A thatch is a type of partial ordering which is most naturally described as a geometric construction. The geometry of these constructions is outlined in this note. A wait-and-see concurrent process may include any number of sub-processes, each regarded as a separate thread of execution. However if at any point the process launches a collection of concurrent threads it must wait and see how they terminate before proceeding. Each sub-process may also launch further sub-processes, which are also subject to the wait and see restriction. A wait-and-see process can be regarded as a thatch. This note describes some interesting mathematical properties of thatching that are relevant to these concurrent processes

Mitchell, Bill

English

Southampton (e-Prints Soton)

Thatched Partial OrdersBill Mitchellw.mitchell@surrey.ac.ukDepartment of ComputingUniversity of SurreyGuildford, Surrey GU2 7XH, UKMay 28, 20091 Geometric ThatchingA thatch is a type of partial ordering which is most naturally described as a geometricconstruction. A thatching consists of a set of vertical and horizontal lines where the ver-tical lines can have any nite number of nodes attached to them. A thatch is constructedinductively as follows.The initial thatch is any number of parallel vertical lines. Any open interval in avertical line can be replaced by two parallel horizontal lines, placed so that the upper lineoccurs at the beginning of the interval, and the lower line occurs at the end of the interval.The new lines must not intersect any other line already in the thatch. Next thesehorizontal lines must be connected by a nite non-zero number of new parallel verticallines which start at the upper horizontal line and terminate at the lower horizontal line.They must not intersect any other lines in the thatch. The two horizontal lines aredened to be dual to each other.This process may iterate any nite number of times on any of the vertical lines inthe current thatch. Finally attach nitely many nodes to distinct points on the verticallines.For example if we start with a single vertical line we may choose to insert four newvertical lines as shown here.We may then choose to add a further two vertical lines as shown here1Bill Mitchell Thatched Partial OrdersThen we may choose to attach nodes as described here:abcde fghijklThatch 1A thatch describes a partial order of the nodes attached to the vertical lines. Denex < y for nodes x and y if there is a continuous path from x to y along the verticaland horizontal lines contained in the thatch which never travels upwards when traversedfrom x to y. That is the direction of travel must always be downwards or sideways. Sucha path is called a non-upwards path. Any partial order that can be dened by a thatchis called a thatch ordering.The region dened by a dual pair of horizontal lines is the set of nodes x, where x ispassed through by some non-upwards path from the top horizontal line to the bottomhorizontal line.In the above example the partial order given by the thatch is described in the moreusual way by the following directed acyclic graph. The ordering is depicted with theorder increasing down the page.2Bill Mitchell Thatched Partial Ordersabcde fghijklFor example, a < c < d < l, and a < b < g < h < k < l.Note that not any arrangement of vertical and horizontal lines forms a thatch. Inparticular note that the horizontal lines have to pair up in the correct way. Here is anexample which is not a thatch.abcdefgDene horizontal line l1 to be contained between horizontal lines l2 and l3 if there is anon-upwards path which leads from some point on l2 to some point on l3 which passesthrough some point on l1.By construction horizontal lines are introduced in pairs. Moreover this is done insuch a way that each pair contains an even number of other horizontal lines betweenthem. Also if a pair contains no horizontal lines (which is still even of course) thenevery vertical line descending from the top horizontal line must terminate at the lowerhorizontal line. By inspection of the above diagram we can see that it is not possible topair up the horizontal lines in this way, and so it does not constitute a thatch.2 Set Theoretic ThatchingWe can characterise a thatch partial order in a set theoretic manner. That is we can givea purely set theoretic characterisation of those partial orders which can be constructedas a thatch. Note that there may not be a unique thatch that corresponds to a particularpartial order. To simplify matters we assume that every partial order has a unique topand bottom element. Given a partial order which does not, we can simply add two newelements to make this true. Without this restriction it becomes much more complex tosuccinctly characterise a thatch in a set theoretic manner.In order to give the set theoretic characterisation we must rst introduce some no-tation. Let < be a partial order on a set S.Denition 2.13Bill Mitchell Thatched Partial Orders X = fy j 8x 2 X; y > xg X = fy j 8x 2 X; y < xg Xc = fy 2 X j6 9z 2 X : y > zg, this set is the ceiling of X Xf = fy 2 X j6 9z 2 X : y < zg, this set is the oor of X For x 2 S, let nxt(x) = fxgc, these are the next elements after x. For x 2 S, let lst(x) = fxgf these are the last elements before x. For a set S, dene X to be a region of S if there are Y , Z  S such that X = Y \Z Xub = fy 62 X j 9x 2 X : y 2 nxt(x)g, this is the upper boundary of X. Xlb = fy 62 X j 9x 2 X : y 2 lst(x)g, this is the lower boundary of X.Theorem 2.2A partial order < over a set S is a thatch ordering if and only if for every region X  S,Xc = XubXf = XlbGiven a thatch partial order, each region corresponds to a geometric region of the thatch.The above characterisation therefore gives a mechanism for constructing a particularthatch which describes the partial order. The ceilings and oors together with theregions describe how to collect together the nodes to form a thatch. If we are givena partial order which is not a thatch, it will be dicult to prove that this is the caseusing the above characterisation. However there is another characterisation, which wewill discuss later, which is appropriate when disproving that a partial order is a thatch.Here is a partial order which is a thatch. Note it has a unique top and bottomelement.n1n2 n3 n4n5n6The regions for this ordering arefn2;n3;n4;n5g = fn1g \ fn6gfn3;n4g = fn1g \ fn5gfn5g = fn3;n4g \ fn6gBy direct calculation we can see thatfn2;n3;n4;n5gc = fn6g = fn2;n3;n4;n5gubfn2;n3;n4;n5gf = fn1g = fn2;n3;n4;n5glbfn3;n4gc = fn5g = fn3;n4gubfn3;n4gf = fn1g = fn3;n4glbfn5gc = fn6g = fn5gubfn5gf = fn3;n4g = fn5glb4Bill Mitchell Thatched Partial OrdersHence this is a thatch. Using the regions together with their oor and ceilings to recon-struct the thatch we can describe the partial order as follows.n1n2n3 n4n5n6As another example consider Thatch 1. Here the regions arefb;c;d;e;f;g;h;i;j;kg = fag \ flgfg;h;i;jg = fbg \ fkgAs a nal example illustrating theorem 2.2 consider this thatch.n1n2n3 n4n5 n6n7Here the regions arefn2;n3;n4;n5;n6g = fn1g \ fn7gfn3;n4g = fn1g \ fn5;n6gfn5;n6g = fn3;n4g \ fn7g5Bill Mitchell Thatched Partial OrdersThe oors and ceilings for these regions are:fn2;n3;n4;n5;n6gc = fn7g = fn2;n3;n4;n5;n6gubfn2;n3;n4;n5;n6gf = fn1g = fn2;n3;n4;n5;n6glbfn3;n4gc = fn5;n6g = fn3;n4gubfn3;n4gf = fn1g = fn3;n4glbfn5;n6gc = fn7g = fn5;n6gubfn5;n6gf = fn3;n4g = fn5;n6glbThe interesting point in this example is that fn3;n4g is the oor of fn5;n6g, and fn5;n6gis the ceiling of fn3;n4g.There is an alternative characterisation of a thatch ordering which is easier to applywhen proving that a given ordering is not a thatch ordering. This characterisation ismore dicult to use when we wish to prove that a given partial order is a thatch ordering.Again we introduce some notation before giving the characterisation.Denition 2.3 For a pair a, b write ajb when a 6< b and b 6< a. We say that a and b are disjoint. A pair a, b are capped above if for every c 2 fa;bg, for each d such that c 6> d, andd > a, then d > b. A pair a, b are capped below if for every c 2 fa;bg, for each d such that c 6< d, andd < a, then d < b.Theorem 2.4A partial order < on a set S is a thatch ordering if and only if every disjoint pair is cappedbelow and above.Here is a partial order, which is also a lattice. This lattice is not a thatch.n1n2 n3n4 n5n6We can prove this using theorem 2.4. Consider n4 and n5, they are disjoint. Note n32 f n4, n5 g, also n3 6< n2, but n2 < n4. If this were a thatch then theorem 2.4 statesthat n2 < n5, however in the lattice n2 6< n5 which is a contradiction. Hence the latticecan not be a thatch ordering.In the next example we consider another lattice which is not a thatch. With this weillustrate that theorem 2.4 is simpler to use than theorem 2.2 when disproving that apartial order is a thatch.6Bill Mitchell Thatched Partial Ordersn1n2n3n4 n5n6n7Notice that n6 2 f n2, n3 g, that n2jn3, n6 6> n5, and n5 > n3. If this were a thatchthen by theorem 2.4 n5 > n2. This is not the case so this lattice can not be a thatch.We can also use theorem 2.2 to disprove that this is not a thatch. Considerfn4;n5;n6g = fn3g \ fn7gTherefore fn4;n5;n6g is a region. However,fn4;n5;n6gf = fn3gfn4;n5;n6glb = fn2;n3gTherefore by theorem 2.2 the lattice is not a thatch. This calculation has required us toconstruct a region, its oor and its lower boundary. This is much more work than therst calculation.3 Wait-and-see concurrent threadsA wait-and-see concurrent process may include any number of sub-processes, each re-garded as a separate thread of execution. However if at any point the process launchesa collection of concurrent threads it must wait and see how they terminate before pro-ceeding. Each sub-process may also launch further sub-processes, which are also subjectto the wait and see restriction.A wait-and-see process can be regarded as a thatch. Each atomic line of code canbe thought of as a node. Each sub-process corresponds to a vertical line in the thatch.When a process launches some sub-processes this is represented by a horizontal line,where each of the subprocesses corresponds to one of the vertical lines hanging downfrom the horizontal line. The termination of the subprocesses is represented by theirrespective thatches being attached at their ends to a common horizontal line.For example Thatch 1 regarded as a wait-and-see process behaves as follows. After ais executed the process launches four sub-processes. The forth of these after executionof b launches a further two sub-processes. The rst of these executes g then h thenterminates. The second executes i then j then terminates. After both terminate sub-process four executes k before terminating. After all four sub-processes have terminatedthe main process executes l and then terminates.Given a set of executable expressions and a specication of their inter-dependency inthe form of a partial order on them, when is it possible to implement this specicationin the form of a set of wait-and-see processes? Theorems 2.2 and 2.4 describe exactlywhen this can occur, and when it can not.7

Thatched Partial Orders

Abstract

Similar works

Full text

Available Versions

Southampton (e-Prints Soton)