Lauranne Chaignon

Daniel Egret
email: daniel.egret@psl.eu

Identify scientific publications country-wide and measure their open access: the case of the French Open Science Barometer (BSO)

Keywords:

We use several sources to collect and evaluate academic scientific publication on a country scale, and we apply it to the case of France for the years 2015-2020, while presenting a more detailed analysis focused on the reference year 2019. These sources are diverse: databases available by subscription (Scopus, Web of Science) or open to the scientific community (Microsoft Academic Graph), the national open archive HAL, and databases serving thematic communities (ADS and PUBMED). We show the contribution of the different sources to the final corpus. These results are then compared to those obtained with another approach, that of the French Open Science Barometer (Jeangirard, 2019) for monitoring open access at the national level. We show that both approaches provide a convergent estimate of the open access rate. We also present and discuss the definitions of the concepts used, and list the main difficulties encountered in processing the data. The results of this study contribute to a better understanding of the respective contributions of the main databases and their complementarity in the broad framework of a country-wide corpus. They also shed light on the calculation of open access rates and thus contribute to a better understanding of current developments in the field of open science.

Introduction

Open access to publications (see e.g. [START_REF] Laakso | Anatomy of open access publishing: a study of longitudinal development and internal structure[END_REF][START_REF] Piwowar | The State of OA: A Large-Scale Analysis of the Prevalence and Impact of Open Access Articles[END_REF] within the general framework of Open Science is now an issue shared by many institutions, universities and research organizations, or funders. France is no exception: two national plans for Open Science have been successively launched, in 2018 and 2021, by the Ministry of Higher Education, Research and Innovation (MESRI). Generalizing open access to publications is the first axis of these two plans, with a goal of 100% of French scientific publications in open access by 2030 1 , either through a publication natively in open access or through a deposit in an open archive. This national plan is in line with the European Plan S 2 . accompanied by a very specific requirement: "a transparent methodology and reproducible results". It is with this in mind that the French Open Science Barometer (BSO) was carried out5 , as described by Eric Jeangirard (2019). For the BSO, the constitutive choice is to use only open sources. The methodology used consists in scanning all the papers references in Unpaywall and in the national open archive HAL (see below), in order to identify either the French authors or the presence of the mention of France in the affiliation. The publications thus identified were then enriched with information on their scientific discipline, using natural language processing (NLP), also based on an open source, to determine, from the title, the discipline to which a document belongs. Finally, the open access status was determined using the Unpaywall database. The corpus obtained by this strategy is available in open access from the MESRI OpenData portal6 . In accordance with the recommendations made at the European level (Open Access Monitoring: [START_REF] Philipp | Open Access Monitoring: Guidelines and Recommendations for Research Organisations and Funders[END_REF], the French National Open Science Barometer is published on an annual basis. About 150,000 publications are thus identified each year by the BSO. The purpose of this study is to consider an alternative approach, this time based on the use of the main open or non-open bibliographic databases, and to analyse the extent to which this new corpus differs from that of the BSO. Our approach is based on the use of six complementary sources, namely WoS, Scopus, Microsoft Academic Graph, PubMed, NASA/ADS and the HAL open archive, to identify and assess academic scientific publication at the scale of a country, in this case France, for publications released during the six years 2015-2020. As the year scale seemed to us more relevant to characterize scientific production, we chose to highlight, in the context of this article, the data related to the year 20197 . We then compare the corpus obtained with that of the BSO, and we show to what extent the diversity of the sources used makes it possible to refine the identification and characterization of French scientific production, as well as the estimation of the open access rate.

While there is an abundant literature on the comparison between Scopus, WoS and other generalist databases (see, for example, in a national production context: [START_REF] Bartol | Assessment of research fields in Scopus and Web of Science in the view of national research evaluation in Slovenia[END_REF][START_REF] Moed | Trends in Russian research output indexed in Scopus and Web of Science[END_REF][START_REF] Archambault | Comparing bibliometric statistics obtained from the Web of Science and Scopus[END_REF] or for a statistical comparison of large reference databases: [START_REF] Mongeon | The journal coverage of Web of Science and Scopus: A comparative analysis[END_REF]Pranckuté, 2021;[START_REF] Visser | Large-scale comparison of bibliographic data sources: Scopus[END_REF], our study provides a detailed quantitative view in the specific context of French research. Far from identifying a source that would be optimal, our study shows the importance of diversifying the sources used to provide complementary views on a country's publication.

Constitution of the France 2015-2020 corpus: data and methods

Definitions

Before describing in detail the methodology used to establish our corpus, we present and discuss here the main concepts used.

DOI (Digital Object Identifier):

The DOI8 is a persistent identifier that can be assigned to any type of content, be it text, software, datasets, etc. [START_REF] Simmonds | The Digital Object Identifier (DOI)[END_REF]. It will be used as a common metadata for the entire study.

Scientific publications: We consider here scientific publications indexed in databases (private or public) and accessible in open archives. All types of documents are taken into account. This primarily concerns articles, generally published in international peer-reviewed journals, but also conference proceedings, book chapters, or any other publication, provided that it has a DOI. However, the restriction to only documents with a DOI is an important restriction, which we must explain here.

In order to facilitate the aggregation of results, and to avoid duplication, we have chosen, as does the BSO (French Open Access Monitoring), to restrict the cross-referencing of data to publications identified by a DOI number. This step is necessary to allow the efficient crossreferencing of documents identified in each database by their DOI identifier, common to all databases. In addition, the Unpaywall database, which will inform us about open access in the next step, only lists publications with a DOI.

Let us note that the requirement of the presence of a DOI immediately rules out a certain number of journals that do not adhere to this very general technology of persistent identifiers [START_REF] Gorraiz | Availability of digital object identifiers (DOIs) in Web of Science and Scopus[END_REF]; some of these journals may be, as [START_REF] Wang | Microsoft Academic Graph: When experts are not enough[END_REF] point out, key journals in their discipline with the example, for the field of Artificial Intelligence of the Journal of Machine Learning Research.

Moreover, grey literature, under which we can group preprints, reports, theses and in some cases conference proceedings (Schöpfel & Prost, 2019), is often ignored by open access measurement tools, mainly for two reasons: the first corresponds to a concern to discard literature whose scientific relevance cannot be sufficiently controlled (lack of peer review); the second is rather related to technical considerations, in particular a difficulty in identifying these publications in the absence of complete and standardized metadata, and in particular persistent identifiers. In practice, this leads to ignoring a large proportion of the work published in certain disciplines where the thematic field, the regional vocation or the applicative nature of the publications take precedence over international referencing.

Our methodology, based on the use of the DOI, therefore effectively excludes some of the documents that might be of interest to us. This is why we will come back to publications without DOIs at the end of our study, by proposing an estimate of the share of grey literature in French national production (part 5.2).

Finally, it should be noted that the publications taken into account to establish our corpus are exclusively those that have a digital version: it is this digital version that we will try to measure the degree of accessibility. Thus, peer-reviewed research published in books/monographs is only covered when it is in digital format and has a DOI. For this reason, non-academic publishing generally falls outside the scope of our study.

Open access: A scientific article that is only available on payment of a subscription or a toll (price per article) is considered closed. In contrast, a scientific article that is freely available, either on a publisher's website or after the deposit of the full text (in its final layout or not) on an open archive, is deemed open.

Our source of information for the open access status of an article will be the Unpaywall database [START_REF] Piwowar | The State of OA: A Large-Scale Analysis of the Prevalence and Impact of Open Access Articles[END_REF], specifically the data in the "is_oa" field. If the value returned for a given publication is equal to "True", the publication will be considered open. If this value is "False", the publication will be considered closed. The so-called "bronze" status is considered open.

Note that the open access status may vary over time, since a closed publication may have its embargo lifted or be subsequently deposited in an open archive. Thus, in our study, it will be the status observed in February 2021, as recorded in the Unpaywall database snapshot for that date.

Let us recall that for France, the Law for a Digital Republic of October 7, 20169 establishes the possibility of deposit on an open archive of the postprint of any scientific article resulting from research funded at least for half by the State or public authorities, at the expiration of a period of 6 months to 12 months depending on the scientific field (respectively, STM or Humanities & Social Sciences).

Sources used to constitute the FR-2015-2020 corpus

The collection of metadata related to a large set of publications is facilitated by the use of databases that systematically, if not exhaustively, collect a large part of the millions of scientific publications published each year worldwide.

In this article, we have privileged the databases providing a search capability for the mention of the country in the affiliation, and we have collected the publications whose affiliation mentions the country considered in our study, France, using the corresponding query modes of six databases that, to our knowledge, effectively cover the French scientific production.

We did not use the Dimensions database, as it is not considered to be a reliable source for establishing a corpus on a country scale [START_REF] Guerrero-Bote | Comparative Analysis of the Bibliographic Data Sources Dimensions and Scopus: An Approach at the Country and Institutional Levels[END_REF].

We use the following databases in our study: § Scopus [START_REF] Baas | Scopus as a curated, highquality bibliometric data source for academic research in quantitative science studies[END_REF] references more than 25,000 journals and is considered one of the most comprehensive databases for international peer-reviewed journals. Query by country is possible. Metadata extraction is limited to batches of 20,000 documents. This database is available by subscription from Elsevier.

§ Web of Science [START_REF] Birkle | Web of Science as a data source for research on scientific and scholarly activity[END_REF] has been the reference database for scientometrics since the pioneering work of [START_REF] Garfield | Science Citation Index -A new dimension in indexing science[END_REF]. The query by country is provided in the advanced query mode. This database is available by subscription from Clarivate Analytics. In this study, we use all the indexes (including ESCI: Emerging Sources) except for the Book Citation Index which was not available to us.

§ The HAL open archive https://hal.archives-ouvertes.fr/ [START_REF] Charnay | L'archive ouverte HAL[END_REF]) is a national multidisciplinary open archive intended for the deposit and dissemination of researchlevel scientific articles (published or not), theses and other objects emanating from French or foreign teaching and research establishments, public or private laboratories. Created in 2001 with ArXiv as a model, this platform has gradually become one of the main tools for reporting French research. A partnership agreement in favour of this archive was signed in 2013 by the Conference of University Presidents (CPU) and 22 institutions. In July 2021, the MESRI also committed to supporting the development of this archive, both in terms of technical aspects and governance, as part of its second national plan for open science 2021-2024.

French researchers are invited to deposit on this platform the products of their research, whether they are publications (article in a journal, communication in a conference, chapter of a book, book, poster, file, patent), unpublished documents (pre-publication, working document, report), academic works (thesis, HDR, course) or research data (image, video, software, map or sound). The recorded documents are either in the form of a notice only, or accompanied by the full text of the article. This production can be grouped within different collections or portals relating to a theme (SHS for example), a medium (images and videos) or a research structure (university, laboratory or research team), but it remains possible to carry out queries covering all portals and collections. After 20 years of use [START_REF] Berthaud | Diffuser et pérenniser le savoir scientifique : 20 ans d'histoire de HAL[END_REF], more than 2,700,000 works are now recorded in this archive.

HAL data can be queried using an advanced query or the API. The latter, which is available free of charge, allows the identification of the country of affiliation.

§ The NASA/ADS database [START_REF] Kurtz | The NASA Astrophysics Data System: Overview[END_REF] is one of the most recognized examples of a bibliographic database covering a research field: astrophysics and physics. Its query mode allows the query by country. Access is free.

§ The PUBMED database is one of the preferred and free access points for metadata related to biomedical science research. A query by affiliation is possible [START_REF] Ibarra | Changes in PubMed affiliation indexing improved publication identification by country[END_REF].

§ The Microsoft Academic Graph database [START_REF] Wang | A Review of Microsoft Academic Services for Science of Science Studies[END_REF][START_REF] Herrmannova | An analysis of the Microsoft Academic Graph[END_REF], one of the three products of the Microsoft Research project, is one of the largest open publication and citation data sets. It is populated automatically, using bibliographic data from web pages crawled by the Bing search engine, also a Microsoft product. The data can be accessed using the Academic Knowledge API. It should be noted that MAG does not contain structured data on affiliation country. Identification of French outputs (provided by the Curtin Open Knowledge Initiative team) was by applying a query to the affiliation string (OriginalAffiliation data element from the MAG PaperAuthorAffiliations table, linked via the PaperID to the DOI) that sought to determine whether the affiliation string finished with "France" (or one of a small set of non-English names). This number may not match that in the online COKI country dashboard, which maps affiliation country from GRIDs in MAG to the country of organisation in the GRID database10 . Some of the characteristics of these databases as well as the number of documents obtained for one year (the year 2019), in the framework of the query "France 2015-2020" carried out in October 2021 are presented in Table 1.

Aggregation of results for publications identified by a DOI

As mentioned above, in order to facilitate the aggregation of results and to avoid duplication, we have chosen, as does the BSO (French Open Access Monitoring), to restrict the crossmatching of data to publications identified by a DOI number.

Table 2 shows the counts obtained for the year 2019: DOIs are available for 94% of the documents indexed in Scopus and 85% of those in the Web of Science. We can notice, in addition, that a major part of the documents without DOI corresponds to communications to conferences (for France and the year 2019: 54% of the documents without DOI in Scopus are communications; 78% in the Web of Science). For ADS the documents without DOI are mainly conference abstracts, while documents without DOI represent only 1% in PubMed.

For the HAL archive, the situation is different: the fact is that the DOI identifier is not systematically filled in because it is not a compulsory metadata during the deposit. While only 2 to 3% of the documents characterized as articles in WoS or Scopus do not have a DOI recorded, this proportion rises to 22% for the documents characterized as articles in HAL. In addition, the open archive contains many unpublished documents, preprints, reports or theses that do not have (or not yet) a DOI: with the book chapters, these documents represent half of the publications without DOI, which will not be considered for the rest of the study.

However, we will return to HAL in Section 5 for a discussion of the grey literature. Note that for MAG, we had direct access to the DOI lists through the COKI team, whom we thank for their help.

Query

Open access and external validation: using Unpaywall

One of the objectives of this study is the measurement of the share of open access to publications. For this we use the Unpaywall database 11 which is the leading database in this field [START_REF] Piwowar | The State of OA: A Large-Scale Analysis of the Prevalence and Impact of Open Access Articles[END_REF][START_REF] Holly | The rise and rise of Unpaywall[END_REF].

This database offers a simplified access mode (by batches of 1,000 DOIs) which allows to easily obtain the status of a publication (open or closed access, with the publisher and/or in an open archive) at the time of the query. It is also possible to download a complete version of the database (called a Snapshot). For this study, we used the version dated February 2021. For the year 2019, this version lists more than 6 million publications.

Querying the Unpaywall database also allows us to validate the DOIs identified in the previous step: we consider that DOIs not found in Unpaywall generally correspond to identifiers that have not been confirmed by Crossref, the agency that certifies their quality and continuity.

Moreover, it is not uncommon to find differences in the date of publication from one database to another (often due to the time lag between the version published online (early access) and the "final" publication). We have chosen to use the year of publication provided in the Unpaywall database as the reference year (see Table 3), whether or not it is consistent with the year of publication mentioned in the source database. This choice is also the one adopted by the BSO (French Open Access Monitoring). Table 3 presents the results of the cross-matching between the six sources, and their validation with Unpaywall.

The first column recalls the number of DOIs obtained from each source, already presented in Table 2. The second column presents the numbers of DOIs found in Unpaywall and recorded in this database as published in 2019.

Note that to obtain the counts in Table 3 we cross-referenced the results of queries covering for the six sources the whole of the years 2015-2020, with the year 2019 from Unpaywall. Discrepancies in publication dates affect about 8% of the documents. Because of the reassignment of publication dates, the number of DOIs with confirmed output (second column of Table 3) for a given year, may be larger than the original number of DOIs for this year (case of MAG), despite a small loss of unidentified DOIs.

In the following section, the 139,514 records described in column 2 will be cross-referenced with the BSO.

Comparison of the FR-2019 and BSO datasets

3.1

Overlap of the two sets The corpus thus constituted (FR-2019) can now be compared with that of the French Open Science Barometer (BSO), which also aims to cover all French production, for several years including 201912 .

Since the BSO data are also restricted to publications with a DOI and have benefited from the Unpaywall query, it is easy to cross-reference the two sets of DOIs. The result is summarized in Table 4 Table 4 shows that, if we restrict ourselves to the data validated after querying Unpaywall, 8% of the total data set (i.e., 13,707 DOIs) are not identified in the BSO, while conversely 17% of the documents (i.e., 27,898 DOIs) had not been identified in our FR-2019 corpus 3.2 Data from our FR-2019 corpus that are not part of the BSO corpus The data from our sources not included in the BSO corpus seem to correspond mainly to a failure to identify the France affiliation in the algorithm developed by [START_REF] Jeangirard | Monitoring Open Access at a national level: French case study[END_REF]. This was expected and corresponds to what Jeangirard calls false negatives -which he says he cannot estimate and which we estimate here at 9% of the BSO corpus.

In our study, the main sources contributing to this subset not identified by the BSO are Scopus (63%), WoS (41%) and MAG (23%). We believe that these documents come from the less represented publishers, for whom it is likely that specific algorithms for extracting the country of affiliation have not been developed for BSO.

Data from the BSO corpus absent from the FR-2019 corpus

The data from the BSO corpus not included in our sources come mainly from humanities and social sciences journals (44%), biomedical journals (24%) and basic biology journals (12%). We note a significantly higher proportion of articles in French in this BSO-only subset: 31% compared to the average of 15% for the global corpus (the language analysis methodology will be presented below, in section 4.4).

These are mainly journals or resources not covered by the databases we have used, in particular documentary resources and journals with a national scope, in French or English: for example, the most represented sources in this set are:

-Case Medical Research: international database of clinical trials -Faculty Opinions -Post-Publication Peer Review of the Biomedical Literature -SSRN Electronic Journal: database of social science preprints.

This set of documents also includes the "false positives" reported by Jeangirard (2019), i.e., documents that their algorithm wrongly identified as publications from the France set. These are publications, for which none of the authors has an affiliation in France, but which the BSO algorithm nevertheless retained. Jeangirard estimates the false positive rate at 4% (which would correspond to about 6,000 publications for the year 2019).

We can try to estimate more precisely this share of "false positives": the search in Scopus of DOIs corresponding to publications collected for the BSO but not confirmed by our other sources sheds light on this subject:

Search in Scopus

Number This search allows us to identify 3,616 probable false positives: the Scopus database recognizes the DOI, the year is indeed 2019, but the article does not include, according to Scopus, an affiliation in France. This corresponds to 3.5% of the DOIs common to BSO and Scopus: this count thus seems compatible with the 4% estimated by [START_REF] Jeangirard | Monitoring Open Access at a national level: French case study[END_REF]. Let us note once again that the cross-referencing of the different sources highlights divergent assessments of the date of the articles. Table 7 presents the cross-referenced contributions of the sources to the overall corpus. It should be noted that the fact that a publication is identified in database A and is not identified in database B as being part of the corpus does not necessarily mean that it is absent from database B: it may be present in database B, but with a DOI that has not been filled in or is incorrect, or a failure to identify the country (no affiliation with France). Table 8 illustrates the results obtained, depending on the sources used, to determine the Open Access rate (%OA) observed in February 2021: overall we find 54% both for the BSO corpus, and for our FR-2019 corpus. The aggregation of the two results gives a slightly lower overall rate of 53% for all 167,412 publications.

Contribution of the different sources to the overall aggregated corpus

Estimated rate of Open Access publications

The reader is referred to [START_REF] Aliakbar | Merits and Limits: Applying Open Data to Monitor Open Access Publications in Bibliometric Databases[END_REF] for a discussion of the merits and limitations of these rate calculations. In their conclusions the authors recommend the use of multiple sources to reduce errors and gaps, and this is clearly a view we share. Cross-matching all these datasets allowed us to correct, at least in part, the problem of false negatives and to obtain a refined estimate of the open access rate.

Variation in open access rate by document type

The calculation for the articles alone, using the journal-article nomenclature proposed by Unpaywall, shows, as expected, a significantly higher rate of opening: 57% for the BSO corpus, and for our corpus, and 56% for the corpus resulting from the aggregation of the two sets.

This category is interesting insofar as the national policy enacted by Article 30 of the 2016 law mentioned above concerns a "scientific writing [...] published in a periodical appearing at least once a year", i.e., in our terminology, a scientific journal article.

In this context, it is worth mentioning that the approaches presented here do not distinguish between publicly funded research articles and other articles from private and industrial research, for which the open science commitments do not apply.

The details of the types of documents identified for both approaches are given in Table 9. The percentages observed are very similar in the two datasets (FR-2019 and BSO) for articles and conference proceedings. The differences are more noticeable for book chapters and can be explained by a significantly wider coverage in the case of the BSO. The 'other' category covers too many different situations for the differences in the observed rate to be significant.

Observation of annual trends (2015-2020)

In order to detect the ability to measure annual changes, we extracted the data -and present the annual counts in Table 10a-for each of the years 2015 to 2020, following the same methodology as outlined for 2019. For 2019 the counts are identical to those in Tables 3 and7, above. Table 10b provides for the years 2015 to 2019 the data from Table 4, above (the BSO does not cover the year 2020). A comparison of the rates obtained for the French corpus with those obtained on an international scale would go beyond the limits of this article: the interested reader may refer to the recent study by [START_REF] Robinson-Garcia | Open Access uptake by universities worldwide[END_REF], which also presents a discussion of the different modes of open access mentioned here (Gold, Bronze, Hybrid, Green).

Are French articles more often in open access?

It is possible to cross-reference the observations presented above with information on the language in which the article is written: are articles in French, for example, more often, or less often, in open access? To examine this, as this information is not systematically provided by all databases, we analyzed the title of the article as provided by Unpaywall, by applying a simple language detection software langdetect13 . Only detections assigned with a displayed probability greater than 0.99 were retained.

In the framework of our study of French national scientific production, for the year 2019, the two main languages concerned are English (83% of the detected documents) and French (15%), the rest of the detected languages not exceeding 3% in total. The distribution is not identical according to the document type, in particular the communications to -mostly international-conferences (labelled proceedings-article in Unpaywall) are almost always in English. Most of the French language material without Open Access comes from three areas: medical research, including journals for practitioners; and the humanities and social sciences.

% English

Results and Discussion

Discussion of the sources used

The six sources we have chosen to use actually provide three different insights:

(1) Scopus and Web of Science provide extensive coverage of the literature in peer-reviewed journals and international conference proceedings; while Scopus has a slightly wider coverage, the use of the two databases together provides a 10 to 20% improvement over what would be obtained with a single database. The MAG database, which will soon be discontinued, brings, as a complement, a set of documents not indexed by WoS and Scopus, contributing to a further increase of about 10% of the corpus identified in our study.

(2) The HAL open archive is filled at the initiative of the authors who deposit the bibliographic record (metadata) and, if applicable, the full text in its preprint or editor version. Part of the archive contains grey literature [START_REF] Schöpfel | Going Green. Publishing Academic Grey Literature in Laboratory Collections on HAL[END_REF] and moreover the DOI is filled in irregularly and not systematically. The metadata and DOI do not seem to be thoroughly quality controlled: for this reason, this source should be considered with caution for bibliometric studies. However, it is a reference source for French research and a cornerstone of the national open science policy.

(3) The ADS and PUBMED databases are thematic databases and are therefore only intended to cover parts of the research field. On the other hand, both databases are deep in their field and cover grey literature and sources not indexed by the large generalist databases.

This study sheds new light on the coverage of French scientific production by the various databases. While the Web of Science and Scopus voluntarily restrict themselves to the perimeter of peer-reviewed publications appearing in referenced journals or books [START_REF] Birkle | Web of Science as a data source for research on scientific and scholarly activity[END_REF][START_REF] Baas | Scopus as a curated, highquality bibliometric data source for academic research in quantitative science studies[END_REF], the use of complementary databases, whether thematic or not, allows us to have a more complete view of the share of literature that is not or poorly referenced, and that may be less general in scope, geographically, linguistically, or thematically. We observe that the strategy adopted by the BSO allows for the systematic collection of data on a significant quantity of these publications -often neglected in bibliometric studies. Far from identifying an optimal source, our study shows the importance of diversifying the sources used to provide complementary views on a country's publication.

Characteristics of excluded national production without DOI

Publications without DOI form a heterogeneous group of peer-review and grey literature. The share of unreferenced grey literature can be approached in particular through the HAL open archive, by considering documents without DOI, which were not taken into account in our study. However, it is advisable to make sure beforehand that the absence of DOI is not due to a lack of information, but corresponds to articles from journals that do not use this identification mode. Since the open archive, which is mainly fed by author deposits, is not fed in a complete and systematic way, this approach can only be qualitative.

We note, first of all, without surprise, a very strong disciplinary variation: only 15% of the documents in the field of humanities and social sciences (SSH) deposited in HAL have a DOI, while the proportion is 70% in Chemistry or Physics, the global average being 42% for the year 2019 considered here (see Table 2). This rate reaches 50% in the field of Computer Science.

Among the records without DOI the share of records from the SSH fields is 52%, compared to an SSH share of 12% of publications with DOI.

We also note that the full text is deposited significantly less frequently for documents without a DOI: 39%, whereas the average is 44%.

We can also note, for HAL (year 2019) a strong differentiation according to the language (we use here the language informed in the archive):

-Among the documents without DOI, the proportion of articles in French is 57% (49% for articles in English), while for articles with DOI it is only 8%. -91% of the documents in French have no DOI (or no DOI indicated). In total, we found nearly 90,000 records without a DOI in HAL (Table 2). If we restrict ourselves to documents classified as Articles, book chapters or conference papers, nearly 56,000 records without a DOI (or without a DOI indicated) listed in HAL had to be excluded from this study.

• For the journal articles (category ART in HAL) we tried to estimate the part which corresponds to not informed DOI: if we consider the articles without DOI published in a journal for which other articles have DOI, we note that it concerns 31% of the articles without DOI (in HAL in 2019). We therefore estimate that at least 30% of DOIs are missing in HAL due to DOIs that are not filled in. Most of this 30% can be expected to be covered by the other sources. If this assumption is correct, it would mean that out of the 56,000 records without DOI entered in HAL, we can estimate that there are around 40,000 articles or communications without DOI, which were therefore not taken into account. This point will be the subject of further study.

Validation of the open strategy used for the BSO

The comparison between the result obtained with our sources and the open strategy of the BSO validates the use of the latter: this strategy, if we summarize it in a few words, consists in scanning all the DOIs available by Unpaywall, and also by HAL, to identify either the French authors, or the presence of the mention of France in the address.

We observe that this strategy makes it possible to identify more than 20,000 records (if we exclude the false positives) not found by our approach, i.e., about 17% of the total: these are mainly journals that are not indexed in the major international databases, and more particularly in the biomedical and social science fields.

Our approach also identified approximately 13,000 DOIs not included in the BSO and thus estimated the false negative rate in the BSO strategy to be close to 9% (see Table 4

above).

Recurrent sources of error include conflicting approaches to publication date (with the usual confusions between the first online publication and the final date of the reference; see for example [START_REF] Liu | A matter of time: publication dates in Web of Science Core Collection[END_REF].

Conclusions

The main results of our Study are as follows:

• Our study validates a strategy of determining a collection of scientific publications with an affiliation in France, for a given year. This corpus is deliberately restricted by the use of DOI. We present the details of the counts for the year 2019. We estimate that the corpus of outputs with DOI covers around 80% of the French national scholarly production in 2019, with an additional set of 40,000 articles or communications without DOI not taken into account here.

• Our determination of cross-coverage by the various databases provides useful insight for users of these databases. We believe that these counts can help users of these databases to identify overlaps and complementarities, in a context comparable to that of our study.

• The use of multiple sources ensures validation at a sufficiently fine level to shed light on the geographical, thematic, linguistic, etc. disparities that affect bibliometric studies. Our study confirms the relevance of adopting a multi-source approach.

• The open-source strategy used by the BSO effectively identifies the vast majority of publications with a persistent identifier (DOI) for Open Science monitoring.

• The determination of the open access rate has been refined. It should be remembered that this rate depends on the date of observation and may differ depending on the type of documents we wish to consider. Our objective is not to comment here on the 54% or 53% rate reached for the opening of publications in 2019 (observed in February 2021), but to note the convergence of two different methodologies that allow us to accurately draw the shifting landscape of open science at the country level.

The question of the place of the national open archive HAL, and of other open archives, in the strategy of Open Science deserves a specific development which should be the subject of a further study. The objective of such a study would be to examine the possibilities of convergence between on the one hand the specific challenges of open archives, allowing an easy deposit at the disposal of the authors, and on the other hand the requirements of referencing and query environment which should not only provide open access to scientific knowledge produced by French research, but also support the most diverse possible readership in their consultation process.

4. 1

 1 Unpaywall results: Share of open access publications (year 2019)

Table 1 -

 1 Sources used: queries, number of records returned for the year 2019

			Number of
	Base	Sample Query (France, year 2019)	documents France	Types of documents	Domains	Practical limitations
			2019
		AFFILCOUNTRY				Export in
	Scopus	(france) and	123,181	All	All areas	batches of
		PUBYEAR = 2019				20,000
	Web of Science	CU = FRANCE AND PY =2019	124,790	All	All areas	Export in batches of 5000
	HAL (Open Archive, France)	Via API: producedDateY_i:2019 structCountry_s:fr	158,937	Open archive of French laboratories	All areas	Export in batches of 10,000
	NASA/ADS	aff: "France" AND year:2019-2019	19,997	All	Physics and Astrophysics	Export in batches of 500
		(France[Affiliation])			Medicine,	Export in
	PUBMED	AND ("2019"[Date -	56,038	All	Biology,	batches of
		Publication])			Health	10,000
		mag.Year = 2019 AND
		((SELECT COUNT(1)
		FROM
	MAG	UNNEST(mag.authors) as auth WHERE REGEXP_EXTRACT	101,885	All (with DOI)	All areas	(COKI, private communication)
		(auth.OriginalAffiliation,
		r'Fran(ce|kreich|cia)(?:\W|\s
		+|$)') is not null) > 0

Table 2 -

 2 DOI counts in the 6 sources for the year 2019. The last column shows the numbers of documents without DOIs in the Article category alone.

Table 3 -

 3 Unpaywall Cross-Reference: DOI and Year of Publication

  .

	BSO only	27,898	17%
	FR-2019 only	13,707	8%
	Global corpus
	FR-2019 + BSO	167,412	100%
	(without duplicates)
	Table 4 -Cross-referencing of FR-2019 sources with BSO data
	(Source BSO: Jeangirard, 2019)
		France 2019 (# DOI)	Contribution to the global corpus
	Corpus FR-2019	139,514	83%
	BSO 2019	153,705	92%
	In common	125,807	75%

Table 5 -

 5 Search in Scopus for false positives of BSO

Table 6

 6 presents the contributions of each source to the overall corpus (aggregating the two approaches: our FR-2019 corpus and the one collected for the BSO):

		Scopus	WoS	HAL	ADS	PUBMED	MAG	BSO
	Share of Total	67%	58%	38%	9%	29%	61%	92%
	In one source	7,211	4,009	6,335	155	230	11,665	27,898

Table 6 -

 6 Share of each source in the overall aggregated corpus (FR-2019 + BSO) The second line gives the number of documents found in only one source(year 2019)

		Scopus	WoS	HAL	ADS	PUBMED	MAG	BSO
	Scopus	111,422	88,327	54,611	14,851	46,503	85,873	102,736
	WoS	88,327	96,712	49,664	14,507	44,493	76,286	91,159
	HAL	54,611	49,664	63,413	10,521	22,934	45,608	61,440
	ADS	14,851	14,507	10,521	15,410	3,243	11,270	14,780
	PUBMED	46,503	44,493	22,934	3,243	48,047	44,071	47,696
	MAG	85,873	76,286	45,608	11,270	44,071	102,338	98,604
	BSO	102,736	91,159	61,440	14,780	47,696	98,604	153,705

Table 7 -

 7 Cross contributions from each source to the overall France 2019 corpus

Table 8

 8 presents the main results of the Open Access (OA) rate estimate observed in February 2021, based on Unpaywall.org, for each of the sources.Note that we do not use here the original BSO open access observations, which were made at a different date, and thus could not be directly compared to ours. We have chosen to report all the calculations to the same observation date: that of the production of the Unpaywall snapshot in February 2021.

	Publications France 2019	# DOI	Total OA	% OA	OA Articles	%OA Articles
	Scopus	111,422	61,854	56%	56,538	59%
	WoS	96,712	56,975	59%	54,473	60%
	HAL	63,413	42,316	67%	38,513	69%
	ADS	15,410	11,981	78%	11,608	80%
	PUBMED	48,047	29,907	62%	29,818	63%
	MAG	102,338	53,392	52%	48,647	55%
	FR-2019	128,344	75,070	54%	67,285	57%
	BSO	153,953	82,267	54%	70,197	57%
	FR-2019 + BSO	167,412	88,365	53%	75,413	56%

Table 8 -

 8 Share of open access for each source (OA calculation: Unpaywall) For all sources, including the BSO: open access as of February 2021

Table 9 -

 9 Share of Open Access by document type (Overall dataset FR-2019 + BSO)

Table 10b -

 10b Results of the two approaches for the years 2015 to 2019. See Table 4 above. For the observation of open access, the reference remains Unpaywall (snapshot of February 2021). The results are shown in Table 11. As expected, they show a steady increase in the open access rate from 2015 to 2019.The year 2020, observed in February 2021, has a different character since the observation is made before the 6-month, 1-year, or in some cases longer embargoes have expired.

	Open access rate / Year of publication	FR2015-2020	BSO	Global corpus
	2015	45,4%	45,5%	44,5%
	2016	47,8%	47,6%	46,6%
	2017	50,0%	50,0%	48,9%
	2018	51,7%	50,6%	49,9%
	2019	53,8%	53,5%	52,8%
	2020	52,6%

-52,6%

Table 11 -

 11 Change in open access rate, observed in February 2021 for publications dated from 2015 to 2020 (The global corpus is the aggregation of the two datasets FR-2015-20 and BSO)In Table12, we give examples of observations of the open access status (Gold, Green, etc.) as provided by Unpaywall for two distinct years. These few examples allow us to affirm the absence of significant bias between the two datasets: the two strategies lead to quite similar estimates.

	Open access status	FR-2015	BSO 2015	FR-2019	BSO 2019
	Gold	12%	13%	18%	18%
	Hybrid	12%	12%	9%	10%
	Bronze	4%	4%	7%	6%
	Green	18%	16%	20%	20%
	Closed	55%	54%	46%	46%

Table 12 -

 12 Open access status, observed in February 2021 for publications dated 2015 and 2019.

Table 13 -

 13 France 2019: language by document type

Table 14 -

 14 Table14shows that the rates of open access observed vary greatly according to the disciplines (extracted here from the BSO). As a general rule, documents detected as being written in French are much less frequently in Open Access. France 2019: Open access rate by language and discipline.Calculations are restricted to documents for which the language can be determined and whose discipline is assessed in the BSO.

		Number of documents	% documents in French	% OA documents in English	% OA documents in French
	Total with language and discipline detected	153,272	15%	58%	26%
	Chemistry	7,050	5%	53%	50%
	Computer and information sciences	10,225	8%	55%	37%
	Mathematics	3,914	11%	73%	55%
	Medical research	48,191	24%	57%	8%
	Biology (fond.)	21,535	12%	69%	57%
	Social sciences	8,020	69%	40%	37%
	Physical sciences, Astronomy	15,701	7%	64%	73%
	Earth, Ecology, Energy and applied biology	12,222	16%	59%	42%
	Engineering	4,402	24%	40%	40%

http://bso.esr.gouv.fr

https://data.enseignementsup-recherche.gouv.fr/explore/dataset/open-access-monitor-france/

The counts for each of the six years are available in the supplementary data file.

DOIs are managed by the non-profit association CrossRef[START_REF] Hendricks | Crossref: The sustainable source of community-owned scholarly metadata[END_REF].

Law for a Digital Republic: see in particular its article 30 https://www.legifrance.gouv.fr/dossierlegislatif/JORFDOLE000031589829/

https://openknowledge.community/dashboards/coki-open-access-dashboard/

The BSO data have been produced in December 2020 and are made available on the Open Data portal of the Ministry of Higher Education (MESRI) : https://data.enseignementsup-recherche.gouv.fr/explore/dataset/open-access-monitor-france/

Langdetect (https://pypi.org/project/langdetect/) is a python-port of Nakatani Shuyo's languagedetection library (https://github.com/shuyo/language-detection). When published (in 2010), it claimed to reach 99%+ accuracy on 49 supported languages.

Acknowledgements

We thank the two reviewers for their stimulating comments, which we believe have significantly helped to improve our work.

.fr/national-plan-for-open-science-4thjuly-2018/; https://www.ouvrirlascience.fr/second-national-plan-for-open-science/ 2 Plan S: https://www.coalition-s.org/ 2 for this purpose, in different contexts, such as the European Open Science Monitor (OSM), the German Open /synoa/oam-dokumentation/-/wikis/Quelldatenbanken/Quelldatenbanken 4 https://www.universitiesuk.ac.uk/sites/default/files/field/downloads/2021-09

Funding Information

No specific funding has been received for this research.

https://www.ouvrirlascience//jugit.fz-juelich.de

Data Availability

Data tables providing the detailed number of records for each year, as well as a notebook describing the whole procedure, are available as supplementary data files (HAL Open Archive). Subscriptions to Scopus and Web of Science are required to replicate the research, with the methods described above.

Author Contributions

Lauranne Chaignon: Writing-review & editing, Validation. Daniel Egret: Supervision, Data curation, Writing-original draft, Writing-review & editing.

Competing Interests

The authors have no competing interests.


Andreas Herzig
email: herzig@irit.fr

Laurent Perrussel
email: perrusse@irit.fr

Ivan Varzinczak

Elaborating domain descriptions

In this work we address the problem of elaborating domain descriptions (alias action theories), in particular those that are expressed in dynamic logic. We define a general method based on contraction of formulas in a version of propositional dynamic logic with an incorporated solution to the frame problem. We present the semantics of our theory change and define syntactical operators for contracting a domain description. We establish soundness and completeness of the operators w.r.t. the semantics for descriptions that satisfy a principle of modularity that we have proposed elsewhere. We also investigate an example of changing non-modular domain descriptions.

Introduction

Suppose a situation where an agent has always believed that if the light switch is up, then there is light in the room. Suppose now that someday, she observes that even if the switch is in the upper position, the light is off. In such a case, the agent must change her beliefs about the relation between the propositions "the switch is up" and "the light is on". This example is an instance of the problem of changing propositional belief bases and is largely addressed in the literature about belief change [START_REF] Kaufmann | Knowledge in Flux: Modeling the Dynamics of Epistemic States[END_REF] and belief update [START_REF] Katsuno | On the difference between updating a knowledge base and revising it[END_REF].

Next, let our agent believe that whenever the switch is down, after toggling it, there is light in the room. This means that if the light is off, in every state of the world that follows the execution of toggling the switch, the room is lit up. Then, during a blackout, the agent toggles the switch and surprisingly the room is still dark.

Imagine now that the agent never worried about the relation between toggling the switch and the material it is made of, in the sense that she ever believed that just toggling the switch does not break it. Nevertheless, in a stressful day, she toggles the switch and then observes that she had broken it.

Completing the wayside cross our agent experiments in discovering the world's behavior, suppose she has believed that it is always possible to toggle the switch, provided some conditions like being close enough to it, having a free hand, the switch is not broken, etc, are satisfied. However, in a £ Supported by a fellowship from the government of the FED-ERATIVE REPUBLIC OF BRAZIL. Grant: CAPES BEX 1389/01-7.

beautiful April fool's day, the agent discovers that someone has glued the switch and, consequently, it is no longer possible to toggle it.

The last three examples illustrate situations where changing the beliefs about the behavior of the action of toggling the switch is mandatory. In the first one, toggling the switch, once believed to be deterministic, has now to be seen as nondeterministic, or alternatively to have a different outcome in a specific context (e.g. if the power station is overloaded). In the second example, toggling the switch is known to have side-effects (ramifications) one was not aware of. In the last example, the executability of the action under concern is questioned in the light of new information showing a context that was not known to preclude its execution. Carrying out modifications is what we here call elaborating a domain description, which has to do with the principle of elaboration tolerance [START_REF] Mccarthy | Mathematical logic in artificial intelligence[END_REF].

Such cases of theory change are very important when one deals with logical descriptions of dynamic domains: it may always happen that one discovers that an action actually has a behavior that is different from that one has always believed it had.

Up to now, theory change has been studied mainly for knowledge bases in classical logics, both in terms of revision and update. Only in a few recent works it has been considered in the realm of modal logics, viz. in epistemic logic [START_REF] Hansson | A Textbook of Belief Dynamics: Theory Change and Database Updating[END_REF], and in action languages [START_REF] Eiter | Updating action domain descriptions[END_REF]. Recently, several works [START_REF] Shapiro | Iterated belief change in the situation calculus[END_REF][START_REF] Jin | Iterated belief revision, revised[END_REF] have investigated revision of beliefs about facts of the world. In our examples, this would concern e.g. the current status of the switch: the agent believes it is up, but is wrong about this and might subsequently be forced to revise his beliefs about the current state of affairs. Such belief revision operations do not modify the agent's beliefs about the action laws. In opposition to that, here we are interested exactly in such modifications. The aim of this work is to make a step toward that issue and propose a framework that deals with the contraction of action theories.

Dynamic logic, more specifically propositional dynamic logic (PDL [START_REF] Harel | Dynamic logic[END_REF])), has been extensively used in reasoning about actions in the last years [START_REF] Castilho | Formalizing action and change in modal logic I: the frame problem[END_REF][START_REF] Castilho | It depends on the context! a decidable logic of actions and plans based on a ternary dependence relation[END_REF][START_REF] Zhang | EPDL: A logic for causal reasoning[END_REF]Foo & Zhang 2002;[START_REF] Zhang | Consistency of action descriptions[END_REF]. It has shown to be a viable alternative to situation calculus approaches because of its simplicity and existence of proof procedures for it. In this work we investigate the elaboration of domain descriptions encoded in a simplified version of such a logical formalism, viz. the multimodal logic ¢¡ . We show how a theory expressed in terms of static laws, effect laws and executability laws is elaborated: usually, a law has to be changed due to its generality, i.e., the law is too strong and has to be weakened. It follows that elaborating an action theory means contracting it by static, effect or executability laws, before expanding the theory with more specific laws.

The present text is organized as follows: in the next section we define the logical framework we use throughout this work and show how action theories are encoded. Then we present our semantics of theory change and its syntactical counterpart. After that we establish soundness and completeness of our change operators w.r.t. the semantics, where completeness is conditioned by a notion of modularity that we have proposed in previous work. We then analyse an example of correcting a non-modular theory. Before concluding, we address related work on the field and discuss on how elaboration tolerant the framework here proposed is.

Background

Following the tradition in the reasoning about actions community, action theories are going to be collections of statements that have the particular form: "if context, then effect after every execution of action" (effect laws); and "if precondition, then action executable" (executability laws). Statements mentioning no action at all represent laws about the world (static laws). Besides that, statements of the form "if context, then effect after some execution of action" will be used as a causal notion to solve the frame and the ramification problems.

Logical preliminaries

Let £ ¥¤ §¦ ©¨¡ a a ¢ be the set of all atomic action con- stants of a given domain. An example of atomic action is toggle. To each atomic action a there is associated a modal operator a .

"! $# % ¨¡ p p ¢ denotes the set of all propositional constants, also called fluents or atoms. Examples of those are light ("the light is on") and up ("the switch is up"). The set of all literals is & ¢' (¦ ¢¨¡ l l ¢ , where each l) is either p or 0 p, for some p 1 2! $# % . If l 30 p, then we identify 0 l with p.

We use small Greek letters 4 5 76 8 to denote classical formulas. They are recursively defined in the usual way:

4 @9 (9 ACB ED F 3D G 3D H0 I4 PD Q4 SR S4 PD Q4 UT S4 PD Q4 WV X4 PD 4 CY `4 a cb ed is the set of all classical formulas. An example of a classical formula is up V light. By valf g4 Ih we denote the set of valuations making 4 true. We view a valuation as a maximally-consistent set of literals. For 2! i# % ¨¡ light up¢ , there are four valuations: ¡ light up¢ , ¡ light p0 up¢ , ¡ 0 light up¢ and ¡ 0 light p0 up¢ . Given a set of formulas q , by litf §q rh we denote the set of all literals appear- ing in formulas of q .

We denote complex formulas (with modal operators) by s 7t u They are recursively defined in the following way: s 9 v9 wẍ4 PD y a s D H0 s D s R s D s T s D s V s D s Y s a is the dual operator of a , defined as a s ¨def 0 r a 0 s . An example of a complex formula is 0 up V toggle up.

The semantics is that of multimodal logic K [START_REF] Popkorn | First Steps in Modal Logic[END_REF].

Definition 1 A ¢¡ -model is a tuple ¨ W R

Describing the behavior of actions in f ©g

Given a domain, we are interested in theories whose statements describe the behavior of actions. ¡ allows for the representation of such statements, that we call action laws.

Here we distinguish several types of them. The first kind of statement represents the static laws, which are formulas that must hold in every possible state of the world.

Definition 4 A static law is a formula 4 h1 a cb ed .

An example of a static law is up V light, saying that if the switch is up, then the light is on. The set of all static laws of a domain is denoted by i a cb ed .

The second kind of action law we consider is given by the effect laws. These are formulas relating an action to its effects, which can be conditional.

Definition 5 An effect law for action a is of the form 4 V a j6 , where 4 5 76 x1 a cb ed .

The consequent 6 is the effect which always obtains when action a is executed in a state where the antecedent 4 holds. If a is a nondeterministic action, then the consequent 6 is typically a disjunction. The set of effect laws of a domain is denoted by k . An example of an effect law is 0 up V toggle light, saying that whenever the switch is down, after toggling it, the room is lit up. If 6 is inconsistent, we have a special kind of effect law that we call an inexecutability law. For example, broken V e toggle G expresses that toggle cannot be executed if the switch is broken. Finally, we also define executability laws, which stipulate the context where an action is guaranteed to be executable.

In l¡ , the operator a is used to express executability.

a pF thus reads "the execution of a is possible".

Definition 6 An executability law for action a is of the form 4 V a pF , where 4 h1 a b d .

For instance, 0 broken V toggle pF says that toggling can be executed whenever the switch is not broken. The set of all executability laws of a given domain is denoted by ¡ .

The rest of this work is devoted to the elaboration of action models and theories.

Models of contraction

When an action theory has to be changed, the basic operation is that of contraction. (In belief-base update [START_REF] Winslett | Reasoning about action using a possible models approach[END_REF][START_REF] Katsuno | On the difference between updating a knowledge base and revising it[END_REF]) it has also been called erasure.) In this section we define its semantics.

In general we might contract by any formula

s . Here we focus on contraction by one of the three kinds of laws. We therefore suppose that s is either 4 , where 4 is classical, or 4 V e a (6 , or 4 V a F .

For the case of contracting static laws we resort to existing approaches in order to change the set of static laws. In the following, we consider any belief change operator such as Forbus' update method [START_REF] Forbus | Introducing actions into qualitative simulation[END_REF], or the possible models approach [START_REF] Winslett | Reasoning about action using a possible models approach[END_REF][START_REF] Winslett | Updating logical databases[END_REF], or WSS [START_REF] Herzig | Propositional belief base update and minimal change[END_REF] or MPMA [START_REF] Doherty | The PMA and relativizing change for action update[END_REF].

Contraction by 4 corresponds to adding new possible worlds to W. Let ¢ be a contraction operator for classical logic.

Definition 7 Let

W R be a ¢¡ -model and 4 a classical formula. The set of models resulting from contracting by

4 is the singleton W R ¤£ ¥ ¨¡ y W R ¢ such that W Ẅ ¢ valf g4 Ih .
Observe that R should, a priori, change as well, otherwise contracting a classical formula may conflict with ¡ .1 For instance, if 0 I4 PV a F 1 ¡ and we contract by 4 , the result may make ¡ untrue. However, given the amount of information we have at hand, we think that whatever we do with R (adding or removing edges), we will always be able to find a counter-example to the intuitiveness of the operation, since it is domain dependent. For instance, adding edges for a deterministic action may render it nondeterministic. Deciding on what changes to carry out on R when contracting static laws depends on the user's intuition, and unfortunately this information cannot be generalized and established once for all. We opt for a priori doing nothing with R and postponing correction of executability laws.

Action theories being defined in terms of effect and executability laws, elaborating an action theory will mainly involve changes in these two sets of laws. Let us consider now both these cases.

Suppose the knowledge engineer acquires new information regarding the effect of action a. Then it means that the law under consideration is probably too strong, i.e., the expected effect may not occur and thus the law has to be weakened. Consider e.g. 0 up V e toggle light, and suppose it has to be weakened to the more specific f g0 up R 0 blackouth ¥V toggle light.2 In order to carry out such a weakening, first the designer has to contract the set of effect laws and second to expand the resulting set with the weakened law.

Contraction by 4 WV a (6 amounts to adding some 'coun- terexample' arrows from 4 -worlds to 0 ¢6 -worlds. To ease such a task, we need a definition. Let PIf g4 h denote the set of prime implicates of 4 . Definition 8 Let 4 p4 I 1 a cb ed . NewCons ¥ §¦ f g4 I Qh XP If g4 R U4 h ©¨PI f 4 h computes the new consequences of 4 ¢ w.r.t. 4 : the set of strongest clauses that follow from 4 uR W4 ¢ , but do not follow from 4 alone (cf. e.g. (Inoue 1992)).

For example, the set of prime implicates of p is just

¡ p ¢ , that of the formula p R f 0 p T p h eR f 0 p T p T p Qh is ¡ p p p T p ¢ , hence we have that NewCons p ¦ f 7f 0 p T p h cR f g0 p T p T p h 7h X¨¡ p p T p ¢ .

Definition 9 Let

W R be a I¡ -model and 4 V a j6 an effect law. The models resulting from contracting by 4 V

a j6 is W R £ ¥ a ¨¡ y W R R a 9 R a ¡ f 7 8 h 9 D ! W " R # 4 5 QD ! W " R # 0 ¢6 and ¨ litf NewCons$ If 0 ¢6 5h ph ¢ ¢ .
In our context, lit f NewCons$ f g0 ¢6 5h ph corresponds to all the ramifications that action a can produce.

Suppose now the knowledge engineer learns new information about the executability of a. This usually occurs when there are executability laws that are too strong, i.e., the condition in the theory guaranteeing the executability of a is too weak and has to be made more restrictive. Let e.g. toggle F be the law to be contracted, and suppose it has to be weakened to the more specific 0 broken V toggle F .

To implement such a weakening, the designer has to first contract the set of executability laws and then to expand the resulting set with the weakened law.

Contraction by 4 WV a F corresponds to removing some arrows leaving worlds where 4 holds. Removing such ar- rows has as consequence that a is no longer always executable in context 4 .

Definition 10 Let

W R be a ¢¡ -model and 4 V a F an executability law. The set of models that result from the

contraction by 4 V a pF is W R £ ¥ % a# '& ¨¡ y W R 9 R ¨R ¨R a R a ¡ f 7 ¥ h W9 8 R a 8 and D ( W " R # 4 ¢ ¢ .
In the next section we make a step toward syntactical operators that reflect the semantic foundations for contraction.

Contracting an action theory

Having established the semantics of action theory contraction, we can turn to its syntactical counterpart. Nevertheless, before doing that we have to consider an important issue. As the reader might have expected, the logical formalism of ¡ alone does not solve the frame problem. For instance,

¡ ¢ ¡ £ up V light 0 up V toggle up up V toggle 0 up toggle pF ¤ ¡ ¥ ¡ ¦ ¨ § D broken V toggle broken
Thus, we need a consequence relation powerful enough to deal with the frame and ramification problems. This means that the deductive power of ¡ has to be augmented in or- der to ensure that the relevant frame axioms follow from the theory. Following the logical framework developed in [START_REF] Castilho | Formalizing action and change in modal logic I: the frame problem[END_REF], we consider metalogical information given in the form of a dependence relation:

Definition 11 A dependence relation is a binary relation © £ ¥¤ ¦ S& ¢' j¦ .
The expression a © l denotes that the execution of action a may change the truth value of the literal l. On the other hand, a l 1 © (written a § © l) means that l can never be caused by a. In our example we have toggle © light and toggle © 0 light, which means that action toggle may cause a change in literals light and 0 light. We do not have toggle © 0 broken, for toggling the switch never repairs it.

We assume © is finite.

Definition 12 A model of a dependence relation © is a ¡ - model such that D ¡ 0 l V a 0 l 9 a §
© l¢ . Given a dependence relation © , the associated conse- quence relation in the set of models for © is noted D . For our example we obtain

¡ ¢ ¡ £ up V light 0 up V toggle up up V toggle 0 up toggle pF ¤ ¡ ¥ ¡ ¦ D ¨ broken V toggle broken
We have toggle § © 0 broken, i.e., 0 broken is never caused by toggle. Therefore in all contexts where broken is true, after every execution of toggle, broken still remains true. The consequence of this independence is that the frame axiom broken V e toggle broken is valid in the models of © .

Such a dependence-based approach has been shown (Demolombe, [START_REF] Demolombe | Regression in modal logic[END_REF] to subsume Reiter's solution to the frame problem [START_REF] Reiter | The frame problem in the situation calculus: A simple solution (sometimes) and a completeness result for goal regression[END_REF]) and moreover treats the ramification problem, even when actions with both indeterminate and indirect effects are involved [START_REF] Castilho | It depends on the context! a decidable logic of actions and plans based on a ternary dependence relation[END_REF]Herzig & Varzinczak 2004a).

Definition 13 An action theory is a tuple of the form i © k u ¡ © .

In our example, the corresponding action theory is

i C¨¡ up V light¢ k 0 up V toggle up up V toggle 0 up ¡ ¨¡ toggle F ¢ y © ¨ ¡ ¢ ¡ £ toggle light $ toggle p0 light i toggle up $ toggle 0 up ¤ ¡ ¥ ¡ ¦
And we have i © 7k u ¡ D 0 up V toggle light. (For par- simony's sake, we write i r k u

¡ D s instead of i Ek ¡ D s .) Let i © 7k u
¡ © be an action theory and s a I¡ -formula.

i © 7k u ¡ © £ is the action theory resulting from the con-

traction of i © k u ¡ © by s .
Contracting a theory by a static law 4 amounts to using any existing contraction operator for classical logic. Let ¢ be such an operator. Moreover, based on (Herzig & Varzinczak 2005b), we also need to guarantee that 4 does not follow from k , ¡ and © . We define contraction of a do- main description by a static law as follows:

Definition 14 i r k u ¡ © £ ¥ ¨ i £ 7k 8 ¡ £ © , where i £ ¨i ¢ 4 and ¡ £ ¨¡ f 4 I) lR W4 h V a pF 9 4 I) "V a pF 1 ¡ ¢ .
We now consider the case of contracting an action theory by an executability law 4 V a F . For every executability in ¡ , we ensure that action a is executable only in contexts where 0 I4 is the case. The following operator does the job.

Definition 15 i r k u ¡ © £ ¥ a# '& ¨ i © 7k 8 ¡ £ © , where ¡ £ ¨¡ f g4 ) R S0 I4 h V a pF 9 ¥4 ) V a F 1 ¡ ¢ .
For instance, contracting glued V toggle pF in our example would give us ¡ £ ¨¡ 0 glued V toggle F ¢ . Finally, to contract a theory by 4 WV a (6 , for every effect law in k , we first ensure that a still has effect 6 whenever 4 does not hold, second we enforce that a has no effect in context 0 I4 except on those literals that are consequences of 0 ¢6 . Combining this with the new dependence relation also linking a to literals involved by 0 ¢6 , we have that a may now produce 0 ¢6 as outcome. In other words, the effect law has been contracted. The operator below formalizes this:

Definition 16 i r k u ¡ © £ ¥ a ¨ i © 7k £ ¡ © £ , with © £ ¨© ¡ y a l 9 l 1 lit f NewCons$ If 0 ¢6 5h ph ¢ and k £ ¨¡ f g4 ) R 0 I4 Ih V a (6 9 E4 ) V a j6 1 Ck ¢ © ¡ f 0 I4 R 0 lh V a 0 l 9 a l 51 Cf © £ ¨© h ¢ .
For instance, contracting the law blackout V toggle light from our theory would give us k £ ¨¡ f g0 up R 80 blackouth V toggle up f up R E0 blackouth V toggle 0 up¢ .

Results

In this section we present the main results that follow from our framework. These require the action theory under consideration to be modular (Herzig & Varzinczak 2005b). In our framework, an action theory is said to be modular if a formula of a given type entailed by the whole theory can also be derived solely from its respective module (the set of formulas of the same type) together with the static laws i . As shown in (Herzig & Varzinczak 2005b), to make a domain description satisfy such a property it is enough to guarantee that there is no classical formula entailed by the theory that is not entailed by the static laws alone.

Definition 17 4 1 a cb ed is an implicit static law of i © k u ¡ © if and only if i r k u ¡ D 4 and i § D ẍ4 .

A theory is modular if it has no implicit static laws. Our concept of modularity of theories was originally defined in (Herzig & Varzinczak 2004b;2005b), but similar notions have also been addressed in the literature [START_REF] Cholvy | Checking regulation consistency by using SOL-resolution[END_REF]Amir 2000a;[START_REF] Zhang | Consistency of action descriptions[END_REF][START_REF] Lang | Causal theories of action -a computational core[END_REF]Herzig & Varzinczak 2005a). A modularitybased approach for narrative reasoning about actions is given in [START_REF] Kaufmann | odulark : an elaboration tolerant approach to the ramification and qualification problems[END_REF].

To witness how implicit static laws can show up, consider the quite simple action theory below, depicting the walking turkey scenario (Thielscher 1995):

i C¨¡ walking V alive¢ y k ¨

i : if i © 7k 8 ¡ © is modular, then i © 7k u ¡ D ¨ G if and
only if i D ¨G . Deduction of an effect of a sequence of actions a ¡ ¢ a¡ (prediction) does not need to take into account the effect laws for actions other than a a¡ .

This applies in particular to plan validation when deciding whether a £ ¢ a¡ 4 is the case. Throughout this work we have used multimodal logic ¡ .

For an assessment of the modularity principle in the Situation Calculus, see (Herzig & Varzinczak 2005a).

Here we establish that our operators are correct w.r.t. the semantics. Our first theorem establishes that the semantical contraction of the models of i © 7k 8 ¡ © by s produces models of i © 7k 8 ¡ © £ .

Theorem 1 Let W R be a model of i © 7k u ¡ © , and let s be a formula that has the form of one of the three laws. For all models , if 1

W R £ , then is a model of i © k u ¡ © ¤£ .
It remains to prove that the other way round, the models of i © 7k u Fortunately, we can establish a result for those action theories that are modular. The proof requires three lemmas. The first one says that for a modular theory we can restrict our attention to its 'big' models.

i © 7k u ¡ © £ p a# & ¨ ¤ ¡ p V a G ¢ ¡ 0 p V a F ¢ §¤
Lemma 1 Let i © 7k u ¡ © be modular. Then i © 7k u ¡ D s if and only if D ! W " R # s for every model W R of i © 7k u ¡ © such that W ¨valf i h .
Note that the lemma does not hold for non-modular theories, as

¡ W R 9 W R is a model of i © 7k u ¡ © and W valf i h ¢ is empty then.
The second lemma says that modularity is preserved under contraction.

Lemma 2 Let i © k u ¡ © be modular, and let s be a formula of the form of one of the three laws. Then i © 7k u ¡ © £ is modular.

The third one establishes the required link between the contraction operators and contraction of 'big' models.

Lemma 3 Let

i © k u ¡ © be modular, and let s be a formula of the form of one of the three laws. If X valf i h R is a model of i © k u ¡ © ¤£ , then there is a model of i © 7k u ¡ © such that ` 1 £ .

Putting the three above lemmas together we get:

Theorem 2 Let i © 7k u ¡ © be modular, s be a formula of the form of one of the three laws, and

i £ 7k £ ¡ £ © £ be i © 7k u ¡ © £ . If it holds that i £ k £ ¡
£ D ¨ ©¨t , then for every model of i © 7k u ¡ © and every ` 1 C £ it holds that D t .

Our two theorems together establish correctness of the operators:

Corollary 1 Let i © 7k u ¡ © be modular, s be a formula of the form of one of the three laws, and

i £ 7k £ ¡ £ © £ be i r k u ¡ © £ . Then i £ k £ ¡ £ D ¨ ¨t if and only if for every model of i © 7k u ¡ © and every ` 1 C £ it holds that D ¨ t .
We give a necessary condition for success of contraction:

Theorem 3 Let s be an effect or an executability law such

that i § D s . Let i £ k £ ¡ £ © £ be i © 7k u ¡ © £ . If i © 7k u ¡ © is modular, then i £ k £ ¡ £ § D ¨ ©¨s .

Contracting implicit static laws

There can be many reasons why a theory should be changed. Following (Herzig & Varzinczak 2004b;2005b; ), here we focus on the case where it has some classical consequence 4 the designer is not aware of.

If 4 is taken as intuitive, then, normally, no change has to be done at all, unless we want to keep abide on the modularity principle and thus make 4 explicit by adding it to i .

In the scenario example of last section, if the knowledge engineer's universe has immortal turkeys, then she would add the static law alive to i . The other way round, if 4 is not intuitive, as long as 4 is entailed by i © 7k 8 ¡ © , the goal is to avoid such an entailment, i.e., what we want is i

£ 7k £ ¡ £ § D ¨ ¨4 , where i £ k £ ¡ £ © £ is i © 7k 8 ¡ © £ .
In the men- tioned scenario, the knowledge engineer considers that having immortal turkeys is not reasonable and thus decides to change the domain description to

i £ k £ ¡ £ © £ so that i £ 7k £ ¡ £ § D ¨ ¨alive.
This means that action theories that are not modular need to be changed, too. Such a changing process is driven by the problematic part of the theory detected by the algorithms defined in (Herzig & Varzinczak 2004b) and improved in (Herzig & Varzinczak ).

The algorithm works as follows: for each executability law 4 V a pF in the theory, construct from k and © a set of inexecutabilities

¡ 4 V a G " $ 4 I¡ CV a G ¢ that potentially conflict with 4 V a pF . For each , ¡ £¢ ¤ ¥¢ ¤¦ , if 4 R 4 I) is satisfiable w.r.t.
i , mark 0 rf g4 R 4 ) h as an implicit static law. Incrementally repeat this procedure (adding all the 0 rf g4 R U4 ) h that were caught to i ) until no implicit static law is obtained.

For an example of the execution of the algorithm, consider i © k u ¡ © as above. For the action tease, we have the executability tease pF . Now, from k , and © we try to build an inexecutability for tease. We take tease walking and compute then all indirect effects of tease w.r.t. i . From walking V alive, we get that alive is an indirect effect of tease, giving us tease alive. But tease alive 1 © , which means the frame axiom 0 alive V tease 0 alive holds. To- gether with tease alive, this gives us the inexecutability 0 alive V tease G . As i ¡ F " p0 alive¢ is satisfiable (F is the antecedent of the executability tease F ), we get the implicit static law alive. For this example no other inexecutability for tease can be derived, so the computation stops.

It seems that in general implicit static laws are not intuitive. Therefore their contraction is more likely to happen than their addition.3 In the example above, the action theory has to be contracted by alive. 4 In order to contract the action theory, the designer has several choices: 1) Contract the set i . (In this case, such an operation is not enough, since alive is a consequence of the rest of the theory.)

2) Weaken the effect law tease walking to alive V tease walking, since the original effect law is too strong. This means that in a first stage the designer has to contract the theory and in a second one expand the effect laws with the weaker law. The designer will usually choose this option if she focuses on the preconditions of the effects of actions.

3) Weaken the executability law tease F by rephrasing it as alive V tease F : first the executability is contracted and then the weaker one is added to the resulting set of executability laws. The designer will choose this option if she focuses on preconditions for action execution. The analysis of this example shows that the choice of what change has to be carried out is up to the knowledge engineer. Such a task can get more complicated when ramifications are involved. To witness, suppose our scenario has been formalized as follows: i ¨¡ walking V alive¢ , k ¨¡ shoot 0 alive¢ , ¡ ¨¡ y shoot pF ¢ , and © ¡y shoot p0 alive ¢ . From this action theory we can derive the inexecutability walking V shoot G and thus the im- plicit static law 0 walking. In this case we have to change the theory by contracting the frame axiom walking V shoot walking (which amounts to adding the missing indi- rect dependence shoot © 0 walking).

Elaboration tolerance

The principle of elaboration tolerance has been proposed by [START_REF] Mccarthy | Mathematical logic in artificial intelligence[END_REF]. Roughly, it states that the effort required to add new information to a given representation (new laws or entities) should be proportional to the complexity of the information being added, i.e., it should not require the complete reconstruction of the old theory (Shanahan 1997).

Since then many formalisms in the reasoning about actions field claim, in a more or less tacit way, to satisfy such a principle. However, for all this time there has been a lack of good formal criteria allowing for the evaluation of theory change difficulty and, consequently, comparisons between different frameworks are carried out in a subjective way.

The proposal by Amir (Amir 2000b) made the first steps in formally answering what difficulty of changing a theory means by formalizing one aspect of elaboration tolerance.

The basic idea is as follows: let § ©¨be the original theory and let § and § be two equivalent (and different) theo- ries such that each one results from § ¨by the application of some sequence of operations (additions and/or deletions of formulas). The resulting theory whose transformation from § ¨has the shortest length (number of operations) is taken as the most elaboration tolerant.

Nevertheless, in the referred work only addition/deletion of axioms is considered, i.e., changes in the logical language or contraction of consequences of the theory not explicitly stated in the original set of axioms are not taken into account. This means that even the formal setting given in (Amir 2000b) is not enough to evaluate the complexity of theory change in a broad sense. Hence the community still needs formal criteria that allow for the comparison between more complex changes carried out by frameworks like ours, for example.

Of course, how elaboration tolerant a given update/revision method is strongly depends on its underlying formalism for reasoning about actions, i.e., its logical background, the solution to the frame problem it implements, the hypothesis it relies on, etc. In what follows we discuss how the dependence-based approach here used behaves when expansion is considered. Most of the comments concerning consequences of expansion can also be stated for contraction. We do that with respect to some of the qualitative criteria given in [START_REF] Mccarthy | Elaboration tolerance[END_REF]. In all that follows we suppose that the resulting theory is consistent.

Adding effect laws In the dependence-based framework, adding the new effect law 4 V a j6 to the theory de- mands a change in the dependence module © . In that case, the maximum number of statements added to © is D ¡ l 9 l 1 litf NewCons$ f 6 5h h ¢ D (dependences for all indirect effects have to be stated, too). This is due to the explanation closure nature of the reasoning behind dependence (for more details, see [START_REF] Castilho | Formalizing action and change in modal logic I: the frame problem[END_REF]). Because of this, according to Shanahan [START_REF] Shanahan | Solving the frame problem: a mathematical investigation of the common sense law of inertia[END_REF], explanation closure approaches are not elaboration tolerant when dealing with the ramification problem. In order to achieve that, the framework should have a mechanism behaving like circumscription that automatically deals with ramifications. This raises the question: "if we had an automatic (or even semi-automatic) procedure to do the job of generating the indirect dependences, could we say the framework is elaboration tolerant?". We think we can answer positively to such a question, and, supported by Reiter [START_REF] Reiter | Knowledge in Action: Logical Foundations for Specifying and Implementing Dynamical Systems[END_REF], we are working on a semi-automatic procedure for generating the dependence relation from a set of effect laws.

Adding executability laws Such a task demands only a change in the set ¡ of executabilities, possibly introducing implicit static laws as a side effect.

Adding static laws Besides expanding the set i , adding new (indirect) dependences may be required (see above).

Adding frame axioms

If the frame axiom 0 l V a 0 l has to be valid in the resulting theory, expunging the dependence a © l should do the job.

Adding a new action name Without loss of generality we can assume the action in question was already in the language. In that case, we expect just to add effect or executability laws for it. For the former, at most D & ¢' j¦ iD depen- dences will be added to © . (We point out nevertheless that the requirement made in [START_REF] Mccarthy | Elaboration tolerance[END_REF]) that the addition of an action irrelevant for a given plan in the old theory should not preclude it in the resulting theory is too strong. Indeed, it is not difficult to imagine a new action forcing an implicit static law from which an inexecutability for some action in the plan can be derived. The same holds for the item below.)

Adding a new fluent name In the same way, we can suppose the fluent was already in the language. Such a task amounts thus to one or more of the above expansions. There will be at most 2 CD £ ¥¤ ¦ D new elements added to © .

Related work

Following [START_REF] Kaufmann | What is believed is what is explained[END_REF][START_REF] Liberatore | A framework for belief update[END_REF], [START_REF] Eiter | Updating action domain descriptions[END_REF] have investigated update of action domain descriptions. They define a version of action theory update in an action language and give complexity results showing how hard such a task can be.

Update of action descriptions in their sense is always relative to some conditions (interpreted as knowledge possibly obtained from earlier observations and that should be kept). This characterizes a constraint-based update. In the example they give, change must be carried out preserving the assumption that pushing the button of the remote control is always executable. Actually, the method is more subtle, as new effect laws are added constrained by the addition of viz. an executability law for the new action under concern. In the example, the constraint (executability of push) was not in the original action description and must figure in the updated theory.

They describe domains of actions in a fragment of the action language ¡ (Gelfond & Lifschitz 1998). However they do not specify which fragment, so it is not clear whether the claimed advantages ¡ has over ¢ really transfer to their framework. At one hand, their approach deals with indirect effects, but they do not talk about updating a theory by a law with a nondeterministic action. Anyway, except for concurrency, their account can be translated into ours, as shown in [START_REF] Castilho | Formalizing action and change in modal logic I: the frame problem[END_REF].

Eiter et al. consider an action theory § as comprising two main components: § ¤£ , the part of the theory that must re- main unchanged, and § ¤¥ , the part concerning the statements that are allowed to change. The crucial information to the associated solution to the frame problem is always in § ¦£ .

Given an action theory § ¨ § £ § §¥ , f f § £ § §¥ h i § ©¨h is the problem of updating § by § i k warranting the result satisfies all constraints in ¨ i ¡ . Even though they do not explicitly state postulates for their kind of theory update, they establish conditions for the update operator to be successful. Basically, they claim for consistency of the resulting theory; maintenance of the new knowledge and the invariable part of the description; satisfaction of the constraints in ¨; and minimal change.

In some examples that they develop, the illustrated "partial solution" does not satisfy ¨due to the existence of implicit laws (cf. Example 1, where there is an implicit inexecutability law). To achieve a solution, while keeping ¨, some other laws must be dropped (in the example, the agent gives up a static law). 5Just to see the link between update by subsumed laws and addition of implicit static laws, we note that Proposition 1 in the referred work is the same as Theorem 14 in (Herzig & Varzinczak 2005b): every implicit static law in Herzig and Varzinczak's sense is trivially a subsumed law in Eiter et al.'s sense.

With their method we can also contract by a static and an effect law. Contraction of executabilities are not explicitly addressed, and weakening (replacing a law by a weaker one) is left as future work.

A main difference between the approach in [START_REF] Eiter | Updating action domain descriptions[END_REF] and ours is that we do not need to add new fluents at every elaboration stage: we still work on the same set of fluents, refining their behavior w.r.t. an action a. In Eiter et al.'s proposal an update forces changing all the variable rules appearing in the action theory by adding to each one a new update fluent. This is a constraint when elaborating action theories.

Concluding remarks

In this work we have presented a general method for changing a domain description (alias action theory) given any formula we want to contract.

We have defined a semantics for theory contraction and also presented its syntactical counterpart through contraction operators. Soundness and completeness of such operators with respect to the semantics have been established (Corollary 1).

We have also shown that modularity is a necessary condition for a contraction to be successful (Theorem 3). This gives further evidence that our modularity notion is fruitful. We have analysed an example of contraction of a nonmodular theory by an implicit static law that is unintended.

Because of forcing formulas to be explicitly stated in their respective modules (and thus possibly making them inferable in independently different ways), intuitively modularity could be seen to diminish elaboration tolerance. For instance, when contracting a classical formula 4 from a non- modular theory, it seems reasonable to expect not to change the set of static laws i , while the theory being modular surely forces changing such a module. However it is not difficult to conceive non-modular theories in which contraction of a formula 4 may demand a change in i as well. To witness, suppose i ¨¡ 4 V 4 ¢ in an action theory from whose dynamic part we (implicitly) infer 0 I4 . In this case, a contraction of 0 I4 keeping 0 I4 I would necessarily ask for a change in i . We point out nevertheless that in both cases (modular and non-modular) the extra work in changing other modules stays in the mechanical level, i.e., in the machinery that carries out the modification, and does not augment in a significant way the amount of work the knowledge engineer is expected to do.

What is the status of the AGM-postulates for contraction in our framework? First, contraction of static laws satisfies all the postulates, as soon as the underlying classical contraction operation ¢ satisfies all of them.

In the general case, however, our constructions do not satisfy the central postulate of preservation i © k u ¡ © £ i © k u ¡ © if i © k u ¡ § D ¨ s . Indeed, suppose we have a language with only one atom p, and a model with two worlds ¨¡ p¢ and ¥ ¨¡ 0 p¢ such that R a 8 , 8 R a , and 8 R a 8 . Then D p V a 0 p and § D a 0 p, i.e., is a model of the effect law p V a 0 p, but not of a 0 p. Now the contraction £ a ¡ p yields the model ` such that R a ¨W W. Then § D $ p V a 0 p, i.e., the effect law p V a 0 p is not preserved. Our contraction operation thus behaves rather like an update operation. Now let us focus on the other postulates. Since our operator has a behavior which is close to the update postulate, we focus on the following basic erasure postulates introduced in [START_REF] Katsuno | Propositional knowledge base revision and minimal change[END_REF]. Let Cnf § h be the set of all logical consequences of a theory § . Here we have presented the case for contraction, but our definitions can be extended to revision, too. Our results can also be generalized to the case where learning new actions or fluents is involved. This means in general that more than one simple formula should be added to the belief base and must fit together with the rest of the theory with as little sideeffects as possible. We are currently defining algorithms based on our operators to achieve that.

KM1

  p a# '& .

  alive tease walking¢ D tease alive, second D 0 alive V tease 0 alive (from the independence

			tease walking loaded V shoot 0 alive
	¡ ¨¡ shoot p0 loaded i tease pF " shoot p0 walking $ With this domain description we have i © 7k u ¢ shoot pF © ¨ shoot 0 alive i tease walking ¡ first,	D alive:
	law of	i © 7k u	¡ © .
	Modular theories have several advantages (Herzig & Varzinczak 2004b; ). For example, consistency of a modular action theory can be checked by just checking consistency of

¡ walking V tease § © alive), and then i r k D ¨ 0 alive V tease G . As long as i © 7k 8 ¡ D ¨ tease pF , we must have i © 7k 8 ¡ D ¨ alive. As i § D ¨alive, the formula alive is an implicit static

  © ¤£ h Cnf i © k u ¡ © h Postulate KM1 does not always hold because it is possible to make the formula 4 hV a G valid in the resulting theory by removing elements of R a (cf. Definition 10). © and i 7k ¡ © are modular and the propositional contraction operator ¢ satisfies Postulate KM3, then Postulate KM3 is satisfied for every

	Cnf	i © 7k u
	KM2 Under the condition that s 1 Cnf i © 7k 8 ¡ © £ h i r k u ¡ © is modular, Postu-late KM2 is satisfied (cf. Theorem 3). KM3 If Cnf i k ¡ © h ¨Cnf i 7k ¡ © h and D ¨ s Y s , then Cnf i ¢ 7k ¡ © £ £¢ h Cnf i 7k ¡ © £ ¦ h .
	Theorem 4 If	i ¢ H k
	s s 1 a cb ed .

¡ ¡

We are indebted to the anonymous referees for pointing this out to us.

The other possibility of weakening the law, i.e., replacing it by ) up 0 21 toggle3 '4 light 5 ) light6 looks silly. We were not able to find examples where changing the consequent could give a more intuitive result. In this sense, we prefer to always weaken a given law by strengthening its antecedent.

In all the examples in which we have found implicit static laws that are intuitive they are so evident that the only explanation for not having them explicitly stated is that they have been forgotten by the theory's designer.

Here the change operation is a revision-based operation rather than an update-based operation since we mainly "fix" the theory.

This does not mean however that the updated theory will necessarily contain no implicit law.

Acknowledgments

We are grateful to the anonymous referees for many useful comments on an earlier version of this paper.