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					Special Issue on ICIT 2009 Conference - Applied Computing




                            Least and greatest fixed points of
                               a while semantics function
                                             Fairouz Tchier
                                        Mathematics department,
                                          King Saud University
                                             P.O.Box 22452
                                       Riyadh 11495, Saudi Arabia
                                          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
other operation that occurs in this article is the re-
                                                                 (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
the upper adjoint.
                                                             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
Special Issue on ICIT 2009 Conference - Applied Computing




    (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).

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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 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.