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Least and greatest ﬁxed points of
a while semantics function
Fairouz Tchier
Mathematics department,
King Saud University
P.O.Box 22452
ftchier@hotmail.com
May 1, 2009

Abstract                                                      1    Relation Algebras
Both homogeneous and heterogeneous relation alge-
bras are employed in computer science. In this pa-
The meaning of a program is given by specifying the           per, we use heterogeneous relation algebras whose
function (from input to output) that corresponds to           deﬁnition is taken from [8, 27, 28].
the program. The denotational semantic deﬁnition,
thus maps syntactical things into functions. A re-            (1) Deﬁnition. A relation algebra A is a structure
lational semantics is a mapping of programs to re-            (B, ∨, ∧, −, ◦, ) over a non-empty set B of elements,
lations. We consider that the input-output seman-             called relations. The unary operations −, are total
tics of a program is given by a relation on its set of        whereas the binary operations ∨, ∧, ◦ are partial. We
states. In a nondeterministic context, this relation is       denote by B∨R the set of those elements Q ∈ B for
calculated by considering the worst behavior of the           which the union R ∨ Q is deﬁned and we require that
program (demonic relational semantics). In this pa-           R ∈ B∨R for every R ∈ B. If Q ∈ B∨R , we say that
per, we concentrate on while loops. We will present           Q has the same type as R. The following conditions
some interesting results about the ﬁxed points of the         are satisﬁed.
while semantics function; f (X) = Q ∨ P 2 X where
P < ∧ Q< = Ø, by taking P := t 2 B and Q := t∼ ,                  (a) (B∨R , ∨, ∧, −) is a Boolean algebra, with zero
one gets the demonic semantics we have assigned to                    element 0R and universal element 1R . The
while loops in previous papers. We will show that                     elements of B∨R are ordered by inclusion, de-
the least angelic ﬁxed point is equal to the greatest                 noted by ≤.
demonic ﬁxed point of the semantics function.
(b) If the products P ◦ R and Q ◦ R are deﬁned,
so is P ◦ Q . If the products P ◦ Q and P ◦ R
are deﬁned, so is Q ◦ R. If Q ◦ R exists, so
does Q ◦ P for every P ∈ B∨R .
Keywords: Angelic ﬁxed points, demonic
ﬁxed points, demonic functions, while                         (c) Composition is associative: P ◦ (Q ◦ R) =
loops, relational demonic semantics.                              (P ◦ Q) ◦ R.

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UbiCC Journal – Volume 4 No. 3                                                                               571
Special Issue on ICIT 2009 Conference - Applied Computing

(d) There are elements R id and idR associated           a vector [28] iﬀ x = x ◦ 1. The second way is via
to every relation R ∈ B. R id behaves as a           monotypes [2]: a relation a is a monotype iﬀ a ≤ id.
right identity and idR as a left identity for        The set of monotypes {a | a ∈ B∨R }, for a given R,
B∨R .                                                is a complete Boolean lattice. We denote by a∼ the
monotype complement of a.
(e) The Schr¨der rule P ◦Q ≤ R ⇔ P ◦−R ≤
o                                              The domain and codomain of a relation R can be
−Q ⇔ −R ◦ Q ≤ −P holds whenever one                  characterized by the vectors R ◦ 1 and R ◦ 1, re-
of the three expressions is deﬁned.                  spectively [15, 28]. They can also be characterized
(f) 1 ◦ R ◦ 1 = 1 iﬀ R = 0 (Tarski rule).                by the corresponding monotypes. In this paper, we
take the last approach. In what follows we formally
If R ∈ B∨R , then R is said to be homogeneous. If            deﬁne these operators and give some of their prop-
all R ∈ A have the same type, the operations are all         erties.
total and A itself is said to be homogeneous.
(3) Deﬁnition. The domain and codomain opera-
For simplicity, the universal, zero, and identity ele-       tors of a relation R, denoted respectively by R< and
ments are all denoted by 1, 0, id, respectively. An-         R> , are the monotypes deﬁned by the equations
(a) R< = id ∧ R ◦ 1,
ﬂexive transitive closure R∗ . It satisﬁes the well-
known laws                                                       (b) R> = id ∧ 1 ◦ R.
R∗ =         Ri and R∗ = id ∨ R ◦ R∗ = id ∨ R∗ ◦ R,         These operators can also be characterized by Galois
i≥0                                                  connections(see [2, 2]). For each relation R and each
monotype a,
where R0 = id and Ri+1 = R ◦ Ri . From Deﬁnition
1, the usual rules of the calculus of relations can be                       R< ≤ a ⇔ R ≤ a ◦ 1,
derived (see, e.g., [8, 10, 28]).                                            R> ≤ a ⇔ R ≤ 1 ◦ a.
The notion of Galois connections is very important
in what follows, there are many deﬁnitions of Galois         The domain and codomain operators are linked by
connections [?]. We choose the following                     the equation R> = R < , as is easily checked.
one [2].                                                     (4) Deﬁnition. Let R be a relation and a be a
(2) Deﬁnition. Let (S, ≤S ) and (S , ≤S ) be two             monotype. The monotype right residual and mono-
preordered sets. A pair (f, g) of functions, where f :       type left residual of a by R (called factors in [5]) are
S → S and g : S → S, forms a Galois connections              deﬁned respectively by
iﬀ the following formula holds for all x ∈ S and y ∈             (a) a/R := ((1 ◦ a)/R)> ,
•

S.
f (x) ≤S y ⇔ x ≤S g(y).                                        (b) R\a := (R\(a 2 1))< .
•

The function f is called the lower adjoint and g
An alternative characterization of residuals can
also be given by means of a Galois connection as
follows [1]:
2    Monotypes and Related Op-                                  b ≤ a/R ⇔ (b 2 R)> ≤ a,
•

b ≤ R\a ⇔ (R ◦ b)< ≤ a.
•

erators
We have to use exhaustively the complement of
In the calculus of relations, there are two ways for         the domain of a relation R, i.e the monotype a such
viewing sets as relations; each of them has its own          that a = R< ∼ . To avoid the notation R< ∼ , we adopt
advantages. The ﬁrst is via vectors: a relation x is         the Notation

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UbiCC Journal – Volume 4 No. 3                                                                               572
Special Issue on ICIT 2009 Conference - Applied Computing

R := R< ∼ .                                                of R . Then, I(R) is a monotype. In a concrete
Because we assume our relation algebra to be com-          setting, I(R) is the set of monotypes which are not
plete, least and greatest ﬁxed points of monotonic            the origins of inﬁnite paths (by R):
functions exist. We cite [12] as a general reference             A relation R is progressively ﬁnite iﬀ for a mono-
on ﬁxed points.                                               type a, a ≤ (R ◦ a)< ⇒ a = 0 equivalently
Let f be a monotonic function.            The fol-         ν(a : a ≤ id : (R ◦ a)< ) = 0 equivalently µ(a : a ≤
lowing properties of ﬁxed points are used below:              id : a/R) = id.
•

(a) µf = {X|f (X) = X} = {X|f (X) ≤ X},                          The next theorem involves the function wa (X) :=
(b) νf = {X|f (X) = X} = {X|X ≤ f (X)},                       Q ∨ P ◦ X, which is closely related to the description
(c) µf ≤ νf,                                                  of iterations. The theorem highlights the importance
(d) f (Y ) ≤ Y ⇒ µf ≤ Y,                                      of progressive ﬁniteness in the simpliﬁcation of ﬁxed
(e) Y ≤ f (Y ) ⇒ Y ≤ νf.                                      point-related properties.
In what follows, we describe notions that are useful
for the description of the set of initial states of a         (6) Theorem. Let f (X) := Q ∨ P ◦ X be a func-
program for which termination is guaranteed. These            tion. If P is progressively ﬁnite, the function f has a
notions are progressive ﬁniteness and the initial part        unique ﬁxed point which means that ν(f ) = µ(f ) =
of a relation.                                                P ∗ ◦ Q [1]:
A relation R is progressively ﬁnite in terms of
As the demonic calculus will serve as an algebraic
points iﬀ there are no inﬁnite chains s0 , ..., si such
apparatus for deﬁning the denotational semantics
that si Rsi+1 ∀i, i ≥ 0. I.e there is no points set y
of the nondeterministic programs, we will deﬁne in
which are the starting points of some path of inﬁnite
what follows these operators.
length. For every point set y, y ≤ R ◦ y ⇒ y = 0.
The least set of points which are the starting points
of paths of ﬁnite length i.e from which we can pro-           3     Demonic reﬁnement order-
ceed only ﬁnitely many steps is called initial part
of R denoted by I(R). This topic is of interest in                  ing
many areas of computer science, mathematics and is
We now deﬁne the reﬁnement ordering (demonic in-
related to recursion and induction principle.
clusion) we will be using in the sequel. This ordering
(5) Deﬁnition.                                                induces a complete join semilattice, called a demonic
semilattice. The associated operations are demonic
(a) The initial part of a relation R, denoted             join ( ), demonic meet ( ) and demonic composition
I(R), is given by                                     ( 2 ). We give the deﬁnitions and needed properties
I(R) = {a | a ≤ id : a/R = a} = {a |
•                            of these operations, and illustrate them with simple
a ≤ id : a/R ≤ a} = µ(a : a ≤ id : a/R),
•                         •                examples. For more details on relational demonic
where a is a monotype.                                semantics and demonic operators, see [5, 8, 6, 7, 14].

(b) A relation R is said to be progressively ﬁnite         (7) Deﬁnition. We say that a relation Q reﬁnes a
[28] iﬀ I(R) = id.                                     relation R [23], denoted by Q   R, iﬀ R< ◦ Q ≤
R and R ≤ Q .
<     <

The description of I(R) by the formulation a/R = a
•

shows that I(R) exists, since (a | a ≤ id : a/R) is•          (8) Proposition. Let Q and R be relations, then
monotonic in the ﬁrst argument and because the set
of monotypes is a complete lattice, it follows from the           (a) The greatest lower (wrt ) of Q and R is,
ﬁxed point theorem of Knaster and Tarski that this                    Q R = Q< ◦ R< ◦ (Q ∨ R),
function has a least ﬁxed point. Progressive ﬁnite-                   If Q< = R< then we have        and ∨ coincide
ness of a relation R is the same as well-foundedness                  i.e Q R = Q ∨ R.

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UbiCC Journal – Volume 4 No. 3                                                                               573
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(b) If Q and R satisfy the condition Q< ∧ R< =              (a) S(R) = I(P ) ◦ [(P ∨ Q)< /P ∗ ] ◦ P ∗ ◦ Q., with
•

(Q ∧ R)< , their least upper bound is Q R =                 the restriction
Q ∧ R ∨ Q ◦ R ∨ R ◦ Q, otherwise, the least
upper bound does not exist. If Q< ∧ R< = 0             (b) P < ∧ Q< = 0
then we have and ∧ coincide i.e Q R =
Our goal is to show that the operational semantics
Q ∧ R.
a is equal to the denotational one which is given as
For the proofs see [9, 14].                                 the greatest ﬁxed point of the semantic function Q ∨
P 2 X in the demonic semilattice. In other words,
(9) Deﬁnition. The demonic composition of rela-             we have to prove the next equation:
tions Q and R [5] is Q 2 R = (R< /Q) ◦ Q ◦ R.
•

(a) S(R) =     {X|X      Q∨P    2   X};
In what follows we present some properties of   2   .
(10) Theorem.                                               by taking P := t 2 B and Q := t∼ , one gets the
demonic semantics we have assigned to while loops
(a) (P   2   Q) 2 R = P   2   (Q 2 R),                   in previous papers [14, 35]. Other similar deﬁnitions
of while loops can be found in [19, 25, 29].
(b) R total ⇒ Q 2 R = Q ◦ R,
Let us introduce the following abbreviations:
(c) Q function ⇒ Q 2 R = Q ◦ R.
(12) Abbreviation. Let P , Q and X be relations
See [5, 6, 7, 14, 35].                                      subject to the restriction P < ∧ Q< = 0 (b) and x
Monotypes have very simple and convenient prop-           a monotype. The Abbreviations wd , wa , w< , a and l
erties. Some of them are presented in the following         are deﬁned as follows:
proposition.                                                   wd (X) := Q ∨ P 2 X,
a := (P ∨ Q)< /P ∗ ,
•
(11) Proposition. Let a and b be monotypes. We                 wa (X) := Q ∨ P ◦ X,
have                                                           l := I(P ).
(a) a = a       = a2 ,                                      w< (x) := Q< ∨ (P 2 x)< = Q ∨ (P 2 x)<

(b) a 2 b = a ∧ b = b 2 a,                              (Mnemonics: the subscripts a and d stand for angelic
and demonic, respectively; the subscript < refers to
(c) a ∨ a∼ = id and a ∧ a∼ = 0,                         the fact that w< is obtained from wd by composi-
tion with <; the monotype a stands for abnormal,
(d) a ≤ b ⇔ b∼ ≤ a∼ ,
since it represents states from which abnormal ter-
(e) a∼ 2 b∼ = (a ∨ b)∼ ,                                mination is not possible; ﬁnally, l stands for loop,
since it represents states from which no inﬁnite loop
(f ) (a ∧ b)∼ = (a 2 b)∼ = a∼ ∨ b∼ ,                    is possible.)
(g) a 2 b∼ ∨ b = a ∨ b,                                    In what follows we will be concerned about the
ﬁxed point of wa , w< and wd .
(h) a ≤ b ⇔ a 2 1 ≤ b 2 1.
(13) Theorem. Every ﬁxed point Y of wa (Abbre-
In previous papers [14, 13, 31, 35], we found the         viation 12) veriﬁes P ∗ ◦ Q ≤ Y ≤ P ∗ ◦ Q ∨ l∼ 2 1,
semantics of the while loop given by the following          and the bounds are tight (i.e. the extremal values are
P

                                              ﬁxed points).
graph: - e
- s -


The next lemma investigates the relationship be-
Q
tween ﬁxed points of w< and those of wd (cf. Abbre-
viation 12).

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UbiCC Journal – Volume 4 No. 3                                                                              574
Special Issue on ICIT 2009 Conference - Applied Computing

(14) Lemma. Let h(X) := (P ∨ Q) ∨ (P ◦ X)< and               4    Application
h1 (x) := (P ∨ Q) 2 1 ∨ P ◦ x.
In [6, 7], Berghammer and Schmidt propose abstract
(a) Y = wd (Y ) ⇒ w< (Y < ) = Y < ,                       relation algebra as a practical means for the speciﬁ-
(b) w< (Y < ) = Y < ⇒ h(Y ) = Y ,                         cation of data types and programs. Often, in these
speciﬁcations, a relation is characterized as a ﬁxed
(c) h(Y ) = Y      ⇒ h1 (Y    2   1) = Y   2   1,         point of some function. Can demonic operators be
used in the deﬁnition of such a function? Let us now
(15) Lemma. Let Y be a ﬁxed point of wd and b be             show with a simple example that the concepts pre-
a ﬁxed point of w< (Abbreviation 12). The relation           sented in this paper give useful insights for answering
b 2 Y is a ﬁxed point of wd .                                this question.
(16) Lemma. If Y and Y are two ﬁxed points of                   In [6, 7], it is shown that the natural numbers can
wd (Abbreviation 12) such that Y < = Y < and Y < ◦P          be characterized by the relations z and S (zero and
is progressively ﬁnite, then Y = Y .                         successeur ) the laws

The next theorem characterizes the domain of the                 (a) Ø = z = zL ∧ zz ⊆ I          (z is a point),
greatest ﬁxed point, wrt , of function wd . This                     SS      =     I         ∧          S S      ⊆
domain is the set of points for which normal ter-                    I   (S is a one to one application.),
mination is guaranteed (no possibility of abnormal                   Sz = Ø (z has a predecessor),
termination or inﬁnite loop).                                        L       =          {x|z ∪ S x               =
x} (generation principle).
(17) Theorem. Let W be the greatest ﬁxed point,
wrt to , of wd (Abbreviation 12). We have W < =                For the rest of this section, assume that we are
a 2 l.                                                       given a relation algebra satisfying these laws. In this
algebra, because of the last axiom, the inequation
The following theorem is a generalization to a non-
deterministic context of the while statement veriﬁ-              (a) z ∪ S X ⊆ X
cation rule of Mills [24]. It shows that the greatest
ﬁxed point W of wd is uniquely characterized by con-         obviously has a unique solution for X, namely, X =
ditions (a) and (b), that is, by the fact that W is a        L. Because the functiong(X) := z ∪ S X is ∪-
ﬁxed point of wd and by the fact that no inﬁnite loop        continuous, this solution can be expressed as
is possible when the execution is started in a state
that belongs to the domain of W . Note that we also              (a) L =    n≥0   g n (Ø) =   n≥0   S   n
z,
have W < ≤ a (see Theorem 17), but this condition
where g 0 (Ø) = Ø, g n+1 (Ø) = g(g n (Ø)), S 0 = I
is implicitly enforced by condition (a). Half of this
and S n+1 = S S n . However, it is shown in [6, 7]
theorem (the ⇐ direction) is also proved by Sekerin-
that z S 2 X ⊆ X, obtained by replacing the
ski (the main iteration theorem [29]) in a predicative
join and composition operators in a by their demonic
programming set-up.
counterparts, has inﬁnitely many solutions. Indeed,
(18) Theorem. A relation W is the greatest ﬁxed                                           o
from Sz = Ø and the Schr¨der rule, it follows that
point, wrt , of function wd (Abbreviation 12), iﬀ
the following two conditions hold:                               (a) z ∩ S L = Ø,

(a) W = wd (W ),                              so that, by deﬁnition of demonic join (8(a))
(b) W < ≤ l.                                and demonic composition (9), z S 2 X = (z ∪
S 2 X) ∩ z ∩ (S 2 X)L ⊆ z ∩ S L = Ø. Hence,
In what follows we give some applications of our           any relation R is a solution to z S 2 X ⊆ X.
results.                                                     Looking at previous papers [14, 32, 33, 34, 31], one

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UbiCC Journal – Volume 4 No. 3                                                                                   575
Special Issue on ICIT 2009 Conference - Applied Computing

immediately sees why it is impossible to reach L by          how the universal relationL arises as the greatest
joining anything to z (which is a point and hence is         lower bound n≥0 S n 2 z of this set of points. Note
an immediate predecessor of Ø), since this can only          that, whereas there is a unique solution to a, there
lead to z or to Ø.                                           are inﬁnitely many solutions to 4 (equivalently, to a),
Let us now go ‘fully demonic’ and ask what is a           for example n≥k S n (= n≥k S n ), for any k.
solution to z S 2 X X. By the discussion above,                For the upward approach, consider
this is equivalent to Ø       X, which has a unique
solution, X = Ø. This raises the question whether                              z     X 2S     X.
it is possible to ﬁnd some fully demonic inequation          Here also there are inﬁnitely many solutions to this
similar to (a), whose solution is X = L. Because L is        inequation; in particular, any vector v, including
in the middle of the demonic semilattice, there are in       Ø and L, is a solution to 4. Because (BL , ) is
fact two possibilities: either approach L from above         only a join semilattice, it is not at all obvious that
or from below.                                               the least ﬁxed point of h(X) := z           X 2 S ex-
For the approach from above, consider the inequa-         ists. It does, however, since the following deriva-
tion                                                         tion shows that n≥0 z 2 S n (=                 n
n≥0 h (z ),
0
X    z   S   2   X.                        where h (z ) = z ) is a ﬁxed point of h and hence
is obviously the least solution of 4: Because z
Using Theorem 10(c), we have z      S 2X =                   and S are mappings, property 10(c) implies that
z S X, since S is deterministic (axiom a(b)).                z 2 S n = z S n , for any n ≥ 0. But z S n is
From a, z ⊆ S L; this implies z ⊆ S XL and                   also a mapping (it is the inverse of the point S n z)
S X ⊆ z, so that, by deﬁnition of ,                          and hence is total, from which, by Proposition 8(a)
z S X = z ∩ S X ∪ z ∩ S XL ∪ z ∩ S X =                       and equation a, n≥0 z 2 S n =            n≥0 z S
n
=
n                n
z ∪ S X.                                                       n≥0 z S = ( n≥0 S z)˘ = L = L. This
This means that 4 reduces to                               means that L is the least upper bound of the set
of mappings {z 2 S n |n ≥ 0}. Again, a look at
(a) X     z ∪ S X.                                        [31] gives some intuition to understand this result,
after recalling that mappings are minimal elements
By deﬁnition of reﬁnement (7), this implies that
in (BL , ) (though not all mappings have the form
z ∪ S XL ⊆ XL; this is a variant of (a), thus
z 2 S n ).
having XL = L as only solution. This means that
Thus, building L from below using the set of map-
any solution to 4 must be a total relation. But L
pings {z 2 S n |n ≥ 0} is symmetric to building it
is total and in fact is the largest (by ) total rela-
from above using the set of points {S n 2 z|n ≥ 0}.
tion. It is also a solution to 4 (since by axiom a(d),
z ∪ S L = L) so that L = {X|X z S 2 X};
that is, L is the greatest ﬁxed point in (BL , ) of
n2
5    Conclusion
f (X) := z S 2 X. Now consider            n≥0 S     z,
n
where S       is a n-fold demonic composition deﬁned         We presented a theorem that can be also used to ﬁnd
by S 0 = I and S n+1 = S 2 S n . By axiom                    the ﬁxed points of functions of the form f (X) :=
a(b), S is deterministic, so that, by 10(c) and asso-        Q ∨ P 2 X (no restriction on the domains of P and
ciativity of demonic composition, conS n 2 z = S n z.        Q). This theorem can be applied also to the program
Hence,                                                       veriﬁcation and construction (as in the precedent ex-
It is easy to show that for any n ≥ 0, S n z is           ample). Half of this theorem (the ⇐ direction) is
a point (it is the n-th successor of zero) and that          also proved by Sekerinski (the main iteration theo-
m = n ⇒ S m z = S n z. Hence, in (BL , ),                    rem [29]) in a predicative programming set-up. Our
{S n z|n ≥ 0} (i.e. {S n 2 z|n ≥ 0}) is the set of           theorem is more general because there is no restric-
immediate predecessors of Ø; looking at [31] shows           tion on the domains of the relations P and Q.

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UbiCC Journal – Volume 4 No. 3                                                                              576
Special Issue on ICIT 2009 Conference - Applied Computing

The approach to demonic input-output relation                 [6] Berghammer, R.: Relational Speciﬁcation of
presented here is not the only possible one. In                     Data Types and Programs. Technical report
[19, 20, 21], the inﬁnite looping has been treated by                           a u                         a
9109, Fakult¨t f¨r Informatik, Universit¨t der
adding to the state space a ﬁctitious state ⊥ to de-                              u
Bundeswehr M¨nchen, Germany, Sept. 1991.
note nontermination. In [8, 18, 22, 26], the demonic
input-output relation is given as a pair (relation,set).        [7] Berghammer, R. and Schmidt, G.: Relational
The relation describes the input-output behavior of                 Speciﬁcations. In C. Rauszer, editor, Algebraic
the program, whereas the set component represents                   Logic, 28 of Banach Center Publications. Polish
the domain of guaranteed termination.                               Academy of Sciences, 1993.
We note that the preponderant formalism em-
ployed until now for the description of demonic                 [8] Berghammer, R. and Zierer, H.: Relational Al-
input-output relation is the wp-calculus. For more                  gebraic Semantics of Deterministic and Nonde-
details see [3, 4, 17].                                             terministic Programs. Theoretical Comput. Sci.,
43, 123–147 (1986).

References                                                      [9] Boudriga, N., Elloumi, F. and Mili, A.: On
the Lattice of Speciﬁcations: Applications to a
[1] Backhouse, R. C., and Doombos, H.: Math-                       Speciﬁcation Methodology. Formal Aspects of
ematical Induction Made Calculational. Com-                    Computing, 4, 544–571 (1992).
puting science note 94/16, Department of Math-
ematics and Computer Science, Eindhoven Uni-              [10] Chin, L. H. and Tarski, A.: Distributive and
versity of Technology, The Netherlands, 1994.                  Modular Laws in the Arithmetic of Relation Al-
gebras. University of California Publications, 1,
[2] Backhouse, R. C., Hoogendijk, P., Voermans,                    341–384 (1951).
E. and van der Woude, J.:. A Relational The-
ory of Datatypes. Research report, Department             [11] Conway, J. H.: Regular Algebra and Finite Ma-
of Mathematics and Computer Science, Eind-                     chines. Chapman and Hall, London, 1971.
hoven University of Technology, The Nether-
lands, 1992.                                              [12] Davey, B. A. and Priestley, H. A.: Introduction
to Lattices and Order. Cambridge Mathematical
[3] R. J. R. Back. : On the correctness of reﬁnement               Textbooks. Cambridge University Press, Cam-
in program development. Thesis, Department of                  bridge, 1990.
Computer Science, University of Helsinki, 1978.
o
[13] J. Desharnais, B. M¨ller, and F. Tchier. Kleene
[4] R. J. R. Back and J. von Wright.: Combining                    under a demonic star. 8th International Con-
angels, demons and miracles in program spec-                   ference on Algebraic Methodology And Software
iﬁcations. Theoretical Computer Science,100,                   Technology (AMAST 2000), May 2000, Iowa
1992, 365–383.                                                 City, Iowa, USA, Lecture Notes in Computer
Science, Vol. 1816, pages 355–370, Springer-
[5] Backhouse, R. C. and van der Woude, J.: De-                    Verlag, 2000.
monic Operators and Monotype Factors. Math-
ematical Structures in Comput. Sci., 3(4), 417–           [14] Desharnais, J., Belkhiter, N., Ben Mo-
433, Dec. (1993). Also: Computing Science Note                 hamed Sghaier, S., Tchier, F., Jaoua, A., Mili,
92/11, Department of Mathematics and Com-                      A. and Zaguia, N.: Embedding a Demonic Semi-
puter Science, Eindhoven University of Technol-                lattice in a Relation Algebra. Theoretical Com-
ogy, The Netherlands, 1992.                                    puter Science, 149(2):333–360, 1995.

7

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Special Issue on ICIT 2009 Conference - Applied Computing

[15] Desharnais, J., Jaoua, A., Mili, F., Boudriga,                                    o
[28] Schmidt, G. and Str¨hlein, T.: Relations and
N. and Mili, A.: A Relational Division Oper-                 Graphs. EATCS Monographs in Computer Sci-
ator: The Conjugate Kernel. Theoretical Com-                 ence. Springer-Verlag, Berlin, 1993.
put. Sci., 114, 247–272 (1993).
[29] Sekerinski, E.: A Calculus for Predicative Pro-
[16] Dilworth, R. P.: Non-commutative Residuated                  gramming. In R. S. Bird, C. C. Morgan, and
Lattices. Trans. Amer. Math. Sci., 46, 426–444               J. C. P. Woodcock, editors, Second Interna-
(1939).                                                      tional Conference on the Mathematics of Pro-
gram Construction, volume 669 of Lecture Notes
[17] E. W. Dijkstra. : A Discipline of Programming.
in Comput. Sci. Springer-Verlag, 1993.
Prentice-Hall, Englewood Cliﬀs, N.J., 1976.
[18] H. Doornbos. : A relational model of programs           [30] Tarski, A.: On the calculus of relations. J.
without the restriction to Egli-Milner monotone              Symb. Log. 6, 3, 1941, 73–89.
constructs. IFIP Transactions, A-56:363–382.            [31] F. Tchier.:         e
S´mantiques relationnelles
North-Holland, 1994.                                          e                 e
d´moniaques et v´riﬁcation de boucles non
[19] C. A. R. Hoare and J. He. : The weakest                       e                                  e
d´terministes. Theses of doctorat, D´partement
prespeciﬁcation. Fundamenta Informaticae IX,                         e                                    e
de Math´matiques et de statistique, Universit´
1986, Part I: 51–84, 1986.                                   Laval, Canada, 1996.

[20] C. A. R. Hoare and J. He. : The weakest                 [32] F. Tchier.:    Demonic semantics by mono-
prespeciﬁcation. Fundamenta Informaticae IX,                 types. International Arab conference on In-
1986, Part II: 217–252, 1986.                                formation Technology (Acit2002),University of
Qatar, Qatar, 16-19 December 2002.
[21] C. A. R. Hoare and al. : Laws of programming.
Communications of the ACM, 30:672–686, 1986.            [33] F. Tchier.: Demonic relational semantics of
compound diagrams. In: Jules Desharnais,
[22] R. D. Maddux. : Relation-algebraic semantics.                Marc Frappier and Wendy MacCaull, editors.
Theoretical Computer Science, 160:1–85, 1996.                Relational Methods in computer Science: The
[23] Mili, A., Desharnais, J. and Mili, F.: Relational                e
Qu´bec seminar, pages 117-140, Methods Pub-
Heuristics for the Design of Deterministic Pro-              lishers 2002.
grams. Acta Inf., 24(3), 239–276 (1987).                [34] F. Tchier.: While loop d demonic relational
[24] Mills, H. D., Basili, V. R., Gannon, J. D. and               semantics monotype/residual style. 2003 In-
Hamlet,R. G.: Principles of Computer Pro-                    ternational Conference on Software Engineer-
gramming. A Mathematical Approach. Allyn                     ing Research and Practice (SERP03), Las Ve-
and Bacon, Inc., 1987.                                       gas, Nevada, USA, 23-26, June 2003.

[25] Nguyen, T. T.: A Relational Model of Demonic            [35] F. Tchier.: Demonic Semantics: using mono-
Nondeterministic Programs. Int. J. Founda-                   types and residuals. IJMMS 2004:3 (2004) 135-
tions Comput. Sci., 2(2), 101–131 (1991).                    160. (International Journal of Mathematics and
Mathematical Sciences)
[26] D. L. Parnas. A Generalized Control Structure
and its Formal Deﬁnition. Communications of             [36] M. Walicki and S. Medal.: Algebraic approches
the ACM, 26:572–581, 1983                                    to nondeterminism: An overview. ACM compu-
tong Surveys,29(1), 1997, 30-81.
[27] Schmidt, G.: Programs as Partial Graphs I:
Flow Equivalence and Correctness. Theoretical           [37] L.Xu, M. Takeichi and H. Iwasaki.:     Rela-
Comput. Sci., 15, 1–25 (1981).                               tional semantics for locally nondeterministic

8

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Special Issue on ICIT 2009 Conference - Applied Computing

programs. New Generation Computing 15, 1997,
339-362.

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Description: UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.
UbiCC Journal Ubiquitous Computing and Communication Journal www.ubicc.org
About UBICC, the Ubiquitous Computing and Communication Journal [ISSN 1992-8424], is an international scientific and educational organization dedicated to advancing the arts, sciences, and applications of information technology. With a world-wide membership, UBICC is a leading resource for computing professionals and students working in the various fields of Information Technology, and for interpreting the impact of information technology on society.