J.-L. Lassez, V. L. Nguyen, and E. A. Sonenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14:112--116, 1982.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....[Tar55] Tarski s theorem considers a monotone function and guarantees the existence of its least fixpoint with respect to a complete partial order. This setup, however, turned out to be too restrictive for a lot of practically relevant applications which led to a number of generalizations. See [LNS82] for a survey of the history of fixpoint theory. Vector iteration [Rob76] provides such a generalization, where one computes the least fixpoint x = x ; x ) 2 D of a monotone vector function f = f ) Liberalizing Tarski s iteration x 0 = x 1 = f(x 0 ) x 2 = f(x 1 ) ....

J.-L. Lassez, V.L. Nguyen, and E.A. Sonnenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112--116, 1982.

....usually easier to do. The usefulness of Theorems 2 and 3 is due to this fact. Remark 4. Theorems 1, 2, and 3 compute the supremal element as the greatest fixpoint of monotone operators. This is a standard technique (see, e.g. 22] 27] which relies on the Knaster Tarski fixpoint theorem [26] [15]. In this paper we are presenting a method for obtaining these monotone operators, i.e. defining a suitable S reflexive relation # and computing #( or#( We shall refer to this method as the # method. Examples given in the following section will show that the method is fairly general. Note ....

J.-L. Lassez, V.L. Nguyen, and E.A. Sonenberg, Fixed point theorems and semantics: A folk tale, Inform. Process. Lett., 14 (1982), pp. 112--116.

....it is obviously the unique morphism such that hfi ffi hhi = g and N(hfi) f . QED We denote the codomain of the cocartesian morphism defined above by h(hC; M;Ai) 3. 2 Unconstrained fibring We need to use the following (well known) results on fixed points originally due to Tarski and Kleene [14]. Proposition 3.14 Let hU; i be a complete lattice, u 2 U and f : U U a monotonic map such that u f(u) Then, the set fv 2 U : u v = f(v)g has a minimum. Moreover, if f is continuous then, letting 15 ffl f 0 (u) u; ffl f n 1 (u) f(f n (u) the minimum element is W n2IN f n ....

J.-L. Lassez, V. L. Nguyen, and E. A. Sonenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112--116, 1982. 33

....any j k 2. T k (I ) P j=j= Gamma G ) T j (I ) P j=j= Gamma G for any j k Thus our construction preserves important properties. In fact, it follows from the monotonicity of T and the fact that v is a chain complete partial order that the least fixed point of T is T (I ) [1, 11]. However it is not hard to establish the same result directly, and as it is somewhat informative, we do so below. Lemma 6.4 Let T be defined as above, and so T (I ) 1 G i=1 T i (I ) T 1 (I ) t T 2 (I ) t T 3 (I ) t : Then T (I ) is a fixed point of T , ....

J-L. Lassez, V.L. Nyugen and E.A. Sonenberg, Fixed Point Theorems and Semantics: A Folk Tale, Information Processing Letters:14:112-116.

....as suggested by the wording of theorem 9, least xed points need not exist. A well known theorem is that a monotonic function on a complete poset is guaranteed to have a least xed point. The theorem is known sometimes as Tarski s theorem and sometimes as the Knaster Tarski theorem see [LNS82] for historical information. The posets we consider in this paper are always complete, allowing us to ignore the niceties of existence problems in the statement of theorems and lemmas. The most powerful of the two rules characterising least pre x points is the induction rule. Its power is, ....

J.-L. Lassez, V.L. Nguyen, and E.A. Sonenburg. Fixed point theorems and semantics: a folk tale. Information Processing Letters, 14(3):112-116, 1982.

.... Gamma for every Gamma such that Gamma 2 L. QED As expected there is also a very close relationship between the categories Dsy and Csy: a reflection. But first we need to recall the following fixed point result originally due to Tarski (for the proof see [22] and for further details see [17]) Theorem 2.16 Let hU; i be a complete lattice and f : U U a monotonic map such that u f(u) Then, for each u 2 U there is an ordinal such that f(f (u) f (u) and u f (u) In the sequel, we denote f (u) by lfp(f; u) where is the least ordinal such that f(f (u) f (u) One might ....

J.-L. Lassez, V. L. Nguyen, and E. A. Sonenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112--116, 1982.

....is a xpoint of T if T (x) x. Given an operator T on X, de ne a sequence of elements T , where is an ordinal, as follows. For the base case, let T 0 = For successor ordinals 1, de ne T 1 = T (T ) For limit ordinals , de ne T = tfT : g. A well known result (see [LNS82] for a discussion of its history) states that if T is continuous then then this sequences converges to the least pre xpoint of T , that convergence has taken place by = and that T is in fact a xed point of T . Thus, we obtain as a corollary of Lemma A.2 and Lemma A.3 that there exists a ....

J.-L. Lassez, V. L. Nguyen, and E. A. Sonenberg. Fixed point theorems and semantics: a folk tale. Information Processing Letters, 14(3):112-116, 1982.

....Tarski s theorem considers a monotonic function and guarantees the existence of its least fixed point with respect to a complete partial order. This setup, however, turned out to be too restrictive for a lot of practically relevant applications which led to a number of generalizations. See [LNS82] for a survey of the history of fixed point theory. In numerical analysis one is interested in computing the least fixed point of x = f(x) where x is a vector (x 0 ; x n Gamma1 ) 2 D n and f : D n D n is a monotonic vector function (f 0 ; f n Gamma1 ) where f k ....

J.-L. Lassez, V.L. Nguyen, and E.A. Sonnenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112--116, 1982.

....is due to Knaster Tarski [24] It states that every monotone function possesses an infimal as well as a supremal fixed point. Another result provides methods of computing the infimal and supremal fixed points under stronger conditions than monotonicity. The paper by LassezNguyen Sonenberg [15] provides a nice historical account of these fixed point theorems. Several other fixed point results have since been discovered and are reported in papers such as [4, 18, 23, 7, 1, 6] The notion of optimal fixed points and their properties are discussed in [17, 14] In this paper we study the ....

J. L. Lassez, V. L. Nguyen, and E. A. Sonenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112--116, 1982.

....the proofs of properties (with respect to the formal specification) become simpler than otherwise. Unfortunately, we cannot apply any of the standard fixpoint theorems, since not all transfer functions of the iteration are monotone (see Section 4. 1) as required by these theorem (cf. e.g. 6] and [14] for a survey) The nonmonotonicity property stems from the requirements on instructions for JVM subroutines. To avoid the problem, we choose to follow the example of a non standard fixpoint theorem, which requires that all transfer functions are increasing, and monotone in case the bigger element ....

....from the high level definition. But they have not considered any properties of their bytecode verifier like those discussed in the current paper 10.3 Fixpoint theorems and chaotic fixpoint iteration Fixpoint theorems have taken many forms and been (re )proved many times in the literature (see [14] for a survey) There has been much work on applications of fixpoint theorems in dataflow analysis (see e.g. 13,18] and abstraction interpretation (see e.g. 5] From these perspectives, our chaotic iteration is not something that is substantially new. However, the result of the current paper ....

J.-L. Lassez, V. Nguyen, and E. Sonnenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112--116, 1982.

....Let us recapitulate (set theoretical versions) of both theorems because some axioms of a standard specifications may be in contrast to pure logic programs Horn clauses with (universal) quantifiers in premises and even non Horn clauses. Theorem 10.13 (fixpoint theorems) see, e.g. LNS82] Let A be a set and Phi : A) A) be a monotonic function (with respect to set inclusion) B A is a fixpoint of Phi if Phi(B) B. Phi is upward continuous if for all increasing chains B 1 B 2 B 3 : of subsets of A, Phi( i2N B i ) is a subset of [ i2N Phi(B i ) Phi is ....

J.-L. Lassez, V.L. Nguyen, and E.A. Sonenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14:112--116, 1982.

....D) P (D Theta D) Phi D (R) f(d leaf a; d leaf a 0 ) j a A a 0 g [ f(d sup f; d sup f 0 ) j f B R f 0 g with the following properties: 4.2.8. Lemma. 1. If d Phi D (R) d 0 then one of the following cases applies: a) d = d leaf a d 0 = d leaf a 0 a A a 0 5 See [LNS82] for a historical account of this folk theorem. Chapter 4. Inductive types 83 (b) d = d sup f d 0 = d sup f 0 f B R f 0 2. Phi D (R 2 PER(D) 2 PER(D) 3. R R 0 Phi D (R) Phi D (R 0 ) Proof. 1. Just apply lemma 4.2.2. 2. We use the previous clause and the fact that A, B R ....

J.-L. Lassez, V.L. Nguyen, and E.A. Sonneberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3), 1982.

....F i (E) are closed, too. Thus the sequence (F i (E) is a decreasing sequence of closed subsets of E, and we can use and continuity (lemma 2. 2) 1 i=0 F (F i (E) F ( 1 i=0 F i (E) 2 For a funny introduction to different lattice fixed point theorems, see for example [LNS82] the authors try to clarify the origin of fixed point theorems in the context of semantics and present different versions with an extensive bibliography. The left hand side gives: 1 i=0 F i 1 (E) 1 i=1 F i (E) E 1 i=1 F i (E) 1 i=0 F i (E) and we have the ....

Lassez, J.L., Nguyen, V.L., and Sonenberg, E.A. Fixed point theorems and semantics: a folk tale. Information Processing Letters, 14(3):112--116, 1982.

....is misleading about the usefulness of proof checking systems based on set theory. Next, we prove the existence of the least and the greatest fixpoints for monotone functions from a powerset to a powerset of a set. Scheme Knaster is the Knaster theorem about the existence of fixpoints, cf. [14]. Theorem (11) is the Banach decomposition theorem which is then used to prove the Schroder Bernstein theorem (12) we followed Paulson s development of these theorems in Isabelle [16] It is interesting to note that the last theorem when stated in Mizar in terms of cardinals becomes trivial to ....

....(43) that every monotone function f over a complete lattice L has a complete lattice of fixpoints. As the consequence of this theorem we get the existence of the least fixpoint equal to f fi ( L) for some ordinal fi with cardinality not bigger than the cardinality of the carrier of L, cf. [14], and analogously the existence of the greatest fixpoint equal to f fi ( L ) Section 5 connects the fixpoint properties of monotone functions over complete lattices with the fixpoints of monotone functions over the lattice of subsets of a set (Boolean lattice) MML Identifier: KNASTER. WWW: ....

J.-L. Lassez, V. L. Nguyen, and E. A Sonenberg. Fixed point theorems and semantics: a folk tale. Information Processing Letters, 14(3):112--116, 1982.

....is usually easier to do. The usefulness of Theorems 2 and 3 is due to this fact. Remark 4 Theorems 1, 2 and 3 compute the supremal element as the greatest fixpoint of monotone operators. This is a standard technique, e.g. 22] 27] which relies on the KnasterTarski fixpoint theorem [26] [15]. In this paper we are presenting a method for obtaining these monotone operators, i.e. defining a suitable S reflexive relation Delta and computing Psi( Delta) or Psi( Delta) We shall refer to this method as the Delta method. Examples given in the following section will show that the ....

J.-L. Lassez, V.L. Nguyen and E.A. Sonenberg, Fixed point theorems and semantics: a folk tale, Inform. Process. Lett., 14 (1982), pp. 112-116.

....Tarski s theorem considers a monotonic function and guarantees the existence of its least fixed point with respect to a complete partial order. This setup, however, turned out to be too restrictive for a lot of practically relevant applications which led to a number of generalizations (see [LNS82] for a survey of the history of fixed point theory) In numerical analysis one is interested in computing the least fixed point of x = f(x) where x is a vector (x 0 ; x n Gamma1 ) 2 D n and f : D n D n is a monotonic vector function (f 0 ; f n Gamma1 ) where f k ....

J.-L. Lassez, V.L. Nguyen, and E.A. Sonnenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112--116, 1982.

....states that every monotone function possesses an infimal as well as a supremal fixed point. Another result provides methods of computing the infimal and supremal fixed points under stronger conditions than monotonicity. Refer to Theorems 1 and 2 in section 2. The paper by Lassez Nguyen Sonenberg [18] provides a nice historical account of these fixed point theorems. Several other fixed point results have since been discovered and are reported in papers such as [4, 23, 30, 9, 1, 6] The notion of optimal fixed points and their properties are discussed in [22, 17] In this paper we study the ....

J.L. Lassez, V.L. Nguyen, and E.A. Sonenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112--116, 1982.

....If (X; is a partial order, we say f : X 7 X is monotonic iff 8x; y 2 X : x y ) f(x) f(y) Theorem 2.3 (Tarski Knaster) Suppose (X; is a chain complete partial order, and suppose f : X 7 X is monotonic. Then there exists a least fixed point fix f of f . Proof This theorem is well known [13]. Transfinite induction can be used to define the fixed point. The following more elementary proof is adapted from Lang[12] Say Y X is admissible if 8y 2 Y : f(y) 2 Y and 8C Y : if C is totally ordered, then F C 2 Y . Let B be the intersection of all admissible subsets of X. Then B X and ....

J. L. Lassez, V. L. Nguyen, and E. A. Sonenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14(3):112--116, May 1982.

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J.-L. Lassez, V. L. Nguyen, and E. A. Sonenberg. Fixed point theorems and semantics: A folk tale. Information Processing Letters, 14:112--116, 1982.

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J.-L. Lassez, V.L. Nguyen, and E.A. Sonenburg. Fixed point theorems and semantics: a folk tale. Information Processing Letters, 14(3):112-116, 1982.

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J.-L. Lassez, V.L. Nguyen, E.A. Sonenberg, Fixed Point Theorems and Semantics: A Folk Tale, Information Processing Letters 14 (1982) 112-116

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J. Lassez, V.L. Nguyen, and L. Sonenberg. Fixed point theorems and semantics: a folk tale. Information Processing Letters, 14(3):109--111, 1982.

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J.-L. Lassez, V.L. Nguyen, and E.A. Sonenberg. Fixed point theorems and semantics: a folk tale. Information Processing Letters, 14(3):112--116, May 1982.

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J.L. Lassez, V. L. Nguyen, and Sonnenberg E. A. Fixed points theorems and semantics: a folk tale. Information processing letters, 14(3):112 -- 116, May 1982.

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J.-L. Lassez, V.L. Nguyen, and E.A. Sonenberg. Fixed point theorems and semantics: a folk tale. Information Processing Letters, 14(3):112--116, May 1982.

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