## Constructive Versions Of Tarski's Fixed Point Theorems (1979)

Venue: | Pacific Journal of Mathematics |

Citations: | 46 - 8 self |

### BibTeX

@ARTICLE{Cousot79constructiveversions,

author = {Patrick Cousot and Radhia Cousot},

title = {Constructive Versions Of Tarski's Fixed Point Theorems},

journal = {Pacific Journal of Mathematics},

year = {1979},

volume = {82},

pages = {43--57}

}

### Years of Citing Articles

### OpenURL

### Abstract

this paper is to give a constructive proof of Tarski's theorem without using the continuity hypothesis. The set of fixed points of F is shown to be the image of L by preclosure operations defined by means of limits of stationary transfinite iteration sequences. Then the set of common fixed points of a family of commuting monotone operators on a complete lattice into itself is characterized in the same way. The advantage of characterizing fixed points by iterative schemes is that they lead to practical computation or approximation procedures. Also the definition of fixed points as limits of stationary iteration sequences allows the use of transfinite induction for proving properties of these fixed points

### Citations

1879 |
Abstract Interpretation: A Unified Lattice Model for Static Analysis of Programs By Construction or Approximation of Fixpoints
- Cousot, Cousot
- 1977
(Show Context)
Citation Context ...ice {XsL: ((JS) G X}. An application of the duality principle completes the proof. This definition is not constructive and many applications of Tarski's theorem (specially in computer science (Cousot =-=[5]-=-) and numerical analysis (Amann [2])) use the alternative characterization of lfp(F) as J {F(_): i e N}. This iteration scheme which originates from Kleene [10]'s first recursion theorem and which was... |

320 |
The Calculi of Lambda Conversion
- Church
- 1941
(Show Context)
Citation Context ...all be used in the form stated by Birkhoff [3]). The set of prefixed points of F is prefp(F) = {X L: XsF(X)}. Dually postfp(F) = {XsL: F(X)sX). Therefore fp(F) = prefp(F) fq postf p(F). We use Church =-=[4]-=-'s lambda notation (so that F is ;X. F(X)). /I CONSTRUCTIVE VERSIONS OF TARSKI'S FIXED POINT THEOREMS 45 3. Behavior of an upper iteration sequence. LEMMA 3.1. Let {X , 3sOrd) be the Ord-termed uppe' ... |

32 |
The closure operators of a lattice
- Ward
- 1942
(Show Context)
Citation Context ...) so that the upper iteration sequence {X ,s) for Z. Z LJ F(Z) starting with P is such that (/, P = X). Hence ,5(P) --P that is postfp (F) fi(L) and by antisymmetry we have postfp (F) = ,(L). By Ward =-=[21]-=-'s theorem (L) is a nonempty complete lattice (, (2_), T, ;S. 5(US), [2). Also by 4.1 luis (Z. Z U F(Z))() = luis (Z.sU F(Z))(_) = luis (F)() = postfp(F) by definition of the infimum of a complete lat... |

11 |
A theorem on partially ordered sets with applications to fixed point theorems
- Abian, Brown
- 1961
(Show Context)
Citation Context ...ds of arbitrary sets that is crucial. The same remark was made by numerous authors who generalized Tarski's fixed point theorem to weaken the completeness hypothesis (see among others Abian and Brown =-=[1]-=-, HSft [9], Pasini [15], Pelczar [16], Markowsky [14], Ward [20], Wolk [22]). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr [6], Markowsky [14], Pelczar [... |

3 |
Induction rules and proofs of termination
- Hitchcoek, Park
- 1973
(Show Context)
Citation Context ... (j F(Z)) is the limit of the sequence Xs= D, Xs= X - LJ F(X -) for successor ordinals and Xs= U X" CONSTRUCTIVE VERSIONS OF TARSKI'S F1XED POINT THEOREMS 51 for limit ordinals) in Hitchcock and =-=Park [8]-=- (where Ifp (F) is the limit of Xs= , Xs= [J, F(X ) for every nonzero ordinal) and in Pasini [15] (where transfinite sequences are defined as in Definition 2.I). COROLLARY 5.2. .Let D be an arbitrary ... |

3 |
Fixed Points in Partially Ordered Sets
- Smithson
- 1973
(Show Context)
Citation Context ...sini [15], Pelczar [16], Markowsky [14], Ward [20], Wolk [22]). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr [6], Markowsky [14], Pelczar [17], Smithson =-=[18]-=-, Wong [23]). Along the same lines our results could be strengthened to be applicable to partially ordered sets which are not complete lattices. ACKNOWLEDGMENT. The authors thank the referee in partic... |

2 |
Common fixed points for isotone mappings
- DeMarr
- 1964
(Show Context)
Citation Context ...ong others Abian and Brown [1], HSft [9], Pasini [15], Pelczar [16], Markowsky [14], Ward [20], Wolk [22]). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr =-=[6]-=-, Markowsky [14], Pelczar [17], Smithson [18], Wong [23]). Along the same lines our results could be strengthened to be applicable to partially ordered sets which are not complete lattices. ACKNOWLEDG... |

2 |
On monotonous mappings of complete lattices
- Devid, V
- 1964
(Show Context)
Citation Context ... llis (F)() and the greatest lower bound S. llis (F)(NS). The construction of extremal fixed points of monotone operators as limits of stationary transfinite iteration sequences may be found in Devid =-=[7] (whe-=-re Ifp (Z. D (j F(Z)) is the limit of the sequence Xs= D, Xs= X - LJ F(X -) for successor ordinals and Xs= U X" CONSTRUCTIVE VERSIONS OF TARSKI'S F1XED POINT THEOREMS 51 for limit ordinals) in Hi... |

2 |
Some fixed point theorems of the mappings of partially ordered sets
- Pasini
- 1974
(Show Context)
Citation Context ... X" CONSTRUCTIVE VERSIONS OF TARSKI'S F1XED POINT THEOREMS 51 for limit ordinals) in Hitchcock and Park [8] (where Ifp (F) is the limit of Xs= , Xs= [J, F(X ) for every nonzero ordinal) and in Pa=-=sini [15]-=- (where transfinite sequences are defined as in Definition 2.I). COROLLARY 5.2. .Let D be an arbitrary element of L. luis (F) o Ills (,Z. Z f) F(Z))(D) and Ills (F) o luis (,Z. Z U F(Z))(D) are fixed ... |

2 |
On the invariant points of a transformation
- Pelczar
- 1961
(Show Context)
Citation Context .... The same remark was made by numerous authors who generalized Tarski's fixed point theorem to weaken the completeness hypothesis (see among others Abian and Brown [1], HSft [9], Pasini [15], Pelczar =-=[16]-=-, Markowsky [14], Ward [20], Wolk [22]). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr [6], Markowsky [14], Pelczar [17], Smithson [18], Wong [23]). Along... |

2 |
on commutings mappings in partially ordered spaces, Zeszyty Nauk
- Remarks
- 1971
(Show Context)
Citation Context ...], HSft [9], Pasini [15], Pelczar [16], Markowsky [14], Ward [20], Wolk [22]). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr [6], Markowsky [14], Pelczar =-=[17]-=-, Smithson [18], Wong [23]). Along the same lines our results could be strengthened to be applicable to partially ordered sets which are not complete lattices. ACKNOWLEDGMENT. The authors thank the re... |

2 |
Dedekind completeness and a fixed point theorem
- Wolk
- 1957
(Show Context)
Citation Context ... authors who generalized Tarski's fixed point theorem to weaken the completeness hypothesis (see among others Abian and Brown [1], HSft [9], Pasini [15], Pelczar [16], Markowsky [14], Ward [20], Wolk =-=[22]-=-). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr [6], Markowsky [14], Pelczar [17], Smithson [18], Wong [23]). Along the same lines our results could be s... |

2 |
Common fixed points of commuting monotone mapping
- Wong
- 1967
(Show Context)
Citation Context ...Pelczar [16], Markowsky [14], Ward [20], Wolk [22]). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr [6], Markowsky [14], Pelczar [17], Smithson [18], Wong =-=[23]-=-). Along the same lines our results could be strengthened to be applicable to partially ordered sets which are not complete lattices. ACKNOWLEDGMENT. The authors thank the referee in particular for st... |

1 |
Fixed point equations and non-linear eigenvalue problems in ordered Banach spaces
- Areann
- 1976
(Show Context)
Citation Context ...n of the duality principle completes the proof. This definition is not constructive and many applications of Tarski's theorem (specially in computer science (Cousot [5]) and numerical analysis (Amann =-=[2]-=-)) use the alternative characterization of lfp(F) as J {F(_): i e N}. This iteration scheme which originates from Kleene [10]'s first recursion theorem and which was used by Tarski [19] for complete m... |

1 |
Some fixed point theorems for partially ordered sets
- HSft, HSft
- 1976
(Show Context)
Citation Context ...trary sets that is crucial. The same remark was made by numerous authors who generalized Tarski's fixed point theorem to weaken the completeness hypothesis (see among others Abian and Brown [1], HSft =-=[9]-=-, Pasini [15], Pelczar [16], Markowsky [14], Ward [20], Wolk [22]). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr [6], Markowsky [14], Pelczar [17], Smith... |

1 |
fntroduction to metamathematics, North-Holland Pub
- Kleene
- 1952
(Show Context)
Citation Context ...m (specially in computer science (Cousot [5]) and numerical analysis (Amann [2])) use the alternative characterization of lfp(F) as J {F(_): i e N}. This iteration scheme which originates from Kleene =-=[10]-=-'s first recursion theorem and which was used by Tarski [19] for complete morphisms, has the drawback to require the additional assumption that F is semi-continuous (F((JS) = (J F(S) for every increas... |

1 |
On completeness of parially ordered sets and fixpoints theorems for isotone mappings
- Kolodner
- 1968
(Show Context)
Citation Context ...used by Tarski [19] for complete morphisms, has the drawback to require the additional assumption that F is semi-continuous (F((JS) = (J F(S) for every increasing nonempty chain S, see e.g., Kolodner =-=[11]-=-). 43 44 PATRICK COUSOT AND RADHIA COUSOT The purpose of this paper is to give a constructive proof of Tarski's theorem without using the continuity hypothesis. The set of fixed points of F is shown t... |

1 |
Prefermeture sur un ensemble ordonn, RAIR0, i (Avril
- LadegailIerie
- 1973
(Show Context)
Citation Context ...Ills (Z. Z ( F(Z)) is a lower preclosure operator since it is the composition of the upper closure operator luis (F) and the !over closure operator llis (Z. Z ( F(Z)) (3.4, 4.1 and 3.5, Ladegaillerie =-=[12]-=-). By duality llis (F)oluis(Z. ZsF(Z)) is an upper preclosure operator. Cousot and Cousot [5] already used the idea of constructing (or approximating) the fixed points of monotone operators by means o... |

1 |
The convergence o f functions to fixed points of recurslye definitions
- Manna, Shamir
- 1978
(Show Context)
Citation Context ... the fixed points of monotone operators by means of 52 PATRICK COUSOT AND RADHIA COUSOT an upper iteration sequence followed by a lower iteration sequence. This idea was also used by Manna and Shamir =-=[13]-=- and our results 3.3, 4.1, 5.2, and 5.3 improve their results obtained on the more restricted model of continuous functional equations on functions of fiat lower semi-lattices. 6. Constructive charact... |

1 |
Chain-complete posers and directed sets with applications
- Markowsky
- 1976
(Show Context)
Citation Context ...k was made by numerous authors who generalized Tarski's fixed point theorem to weaken the completeness hypothesis (see among others Abian and Brown [1], HSft [9], Pasini [15], Pelczar [16], Markowsky =-=[14]-=-, Ward [20], Wolk [22]). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr [6], Markowsky [14], Pelczar [17], Smithson [18], Wong [23]). Along the same lines ... |

1 |
A lattice theoretical fixpoint theorem ad its applications
- Tarski
- 1955
(Show Context)
Citation Context ...nclusion, union and intersection are respectively denoted by _, J and Q. Let F be a monotone operator on L(, l, , J, N) into itself (i.e., vX, YL, {X_Y}{F(X) F(Y)}). The fundamental theorem of Tarski =-=[19]-=- states that the set fp(F) of fixed points of F (i.e., fp(F) = {X L: X = F(X)}) is a nonempty complete lattice with ordering _. The proof of this theorem is based on the definition of the least fixed ... |

1 |
Completeness i semi-lattices
- Jr
- 1957
(Show Context)
Citation Context ...by numerous authors who generalized Tarski's fixed point theorem to weaken the completeness hypothesis (see among others Abian and Brown [1], HSft [9], Pasini [15], Pelczar [16], Markowsky [14], Ward =-=[20]-=-, Wolk [22]). This was also the case for Tarski's fixed point theorem on commuting maps (see a.o., DeMarr [6], Markowsky [14], Pelczar [17], Smithson [18], Wong [23]). Along the same lines our results... |