Towards an Algebra for Genealogy

Towards a Notation for Family Relationships

Natural languages give names for important blood relatives: mother, father, uncle, grandmother, daughter. It is convenient to find a way of codifying these in a notation that is concise and logical. The notation that is used here, consists of two integers, representing steps up and down the family tree, respectively, separated by a binary value (x or y) to represent female or male.

Thus 1y0 is the male person one step up the tree, and no steps down, namely my father. A similar code can be given for each of the other commonly occurring blood relations:

From this notation, it becomes apparent that siblings are zeroth-cousins, uncles and aunts are first-cousins-once-removed-senior, and parents, grandparents and great-grandparents are similarly bar-oneth- cousins once-, twice- or thrice-removed (senior).

Here are a few examples going in the other direction on the tree:

This notation already has the advantage of being concise, and much shorter than the equivalent expression in English (for writing in the cramped space on the family tree, for instance).

Incidentally, the letters x and y were chosen to correspond to the names of the female and male chromosomes. The letters f and m had been considered for female and male, but these led to a great deal of confusion, since the most commonly expressed relationships are mother and father, which have the same initials, but reversed.

There is one other important relationship that needs to be added to the above lists: self (0y0) (or 0x0 for a female speaker). Using this, the next table shows how the notation can be extended to represent various spouse relationships. The notation for aunt-by-marriage, for example, is coded as "my uncle's spouse's self".

When written on the tree, the "self" notation can be omitted, and left implied, as in (2y1s) for my aunt-by-marriage (my uncle's spouse), and (s1x0) for my mother-in-law (my spouse's mother). However, it has been left explicit for the descriptions on this page.

Whenever cousins (or other relations) marry together in the family tree, their descendants end up, from my perspective, with two alternative notations (which might or might not be the same). The notation gives us a way of determining which of the alternative relationships is the stronger.

There is obviously scope to cascade the spouse notation ad infinitum, for example to represent my uncle's spouse's great-aunt's spouse's brother (2y1s3x1s1y1). It is possible, also, to add other pseudo-relationships, such as friend, just for the sake of concise notation in a description on a family tree. For this, the letter "a" stands for "ami", so as to avoid the letter "f" still. Thus, a reference to my grandmother's friend (as a bridesmaid on a wedding photograph, for example) might be indicated as (2x0a0x0).

Narrowing down the Year Ranges

As an aside from the main story, but still in the context of "functionality that a family-tree computer program can incorporate", we can note that one important way of tracing ancestors is to look for birth and marriage certificates. This involves looking in the registers of the appropriate year and place. If the year is not known precisely, but only to a range of ±5 years, then all eleven registers in that range need to be consulted, one by one.

The aim of the "year-ties" notation is to get the computer to narrow down the range of years as tightly as possible, even where the exact year is unknown, to make the process of searching for birth and marriage certificates easier.

Every event (year of birth, YoB; year of marriage, YoM; year of death, YoD) is first represented as a pair of integers, expressing a range: (YoB.min, YoB.max), (YoM.min, YoM.max), (YoD.min, YoD.max). If the date is known exactly, then the min and max fields are set to the same value. Similarly, if the range of possible dates is known, then the min and max fields are set accordingly. Otherwise the fields are set to the default range of 1066 (say) in the min field, and the current year in the max field. (In the case of the YoD, the max field can be left blank (zero) to indicate "not yet dead").

The "tiemax(A.YoE, B.YoF, val)" procedure asserts that the time between two events cannot be greater than "val". The "tiemin(A.YoE, B.YoF, val)" procedure asserts that the time between two events cannot be less than "val". At their hearts, these procedures behave as follows:

tiemax(A.YoE, B.YoF, val) BEGIN
  IF( (B.YoF.min-A.YoE.min) > val )
    THEN A.YoE.min := B.YoF.min-val;
  IF( (B.YoF.max-A.YoE.max) > val )
    THEN B.YoF.max := A.YoE.max+val
END;
tiemin(A.YoE, B.YoF, val) BEGIN
  IF( (B.YoF.min-A.YoE.min) < val )
    THEN B.YoF.min := A.YoE.min+val;
  IF( (B.YoF.max-A.YoE.max) < val )
    THEN A.YoE.max := B.YoF.max-val
END;

In reality, these procedures cannot be quite as simple as this, since various checks and safeguards need to be included; for example, to check that the min value is not allowed to pass the max value, and vice versa.

Having defined these procedures, the whole database can be scanned repeatedly for a number of constraints:

tiemax( Z.YoD, Z.YoB, 111 )

Since no person, Z, in my family is known to have made it into Guinness Book of Records (GBR), I can assume that no-one has lived beyond the age of 111 (say).

tiemin( Z.YoD, Z.YoB, 0 )

No person, Z, has died before his birth-date. (Luckily, family trees do not worry about technical exceptions to this rule).

tiemin( mth_child(Z).YoB, Z.YoB, 13+m )

Again, because of the GBR principle, I can assume that no person, Z, in my family is known to have given birth to a first child before the age of 14, and hence of an m^th child before the age of 13+m. This further assumes that consecutive children are not born in the same year. This is not strictly true, since consecutive children can be born in January and December (say) of the same year, but the rule is a good enough for it not to need to be overridden too often. The computer program does, though, need to be sophisticated enough to cope with twins, and parts of the tree where the order of births in a family is presently unknown (one of five children is known to have been the third one born, but the others are in unknown order).

tiemax( mth_child(X).YoB, X.YoB, 55-numbofchildren(X)+m )

Again, because of the GBR principle, I can say that no female person, X, in my family is known to have given birth to a last child beyond the age of 55.

tiemax( child(Z).YoB, Z.YoD, 0 )

No person, Z, has given birth to a child beyond his or her own year of death. (To be safer, the value 0 applies to mothers, X, and the value 1 to fathers, Y).

tiemin( elder_sibling(Z).YoB, Z.YoB, 1 )

There must be at least one year between consecutive siblings. In fact, we can be a lot more adventurous than this. If we know the dates of birth of the m^th and n^th children, and Z is known to be the z^th child in between them, then we can assert tiemin(M.YoB, Z.YoB, z-m) and tiemin(Z.YoB, N.YoB, n-z).

There are a number of other ties, including assumptions about how to narrow down a year of marriage. Most constraints occur in pairs, so as to express the min and max values for the constraint. Events, such as getting married, or being present at someone else's wedding, or appearing on a photograph (at a certain age, plus or minus a range), can all be used to constrain the dates of birth and death of the person.

This is no magic panacea, of course. If one date of birth is known exactly in one portion of the tree, that person's mother's date of birth can be inferred within a range of 41 years, and the maternal grandmother's within a range of 82 years, and so on, with the range widening linearly at each generation. (Because of the symmetry, this growth rule works in both directions on the tree, so that the dates of births of the aunts are also then known within a range of about 123 years). The ranges grow even faster along the male lines, due to the lack of the male menopause. Luckily, information ripples in from other sources around the tree, and combines in a way that tends to narrow the ranges down from this. The amount of information that the computer program can infer, therefore, though limited, can still be of some help in narrowing down the search in birth, death and marriage registers.

Lastly, the mechanism for printing out of the tree can be described. Each person is represented by a node that records the surname at birth, the forenames, DoB range, DoD range, residential addresses), other notes, and a linked list to zero or more marriage (or more accurately union) nodes. Each marriage node records the DoM range, any notes (such as addresses of the wedding and reception) pointers to the two person nodes, and a linked list of person nodes starting with the first child (if any). By starting at a fairly senior marriage node, a printout can be requested of the family tree of the descendants of that marriage; meanwhile, by starting at a fairly junior person node, a pedigree tree can be requested for that person.