The inverse of this relationship is: P 2 [overlaps À1 ]P 1 . In the superscripted relationship, the first time period is the later one. The predicate for this relationship, as it holds between two time periods expressed as pairs of dates using the closed-open con- vention, is: (eff_beg_dt 1 > eff_beg_dt 2 ) AND (eff_beg_dt 1 < eff_end_dt 2 ) AND (eff_end_dt 1 > eff_end_dt 2 ) It says that P 1 starts after P 2 starts and before P 2 ends, and ends after P 2 ends. Consider the following request for information: which policies began before the Diabetes Management Wellness Pro- gram for 2009, and ended while that program was still going on? The SQL written to fulfill this request is: SELECT * FROM V_Allen_Example WHERE pol_eff_beg_dt < wp_eff_beg_dt AND pol_epis_end_dt > wp_eff_beg_dt AND pol_epis_end_dt < wp_epis_end_dt P 1 [intersects] P 2 This not an Allen relationship. It is the node in our taxonomy of Allen relationships which includes the [starts], [starts À1 ], [finishes], [finishes À1 ], [during], [during À1 ], [equals], [overlaps] and [overlaps À1 ] relationships. In other words, it combines the [overlaps] relationships with the [ fills] relationships. In the non -superscripted relationship, the first time period is the earlier one. The predicate for this relationship, as it holds between two time periods expressed as pairs of dates using the closed-open convention, is: (eff_beg_dt 1 <¼ eff_beg_dt 2 ) AND (eff_end_dt 1 > eff_beg_dt 2 ) It says that P 1 starts no later than P 2 starts, and ends after P 2 starts. The idea behind it is that it includes every relationship in which P 1 and P 2 have at least one clock tick in common and in which P 1 is the earlier time period. The limiting case is that in which P 1 ends at the same time P 2 starts. So let P 1 be [4/15/2010 – 5/13/2010] and let P 2 be [5/12/2010 – 9/18/2010]. The clock tick they share is 5/12/2010. The inverse of this relationship is: P 1 [intersects À1 ]P 2 . The first time period in this non-superscripted relationship is the later one. The predicate for this relationship, as it holds between 326 Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES two time periods expressed as pairs of dates using the closed- open convention, is: (eff_beg_dt 1 >¼ eff_beg_dt 2 ) AND (eff_beg_dt 1 < eff_end_dt 2 ) It says that P 2 starts no later than P 1 starts, and ends after P 1 starts. The idea behind it is that it includes every relationship in which P 1 and P 2 have at least one clock tick in common and in which P 1 is the earlier time period. All pairs of time periods that share at least one clock tick sat- isfy one or the other of these two predicates. So the predicate that expresses the truth condition for all time periods that share at least one clock tick is: ((eff_beg_dt 1 < eff_end_dt 2 ) AND (eff_end_dt 1 > eff_beg_dt 2 )) It says that either one of the clock ticks in P 1 is also in P 2 or that one of the clock ticks in P 2 is also in P 1 . The idea behind it is that it covers all the cases where two time periods have at least one clock tick in common, regardless of which is the later time period. It is interesting to look at this relationship in terms of what falls outside its scope. For any two relationships that share at | | | | Equals | | | | Intersects OverlapsFills Occupies Starts Finishes During | | | | Aligns | | | | | | | | Time Period Relationships Along a Common Timeline Figure 14.10 P 1 [intersects] P 2 . Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES 327 least one clock tick, neither ends before the other begins. Other- wise, they could not share a clock tick. Looking at [ includes] in terms of what falls outside its scope, we can express it as follows: NOT(eff_end_dt 1 <¼ eff_beg_dt 2 ) AND NOT(eff_beg_dt 1 >¼ eff_end_dt 2 ) And for those who like as few NOTs as possible, a logical rule (one of the transformation rules known as the De Morgan’s equi- valences) gives us the following predicate: NOT((eff_end_dt 1 <¼ eff_beg_dt 2 ) OR (eff_beg_dt 1 >¼ eff_end_dt 2 )) In other words, if two things are both not true, then it isn’t true that either of them is true! On such profundities are the rules of logic constructed. Consider the following request for information: which policies share any clock tick with the Diabetes Management Wellness Program for 2009? The SQL written to fulfill this request is: SELECT * FROM V_Allen_Example WHERE pol_eff_beg_dt < wp_epis_end_dt AND pol_epis_end_dt > wp_eff_beg_dt Notice how this SQL is much simpler than the OR’d collection of all of the conditions that make up the leaf nodes of its Allen relationships. P 1 [before] P 2 This is a pair of relationships, one the inverse of the other. In the non-superscripted relationship, the first time period is the earlier one. Time Period Relationships Along a Common Timeline Excludes Before | | | | Figure 14.11 P 1 [before] P 2 . 328 Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES The predicate for this relationship, as it holds betw een two time periods expressed as pairs of dates using the closed-open convention, is: (eff_end_dt 1 < eff_beg_dt 2 ) It says that after P 1 ends, there is at least one clock tick before P 2 begins. For example, consider the case where eff_end_dt 1 is 5/13/2014 and eff_beg_dt 2 is 5/14/2014. Because of the closed- open convent ion, the last clock tick in P 1 is 5/12/2014, and so there is one clock tick gap between the two time periods, that clock tick being 5/13/2014. The inverse of this relationship is: P 1 [before À1 ]P 2 . In the superscripted relationship, the first time period is the later one. The predicate for this relationship, as it holds between two time periods expressed as pairs of dates using the closed-open convention, is: (eff_beg_dt 1 > eff_end_dt 2 ) It says that before P 1 begins, there is at least one clock tick after P 2 ends. For example, consider the case where eff_beg_dt 1 is 5/14/2014 and eff_end_dt 2 is 5/13/2014. Because of the closed-open convention, the last clock tick in P 2 is 5/12/2014, and so there is one clock tick gap between the two time periods, that clock tick being 5/13/2014. Throughout this book, if it isn’t important which time period comes first, we will simply say that the two time periods are non- contiguous. This is a particularly useful pair of relationships because they distinguish episodes of the same object from one another. Two adjacent versions—versions of an object with no other ver- sion of the same object between them—belong to different episodes just in case the earlier one is [before] the later one. Of two adjacent episodes of the same object, one is [before] the other, and the other is [before À1 ] the former. Consider the following request for information: which policies ended at least one date before the Diabetes Manage- ment Wellness Program for 2009 began? The SQL written to fulfill this request is: SELECT * FROM V_Allen_Example WHERE pol_epis_end_dt < wp_eff_beg_dt P 1 [meets] P 2 This is a pair of relationships, one the inverse of the other. In the non-superscripted relationship, the first time period is the earlier one. Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES 329 The predicate for this relationship, as it holds between two time periods expressed as pairs of dates using the closed-open convention, is: (eff_end_dt 1 ¼ eff_beg_dt 2 ) It says that after P 1 ends, P 2 begins on the very next clock tick. There is no clock tick gap between them. Say that both dates are 5/13/2004. This means that the last clock tick in P 1 is 5/12/2004 and the first clock tick in P 2 is 5/13/2004, and so there are no clock ticks between the two time periods. The inverse of this re lationship is: P 2 [meets À1 ]P 1 . In the super- scripted relationship, the first time period is the later one. The p r ed- icate for this relationship , as it holds between two time periods expressed as pairs of dates using the closed-open convention, is: (eff_beg_dt 1 ¼ eff_end_dt 2 ) It says that before P 1 begins, P 2 ends on the previous clock tick. There is no clock tick gap between them. This is a particularly useful relationship because it defines a collection of versions of the same object that belong to the same episode. Every adjacent pa ir of versions of the same object that do not share any clock ticks, i.e. in which neither includes the other, and which also do not have a single clock tick between them, belong to the same episode. The earlier version of the pair meets the later one; the later version is met by the earlier one. Throughout this book, if it isn’t important which of two time periods that meet come first, we will simply say that the two time periods are contiguous. Consider the following request for information: which policies ended immediately before the Diabetes Management Wellness Program for 2009 began? Time Period Relationships Along a Common Timeline Excludes | | | Meets Figure 14.12 P 1 [meets] P 2 . 330 Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES The SQL written to fulfill this request is: SELECT * FROM V_Allen_Example WHERE pol_epis_end_dt ¼ wp_eff_beg_dt P 1 [excludes] P 2 This not an Allen relationship. It is the node in our taxonomy of Allen relationships which includes the [before], [before À1 ], [meets] and [meets À1 ] relationships. In the non-superscripted relationship, the first time period is the earlier one. The predicate for thi s relationship, as it holds between two time periods expressed as pairs of dates using the closed-open convention, is: (eff_end_dt 1 <¼ eff_beg_dt 2 ) It says that P 2 starts either immediately after the end of P 1 ,or later than that. The idea behind it is that it includes every rela- tionship in which P 1 and P 2 have no clock ticks in common and in which P 1 is the earlier time period. The inverse of this relationship is: P 1 [excludes À1 ]P 2 . The first time period in this non-superscripted relationship is the later one. The predicate for this relationship, as it holds between two time periods expressed as pairs of dates using the closed- open convention, is: (eff_beg_dt 1 >¼ eff_end_dt 2 ) It says that P 2 ends either immediately before the start of P 1 , or earlier than that. The idea behind it is that it includes every relationship in which P 1 and P 2 have no clock ticks in common and in which P 1 is the later time period. All pairs of time periods that share no clock ticks satisfy one or the other of these two predicates. So the predicate that Time Period Relationships Along a Common Timeline Excludes Before Meets | | | | | | | Figure 14.13 P 1 [excludes] P 2 . Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES 331 designates all and only those time periods that share no clock ticks is: (eff-end-dt 1 <¼ eff-beg-dt 2 ) OR (eff-beg-dt 1 >¼ eff-end-dt 2 ) It says that either P 2 starts after P 1 ends or ends before P 2 starts. The idea behind it is that regardless of which time period comes first, they share no clock ticks. It should be the case that two time periods [ exclude] one another if and only if they do not [ intersect] one another. If so, then if we put a NOT in front of the predicate for the [ intersects] relationship, we should get a predicate which expresses the [ excludes] relationship. 2 Putting a NOT in front of the [intersects] relationship, we get: NOT((eff_beg_dt 1 < eff_end_dt 2 ) AND (eff_end_dt 1 > eff_beg_dt 2 )) This is a statement of the form NOT(X AND Y). The first thing we will do is transform it, according to the De Morgan’s rules, into (NOT-X OR NOT-Y). This gives us: NOT(eff_beg_dt 1 < eff_end_dt 2 ) OR NOT(eff_end_dt 1 > eff_beg_dt 2 ) Next, we can replace NOT(eff_beg_dt 1 < eff_end_dt 2 ) with (eff_beg_dt 1 >¼ eff_end_dt 2 ), and NOT(eff_end_dt 1 > eff_beg_dt 2 ) with (eff_end_dt 1 <¼ eff_beg_dt 2 ). This gives us: (eff_beg_dt 1 >¼eff_end_dt 2 ) OR (eff_end_dt 1 <¼ eff_beg_dt 2 ) Finally, by transposing the two predicates, we get: (eff_end_dt 1 <¼ eff_beg_dt 2 ) OR (eff_beg_dt 1 >¼ eff_end_dt 2 ) And this is indeed the predicate for the [excludes] relation- ship, demonstrating that [ excludes] is indeed logically equivalent to NOT[ intersects]. Consider the following request for information: which policies either ended before the Diabetes Management Wellness Program for 2009 began, or began after that program ended? The SQL written to fulfill this request is: SELECT * FROM V_Allen_Example WHERE (pol_epis_end_dt <¼ wp_eff_beg_dt OR pol_eff_beg_dt >¼ wp_epis_end_dt) 2 Since, at the time we are writing this paragraph, we haven’t done this, it is an excellent way of finding out if we have made any logical mistakes so far. 332 Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES Point in Time to Period of Time Queries A point in time is a period of time that includes only one clock tick. Thus, using the closed-open convention, a point in time, T 1 , is identical to the period of time [T 1 –T 2 ] where T 2 is the next clock tick after T 1 . The only difference is in the notation. In the following discussions, we will use the simpler notation, T 1 , for the point in time. In this section, we consider periods of time that are longer than a single clock tick. Periods of time that are one clock tick in length are points in time, and we consider Allen relationships between two points in time later. Given that P 1 is longer than a single clock tick, it may or may not share a clock tick with T 1 . If it does, then T 1 [occupies] P 1 . Otherwise, either one is [before] the other, or else they [meet]. In Asserted Versioning databases, all temporal periods are delimited with the same point in time granularities. When compar- ing time periods to time periods, the logic in the AVF does not depend on the granularity of the clock ticks used in temporal parameters, as long as all of them are the same. The clock ticks could be months (as they are in the examples throughout this book), days, seconds or microseconds of any size. As we noted in Chapter3, the AVF can carry out its temporal logic without caring about granularity specifically because of the closed-open convention. However, when comparing a point in time to a period of time, we must be aware of the granularity of the clock tick, and mus t often either add a clock tick to a point or period in time, or sub- tract a clock tick from a point or period in time. Consequently, we need to specify the clock tick duration used in the specific implementation to correctly perform this arithmetic. We will use “fCTD”, standing for “clock tick duration”, as the name of a function that converts an integer into that integer number of clock ticks of the correct granularity. So, for example, in: eff_end_dt – fCTD(1) fCTD takes on the value of one clock tick. If the granularity is a month, as it is in most of the examples in this book, the result will be to subtract one month from the effective end date. If the gran- ularity is a millisecond-level timestamp, it will subtract one milli- second from that date. The fCTD function determines the granularity for a specific Asserted Versioning database from the miscellaneous metadata table, shown as Figure 8.7 in Chapter 8. Different DBMSs use different date formats for date literals. It is also dependent on the def ault language and the date format currently set. These formats are shown in Figure 14.14. Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES 333 We used the USA format in parts of the book, so we will assume that the default date format in our sample DBMS is the same. Different DBMSs use different syntax for date arithmetic. SQL Server would use something like this: AND DATEADD(DAY, -1, pol.eff_end_dt) > ‘07/15/2010’ where DAY is the granularity (which can also be abbreviated as DD or D), while DB2 might use: AND (pol.eff_end_dt - 1 DAY) > ‘07/15/2010’ with the reserved word DAY indicating the granularity. We will use the T-SQL format for our examples, and will assume our clock tick granularity is one month, to keep it in synch with the examples used in the book. However, in real-world databases, the granularity would more likely be a day, a second or a microsecond. This fCTD translation could be built into a reusable database function as part of the framework based on metadata. T 1 [starts] P 1 This is a pair of relationships, one the inverse of the other. In the non-superscripted relationship, the first time period is the point in time, i.e. the single clock-tick time period. Figure 14.15 shows this relationship, and its place in our taxonomy. The two dashed lines in the illustra tion graphically represent T 1 and P 1 , with T 1 being the upper dashed line. The predicate for this relationship, as it holds between a period of time expressed as a pair of dates using the closed-open convention, and a point in time, is: (T 1 ¼ eff_beg_dt) It says that T 1 starts at P 1 . Consider the following request for information: which policies begin on the same date as the 2009 Diabetes Manage- ment Wellness Program? Name Layout Example ISO yyyy-mm-dd 2010-09-25 09/25/2010 25.09.2010 2010-09-25 mm/dd/yyyy dd.mm.yyyy yyyy-mm-dd USA EUR JIS Figure 14.14 Date Formats for Date Literals. 334 Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES The SQL written to fulfill this request is: SELECT * FROM V_Allen_Example WHERE pol_eff_beg_dt ¼ wp_eff_beg_dt T 1 [finishes] P 1 This is a pair of relationships, one the inverse of the other. In the non-superscripted relationship, the first time period is the point in time, i.e. the single clock-tick time period. Figure 14.16 shows this relationship , and its place in our taxonomy. The two dashed lines in the illustration graphically represent T 1 and P 1 , with T 1 being the upper dashed line. The predicate for this relationship, as it holds between a period of time exp ressed as a pair of dates using the closed-open convention, and a point in time, is: (T 1 ¼ eff_end_dt – fCTD(1)) Since the effective end date of a time period is the next clock tick after the last clock tick in that time period, this predicate says that P 1 ends on the clock tick that is T 1 . Fills Occupies Aligns Starts |-| | | Intersects Time Period Relationships Along a Common Timeline Figure 14.15 T 1 [starts] P 1 . Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES 335 [...]... relationship It is the node in our taxonomy of Allen relationships which, when one of the time periods is a point in time, includes the [starts], [finishes], and [during] relationships In other words, it combines the [during] relationships with the [aligns] relationships These are all the relationships in which a time period (of more than one clock tick) includes a point in time The predicate for this... using the closed-open convention, and a point in time, is: (eff_beg_dt < T1) AND (eff_end_dt À fCTD(1) > T1) It says that T1 occurs during P1 just in case P1 starts before T1 and ends after T1 Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES Time Period Relationships Along a Common Timeline Intersects Fills Occupies During |-| | | Figure 14.17 T1 [during] P1 Consider the following request for information:... relationship, as it holds between two points in time, expressed as points in time, is: (T1 ¼ T2) It says that T1 and T2 are equal if and only if they occur on the same clock tick Time Period Relationships Along a Common Timeline Excludes Meets |-|-| Figure 14.23 T1 [meets] T2 Chapter 14 ALLEN RELATIONSHIP AND OTHER QUERIES Time Period Relationships Along a Common Timeline Intersects Fills Equals |-| |-| Figure... predicates, and SQL statements illustrating their use, each corresponding to one of the Allen relationships or one of the nodes in our taxonomy of Allen relationships We have reviewed all possible Allen relationships, and taxonomic groupings of them, between pairs of time periods, between a time period and a point in time, and between two points in time Once again, completeness of coverage has been guaranteed... V_Allen_Example WHERE (pol_epis_end_dt . AND OTHER QUERIES Point in Time to Period of Time Queries A point in time is a period of time that includes only one clock tick. Thus, using the closed-open. a single clock tick. Periods of time that are one clock tick in length are points in time, and we consider Allen relationships between two points in time