 : 294). Yet he no longer speaks of MSA but of CLA (Classical Arabic). Going back to MSA, he states that "the reality is different" (Holes, 2004 : 295). He does record that in is now mainly used in concessive clauses (wa-in) or subordinate clauses like … -in faʿala/lam yafʿal -… (Holes, 2004 : 295). Therefore it is iḏā which replaces the latter in the expression of Potential (Holes, 2004 : 296) 19 , and the author then retains the classic dichotomy iḏā/in-Potential vs law-Unreal (Holes, 2004 : 296). However, he records, in the case of iḏā, apodoses in faʿala, yafʿalu and fa-sawfa-yafʿalu, which is 16. For a look at the possible verbal combinations in the field of the classical operator in, see Zamaḫšarī (Al-), 1999 : 416. 17. Moïnfar, 1973 : 123, 129. 18. According him Real seems to signify Eventual, possible, Potential and unreal Unreal 19. That the author calls "open" conditional.

rarely the case in other grammars, and links it to the Arabic dialects. He notes that in these: a. the colloquial equivalent of in has almost disappeared, relagated to expressions such as in šāʾa llāh, to the benefit of the colloquial equivalent of iḏā, and b. the verbal sequence of Potential systems (Holes, 2004 : 298) 20 is essentially faʿala… yafʿalu (Holes, 2004 : 298) which is ambiguous without a context (Holes, 2004 : 298). Schulz, Krahl, and Reuschel (Schulz et al., 2008 : 362-376) record the relegation of in to second place after iḏā, and retain the classic dichotomy iḏā-Potential (Schulz et al., 2008 : 362 onwards) vs law-Unreal (Schulz et al., 2008 : 366 onwards). Furthermore, if they record structures such as fa-sa/ sawfa-yafʿalu as apodoses of iḏā systems, they only mention pefects in the case of law. On the other hand, their presentation systematizes the emergence of fain the classic cases (lan, qad, laysa, saand sawfa, and inna + [pro]name) (Schulz et al., 2008 : 363-364). The authors thus finally show the non-systematic nature of la-. The examples seem mostly invented rather than authentic.

Buckley (Buckley, 2004 : 540, 668, 731-750) also presents the classic dichotomy in/iḏā-Potential vs law-Unreal, but he paraphrases some iḏā faʿala… faʿala systems in the Present Unreal. Moreover, he records for the iḏā systems apodoses of varied verbal forms (yafʿalu, sa-yafʿalu and faʿala), and points, by use of an example, to the occurrence of a lain an iḏā system (Buckley, 2004 : 737). Concerning law, he only gives examples of law faʿala… faʿala systems (except for two law yafʿalu… la-faʿala, one paraphrased in the Present Unreal, and the other in the Past Unreal), but nevertheless specifies that lais not systematic in these cases. Furthermore, his translations reproduce the usual ambiguity between the Present Unreal and the Past Unreal. It is only under the section devoted to the presence of fathat the author offers three examples of apodosis of law systems that are not in faʿala: one is in fa-lan yafʿala, the second one is in fa-sawfa yafʿalu, and the last is in fa-inna-hu sa-yafʿalu. Regarding law, no apodosis in yafʿalu is therefore reported. Finally, the appearance of faconforms to classical rules on the issue, but it is perceived by the author as non-systematic (Buckley, 2004 : 748).

Badawi, Carter and Gully (Badawi et al., 2004 : 40, 623-624, 632-670) note the disappearance of in not only to the benefit of iḏā but also of law; these authors say that CA law 'if (unreal)' has expanded to cover some of the functions of in 'if (real)' as the latter falls increasingly into disuse" (Badawi et al., 2004 : 636, 647). They record moreover the use of iḏā in syntaxes imitating that of law 21 . In doing so, the authors add nuance to the sacrosanct classic dichotomy. In law systems linked to the Unreal, the verbal forms given by authors for the protasis and apodosis are those of the imperfect. Thus they have the law faʿala… faʿala syntax retain its classic ambiguity between Present and Past Unreal (Badawi et al., 2004 : 645). However, concerning the law systems that are "synonymous of in" the apodoses can be paraphrased in (fa-)sa-yafʿalu (Badawi et al., 2004 : 647). The authors therefore present two verbal forms for law: faʿala and (fa-)sa-yafʿalu. For iḏā, they offer three different verb forms for the apodosis: faʿala, yafʿalu and sa-yafʿalu (Badawi et al., 2004 : 653-654). Fais presented as quasi-systematic with iḏā under the same conditions as with the classical in. Lais, according to the authors, generally present in law systems.

Alosh (Alosh, 2005 : 270-272) reproduces the classic dichotomy iḏā/in-Potential vs law-Unreal, noting that iḏā should be followed by a perfect verb and that its apodosis can be either faʿala or yafʿalu. The author paraphrases the two syntaxes in the same way (Alosh, 2005 : 218). No mention is 20. Called "open" conditional sentence[s].

21. Badawi et al. assert that iḏā then has the "same syntax and sense as law" (Badawi et al., 2004 : 656). Nevertheless, as we will see thanks to the data provided by our corpus, this statement is true only if we add "when considered in classical Arabic where law faʿala… la-faʿala is neutral as to meaning between Present and Past Unreal "or if we specify "has the same syntax as the Classical Arabic Unreal's law and the same sense as the Modern Standard Arabic Present Unreal's law". made of fa-. As to law, the system is presented as fixed in the form law faʿala… la-faʿala with the occurrence of laobligatory. Thus the author does not record apodoses in sa-yafʿalu, neither for iḏā nor for law.

Ryding (Ryding, 2005 : 671-676) also remains very "classic" in her presentation of the traditional dichotomy iḏā/in-Potential vs law-Unreal. She writes: "Arabic uses different particles to express possible conditions and impossible conditions" (Ryding, 2005 : 671) with supporting references in her footnotes that border on modernity: Peled, Cantarino, Blachère and Gaudefroy-Demombynes, Fischer… [START_REF] Ryding | A Reference Grammar of Modern Arabic[END_REF] footnote 2 : 671). She only offers, for law, faʿala/lam yafʿal structures, in protasis as well as in apodosis (Ryding, 2005 : 675). She indicates the general, but not the systematic, nature of la-. Nothing is said about the negation of the apodosis, nor about the presence -or not -of lain this case. Iḏā is presented as having nowadays replaced in. Concerning iḏā, whose protasis is in faʿala, she specifies that a rupture (Ryding, 2005 : 672) may appear in the apodosis, i.e. a tense other than faʿala. However she does only give three examples: a defensive, a injunctive and a prepositional phrase introduced by fa-(fa-ʿalay-ka an). In doing so, she does not present, for iḏā, apodoses in yafʿalu nor in sa-yafʿalu.

Hassanein (Hassanein, 2006 : 98-100) certainly registers a syntax iḏā faʿala… fa-sa-yafʿalu, but translates it in the same manner as iḏā faʿala… faʿala. In addition to replicating the strict dichotomy iḏā/in-Potential vs law-Unreal, she does not mention, in the case of law, any apodosis in yafʿalu nor in sa-yafʿalu. She contents herself with saying that law is followed by a perfect verb (protasis), but does not specify anything for the apodosis while her examples, nevertheless, only show for the latter faʿala. Still in the case of law, lais presented as necessary before a perfect verb and as optional before a negative form which, according to her, must be in mā faʿala, not in lam yafʿal. Finally, fais presented as systematically used in the case where the apodosis is neither a perfect nor a jussive22 and as obligatory where there are no perfect verbs for iḏā (Hassanein, 2006 : 98). We should note that her examples also do not seem genuine.

McCarus (McCarus, 2007 : 149-152) does not, for his part, record the apodosis in anything other than faʿala for iḏā/in and law, and adheres to the strict dichotomy iḏā/in-Potential vs law-Unreal. Like others, he notes that iḏā has taken over in; the former signifies more realisable conditions, while in would suggests a hypothesis in the true sense of the word. Here again, the examples are not authentic.

Conclusion:

The authors of these Modern Arabic grammars 1) retain more often the classical dichotomy iḏā/in-Potential vs law-Unreal, 2) show, in the vast majority of cases, apodoses only in faʿala and almost never in yafʿalu nor in sa-yafʿalu, or if it is so, almost never for law, but only for iḏā, and 3) consider, more generally, that lais not systematic and that fa-, on the contrary, is in cases where the verb of the apodosis is neither a perfect nor a jussive introduced by the lam of negation. Of these grammars, Buckley and Badawi et al. especially stand out due to the number and authenticity of their examples, which allow us to see a reality that is far more complex than any of the others show. Most of the latter simply content themselves to recording the replacement of in by iḏā without offering any system that could be descriptively adequate to the reality of the uses. Nonetheless, Buckley, Badawi et al. do not actually organize these structures into a coherent system and are thus forced to interpret identical structures in various different ways 23 .

As we see, the first observations from our corpus do not correspond to the majority of the descriptions of hypothetical systems of Modern Arabic given by the recent grammars of the Modern Standard language. We will now detail our observations in order to identify a system that seems to us coherent.

Data Presentation

I. The Eventual

For the Eventual, iḏā (33 cases out of 37), with iḏā mā (4/37), is indeed the majority operator, and thus continues to express the Eventuals classically.

Present Eventual: iḏā faʿala… yafʿalu

That this syntax describes the Eventual will be highlighted by a first example with iḏā mā, which operates in the same way as iḏā, iḏā mā faʿala… yafʿalu in being presented as the equivalent of ʿinda-mā yafʿalu… yafʿalu which is, itself, only interpreted as an Eventual:

(1) wa-id̠ ā mā faʿala aḥ adu-hum, fa-inna-hu yuḍ ṭ arru bi-l-ṭ abʿi ilā rtiqāʾi l-daraği, wa-ʿinda-mā yablugu̇l-ṭ ābiqa l-aḫ īra takūnu ḫ uṭ uwātu-hu qad abṭ aʾat mina l-taʿabi (Al-Laǧna : 53) "As soon as one of them does/has done it then he is naturally obliged to climb the stairs, and when he reaches the top floor, his steps are slowed with fatigue"

(2) iḏā ʿalā ṣawtu raǧulin yaṭlubu l-isrāʿa li-talbiyati ṭalabi-hi, huna yanẓuru ilay-hi wa-yušīru bi-raʾsi-hi išaratan wāḥidatan mūgazatan : " -mšī.." (Al-Zaynī : 240) "If/when someone raises his voice urging him to serve him, he looks at him at once and makes a simple gesture of the head: "Back off. . . ! ""

(3) ammā iḏā qārabati l-intihāʾa fa-inna l-šarikata taqūmu bi-iʿādati taʿbiʾati-hā dāḫila akyāsin taḥmilu -smahā wa-taʾrīḫa iʿādati l-taʿbiʾati (Ḏat : 271) "And when they are close to being outdated, the company repackages them in bags bearing its name and the date of the repackaging" (4) wa-iḏā fuṣila l-hindī lā yaškū wa-inna-mā yabḥaṯu fī hudūʾin ʿan ʿamalin āḫara (Warda : 17) "And when the Indian is dismissed, he does not complain and simply searches quietly for another job"

(5) iḏā tašāǧarat imraʾatun maʿa zawǧi-hā lā taqṣidu bayta ahli-hā šākiyatan inna-mā talǧaʾu ilay-ka wāṯiqatan bi-anna mā laḥiqa bi-hā min ẓulmin sa-yazūlu (Al-Numūr : 33) "When a woman quarrels with her husband, she does not go to his parents complaining, but she takes refuge with you, confident that the injustice towards her will pass" (6) iḏā takallamat tanfatiḥu l-nafsu kulliyyatan (Al-Ǧāziya : 70) "When she speaks, the soul opens itself up completely" If Badawi et al. have noted the existence of this syntax, they have at least suggested that its meaning is not obvious. Thus, they translate this syntax as both as a Potential (Badawi et al., 2004 : 654) and as an Eventual (Badawi et al., 2004 : 661). For some of their examples, we would have opted for an Eventual. This is particularly the case with maʿa ḍālika (sic) iḏā ḫaraǧa minhā fī riḥlatin aw ziyāratin ilā makānin mā min alʿālami narāhu lā yaḫšā l-iḫtināqa ka-asmāki l-baḥri min hāḏā l-ḫurūǧi, however, if he leaves it for traveling or to visit any place in the world we find [lit. "see"] that he does not fear being stifled like the fish of the sea do in this way (Badawi et al., 2004 : 653-654). For our part we translate as follows: "however, when he leaves it for traveling or to visit any place in the world, we see that he does not fear being stifled like the fish of the sea do". Nevertheless, it remains that of the 35 systems in iḏā whose apodosis is in (fa-)yafʿalu form, 22 actually designate the Eventual, but 13, that is to say 37.14%, designate the Potential. It seems here that only the context can help to distinguish between iḏā faʿala… yafʿalu for the Present Eventual and iḏā faʿala… yafʿalu for the Potential.

Past Eventual: iḏā faʿala… faʿala

Before considering the majority syntax for the expression of the Past Eventual, and to link this point to the previous point, we should note that this expression can be formed, as in Classical Arabic, with the syntax of a present Eventual placed in the field of a perfect verb (most commonly kāna). This is shown by the following example, which is in parallel here also with ʿinda-mā but, this time, in a q if p sequence:

(7) wa-kānati l-bahǧatu tušiʿʿu fī kiyānī ʿinda-mā yaqaʿu naẓarī ʿalay-hā wa-yaǧtāḥu-nī l-yaʾsu iḏā lam aǧidhā (Warda : 56) "Delight would irradiate in my being whenever my eye fell on her and despair would overwhelm me when I could not find her" Another example, this time from Badawi et al., also puts into play kāna and its sisters to express habitual actions: kuntu iḏā waṣaltu ilā l-munḥanā ʿinda furni l-ḥaǧǧi Nāṣif altafitu ilā l-ḫalfi, whenever I used to reach the corner at Hajj Nasif's bakery I would look behind me (Badawi : 662). Now let us see the alternative offered by the system: (8) raʾaytu-hum yuzīḥūna l-aṯqāla ʿan dawābbi-him kulla-mā tawaqqafa bi-him al-sabīlu l-abadī ḥattā iḏā faraġū min dawābbi-him wa-ṭmaʾannū ʿalā ḏawī-him habbū li-naǧdati aqraba man ǧāwara-hum li-yuʿīnū ʿalā amri-hi (Malakūt : 130) "I saw them removing the loads from their animals at each time they encountered an interruption along the Everlasting Way. So much so that when they had finished with their cattle and were assured of their relatives, they rose to rescue the nearest neighbor and help him with his business" (9) wa-kāna iḏā arāda l-ṭaʿāma taṭahhara la-hu ka-taṭahhuri-hi li-l-iḥrāmi (Ḥaddaṯa : 96) where the auxiliary verb kāna provides the interpretation of the Past Eventual 24 : "And when he wanted food, he would purify himself for it as he would for the state of ritual consecration" (10) iḏā taḥaddaṯa l-sukkānu ʿan buṭūlati-him taḥaddaṯū bi-basāṭatin wa-tawāḍuʿin muḍhilayni ! maʿa annahum samaw bi-buṭūlati-him ilā mustawā l-maṯali l-sāʾiri (Al-Ǧāziya : 37) "When people spoke of their heroism, they did so with disconcerting simplicity and humility! And this despite the fact that they have raised their heroism to the level of a proverb!" 25 In modern Arabic, however, the emergence of a new layer does not necessarily cancel de facto the previous one. Then, interpretation of data remains difficult and some iḏā faʿala… faʿala may be interpreted as Present Eventual. Thus, Badawi et al. interpret iḏā āʿǧaba-nī kitābun tamannaytu law iqtanaytu-hu as a Present Eventual: "if a book pleases me I wish I owned it" (Badawi et al., 2004 : 653); we would have read it in the past: "When a book pleased me I wished I owned it" 26 . Similarly they interpret as Present Eventual both iḏā faʿala… yafʿalu and iḏā faʿala… faʿala: iḏā ḫaraǧati l-asmāku mina l-baḥri tamūtu bi-l-iḫtināqi, "when the fish come out out the sea they die of asphyxiation" (Badawi : 661) and iḏā samiʿa aḏāna l-faǧri fī hudūʾi l-layli ṭariba l-qalbu, "when he hears the dawn prayer call in the calm of the night his heart rejoices". For our part we would have interpreted the second as having a Past Eventual meaning: "Whenever he heard the dawn prayer call in the calm of the night his heart would rejoice".

II. The Potential: iḏā faʿala… (fa-)sa-yafʿalu and law faʿala… yafʿalu

In, despite a tentative incursion into the Past Unreal field, continues only to express the Potential, regardless of the syntax of the verb forms that follow it. Concerning the latter, it is essentially this that sets the tone, and not the syntax of the apodosis. It represents nevertheless only 19.23% of Potential systems; iḏā, as was pointed out long ago, now mostly expresses the Potential with 67.95%. However, what to our knowledge is almost never mentioned, except quite remarkably by Badawi et al., is the significant appearance of law in this hypothetical status (12.82%).

1. In: variable syntax 24. The French version presents a printmistake saying Present Unreal.

25. The context is that of a narrative in the past: maʿa anna l-qaryata kāfaḥat, ṣamadat, waqafat fī waǧhi l-ẓulmi, baytan baytan, fardan fardan, lākin bi-dūni ḥiqdin. Al-šāmbiṭu nafsu-hu ʿinda-mā umira bi-l-istiqālistaqāla. Wa-lammā ǧāʾa l-istiqlālu wa-umira bi-l-ʿawdati ʿāda… Iḏā taḥaddaṯa l-sukkānu… ("despite the fact that the village fought, resisted, stood up in front of the oppression, house after house, individual after individual, but without hatred. The « garde-champêtre » himself when ordered to resign resigned. And when independence came and they ordered him to return he returned… When people were speaking… 26. As we shall see, it could also be interpreted as a Present Unreal, meaning "If I liked a book, I would wish I owned it".

The syntax remains classic, with faʿala/lam yafʿal or yafʿal/lā yafʿal27 in protasis as well as in apodosis and with the appearance of fain the beginning of the apodosis as it appeared in Classical Arabic, except that an innovation has shown up: an apodosis in sa-yafʿalu and its negative counterpart lan yafʿala both juxtaposed with the protasis without being separated from the latter by fa-. This innovation is certainly imitated from the syntax of European languages.

(11) in tazawwaǧtu bi-ki uʿṭi-ki kulla mā yumkinu an yaḍumma qalbī min ḥubbin (Al-Ǧāziya : 71) = in faʿala… yafʿal "If I marry you, I'll give you all the love my heart can contain" (12) wa-in lam yastaṭiʿ kabīru l-baṣṣāṣīna l-ʿuṯmānliyyīna hāḏā fa-lā yastaḥiqqu manṣiba-hu (Al-Zaynī : 230) = in faʿala… fa-yafʿalu "And if the Ottoman Grand Master can not do this, then he does not merit his title" (13) yataḫallā ʿani l-dunyā li-ann-hu in lam yataḫalla ʿani l-dunyā fa-inna l-dunyā sawfa tataḫalla ʿan-hu (Malakūt : 61) = in faʿala… fa-inna-hu sa-yafʿalu "He gives up life for if he does not give it up, it will give him up" (14) fa-in lam yataḥaqqaqi l-amalu hullu-hu baʿda ḏālika fa-l-masīratu lam tantahi (Ḥaddaṯa : 31) = in faʿala… fa-inna-hu faʿala "If all hope is not realized after that, then the march is not over" fa-, while assuming a role of focalisation on the logical subject of the verb, forbids the interpretation of the verb in a future sens. If we had have lam tantahi l-masīratu, the meaning would have been "the march will not end", but in the case of fa-lam tantahi l-masīratu or of fa-l-masīratu lam tantahi, we then have "the march is not ended/the march, it is not ended" (15) in arkaba bna-hu ʿalā baġlatin uḫrā fa-lā yaʾmanu ʿalay-hi min ʿaṯratin aw šayʾin yuḫīfu-hā fa-taqfiza watarmiya bi-hi ilā l-hāwiyati (Al-Ǧāziya : 189) "If he raises his son onto another mule, then there is nothing to prevent it stumbling or being frightened by something, and then falling and rushing him toward the abyss" (16) in naǧā ʿinda bābin lan yanǧuwa ʿinda l-abwābi l-uḫrā llatī taḥrusu-hā l-fataḥātu l-mawǧūdatu ʿinda labwābi l-tāliyati la-hā (Warda : 36) = in faʿala… sa-yafʿalu where fadoes not occur.

"If he escapes from one door, he will not escape from the others which are guarded by the loopholes that are at the doors which follow them" (17) in ḏahaba huwa sa-yaʾtī man yaḫlufu-hu (Al-Ǧāziya : 187) = in faʿala… sa-yafʿalu where fadoes not occur.

"If he goes, there will come one who will succeed him"

iḏā: iḏā faʿala… (fa-)sa-yafʿalu

If Badawi et al. clearly show that iḏā can be a "pure conditional" that is to say a Potential, they offer only a single syntax, iḏā faʿala… faʿala (Badawi et al., 2004 : 653). However, in our corpus, we have already reported that iḏā faʿala… faʿala belonged, according to the context, either to the Past Eventual or to the Present Unreal. Thus, none of the examples placed under the heading "iḏā 'if' as a pure conditional" account what we see in our corpus, and these examples should, we believe, be reinterpreted in the Present Unreal (see below). It is in the next section, entitled "variant forms of verb in apod.", a section which appears to complete the first, that we find two other syntaxes: iḏā faʿala… yafʿalu and iḏā faʿala… sa-yafʿalu which also are paraphrased by Potentials. Concerning iḏā faʿala… yafʿalu, if it indeed expresses the Potential (37.14%), it also and more importantly expresses, as we have shown, the present Eventual (62.86%). There is therefore, on pages 653-654, only one example which is indeed a Potential and whose English translation respects the meaning. This example also happens to have an iḏā faʿala… sa-yafʿalu syntax, in accordance with what our corpus has allowed us to demonstrate. This is the statement: wa-iḏā saʾalta-nī l-āna li-māḏā wāṣaltu l-taradduda ʿalā ʿiyādati-hi baʿda an ṣāraḥa-ni bi-ḏālika wa-li-māḏā lam aḏhab l-ʿiyādata ṭabībin āḫara sa-yakūnu ǧawābī…, and if you ask me now why I continued to go to his clinic after he clearly told me about that and why I did not go to some other doctor's clinic, my answer will be… That we understand the same way.

(18) wa-iḏā taǧāhala-hu fa-la-sawfa yaʿrifu kayfa yahtadī ilā awwali l-ṭarīqi ka-mā htadā l-kaṯīrūna (Riǧāl : 46). Note the presence of an infix lām between faand the future particle sawfa. This lām probably has a corroborative value (lām al-tawkīd) "And if he pretends to ignore it, then he will necessarily find how to reach the beginning of the road as did so many others 28 " (19) wa-iḏā ṯabata anna-hu ẓalama maḫlūqan, sa-yaqbalu ayya qiṣāṣin yaqaʿu ʿalay-hi ka-ayyi maḫluqin (Al-Zaynī : 249) "And if it turns out that he has been unjust towards anyone, he will be ready to pay for it as a simple mortal 29 " (20) wa-iḏā kunta ṭayyiban (Al-ʿĀšiq : 40) "And if you're good, then you'll love" (21) iḏā ḏahabtumā l-laylata li-taḫrībi qabri l-waliyyi wa-šaǧarati-hi fa-sa-yaḥkumu ʿalay-kumā bi-l-maḥqi (Al-ʿĀšiq : 92) 30 28. It is not a question here of a concessive clause, which might imply the presence of the prefixed wato iḏā, but the context requires us not to consider this waas anything other than a coordination conjunction between two hypothetical statements: "If Zakariyā helped him it would be better, and if he pretends to ignore it, then he will necessarily find how to reach…" 29. Same remark as for example (18). "If you both go tonight to destroy the tomb of the saint and his tree, then he will condemn you to death" (22) wa-iḏā lam taḏhabi l-āna fa-sa-aṭlubu mina l-mumarriḍi an yulqiya bi-ka ilā l-šāriʿi (Al-Laǧna : 117) "And if you do not leave now, I'll ask the nurse to put you out on the street" (23) iḏā aradtum tarkanā lan aʿtariḍa (Warda : 313) "If you want to leave, I will not be opposed to it" (24) iḏā lam taʿtarif sa-aḍribu-ka bi-l-ḥiḏāʾi wa-aḍribu ahla ḥārati-ka (Al-Numūr : 37) "If you do not confess, I will smite thee with the shoe and beat the people of your neighborhood" Note that the second verb, also interpreted as a future, is not introduced by a particle of the future. The latter is unnecessary because the verb is placed in the field of the future particle which introduces the first one and is coordinated with it by wa-.

(25) iḏā lafafta-hu ḥawla ʿunqi-ka sa-yakūnu rāʾiʿan (Ḥiwār : 26) "If you tie it around your neck, it will be great"

The hypothetical system is in the field of a past tense verb:

(26) awhama-hu baʿḍu l-sukkāni anna l-Ǧāziyata sa-taḥḍuru l-zaradata, wa-anna-hā iḏā raʾati bna-hu sa-taʿšaqu-hu fī l-ḥāli (Al-Ǧāziya : 26) "Some residents made him believe that Al-Ǧāziya would be present at the party, and that if she saw her son, she would fall in love with him immediately" (27) fa-l-buyūtu fī sīrati-hā l-ūlā lam tanhaḍ ʿani l-arḍi li-tataṭallaʿa ilā l-samāwāti li-takūna muǧarrada maʾwan li-anna-hā iḏā faqadat huwiyyata-hā l-aṣliyyata llatī anwī an uḥaddiṯa-kum ʿan-hā fa-lan yakūna bayna-hā wa-bayna l-maṯwā farqun (Malakūt : 99) "The houses in their early development did not rise from earth to aspire to heaven to be simple shelters, because if they lost their original identity, about which I want to talk to you, there would be no difference between them and a mere dwelling-place"

Law: law faʿala… yafʿalu

Badawi et al. note, in a specific section, that law may now have, in Modern Written Arabic (MWA), the meaning of Potential: law 'if' can occur with the sense of 'in 'if', thus losing its conterfactual quality 31 . But of the five quoted examples, we exclude the first, non doubly verbal, which does not purtain to our purpose. Of the remaining four, the authors present to us the following syntaxes: 1) law faʿala… fa-lan yafʿala, 2) law faʿala… lan-yafʿala, 3) law faʿala… yafʿalu and 4) law faʿala… la-faʿala. We have, that is to say, three verbal forms of apodosis: (fa-)sa-yafʿalu, yafʿalu and faʿala. Howev-er, only examples 1) and 2) are actually paraphrased as Potential while examples 3) and 4) are put in the Present Unreal. We agree with these authors for examples 1) and 2), even though we shall insist that the law faʿala… (fa-)sa-yafʿalu syntax, as it is the case for its English and French equivalents, presents a case of ambiguity between the Potential and the Present Unreal (See below, footnote 121 and Table 3, footnote 156) [New footnote : This is not in the French version and presents a nuance of what was said there]. Then we propose to reinterpret the syntax law faʿala… (fa-)sa-yafʿalu of examples 1) and 2) as Present Unreal, as we shall see 32 .

The authors only distinguish between Eventual (Temporal), Potential (Conditional) and Past Unreal (Counterfactual). They are therefore led, in case 4), to classify as Conditional, "with an unlikely future sens, retaining the syntaxe used in contrefactual sentences (Badawi : 647), a sentence like law utīḥa li-l-šuʿūbi l-ʿarabiyyati an tusammiya l-zaʿīma l-akṯara šaʿbiyyatan fī hāḏihi l-āwinati la-aǧābat… which they translate as a Present Unreal: if the Arab people were given [the opportunity] to name the most popular leader during these times, [then] they would answer… (Badawi et al., 2004 : 648) 33 .

This leaves then only one example that Badawi et al. classify as Conditional, and which we believe is, both syntactically and semantically, actually a Potential, but which the authors translate as if it were an Present Unreal. This example is the following: law ʿalimat bi-l-amri yumkinu an taṭluba min-hu an yuṭalliqa-hā, if she were to find out about the matter she could [lit."it would be possible that"] ask him to divorce her, which we translate for our part as a Potential: " if she finds out about the matter she will [lit."it will be possible that"] ask him to divorce her". Let us now give a few examples from our corpus.

(28) law wuǧida bayna l-ḥabībi wa-mubtaġā-hu ʿaqābātun yaʾmulu hadma-hā, aǧʿalu min-hā mustaḥīlan lā yumkinu taḫaṭṭī-hi (Al-Zaynī : 287) "If there should occur between the lover and his desire obstacles which he hopes to destroy, I'll create an obstacle that can not be overcome" (29) law naẓarnā ilā dāʾirati l-mašriqi l-ʿarabiyyi l-muntiǧati l-raʾīsiyyati li-l-bitrūli, wa-llatī bi-hā 60 bi-l-miʾati min kulli mā ladā l-ʿālami mina l-bitrūli naǧidu anna asāsan raʾīsiyyan min istrātīǧiyyati-hā huwa… (Ḏāt : 32) "If you look at the Arab East, the leading oil producer and holder of 60 per cent of the world reserves, we see that the foundation of its strategy is…" (30) law qaṭaʿnā l-šaǧarata wa-dafannā raʾsa l-waliyyi fa-laʿalla-nā nastariddu abṣāra-nā wa-asmāʿa-nā (Al-ʿĀšiq : 92) 32. We propose then to translate wa-law saʾalta aḥada-hum min ayna atayta bi-hāḏā l-ḥaqqi… fa-lan taǧida raddan, and if you ask one of them where did he get this truth from… [then] you will not find an answer (Badawi : 647) as and if you asked one of them where did he get this truth from… [then] you would not find an answer and to translate law daḫalnā sibāqa tasalluḥin nawawiyyin fī l-minṭaqati lan yantahiya, if we enter a nuclear arms race in the region it will never [lit. will not] end (Badawi et al., 2004 : 647) as if we entered a nuclear arms race in the region it would never [lit. would not] end.

33. However, given the syntactic regularities offered by our corpus, it seems possible to infer that this sentence, whose syntax is law faʿala… la-faʿala falls within the Past Unreal (and not the Present) and therefore its meaning is: "if the Arab people had been given [the opportunity] to name the most popular leader during these times, [then] they would have answered… See below.

"If we cut down the tree and bury the saint's head, then maybe we will regain our vision and our hearing" Here, laʿalla, which marks the modality of possibility, confirms that law does mark the Potential.

(31) law ṯabata hāḏā taqaʿu kāriṯatun (Warda : 112) "If it actually happens, it will be a disaster" (32) law kāna raʾsu l-ǧanīni fī l-aysari yakūnu waladan (Warda : 361) "If the head of the foetus is on the left side, it will be a boy"

III. The Unreal

Concerning the Past Unreal it is law that classically expresses it (98.86%). As for the Present Unreal, if it is essentially and classically expressed through its historic operator law (40.74%), it is also through iḏā (mā) which appears in force in this hypothetical category (59.26%), as Badawi et al. timidly note (Badawi et al., 2004 : 656) 34 . Moreover, it appears that of the 47 systems in iḏā faʿala… faʿala, which are classically linked to Eventual and Potential, only 8 really belong there, and then only for the Potential. Therefore only 17.02% of iḏā faʿala… faʿala actually describe a hypothetical category conforming with that described in the Classical Arabic grammars, while 82.98% of them describe the Present Unreal -which does not conform! Further, as we reported earlier, lais not systematic with iḏā, appearing only in 4.16% of the cases without further input than to emphasise the beginning of the apodosis. On the other hand, it appears much more regularly in the case of the law Past Unreal , in fact in 87.35% of the occurrences.

1. Present Unreal: iḏā faʿala… (la-) faʿala and law faʿala… (fa-)sa-yafʿalu a. iḏā Of examples given by Badawi et al., three must be, according to us, reinterpreted as Present Unreal. It requires only one example to show this: iḏā kāna kātibu l-inšāʾi mulimman bi-miṯli haḏihi (sic) l-luġāti kāna aqdara ʿalā murāsalati-him, if the secretary of the chancellery was conversant with such languages as these he was more able to correspond with them [namely foreign people] (Badawi et al., 2004 : 653) 35 . This, as we see, makes no sense. The translation given by Badawi et al. indeed does not work, for a semantic reason on one hand (and with regard to common sense, because what the secretary of the chancellery has been reproached about is precisely his lack of any foreign language…) and for a syntactic reason on the other hand. This is because, in order to interpret (despite the meaning) the second kāna as an imperfect or preterite, it would necessarily have required that fabe prefixed to it; fawhich in these cases, never fails to distinguish a Perfect form verb/[or a] jussive introduced by lam from a verb with a Perfect form and sense. Thus this sentence can not be understood as anything other than the following: "If the secretary of the chancellery was more conversant with such lan-guages [but this is not the case] 36 , he would be more able to correspond with them". We will give other examples of this Present Unreal syntax in iḏā.

(33) al-wāḥidu hunā lā yarā zawǧata ṣadīqi-hi lākin iḏā qābala-hā fī Lunduna saharū maʿan (Warda : 241) "Here, a man does not see the wife of his friend, but if he met her in London, they would go out together"

Here, the restriction implied by lākin ("but"), that is to say the refutation of a present or future reality, indicates the interpretation as an Unreal.

(34) iḏā waḍaʿta fī-hā gamalan taḥawwala ilā ṭayrin min maʿdinin yaṭīru afḍala min ayyi ṭayrin (Al-Numūr : 60) "If you put a camel there, it would turn into an iron bird flying better than any bird"

In this example, it is the absurdity that proves the Unreal status of the sentence.

(35) fa-iḏā aḍafnā ilā ḏālika anna l-duwala l-ʿarabiyyata bi-ḥukmi īmāni-hā bi-l-risālāti l-samāwiyyati wa-ladyāni yaǧʿalu-hā aqraba ilā l-ġarbi mina l-šarqi la-waǧadnā anna ǧamīʿa hāḏihi l-asbābi tanfī… (Ḏāt : 32) 37 "By adding that the Arab States, by virtue of their attachment to the revealed [Abrahamic] religions, feel closer to the West than the East, we would see that all of that would completely deny…" It is clearly here the presence of the lām in imitation of the syntax of law which allows to interpret this as an Unreal, and specifically, because of iḏā, as a Present Unreal.

(36) iḏā sāʿada-hu Zakariyā kāna ḏālika afḍala (Riǧāl : 46) "If Zakariyā helped him, it would be better" 36. We here draw attention to the fact that in languages like French, the phrase "S'il faisait beau, je sortirais" can be interpreted either as Potential ("If the weather is nice I will go out") or as Present Unreal ("If the weather was nice, I would go out"). It is therefore a case of ambiguity that only the context can remove. The intrinsic difference between Potential and Unreal indeed lies in the necessary existence of an implicit 'but' in the case of the Unreal ("but the weather is not/will not be nice" for the Present Unreal and "but the weather was not nice" for the Past Unreal). Then it is especially that 'but' that allows us to differentiate between the two meanings of the phrase "If the weather was nice, I would go out": the one of direct speech, with existence of a 'but', which is indeed a Present Unreal, and that of reported speech, which only has a syntactic form of Past Unreal for narrative reasons and because of the concordance of tenses, but not involving 'but', which would make it belong to the Potential. Let us consider the following sequence happening at night or in a closed room without knowledge of the weather: A.: "If the weather is nice, I will go out (Potential); B.: "What did A say? ; "C.: "He said that if the weather was nice he would go out" (Potential, because there is no 'but' that could afford to connect this sentence to Unreal). On the difference between direct speech and reported speech, see among others Abi Aad (Abi Aad, 2001 : especially 49-72).

37. The syntax is not clear. Especially, we do not know what would be the subject of yaǧʿalu-hā. Maybe there is here a misprint and it lackes an indefinite relative pronoun mā ("what makes them closer to West than to East…").

(37) fa-iḏā ḏakarta la-nā asmāʾa-hum, rubba-mā kāna li-ḏālika aṯarun fī taḫfīfi l-amri bi-l-nisbati la-ka (Al-Laǧna : 94) "If you told us their names, maybe it would mitigate your situation" (38) iḏā ḏahabtumā li-taḫrībi qabri l-waliyyi wa-šaǧarati-hi aḫaḏtu ʿalā ʿātiqī iblāġa l-šurṭati (Al-ʿĀšiq : 92) "If you were both going tonight destroy the tomb of the saint and his tree, I would take upon myself the responsibility of contacting the police" (39) fī waqtin ṭawīlin raʾā nafsa-hu ḥāmila l-ṯaqli l-fādiḥi, lā aḥada yuʿīnu-hu ʿalay-hi, ḥattā Manṣūrun ṣāḥibu-hu, iḏā suʾila ʿan aṣḥābi-hi wa-zumalāʾi-hi qāla lā fāʾidata min-hum turǧā (Al-Zaynī : 259) "For a long time he watched himself bearing the crushing weight. Nobody helped him, not even his friend Manṣūr. If anyone asked him what he thought about his friends and colleagues, he would say that there was nothing to be expecting from them" (40) fa-fāḍa qalbu l-abi bi-l-fuḍūli fa-tasāʾala ʿan sirri ʿabaṯi l-aqdāri dūna an yadiya, fa-aǧāba-hu l-ḫafāʾu: sirru ʿabaṯi l-aqdāri laysa hawan wa-lākinna-hu ǧadalun lan yaʿmala-hu illā ṣuḥbānu l-ḫāfiyati, li-anna l-aqdāra iḏā arādat bi-maḫlūqin šarran aḥyat-hu wa-iḏā šāʾati l-aqdāru bi-maḫlūqin ḫayran amātat-hu (Malakūt : 79) "The father's heart was then filled with curiosity and he questioned himself about the secret of the absurdity of fate, which he did not know. The secret then answered him: "The secret of the absurdity of fate is not a caprice but a debate that only companions of the secret will know; because fate, if it wanted to hurt a creature, would make it live, and if it wished it well, would make it die"" (41) inna-hu yā Kahlānu iḏā kariha l-marʾu l-ḥaṣra wa-l-qaṣra ṭalaba kaṯrata l-yawmi wa-štāqa l-ʿadada (Ḥaddaṯa : 150-151) "Kahlān! If Man hated confinement and restriction, he would ask for abundance of the sea and would rejoice in such plentitude" b. law (42) yā Saʿīdu anā maqṭūʿu l-amali mina l-mahdiyyi l-muntaẓari, law qāma nāṭiqu l-zamāmi, law ẓahara, law ǧāʾa mina l-kaʿbati yušhiru sayfa-hu l-ḏahabiyya, sa-yataṣaddā la-hu Zakariyā, sa-yuḥarrimu-hu duḫūla ldiyāri (Al-Zaynī : 256) "Saʿīd, I lost hope of seeing the Messiah. If he rose, if he appeared, if he came from the Kaaba, brandishing his gold saber, Zakariyā would opposed him, he would prevent him from coming to us" (43) lawi -stimarrat sa-yastaḥīlu ʿalay-nā muġādarati l-makāni (Warda : 134) "If ever [the bombing] continued, it would be impossible to go out" 38 (44) law ḍaraba-nī fa-sa-aqūlu li-abī fa-yaʾtī wa-yaḍribu-hu (Al-Numūr : 26) "If he hit me, I would tell my father who would beat him"39 2. Past Unreal a. law faʿala… (la-)(mā) faʿala Lais semi-systematicaly placed at the beginning of the apodosis, and even before the negation there of the latter (which is still, as is conventional, mā faʿala), showing structures in la-mā faʿala. The context given by the following example shows that although this syntax is linked to the Past Unreal, lākin(na) + perfect verb ("but" + past tense verb) leaves no doubt about the possible interpretation.

(45) law saʾalta-nī ʿan ḏālika la-dalaltu-ka ʿalā ṭarīqatin ǧiddi sahlatin, lā tukallifu ṯamanan! lākinna-ka fakkarta bi-ṭarīqati-ka (Al-Ǧāziya : 133) "If you had asked me about this, I would have indicated to you a very easy method, which costs nothing! But you thought according to your method"

In the same manner as above, lākinna comes here to impose an Unreal and, associated with a past tense verb (fakkarta), a Past Unreal.

(46) wa-rakaḍa l-awlādu bi-aqṣā quwwatin wa-lammā btaʿadū waqafū lāhiṯīna muḥmarri l-wuǧūhi. qāla Muḥammadun: law amsaka-nā la-ašbaʿa-nā ḍarban (Al-Numūr : 62) "The children ran as fast as they could. When they were far away, they stopped, panting, faces reddened. Muḥammad said: if he had caught us, he would have beaten us"

The context here shows that the only possible interpretation is that of the Past Unreal.

(47) wa-law kāna muʿaqqadan bi-sababi lawni-hi ka-mā qāla Haykalun mā ǧalasa bi-l-sāʿāti kulla yawmin fīšamsi (Ḏāt : 38) "If [Anwar al-Sadāt] had had a lot of hang ups about the color [of his skin], as Haykal said, he would not have spent hours in the sun every day " (48) la-qad fakkara wālidī bi-l-amri: law aǧǧara ġurfatayni wa-sakana maʿa zawǧati-hi l-kasḥāʾi fī l-ṯāliṯati iḏan la-ʿāša mā tabaqqā la-hu mina l-ḥayāti mustaqirran (Riǧāl : 41) "My father had thought about it: if he had rented two rooms and lived with his lamed wife in the third, then he would have lived out what remained of his life well installed" (49) wa-law ǧāʾa aḥadu-humā qabla l-āḫari aw baʿda-hu, la-marrati l-umūru fawqa saṭḥi ayyāmī miṯla-mā nzalaqa ālāfu l-awliyāʾi ilā nisyāni wa-lākinna-humā ǧāʾā maʿan (Al-ʿĀšiq : 77) "If one of the two had come before the other, or after him, things would have moved on just as thousands of saints have slipped into oblivion; but they came together" (50) law saqaṭat Raḫyūtu ḍāʿati l-minṭaqatu l-muḥarraratu (Warda : 315) "If ever Raḫyūt had fallen, the liberated area would have been lost" (51) wa-lākin law raʾaytumū-hum wa-taʾammaltum waǧūha-humu llatī tanḍaǧu bi-mā tusammūna-hu fī muʿǧami-kum saʿādatan la-ayqantum bi-ǧadwā riḥlati-him wa-āmantum bi-risālati (Malakūt : 128) "But if you had seen them and had seen their faces which were perspiring with what you call happiness in your language, you would have been assured of the success of your journey, you would have believed in the prophecy" (52) wa-law šāʾa la-amkana-hu an yuqaddima qiṣṣata-hu bi-miṯli mā qaddama bi-hi masrḥiyyata-hu l-Sudd (Ḥaddaṯa : 32) "If only he had wanted, it would have been possible for him to present his story as he presented his play al-Sudd [The Dam]" (53) wa-law lam takun qābilatan fī naẓari l-qaryati, la-mā aqdama l-šambīṭu ʿalā ḫiṭbati-hi li-bnihi (Al-Ǧāziya : 28) "And if she had not been fit for marriage in the eyes of the village, the garde-champêtre would not have undertaken to engage her to his son" (54) wa-law taraka-hum la-ǧāʾū-hu bi-suyūfi-him (Al-Zayni : 299) "If he had let them be, they would have come to him with their swords" b. law faʿala… kāna yafʿalu Note finally that we find in Arabic press an alternative to faʿala for the Past Unreal, kāna sa-yafʿalu, as Badawi et al. note (Badawi et al., 2004 : 645-646), solution of which we only have three occurrences in our corpus:

(55) ʿiqābun āḫaru law kāna ʿtāda-hu munḏu ṯalāṯi sanawātin la-mā kāna, al-āna ʿalā l-aqalli, yaktariṯu bi-hi, miṯla-mā yafʿalu hāḏihi l-laḥẓata (Al-ʿĀšiq : 28) "Another punishment: if he had not got used to him over three years, then he would not have been, now at least, preoccupied by him, as he is at the moment"

Data Analysis

It seems thus that we should replace the presentation of Modern Standard Arabic's hypothetical systems with another presentation, quite different from that of the grammars, regarding the language of this period.

Classically there were two particles attributed to the expression of conditional clauses, in and law 40 . Later, the circonstant iḏā, initially reserved for the Eventual, gradually ousted in to supplant it in the expression of the Potential 41 . The classic distinction between operators allowed us to place in their fields verbal forms generally considered neutral from a temporal point of view 42 . It was generally māḍī (perfect) although it appears that the primary form of the in systems was the muḍāriʿ mağzūm (jussive) 43 . All works consulted regarding Modern Standard Arabic continue to reproduce this strict dichotomy iḏā-Potential vs law-Unreal. Buckley puts it as follows: "The temporal meaning of the verb will depend on the meaning of the condition (Buckley, 2004 : 739). To that solely semantic criterion we oppose a syntactic criterion which is expressed thus: "On the temporal form of the verb of the apodosis (and therefore on its meaning) and the conditional particle will depend the significance of the condition". Indeed, concerning Modern Arabic's hypothetical system, and this is the most prominent new feature, we observe at first, probably under the influence of an unique if (si in French, if in English), the synonymization 44 of Classical Arabic conditional operators. Iḏā has come to mean, in addition to its traditional tasks, and standing beside law for whom this role was historical, the Unreal (present), while law, in turn, also expresses the Potential as well as the Unreal.

The result of this synonymization is a necessary upheaval of the traditional system in which operators, and operators only, carried the meaning and permitted a conferral to the considered statement of one of the values involved (Eventual/Potential, Unreal). upheaval is then reflected by the introduction of a kind of sequence of tenses, where the modern apodosis now gains a tense value. We can see that a system similar to our English (or French) sequence of tenses thus began, and have made note of it here 45 : 40. See Zamaḫšarī (Al-), 1999 : 416. Regarding the semantical and not merely syntactical identification of law to in in the expression of the conditional clause, see Versteegh, 1991. 41. So much so that in is now juts in residual use (we find only 16 examples within our study, that is to say 5.65%), most commonly confined to a few idioms like in šāʾa llah or for the expression of concessive clauses (wa-in, but this is beyond our purpose which focuses on hypothetical systems).

42. With the exception that for his examples, Zamaḫšarī (Al-), 1999 : 416, places in the field of in, specifying that it is used in the sense of the future ("in" taǧʿalu l-fiʿla li-l-istiqbāli wa-in kāna māḍiyan) jussives (it is to say muḍāriʿ) and not the perfect: in taḍrib-nī aḍrib-ka ("if you hit me I'll hit you") and places in the field of law, specifying a contrario that it is used to express a condition in the past (wa-"law" tagʿalu-hu li-l-māḍī wa-in kāna mustaqbilan), perfects: law ǧiʾta-nī la-akramtu-ka ("if you came/had come, I honored you/would have honored you"). In doing so, he still seems to retain for these socalled neutral verbal forms a certain tense value. Moreover Ayoub, 2003 notes the semantic implication of the speaker attached to mağzūm contrasting with the neutral utterance value of māḍī.

43. As the first two examples given by Zamaḫšarī (Al-), 1999 : 416 tend to show. Moreover, Pierre Larcher seems to favour a jussive origin of the hypothetical in systems, noting that one finds in fact very few systems in faʿala… faʿala in Quranic Arabic, and actually none in faʿala… lam yafʿal nor in lam yafʿal… faʿala; and the only denial of jussive yafʿal is lā yafʿal, the only negative system present in the Koran being illā [< in + lā] yafʿal… lā yafʿal. See Larcher, 2003a and Larcher, 2008. 44. Which we believe we have shown, in a colloquial form of Arabic, Egyptian, the reality and the syntactic implications in Sartori, 2009. 45. For the French, see [START_REF] Grévisse | DeBoeck et Duculot[END_REF]or Riegel et al., 2004 : 509. Law retains for itself the anteriority in the expression of the Unreal. In imitation of European languages, the Present Unreal acquires the equivalent meaning of the French conditional in s'il faisait beau (mais il ne fait pas beau) je sortirais or English in "if the weather was nice (but it is not) I would go out", whose links with the indicative future form are evident. The form of the Arabic future sa-yafʿalu, equivalent to that future (respectively -rais/would) becomes conditional as the apodosis of law.

The Past Unreal could therefore only be designated by an apodosis of a past verbal form (faʿala or kāna sa-yafʿalu) in the field of law.

It thus became possible to make a system in law express, on the model of iḏā, a third hypothetical status, the Potential, through the use of the last verbal form not yet used in apodosis, the muḍāriʿ marfūʿ (yafʿalu). Badawi et al. have noted this (2004 : 636, 647-648), although, as mentioned, we do not fully agree with them on the topic of their presentation data.

We arrived, so it seems to us, at the next state concerning the apodoses based on the operators observed: Table 3. Verbal forms in the apodosis of the hypothetical system of Modern Literary Arabic

Conclusion

The emergence of a new layer in modern Arabic does not necessarily invalidate the previous layer, which makes data interpretation difficult. It seems, however, that this is no longer only the operator that determines the meaning of the conditional clause, but the operator in connection with the verbal form of the apodosis. This verb is in charge of a temporal value where classically (and as is still the case today for the one verb of the protasis) it was a verbal form tending to be perfect and neutral from a temporal point of view. However this upheaval, even if it is visible over a period of forty years, seems not have allowed the system to regain full coherence, as the three major cases of ambiguity reflect: iḏā faʿala… faʿala (Past Eventual and Present Unreal), iḏā faʿala… yafʿalu (Present Eventual and, less often, the Potential) and law faʿala… (fa-(inna-hu))sa-yafʿalu (Present Unreal and less often the Potential). These cases of ambiguity, as we see, relate essentially to iḏā which also does not cover the whole core of the conditional (Potential and Unreal); in contrast to law, which also offers, it seems, a more stable system. It is therefore possible that we find ourselves in an intermediate phase between two states of the language. Ultimately, iḏā could return to its pre-classical and classical domain, the Eventual, and then law would supplant it in the core of the conditional (Potential and Unreal), as can be observed in some dialects, especially Egyptian. This assumption would be realized only if the development in question were not perceived as too colloquial by speakers inclined to over-correction. In any case, it is already possible to hear on the airwaves uses conforming to what

  30.

→

  Present Unreal → Past Unreal iḏā (fa-)yafʿalu* → (fa-(inna-hu))sa-yafʿalu → faʿala** non-existant law nonexistant yafʿalu → (fa-(inna-hu))sa-yafʿalu*** → (la-)faʿala* case of ambiguity between the Present Eventual and the Potential ** case of ambiguity between the Past Eventual and the Present Unreal *** case of ambiguity between the Present Unreal and the Potential

Table 1 .

 1 The hypothetical system in grammars of Postclassical Arabic

		Protasis	Apodosis
	iḏā	faʿala	faʿala	Past Eventual
	iḏā	faʿala	faʿala/yafʿalu	Present Eventual
	iḏā	faʿala	faʿala	Potential 11
	law	faʿala/kāna yafʿalu	(fa-)yafʿalu/(la-)faʿala	Present Unreal 12
	8.

Case of in. See Hassanein : 99.

Thus, for example, in Buckley(Buckley, 2004 : 739-740), the structure law faʿala… faʿala, classically interpreted as a case containing ambiguity between the Present Unreal and the Past Unreal; here we see only, in what emerges from our study, a Past Unreal. Alternatively there is the structure iḏā faʿala… faʿala interpreted as Present Unreal and Potential(Buckley, 2004 : 734 and 737) where we do not interpret it as Potential.

Not seen, however, in our corpus. The only positive apocopate found in our corpus is in apodosis. See example below (11).

The authors evoke equally timidly the existence of a structure in law faʿala… (fa-)-sa-yafʿalu, which they interpret as law faʿala… yafʿalu, and here we see a difference (see footnote above 109).

We should note that Badawi et al. do not reference their examples.

This sentence could take a Potential interpretation, if there were not an implicit but, but they will not continue, (the heroine, writing her diary at the end of the day, succeeding in stopping the bombing).

Same remark as in (43). Compare again this syntax with that of the immediately preceding: cf. example below (46).

It is indeed likely that under the influence of European languages like French and English, the Arabic system of the expression of the conditional has changed.

The distinction between the different statuses of the conditional seems now to take place through a contrastive syntax, which moves the Arabic system closer to "sequence of tenses" systems like the above. Nevertheless, this sequence is not perfect (read: it is not a perfect imitation of systems such as the French or English for example), since the verb of the protasis retains its neutral temporal value: it is a perfect 47 in the field of an operator of the conditional to indicate that the statement is made in the framework of a hypothetical system. It is then the verb of the apodosis, the form of which is very different from that of hypothetical systems of Classical Arabic, which has a tense value to allow us to distinguish between the hypothetical statuses, and this purtains to the model of languages like French or English. This is no longer the operator only that demonstrates the meaning of a conditional clause, but rather the relationship between the operator and the verbal form of the apodosis. We are therefore no longer in an essentialist system as in CA, but in a relational one.

In the case of iḏā, the latter has retained an Eventual sense. In doing so, and in imitation of European languages, it has become natural, in the case of the Present Eventual, to use a muḍāriʿ which is the equivalent of French's indicative present tense in si/quand il fait beau je sors and the English present in "if/when the weather is nice I go out", and a māḍī in the case of the Past Eventual like quand il faisait beau, je sortais, "when the weather was nice I used to go out".

In this manner, for the Potential it was perhaps seen as more natural now to have, instead of the classic faʿala with its non-temporal value, and by contrast to the latter (faʿala), a semantic and syntactic future (sa-yafʿalu), which is the Arabic equivalent of the French future in s'il fait beau je sortirai and of the English future in "if the weather is nice I will go out".

In return, iḏā could retain in its field in apodosis a māḍī which would come then to designate the Present Unreal, the context here allowing a distinction from the Past Eventual.

46. Case of ambiguity between the Potential and the Present Unreal. See above.

47. Except very few law yafʿalu, the imperfect indicating here a Present Unreal.


Mehdi Kaytoue
email: kaytouem@loria.fr

Sergei O Kuznetsov
email: skuznetsov@hse.ru

Amedeo Napoli
email: napoli@loria.fr

Revisiting Numerical Pattern Mining with Formal Concept Analysis

We investigate the problem of mining numerical data with Formal Concept Analysis. The usual way is to use a scaling procedure -transforming numerical attributes into binary ones-leading either to a loss of information or of efficiency, in particular w.r.t. the volume of extracted patterns. By contrast, we propose to directly work on numerical data in a more precise and efficient way. For that, the notions of closed patterns, generators and equivalent classes are revisited in the numerical context. Moreover, two algorithms are proposed and tested in an evaluation involving real-world data, showing the quality of the present approach.

Introduction

In this paper, we investigate the problem of mining numerical data. This problem arises in many practical situations, e.g. analysis of gene and transcriptomic data in biology, soil characteristics and land occupation in agronomy, demographic data in economics, temperatures in climate analysis, etc. We introduce an original framework for mining numerical data based on advances in itemset mining and in Formal Concept Analysis (FCA, [START_REF] Ganter | [END_REF]), respectively condensed representations of itemsets and pattern structures in FCA [START_REF] Ganter | [END_REF]. The mining of frequent itemsets in binary data, considering a set of objects and a set of associated attributes or items, is studied for a long time and usually involves the so-called "pattern flooding" problem [Bastide et al., 2000]. A way of dealing with pattern flooding is to search for equivalence classes of itemsets, i.e. itemsets shared by the same set of objects (or having the same image). For an equivalence class, there is one maximal itemset, which corresponds to a "closed set", and possibly several minimal elements corresponding to "generators" (or "key itemsets"). From these elements, families of association rules can be extracted. These itemsets are also related to FCA , where a concept lattice is built from a binary context and where formal concepts are closed sets of objects and attributes.

The present work is rooted both in FCA and pattern mining with the objective of extracting interval patterns from numerical data. Our approach is based on "pattern structures" where complex descriptions can be associated with objects. In [START_REF] Kaytoue | [END_REF] we introduced closed interval patterns in the context of gene expression data mining. Intuitively, an interval pattern is a vector of intervals, each dimension corresponding to a range of values of a given attribute ; it is closed when composed of the smallest intervals characterizing a same set of objects.

In the present paper, we complete and extend this first attempt. Considering numerical data, some general characteristics of equivalence classes remain, e.g. one maximal element which is a closed pattern and possibly several generators which are minimal patterns w.r.t. a subsumption relation defined on patterns. We show that directly extracting patterns data from numerical is more efficient using pattern structures than working on binary data with associated scaling procedures. We also provide a semantics to interval patterns in the Euclidean space, design and experiment algorithms to extract frequent closed interval patterns and their generators.

The problem of mining patterns in numerical data is usually referred as quantitative itemset/association rule mining [START_REF] Srikant | [END_REF]. Generally, an appropriate discretization splits attribute ranges into intervals maximizing some interest functions, e.g. support, confidence. However, none of these works covers the notion of equivalence classes, closed patterns, and generators, and this is one of the originality of the present paper.

The plan of the paper is as follows. Firstly, we introduce the problem of mining numerical data and interval patterns. Then, we recall the basics of FCA and interordinal scaling. We pose a number of questions that we propose to answer using our framework of interval pattern structures dealing with numerical data. We then detail two original algorithms for extracting closed interval patterns and their generators. These algorithms are evaluated in the last section on real-world data. Finally, we end the paper in discussing related work and giving perspectives to the present research work. As a complement, an extended version of this paper is given in [START_REF] Kaytoue | [END_REF], providing algorithms pseudo-code and a longer discussion on the usefulness of interval patterns in classification problems and privacy preserving data-mining.

Problem definition

We propose a definition of interval patterns for numerical data. Intuitively, each object of a numerical dataset corre-sponds to a vector of numbers, where each dimension stands for an attribute. Accordingly, an interval pattern is a vector of intervals, and each dimension describes the range of a numerical attribute. We only consider finite intervals and that the set of attributes/dimensions is assumed to bed (canonically) ordered.

Numerical dataset.

A numerical dataset is given by a set of objects G, a set of numerical attributes M , where the range of m ∈ M is a finite set noted W m . m(g) = w means that w is the value of attribute m for object g.

G = {g 1 , ..., g 5 }, attributes in M = {m 1 , m 2 , m 3 }. The range of m 1 is W m1 = {4, 5, 6}
, and we have m 1 (g 1 ) = 5. Here, we do not consider either missing values or multiple values for an attribute. [5,6], [7,8], [4,6] is an interval pattern in Table 1, with image {g 1 , g 2 , g 5 } and support 3. All possible intervals for m 1 are in {[4, 4], [5,5], [6,6], [4,5], [5,6], [4,6]}. Considering also attributes m 2 and m 3 , we have 6 × 6 × 10 = 360 patterns.

The classical problem of "pattern flooding" in data-mining is even worst for numerical data. Indeed, with three attributes, there are only 2 3 = 8 possible itemsets, compared to the 360 interval patterns in the above example with same number of attributes. A solution widely investigated in itemset-mining for minimizing the effect of pattern flooding relies on condensed representations including closed itemsets and generators [Bastide et al., 2000]. By contrast, the analysis of numerical datasets can be considered within the formal concept analysis framework (FCA) [START_REF] Ganter | [END_REF], which is closely related to itemset-mining [START_REF] Stumme | [END_REF]. Accordingly, we are interested in adapting the notions of (frequent) closed itemsets and their generators to interval patterns within the FCA framework, and in providing an appropriate semantics to these patterns.

3 Interval patterns in FCA [START_REF] Ganter | [END_REF]] define a discretization procedure, called interordinal scaling, transforming numerical data into binary data that encodes any interval of values from a numerical dataset. We recall here the basics on FCA and interordinal scaling.

Formal concept analysis

FCA starts with a formal context (G, N, I) where G denotes a set of objects, N a set of attributes, or items, and I ⊆ G × N a binary relation between G and N . The statement (g, n) ∈ I, or gIn, means: "the object g has attribute n". Two operators (•) ′ define a Galois connection between the powersets (P(G), ⊆) and (P(N ), ⊆), with A ⊆ G and B ⊆ N :

A ′ = {n ∈ N | ∀g ∈ A : gIn} and B ′ = {g ∈ G | ∀n ∈ B : gIn}. A pair (A, B), such that A ′ = B and B ′ = A, is called a (formal) concept, while A is called the extent and B the intent of the concept (A, B).
From an itemset-mining point of view, concept intents correspond to closed itemsets, since (.) ′′ is a closure operator. An equivalence class is a set of itemsets with same closure (and same image). For any subset B ⊆ N , B ′′ is the largest itemset w.r.t. set inclusion in its equivalence class. Dually, generators are the smallest itemsets w.r.t. set inclusion in an equivalence class. Precisely,

B ⊆ N is closed iff ∄C such as B ⊂ C with C ′ = B ′ ; B ⊆ N is a generator iff ∄C ⊂ B with C ′ = B ′ .

Interordinal scaling

Given a numerical attribute m with range W m , Interordinal Scaling builds a binary table with 2 × |W m | binary attributes. They are denoted by "m ≤ w" and "m ≥ w", ∀w ∈ W m , and called IS-items. An object g has an IS-item "m ≤ w" (resp. "m ≥ w") iff m(g) ≤ w (resp. m(g) ≥ w). Applying this scaling to our example gives Table 2. It is possible to apply classical mining algorithms to process this table for extracting itemsets composed of IS-items. These itemsets are called ISitemsets in the following.

IS-itemsets can be turned into interval patterns, since an IS-item gives a constraint on the range W m of an attribute m. For example, the IS-itemset 4,8] . We have here the interval [4,8] for attribute m 3 : [4, 8] covers the whole range of m 3 since no constraint is given for m 3 .

{m 1 ≤ 5, m 1 ≤ 6, m 1 ≥ 4, m 2 ≤ 9, m 2 ≥ 7} corresponds to the interval pattern [4, 5], [7, 9], [
Therefore, mining interval patterns can be considered with a scaling of numerical data. However, this scaling produces a very important number of binary attributes compared to the original ones. Hence, when original data are very large, the size of the resulting formal context involves hard computations. Accordingly, this raises the following questions:

(i) Can we avoid scaling and directly work on numerical data instead of searching for IS-itemsets? (ii) Can we adapt the notions of condensed representations such as closed patterns and generators for numerical data, and efficiently compute those patterns? (iii) What would be the semantics that could be provided to closed patterns and generators?

m 1 ≤ 4 m 1 ≤ 5 m 1 ≤ 6 m 1 ≥ 4 m 1 ≥ 5 m 1 ≥ 6 m 2 ≤ 7 m 2 ≤ 8 m 2 ≤ 9 m 2 ≥ 7 m 2 ≥ 8 m 2 ≥ 9 m 3 ≤ 4 m 3 ≤ 5 m 3 ≤ 6 m 3 ≤ 8 m 3 ≥ 4 m 3 ≥ 5 m 3 ≥ 6 m 3 ≥ 8 g 1 × × × × × × × × × × × × × g 2 × × × × × × × × × × × × × g 3 × × × × × × × × × × × × × g 4 × × × × × × × × × × × × × g 5 × × × × × × × × × × × × ×

Revisiting numerical pattern mining

In this section, we answer those questions. First, we show that a closure operator can be defined for interval patterns based on their image. Then, we provide interval patterns with an appropriate semantics for defining the notion of equivalence classes of patterns, closed and generator patterns. After discussing why working with interordinal scaling is not acceptable thanks to the semantics of interval patterns, we propose two efficient algorithms for mining closed interval patterns and generators. Experiments follow in Section 5.

A closure operator for interval patterns

We introduce the formalism of pattern structures [START_REF] Ganter | [END_REF], an extension of formal contexts for dealing with complex data in FCA. It defines a closure operator for a partially ordered set of object descriptions called patterns.

Formally, let G be a set of objects, (D, ⊓) be a semilattice of object descriptions, and δ : G → D be a mapping: (G, (D, ⊓), δ) is called a pattern structure. Elements of D are called patterns, and are ordered as follows c ⊓ d = c ⇐⇒ c ⊑ d. Intuitively, objects in G have descriptions in (D, ⊓). For example, g 1 in Table 1 has description [5, 5], [7,7], [6,6] where D is the set of all possible interval patterns ordered with ⊓, as made precise below. Consider the two operators (.) defined as follows, with A ⊆ G and d ∈ (D, ⊓)

d = {g ∈ G|d ⊑ δ(g)} A = g∈A δ(g)
: c ⊑ d ⇐⇒ [e i , f i ] ⊆ [a i , b i ], ∀i ∈ {1, ..., |M |}.
Consider now a numerical dataset, e.g. Table 1. (D, ⊑) is the finite ordered set of all interval patterns. δ(g) ∈ D is the pattern associated to an object g ∈ G. Then:

[5, 6], [7, 8], [4, 8] = {g ∈ G| [5, 6], [7, 8], [4, 8] ⊑ δ(g)} = {g 1 , g 2 , g 5 } {g 1 , g 2 , g 5 } = δ(g 1 ) ⊓ δ(g 2 ) ⊓ δ(g 3 ) = [5,

Semantics

An interval pattern d is a |M |-dimensional vector of intervals and can be represented by a hyper-rectangle (or rectangle for short) in Euclidean space R |M| , whose sides are parallel to the coordinate axes. This geometrical representation provides a semantics for interval patterns. In formal terms, an interpretation is given by I = (R |M| , (.) I ) where R |M| is the interpretation domain, and (.) I : D → R |M| the interpretation function. Figure 1 gives four interval pattern representations in R 2 , with only attributes m 1 and m 3 of our example. The image of d 1 is given by all objects g whose description δ(g) is included in the rectangle associated with d 1 , i.e. the set {g 1 , g 3 , g 4 , g 5 }. We can interpret the closure operator (.) according to this semantics. The first operator (.) applies to a rectangle and returns the set of objects whose description is included in this rectangle. The second operator (.) applies to a set of objects and returns the smallest rectangle that contains their descriptions, i.e. the convex hull of their corresponding descriptions.

Closed interval patterns and generators

Now, we can revisit the notion of equivalence classes of itemsets as introduced in [Bastide et al., 2000]: an equivalence class of interval patterns is a set of rectangles containing the same object descriptions (based on all rectangles in the search space as given in Section 2). This enables to define the notions of (frequent) closed interval patterns ((F)CIP) and (frequent) interval pattern generators ((F)IPG), adapted itemisets. Among the four patterns in Figure 1, d 1 is the only frequent interval pattern with minSup = 3. Based on the above semantics, an equivalence class is a set of rectangles containing the same set of object descriptions, with a (unique) closed pattern corresponding to the smallest rectangle, and one or several generator(s) corresponding to the largest rectangle(s).

Closed interval pattern (CIP

These definitions are counter-intuitive w.r.t. itemsets: the smallest rectangles subsume the largest ones. This is due to the definition of infimum as set intersection for itemsets while this is convex hull for intervals, which behaves dually as a supremum.

IS-itemsets versus interval patterns

Interordinal scaling allows to build binary data encoding all interval of values from a numerical dataset. Therefore, one may attempt to mine closed itemsets and generators in these data with existing data-mining algorithms. Here we show why this should be avoided.

Local redundancy of IS-itemsets. Extracting all ISitemsets in our example (from Table 2) gives 31, 487 ISitemsets. This is surprising since there are at most 360 possible interval patterns. In fact, many IS-itemsets are locally redundant. For example, {m 1 ≤ 5} and {m 1 ≤ 5, m 1 ≤ 6} both correspond to interval pattern [4, 5], [7,9], [4,8] : the constraint m 1 ≤ 6 is redundant w.r.t. m 1 ≤ 5 on the set of values W m1 . Hence there is no 1-1-correspondence between IS-itemsets and interval patterns. It can be shown that there is a 1-1-correspondence only between closed IS-itemsets and CIP [START_REF] Kaytoue | [END_REF]. Later we show that local redundancy of IS-itemsets makes the computation of closed sets very hard.

Global redundancy of IS-itemset generators. Since ISitemset generators are the smallest itemsets, they do not suffer of local redundancy. However, we can remark another kind of redundancy, called global redundancy: it happens that two different and incomparable IS-itemset generators correspond to two different interval pattern generators, but one subsuming the other. In Table 2, both IS-itemsets

N 1 = {m 1 ≤ 4, m 3 ≤ 5} and N 2 = {m 1 ≤ 4, m 3 ≤ 6}

Algorithms

We detail a depth-first enumeration of interval patterns, starting with the most frequent one. Based on this enumeration, we design the algorithms MinIntChange and MinIntChangeG for extracting respectively frequent closed interval patterns (FCIP) and frequent interval pattern generators (FIPG).

Interval pattern enumeration. Consider firstly one numerical attribute of the example, say m 1 . The semi-lattice of intervals (D m1 , ⊓) is composed of all possible intervals with bounds in W m1 and is ordered by the relation ⊑. The unique smallest element w.r.t. ⊑ is the interval with maximal size, i.e. [4, 6] = [min(W m1 ), max(W m1 )] and maximal frequency (here 5). The basic idea of pattern generation lies in minimal changes for generating the direct subsumers of a given pattern. For example, two minimal changes can be applied to [4,6]. The first consists in replacing the right bound with the value of W m1 immediately lower that 6, i.e. 5, for generating the interval [4,5]. The second consists in repeating the same operation for the left bound, generating the interval [5,6]. Repeating these two operations allows to enumerate all elements of (D m1 , ⊓). i.e. adding m ≤ v to the original IS-itemset. The same applies dually to left minimal changes. These IS-items characterizing minimal changes are drawn on Figure 2. This figure accordingly represents a prefix-tree, factoring out the effort to process common prefixes or minimal changes, and avoiding redundancy problems inherent in interordinal scaling. The generalization to several attributes is straightforward. A lectic order is classically defined on numerical attributes as a lexicographic order, e.g. m 1 < m 2 < m 3 . Then changes are applied as explained above for all attributes respecting this order, e.g. after applying a change to attribute m 2 , one cannot apply a change to attribute m 1 . Extracting FCIP with MintIntChange. The pattern enumeration starts with the minimal pattern w.r.t ⊑ and generates its direct subsumers with lower or equal support. The next problem now is that minimal changes do not necessarily generate patterns with strictly smaller support. Therefore, we should apply changes until a pattern with different support is generated to identify a closed interval pattern (FCIP) but this would not be efficient. We adopt the idea of the algorithm CloseByOne [Kuznetsov and Obiedkov, 2002]: before applying a minimal change, the closure operator (.) is applied to the current pattern, allowing to skip all equivalent patterns. Indeed, the minimal pattern d w.r.t. ⊑ is closed as it is given by d = G . Applying a minimal change returns a pattern c with strictly smaller support, since d ⊑ c and d is closed. If c is frequent, we can continue, apply the closure operator and next changes in lectic order, allowing to completely enumerate all FCIP. Since a FCIP may have several different associated generators, it can be generated several times. Still following the idea of CloseByOne, a canonicity test can be defined according to lectic order minimal changes.

Consider a pattern d generated by a change at attribute m j ∈ M . Its closure is given by d . If d differs from d for some attributes m h ∈ M such as m h < m j , then d has already been generated: it is not canonically generated, hence the algorithms backtracks.

Example.

We is not canonical and has already been generated (previous example), hence the algorithm backtracks.

Extracting FIPG with MintIntChangeG. We now adapt MinIntChange to extract FIPG, following a well-known principle in itemset-mining algorithms [START_REF] Calders | [END_REF]. For any FCIP d, a minimal change implies that the support of the resulting pattern c is strictly smaller than the support of d. Therefore, c is a good generator candidate of the next FCIP. Accordingly, at each step of the depth-first enumeration a FIPG candidate c is generated from the previous one b, by applying a minimal change characterized by b . Then, each candidate c has to be checked whether it is a generator or not. We know that the candidate has no subsumers in its branch with same support. However, it could exist a branch with another FIPG e with same image and resulting from less changes. Considering the lectic order on minimal changes, we use a reverse traversal of the tree (see Figure 2 : 7,8,9,10,1,4,5,2,3,6), as already suggested in the binary case in [START_REF] Calders | [END_REF]. Since generators correspond to largest rectangles, i.e. on which the fewest minimal changes have been applied, if c is not a generator, a generator e associated to its equivalence class has already been generated, and c is discarded. To check the existence of e, we look up in an auxiliary data-structure storing already extracted FIPG. Precisely, if the data structure contains a FIPG e with same support than candidate c, such that e ⊑ c, c is discarded, and the algorithm backtracks. Otherwise c is declared as a FIPG and stored. We have experimented the MinIntChangeG algorithm with two well-known and adapted data structures, a trie and a hashtable.

Experiments

We evaluate the performances of the algorithms designed in Java, namely MinIntChange, MinIntChangeG-h with auxiliary hashtable and MinIntChangeG-t with auxiliary trie. Recalling that closed IS-itemsets and CIP are in 1-1correspondence, we compare the performance for mining interordinal scaled data with the closed-itemset-mining algorithm LCMv2 [Uno et al., 2004]. For studying the global redundancy effect of IS-itemset generators, we use the generator-mining-algorithm GrGrowth [START_REF] Liu | [END_REF]. Both implementations in C++ are available from the authors. All experiments are conducted on a 2.50Ghz machine with 16GB RAM running under Linux 2.6.18-92.e15. We choose dataset from the Bilkent repository1 , namely Bolts (BL), Basketball (BK) and Airport (AP), AP being worst case where each attribute value is different.

First experiments compare MinIntChange for extracting FCIP and LCMv2 for extracting equivalent frequent closed IS-itemsets in Table 3. Second experiments consist in extracting frequent interval pattern generators (FIPG) with MinIntChange-h and MinIntChange-t. We also extract frequent itemset generators (FISG) in corresponding binary data after interordinal scaling with GrGrowth for studying the global redundancy effect in Table 4. In both cases, using binary data is better when the minimal support is high (e.g. 90%). For low supports, a critical issue, our algorithms deliver better execution times. Most importantly, the global redundancy effect discards the use of binary data, e.g. only 1.6% of all FISG are actually FIPG in dataset BL. Finally, the algorithm MinIntChangeG-t outperforms MinIntchangeG-h. MinIntChangeG-t however needs more memory since it stores each closed set of objects as a word in the trie, and to each word the list of associated FIPG.

It is very interesting to analyse the compression ability of closed interval patterns and generators. For that, we compare

Conclusion

We discussed the important problem of pattern discovery in numerical data with a new and original formalization of interval patterns. The classical FCA/itemset-mining settings are adapted accordingly: from a closure operator naturally rise the notions of equivalence classes, closed and generator patterns, and we designed corresponding algorithms. An appropriate semantics of interval patterns shows from a theoretical (redundancy) and practical (computation times) points of view that mining equivalent binary data (encoding all possible intervals) is not acceptable. This is due to the fact that interval patterns are provided with a stronger partial ordering than IS-itemsets (classical set inclusion), hence pattern structures yield significantly less generators w.r.t. their semantics.

Dealing with interval patterns has applications in computational geometry, machine learning and data-mining, e.g. [Boros et al., 2003] and references therein. It is indeed highly related to the actual problem of (maximal) k-boxes which corresponds to interval patterns (generators) with support k. When k = 0, it corresponds to largest empty subspaces of the data. Our contribution to this field is the characterization of smaller subsets (closed and generators).

In data-mining, closed patterns and their generators are crucial for extracting valid and informative association rules [Bastide et al., 2000], while generators can be preferable to closed patterns following the minimum descriptions length principle for so-called itemset-based classifiers [Li et al., 2006]. How these notions can be shifted to interval patterns is an original perspective of research rising questions concerning missing values, fault-tolerant patterns, and interestingness measures that are critical issues even in classical itemset mining: although the compression ability of closed interval patterns and generators is spectacular, the number of patterns remains too high for large datasets. However, bringing the problem of numerical pattern mining into well known settings in favor of these perspectives of research.

  Interval pattern search space. Given a set of attributes M = {m i } i∈{1,|M|} , the search space of interval patterns is the set D of all interval vectors [a i , b i ] i∈{1,...,|M|} , with a i , b i ∈ W mi . The size of the search space is given by |D| = i∈{1,...,|M|} (|W mi | × (|W mi | + 1))/2 Example.

Figure 1 :

 1 Figure 1: Interval patterns in the Euclidean space.

Equivalence class .

 class Two interval patterns c and d with same image are equivalent, i.e. c = d and we write c ∼ = d. ∼ = is an equivalence relation, i.e. reflexive, transitive and symmetric. The set of patterns equivalent to a pattern d is denoted by [d] = {c|c ∼ = d} and called the equivalence class of d.

  have the same image {g 3 } and are generators, i.e. there does not exist a smaller itemset of these itemsets with same image. However, their corresponding interval pattern are respectively c = [4, 4], [7, 9], [4, 5] and d = [4, 4], [7, 9], [4, 6] and we have d ⊑ c, while c = d , hence c is not an interval pattern generator.

  A right minimal change is defined formally as, given a, b, v ∈ W m , a = b, mcr([a, b]) = [a, v] with v < b and ∄w ∈ W m s.t. v < w < b while a left minimal change mcl([a, b]) is formally defined dually. Minimal changes give direct next subsumers and implies a monotonicity property of frequency, i.e. support of [a, v] is less than or equal to support of [a, b]. To avoid generating several times the same pattern, a lectic order on changes, or equivalently on patterns, is defined. After a right change, one can apply either a right or left change; after a left change one can apply only a left change. Figure 2 shows the depth-first traversal (numbered arrows) of diagram of (D m1 , ⊓). Backtrack occurs when an interval of the form [w, w] is reached (w ∈ W m1 ), or no more change can be applied. Each minimal change can be interpreted in term of an IS-item. For example, if [a, b] corresponds to the IS-itemsets {m ≥ a, m ≤ b} then mcr([a, b]) = [a, v] corresponds to {m ≥ a, m ≤ b, m ≤ v},

Figure 2 :

 2 Figure 2: Depth-first traversal of (D m1 , ⊓).

  start from the minimal pattern d = [4, 6], [7, 9], [4, 8] . The first minimal change in lectic order is a right change on attribute m 1 . We obtain pattern c = [4, 5], [7, 9], [4, 8] , and obviously d ⊑ c. However, c = [4, 5], [7, 9], [5, 8] , hence c is not closed. c is stored as FCIP and next changes will be applied to it. Now consider the pattern obtained by minimal change on left border for attribute m 3 , i.e. e = [4, 6], [7, 9], [5, 8] . We have e = [4, 5], [7, 9], [5, 8] . e and e differ for attribute m 1 , but e has been generated from a change on m 3 . Since m 1 < m 3 , e

Table 1 :

 1 A numerical dataset. Interval pattern and support. In a numerical dataset, an interval pattern is a vector of intervals d = [a i , b i ] i∈{1,...,|M|} where a i , b i ∈ W mi , and each dimension corresponds to an attribute following a canonical order on vector dimensions, and |M | denotes the number of attributes. An object g is in the image of an interval pattern [a i , b i ] i∈{1,...,|M|} when m i (g) ∈ [a i , b i ], ∀i ∈ {1, ..., |M |}. The support of d, denoted by sup(d), is the cardinality of the image of d.

		m1	m2	m3
	g1	5	7	6
	g2	6	8	4
	g3	4	8	5
	g4	4	9	8
	g5	5	8	5
	Running example. Table 1 is a numerical dataset with objects
	in

Table 2 :

 2 Interordinally scaled context encoding the numerical dataset from Table1.

  ). A pattern d is closed if there does not exist any pattern e such as d ⊑ e with d ∼ = e.Interval pattern generator (IPG).A pattern d is a generator if there does not exist a pattern e such as e ⊑ d with d ∼ = e. Frequent Interval pattern. A pattern d is frequent if its image has a higher cardinality than a given minimal support threshold minSup.We illustrate these definitions with two dimensional interval patterns, and their representation in Figure1, i.e. considering attributes m 1 and m 3 only.

	[4, 5], [6, 8] ∼ =
	[4, 6], [6, 8] with image {g 1 , g 4 }. [4, 6], [6, 8] is not closed
	as [4, 6], [6, 8] ⊑ [4, 5], [6, 8] , these two patterns having
	same image, i.e. {g 1 , g 3 , g 4 , g 5 }. [4, 5], [5, 8] is closed.
	[4, 6], [5, 8] and [4, 5], [4, 8] are generators in the class of
	the closed interval pattern d 1 = [4, 5], [5, 8] with image
	{g 1 , g 3 , g 4 , g 5 }.

Table 3 :

 3 Execution time for extracting FCIP (in ms).

Table 4 :

 4 Execution time for extracting FIPG and global redundancy evaluation.in each dataset the number of those patterns w.r.t. to all possible interval patterns. It gives the ratio of closed (generators) in the whole search space. In both cases, ratio varies between 10 -7 and 10 -9 . This means that the volume of useful interval patterns, either closed or generators, is very low w.r.t. the set of all possible interval patterns, justifying our interest in equivalence classes for interval patterns.

	Dataset	minSupp	GrGrowth	MinIntChangeG-h	MinIntChangeG-t	|F IP G|	|F ISG|	|F IP G| |F ISG|	|F CIP |	|F IP G| |F CIP |
	BL	90%	< 50	< 50	< 50	176	194	90%	112	1.57
		80%	< 50	< 50	< 50	1,952	2,823	69%	1,130	1.73
		50%	150	1,212	529	66,350	222,088	29%	32,107	2
		25%	3,432	27,988	3,893	411,442	3,559,419	11%	171,192	2.4
		1	123,564	438,214	24,141	1,165,824	69,646,301	1.6%	272,223	4.3
	BK	90%	< 50	1,268	1,207	67,737	75,058	84%	48,847	1.3
		85%	4,565	26,154	12,139	554,956	799,574	69%	403,562	1.37
		80%	Untractable	512,126	107,700	2,730,812	NA	NA	1,938,984	1.40

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