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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. 1 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 2 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. 3 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). 4 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 5 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. 6 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. 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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.

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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.

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