NATIONAL AND KAPODISTRIAN UNIVERSITY OF ATHENS
SCHOOL OF SCIENCE
DEPARTMENT OF INFORMATICS AND TELECOMMUNICATION
BSc THESIS
A SIMPLIFIED PROOF OF A FIXPOINT THEOREM FOR FUNCTIONS OVER STRATIFIED COMPLETE LATTICES
Ioannis S. Chatziagapis
Supervisor: Panos Rondogiannis,Professor
ATHENS JUNE 2018
ΕΘΝΙΚΟ ΚΑΙ ΚΑΠΟΔΙΣΤΡΙΑΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΘΗΝΩΝ
ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ
ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΚΑΙ ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ
ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ
ΜΙΑ ΑΠΛΟΥΣΤΕΥΜΕΝΗ ΑΠΟΔΕΙΞΗ ΕΝΟΣ ΘΕΩΡΗΜΑΤΟΣ ΣΤΑΘΕΡΟΥ ΣΗΜΕΙΟΥ ΓΙΑ ΣΥΝΑΡΤΗΣΕΙΣ ΣΕ
ΣΤΡΩΜΑΤΟΠΟΙΗΜΕΝΑ ΠΛΗΡΗ ΠΛΕΓΜΑΤΑ
Ιωάννης Σ. Χατζηαγάπης
Επιβλέπων: Πάνος Ροντογιάννης,Καθηγητής
ΑΘΗΝΑ ΙΟΥΝΙΟΣ 2018
BSc THESIS
A simplified proof of a fixpoint theorem for functions over stratified complete lattices
Ioannis S. Chatziagapis S.N.:1115201400221
Supervisor: Panos Rondogiannis,Professor
ΠΤΥΧΙΑΚΗ ΕΡΓΑΣΙΑ
Μια απλουστευμένη απόδειξη ενός θεωρήματος σταθερού σημείου για συναρτήσεις σε στρωματοποιημένα πλήρη πλέγματα
Ιωάννης Σ. Χατζηαγάπης Α.Μ.:1115201400221
Επιβλέπων: Πάνος Ροντογιάννης,Καθηγητής
ABSTRACT
I present a simplified and more direct proof of the fixed point theorem for non-monotonic functions developed by Z. Ésik and P. Rondogiannis ([4]). This theorem proves the exis- tence of a least fixed point of (potentially) non-monotonic functions over specially structured complete lattices, which I will define asstratified complete lattices. When the theorem is ap- plied to monotonic functions is equivalent to Knaster-Tarski theorem. The theorem has di- rect applications in the semantics ofnegation-as-failurein logic programming. In particular, it could be used to provide a more direct proof of the least fixed point result of [2].
SUBJECT AREA: Fixed point theory
KEYWORDS: fixed point theory, non-monotonicity, semantics of logic programming, negation-as-failure, infinite-valued logic
ΠΕΡΙΛΗΨΗ
Παρουσιάζω μια απλουστευμένη και πιο άμεση απόδειξη του θεωρήματος ‘A fixed point theorem for non-monotonic functions’ των Z. Ésik και Π. Ροντογιάννης ([4]). Το Θεώρημα αυτό αποδεικνύει την ύπαρξη ενός ελάχιστου σταθερού σημείου (πιθανώς) μη-μονοτονικών συναρτήσεων σε ειδικά δομημένα πλήρη πλέγματα, τα οποία θα ορίσω ως στρωματοποι- ημένα πλήρη πλέγματα. Όταν το θεώρημα εφαρμοστεί σε μονοτονικές συναρτήσεις είναι ισοδύναμο του θεωρήματος Knaster-Tarski. Το θεώρημα έχει άμεσες εφαρμογές στην ση- μασιολογία της άρνησης μέσω αποτυχίας του λογικού προγραμματισμού. Για παράδειγμα με την χρήση του θα γινόταν πιο άμεση η απόδειξη της ύπαρξης ελάχιστου σταθερού ση- μείου στο [2].
ΘΕΜΑΤΙΚΗ ΠΕΡΙΟΧΗ: Θεωρία σταθερού σημείου
ΛΕΞΕΙΣ ΚΛΕΙΔΙΑ: θεωρία σταθερού σημείου, μη-μονοτονικότητα, σημασιολογία λογι- κού προγραμματισμού, άρνηση μέσω αποτυχίας, απειρότιμη λογική
CONTENTS
1. INTRODUCTION 8
2. AN OVERVIEW OF THE INFINITE-VALUED APPROACH 9
3. THE STRATIFIED COMPLETE LATTICES 13
4. SOME CONSEQUENCES 16
5. THE FIXED POINT THEOREM 17
6. THE CASE OF MONOTONIC FUNCTIONS 20
7. CONCLUSION 21
A simplified proof of a ixpoint theorem for functions over stratified complete lattices
1. INTRODUCTION
Negation-as-failure is one of the many attempts that have been made in order to extend logic programming. Intuitively negation-as-failure means that in order to determine ifnot pis valid, one has to examine whetherpcan be proved. Ifpis proved thennot pfails. Otherwise not psucceeds. Even though negation-as-failure is trivial to implement, it is extremely diffi- cult to formalize from a semantic point of view. The primary reason for this is that negation is in general non-monotonic.
A huge leap in capturing the meanings of negation-as-failure was the introduction of thewell- founded semantics[3], which utilizes a three-valued logic in order to formalize the meaning of a logic program with negation. After that, P. Rondogiannis. and W. Wadge presented a new approach [2] based on an infinite-valued logic. This approach is important because it leads to a unique minimum model in a program-independent ordering. Moreover, if we restrict this model to a three-valued logic we get the well-founded model.
It is shown in [2] that the minimum model is theleast fixed pointof a non-monotonic operator with respect to an ordering relation. In [4] only the set-theoretic essence is kept, the logic programming related issues are omitted and some sufficient conditions are presented for a fixed point to exist.
In [4] two ordering relations are being used over a set L, namely ≤ and ⊑. The first one corresponds to a ”pointwise” comparison, whereas the second to a ”lexicographic” one.
(L,≤)is supposed to be a complete a lattice and (L,⊑)is later shown to form a complete lattice. In my proof, I am using only the ”lexicographic” comparison and I suppose that it forms a complete lattice. Omitting the first ordering leads to a simpler proof and it becomes more clear that this theorem is a generalization of the Knaster-Tarski theorem.
The rest of the paper is organized as follows: Section 2 introduces the infinite-valued ap- proach. Section 3 defines the specially structured complete lattices that will be the objects of our study. In Section 4 some properties of these lattices are investigated. Section 5 presents and proves the fixed point theorem. Finally, Section 6 investigates the special case of monotonic functions and Section 7 concludes the study.
In the following, we assume familiarity with the basic notions regarding logic programming (such as for example [1]) and of partially ordered sets and particularly lattices (such as for example [7]).
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2. AN OVERVIEW OF THE INFINITE-VALUED APPROACH
In this section we provide the basic notions and definitions behind the infinite-valued ap- proach; our presentation mostly follows [2]. Some standard technical terminology regard- ing logic programming (such as “atoms”, “literals”, “head/body of a rule”, “ground instance”,
“Herbrand Base”, and so on), will be used without further explanations (see [1] for a basic introduction).
Definition 2.1 A (first-order) normal program rule is a rule with an atom as head and a conjunction of literals as body. A (first-order)normal logic programis a finite set of normal program rules.
We follow a common practice which dictates that instead of studying finite first-order logic programs it is more convenient to study their, possibly infinite, ground instantiations[5]:
Definition 2.2 If P is a normal logic program, its associatedground instantiationP∗ is con- structed as follows: first, put in P∗ all ground instances of members of P; second, if a rule A←with empty body occurs in P∗, replace it with A← true; finally, if the ground atom A is not the head of any member of P∗, add A← false.
Notice that, by construction, P∗ is a propositional program that has a possibly infinite but countable number of rules (since the Herbrand Base of a normal logic program is countable).
To simplify our presentation, in the rest of the paper we will not talk explicitly about “the ground instantiation P∗ of a program P”. Instead, we will assume that we study programs that are propositional and have a countable number of rules. Since we will be dealing with propositional programs, we will often talk about “propositional atoms” and “propositional literals” that appear in the rules of our programs.
The basic idea behind the infinite-valued approach is that in order to obtain a minimum model semantics for logic programs with negation, it is necessary to consider a refined multiple-valued logic which will allow the meaning of negation-as-failure to be expressed properly. Let Ω denote the first uncountable ordinal. Then, the logic of [2] contains oneFα
and one Tα for each countable ordinalα (ie., for allα <Ω), and also an intermediate truth value denoted by 0. The ordering of the truth values is as follows:
F0 <F1 <· · ·<Fω <· · ·<Fα <· · ·<0<· · ·<Tα <· · ·<Tω <· · ·<T1 <T0
Intuitively, F0 andT0 are the classicalFalseandTruevalues and 0 is theundefinedvalue.
The intuition behind the new values is that they express different levels of truthfulness and falsity. Alternatively, the ordinal indices of truth values can be shown to correspond to the level at which truth or falsity of atoms is derived during the well-founded construction (see [2][Proof outline of Theorem 7.6]). In the following we denote byVthe set consisting of the above truth values.
Definition 2.3 Theorder of a truth value is defined as: order(Tα) = α, order(Fα) = α and order(0) = +∞.
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Interpretations of programs are defined as follows:
Definition 2.4 An (infinite-valued) interpretation I of a program P is a function from the set of propositional atoms of P to V.
Interpretations can be extended to apply to literals, to conjunctions of literals and to the two constants true and false. Of special interest is the way that negation is treated (and which intuitively expresses the fact that the more times negation is iterated, the more it approaches to the intermediate value 0).
Definition 2.5 Let I be an interpretation of a program P. Then, I can be extended as follows:
• For every negative atom∼p appearing in P:
I(∼p) =
Tα+1 if I(p) = Fα
Fα+1 if I(p) = Tα
0 if I(p) = 0
• For every conjunction of literals l1, . . . ,lnappearing as the body of a rule in P:
I(l1, . . . ,ln) =min{I(l1), . . . ,I(ln)} Moreover, I(true) =T0 and I(false) = F0.
The notion of satisfiability of a rule can be defined as follows:
Definition 2.6 Let P be a program and I an interpretation of P. Then, I satisfies a rule
p←l1, . . . ,lnof P if I(p)≥I(l1, . . . ,ln). Moreover, I is amodelof P if I satisfies all rules of P.
As it is customary in the theory of logic programming, it would be desirable if we could prove that every program has a unique minimum model with respect to an ordering relation. The first idea that comes to mind is to use the “pointwise” ordering of interpretations, namely I≤Jiff for all propositional symbolspit holdsI(p)≤J(p). Actually, this is the ordering used in classical logic programming in order to establish the minimum model property (see for example [1]), where the set of truth values is {False,True} with False < True. As it can easily be checked, the pointwise ordering of interpretations does not lead to a minimum model result in the case of logic programs with negation.
In order to obtain the minimum model property for logic programs with negation, a more refined ordering on interpretations was defined in [2]. This ordering was actually motivated by the fact that the construction of the well-founded model [6] proceeds in stages and there- fore the atoms that belong to the lower stages must (intuitively) be given a “higher priority”
than those belonging to upper stages. We will need the following definition:
Definition 2.7 Let P be a program, I an interpretation of P and v∈V. Let BP be the set of all propositional symbols that appear in program P. Then I ∥v={p∈BP |I(p) =v}.
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A simplified proof of a ixpoint theorem for functions over stratified complete lattices
We can now proceed to the definition of the ordering relations that are needed in order to obtain the minimum model property:
Definition 2.8 Let I and J be interpretations of a given program P and α be a countable ordinal. We write I =α J, if for all β≤a, I∥Tβ =J∥Tβ and I∥Fβ =J∥Fβ.
Definition 2.9 Let I and J be interpretations of a given program P and α be a countable ordinal. We write I ⊏α J, if for all β <a, I =βJ and either I∥Tα ⊂J∥Tα and I∥Fα ⊇J∥Fα, or I∥Tα ⊆J∥Tα and I∥Fα ⊃J∥Fα. We write I⊑α J if I=α J or I⊏α J.
Definition 2.10 Let I and J be interpretations of a given program P. We write I⊏J, if there exists a countable ordinal α such that I ⊏α J. We write I⊑J if either I=J or I⊏J.
It is easy to see that the relation ⊑ on the set of interpretations of a given program, is a partial order (ie. it is reflexive, transitive and antisymmetric). On the other hand, for every countable ordinalα, the relation⊑α is a preorder (ie. reflexive and transitive).
In order to study the fixed point semantics of the programs we consider, we can easily define an immediate consequence operatorTP:
TP(I)(p) = ∨
{I(l1, . . . ,ln)|(p←l1, . . . ,ln)∈P} where∨
is the obvious least upper bound operation on the setV. Notice that, as discussed in [2][Example 5.7], TP is not in general monotonic with respect to ⊑. This fact makes the proof of the following theorem nontrivial.
Theorem 2.11 [2][Corollary 7.5, page 460] For every program P, TPhas a least fixed point MP (with respect to⊑).
Actually, it is easy to show (see [2]) that MP is a model of P (in fact, the least model of P with respect to ⊑), and therefore it can be taken as the intended meaning ofP. Moreover, MP is directly connected to the well-founded model of P, as the following theorem from [2]
suggests:
Theorem 2.12 [2][Theorem 7.6, page 460] Let NP be the interpretation that results from MP by collapsing all true values to True and all false values to False. Then, NP is the well- founded model of P.
Example 2.13 Consider the program:
p ← ∼q q ← ∼r
s ← p
s ← ∼s r ← false
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One can easily compute the infinite-valued model of the program (see [2][pages 454-455]
for details), which is equal to:
MP ={(p,F2),(q,T1),(r,F0),(s,0)} Moreover, the well-founded model of the program is
NP ={(p,False),(q,True),(r,False),(s,0)} and has been obtained from MPin the obvious way.
The proof of Theorem 2.11 was performed in [2] using techniques that were specifically tailored to the case of logic programs. In the next sections we abstract away from the issues regarding logic programming, and we obtain a general and abstract fixed point theorem from which the above result follows in a very direct way.
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3. THE STRATIFIED COMPLETE LATTICES
In this section we will abstract away from the logic programming issues and we will define the structures presented in the previous section from a set-theoretic point of view. Instead of “the set of interpretations” we will usestratified complete lattices.
Definition 3.1 Let κ be a fixed ordinal. A set L equipped with a set of preorderings{⊑α}α<κ
is called stratified complete lattice if it satisfies the following properties (1, 2, 3, 4).
Note that for allα <κ,⊑α is a preorder (i.e. it is reflexive and transitive).
Definition 3.2 Let α <κ and x,y∈L. If x⊑α y and y⊑α x, we will write x=α y.
Property 1 For all ordinal α <β <κ and for each x,y∈L, if x⊑β y then x=α y.
Property 2 ∩
α<κ =α is the identity relation on L.
Definition 3.3 Let α <κ and x,y∈L. We define x⊏α y, if x⊑α y but x=α y does not hold.
Definition 3.4 Let x,y∈L. We define x⊏y, if x⊏α y for some α<κ.
Definition 3.5 Let x,y∈L. We write x⊑y, if x⊏y or x=y.
Property 3 (L,⊑)is a complete lattice.
We will denote the upper bound operation as⊔
and the least element of the complete lattice as⊥.
Definition 3.6 Given an ordinal α <κ and x∈L, we define(x]α as {y∈L:∀β<α x=β y}.
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Note that x∈(x]α holds. In the following property and also in the rest of the paper,X⊑α y should be interpreted in the standard way, namely as “for every x ∈ X,x ⊑α y”. Similarly, X=α yshould be interpreted as “for every x∈X,x=α y”.
Property 4 For each x ∈ L, for every ordinal α < κ, and for any non-empty X ⊆ (x]α
there is some y∈(x]α such that:
• X⊑α y, and
• for all z∈(x]α, if X⊑α z then y⊑α z and y⊑z.
• y=α
⊔X
I will present 3 properties that can substitute Property 4. These properties are restrictions regarding the relation ⊑ and the operation ⊔
, and they make the structure of a stratified complete lattice clearer.
Property 5 For each x ∈ L, for every ordinal α < κ and for any non-empty X ⊆ (x]α,
⊔X∈(x]α.
Property 6 For each x ∈L, for every ordinal α < κ and for any non-empty X⊆(x]α, if X⊑α x then⊔
X⊑α x.
Definition 3.7 Given an ordinal α <κ and x∈L, we define[x]α as {y∈L:x=α y}.
Note that according to Property 1 we have
{y∈L:x=α y} = {y∈L:∀β ≤α x=β y}. Thus, for each ordinalα <κ and for everyx∈L,[x]α=(x]α+1.
Property 7 Let α <κ. For each x∈L,[x]α has a⊑-least element.
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Now, I will prove that properties 5, 6 and 7 are equivalent to Property 4.
Theorem 3.8 Let (L,{⊑α}α<κ)be a stratified complete lattice. Then (L,{⊑α}α<κ) satisfies properties 5, 6 and 7.
Proof. Let x be an arbitrary element of L and suppose non-empty X ⊆ (x]α. Let y be the element specified by the Property 4.
Property 5: We have y =α ⊔
X. By Property 1 for each β < α we have y =β ⊔
X. Since y∈(x]α,⊔
X∈(x]α must apply.
Property 6: Suppose X ⊑α x. Since x∈(x]α, by Property 4 we have y⊑α x. Also, y=α
⊔X.
Therefore, by the transitivity of⊑α,⊔
X⊑α x.
Property 7: Suppose z ∈ [x]α. Then, since y =α z and y ∈ (x]α, by Property 1 we have z=β x for all β <α, so that z∈(x]α. Since X⊑α y and y=α z, also X⊑α z. Thus, y⊑z, by Property 4. Therefore y is the ⊑-least element of[x]α. □ Lemma 3.9 Suppose that(L,{⊑α}α<κ)satisfies property 1. Let α <κ and suppose x,y∈L such that x=β y for all β<α. If x⊑y then x⊑α y.
Proof. By definition of⊑, we have x =y or x⊏β y for some β. If x=y, by reflexivity of⊑α
we have x ⊑α y. If x ⊏β y for some β, β ≥ α must hold, because x =β y for all β < α. If β = α, then x⊏α y, so x ⊑α y clearly holds. If β > α, by Property 1 x =α y, so that x ⊑α y
again. □
Theorem 3.10 Suppose that (L,{⊑α}α<κ) satisfies properties 1, 2, 3, 5, 6 and 7. Then it also satisfies Property 4.
Proof. For each x ∈L, for every ordinal α <κ, and for any non-empty X⊆(x]α, I will show that the ⊑-least element of[⊔
X]α (I will denote it as y) satisfies Property 4. First of all, we have y ∈ [⊔
X]α, hence y =α
⊔X. By Property 1 we have y =β
⊔X for every β < α. Also
⊔X =β x for every β <α. Thus, y ∈ (x]α. Moreover, X⊑ ⊔
X, so by Lemma 3.9 we have X⊑α
⊔X. Also⊔
X=α y, so by transitivity of⊑α we have X⊑α y. Finally, suppose z∈(x]α
such that X⊑α z. By Property 6 we have⊔
X⊑α z. Also y=α
⊔X. By transitivity of⊑α we have y ⊑α z. Lastly, since y⊑α z we have y ⊏α z or y =α z. In the first case y ⊏z, so that y ⊑ z. In the second case z ∈[⊔
X]α and y is the⊑-least element of [⊔
X]α, so that y ⊑ z
again. □
Suppose x ∈ L and non-emptyX ⊆ (x]α for some α < κ. In the rest of the paper we will denote the⊑-least element of[⊔
X]α(which is also theyspecified by Property 4) by⊔
αX.
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4. SOME CONSEQUENCES
In this section we establish some properties of the stratified complete lattices, based on the above properties. In the following lemmas we suppose that (L,{⊑α}α<κ) is a stratified complete lattice.
Lemma 4.1 Let x,y∈L. If x⊑y then x⊑0 y.
Proof. It follows from Lemma 3.9 for the case α =0. □
Lemma 4.2 Suppose x∈L. Then⊥⊑0 x.
Proof. It follows from Lemma 4.1 for the case x=⊥. □
Lemma 4.3 For each x ∈L, for every ordinal α <κ and for any non-empty X⊆ (x]α, if for some y∈L, y⊑α X then y⊑α
⊔
αX.
Proof. Let z be any element of X. We have y⊑α z and by Property 4, z⊑α
⊔
αX. Thus, by transitivity, y ⊑α
⊔
αX. □
Lemma 4.4 For each x∈L, for every ordinal α <κ and for any non-empty X⊆(x]α,⊔
αX is the⊑-least element and a ⊑α+1-least element of[⊔
αX]α. Proof. Suppose z ∈ [⊔
αX]α. Note that since ⊔
αX =α ⊔ X, [⊔
αX]α = [⊔
X]α. Thus z ∈ [⊔
X]α. By definition ⊔
αX ⊑ z. Since by Property 1, z =β
⊔
αX for any β < α +1, by Lemma 3.9 we have ⊔
αX⊑α+1z. □
Lemma 4.5 Let α ≤κ be an ordinal and(xβ)β<α be a sequence of elements of L such that each xβis the least element of[xβ]βand xβ =βxγ whenever β <γ <α. If x=⊔
{xβ: β<α}, then xβ =βx for all β<α and x is the⊑-least element of(x]α.
Proof. Since xβ =β xγ whenever β <γ <α, we have xγ ∈[xβ]β. Since each xβis the least element of[xβ]β, xβ ⊑xγ whenever β<γ <α.
Let us suppose β < α. We are going to prove xβ =β x. Note that since xβ ⊑ xγ whenever β <γ <α we have ⊔
{xβ : β <α}= ⊔
{xγ : β <γ < α}. Also,{xγ : β <γ <α} ⊆ (xβ]β+1 because for any δ < β +1 and for each y ∈ {xγ : β < γ < α}, y =δ xδ =δ xβ. Thus, by Property 5, x∈(xβ]β+1, so that xβ =β x.
Now suppose y∈ (x]α. We have y=β x =β xβ for all β <α, hence xβ ⊑y for all β <α and x=⊔
{xβ :β <α} ⊑y. □
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5. THE FIXED POINT THEOREM
In this section we develop a fixed point theorem for functionsf:L→L, where(L,{⊑α}α<κ) is a stratified complete lattice. Notice that the functions fwe consider are not necessarily monotonic with respect to⊑(and therefore the traditional theorems of fixed point theory do not apply to them). Instead, we require that the functions we consider areα-monotonicwith respect to⊑α, for allα <κ:
Definition 5.1 Suppose that (L,{⊑α}α<κ) is a stratified complete lattice and let α < κ. A function f:L→L is called α-monotonic if for all x,y∈L, if x⊑α y then f(x)⊑α f(y).
In order to prove the main theorem (Theorem 5.4) we need the following technical lemma.
Lemma 5.2 Let (L,{⊑α}α<κ) be a stratified complete lattice. Suppose that f : L → L is α-monotonic, where α <κ. If x∈L and x⊑α f(x), then there is some y∈L such that:
• x⊑α y=α f(y)
• If x⊑α z and f(z)⊑α z, then y⊑α z
• y is the⊑-least element and a⊑α+1-least element of[y]α. Proof. Let us define x0 = x, xγ = f(xδ) if γ = δ+1and xγ = ⊔
α{xδ : δ < γ}if γ > 0is a limit ordinal. The definition makes sense since we can prove by induction on γ that x⊑α xγ
for all γ. It is clear when γ =0. Suppose that γ >0and our claim holds for all ordinals less than γ. If γ =δ+1, then x⊑α xδ by the induction hypothesis, thus x ⊑α f(x)⊑α f(xδ) =xγ
by the assumption that f is α-monotonic. If γ is a limit ordinal, then xγ = ⊔
α{xδ : δ < γ}, and since x⊑α xδ for all δ <γ, also x⊑α xγ by Lemma 4.3.
Claim: For all γ, xγ ⊑α f(xγ).
We prove this claim by induction on γ. When γ =0, it holds by assumption. Suppose now that γ = δ+1. By the induction hypothesis, we have that xδ ⊑α f(xδ) =xγ, and since f is α-monotonic, we have xγ =f(xδ)⊑α f(xγ). Finally, suppose that γ >0is a limit ordinal, then xγ =⊔
α{xδ :δ <γ}. By Property 4, we have xδ ⊑α xγ for all δ <γ. By the α-monotonicity of f and the induction hypothesis, we have xδ ⊑α f(xδ)⊑α f(xγ)for all δ <γ. By Property 4, it follows xγ ⊑α f(xγ)
Claim: For all ordinals β <γ, xβ ⊑α xγ.
Again, we prove this claim by induction on γ. When γ =0, our claim is trivial. Suppose now that γ >0. If γ =δ+1, then β≤δ and xβ ⊑α xδ ⊑α f(xδ) = xγ by the induction hypothesis and the previous claim. If γ >0is a limit ordinal, then xβ ⊑α xγ by Property 4.
Claim: Suppose that x ⊑α z and f(z)⊑α z. Then xγ ⊑α z for all γ.
We proceed by induction on γ. Since x0 = x, our claim is clear for γ = 0. Suppose that γ > 0and our claim holds for all ordinals less than γ. If γ = δ+1, then since xδ ⊑α z by the induction hypothesis, we have xγ = f(xδ) ⊑α f(z) ⊑α z by assumption and since f is
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α-monotonic. Thus, xγ ⊑α z. If γ is a limit ordinal, then xγ =⊔
α{xδ :δ <γ}. Since xδ ⊑α z for all δ <γ by the induction hypothesis, by Property 4 we have xγ ⊑α z.
Now, there is an ordinal λ0such that xγ =α xδfor all γ,δ ≥ λ0(since otherwise the cardinality of the set {xγ : γ is an ordinal} would exceed the cardinality of L). Let λ denote the least limit ordinal with λ≥ λ0. Let y=xλ. By the definition of y, we have that f(y) =α y and x⊑α y (since x = x0). Suppose that z ∈ L with x ⊑α z and f(z) ⊑α z. Then, as shown above, xγ ⊑α z for all γ, thus y⊑α z. Finally since y=⊔
α{xγ :γ < λ}by Lemma 4.4 we have that y is the⊑-least element and ⊑α+1-least element of[y]α. □ Below for any x ∈ L and ordinal α < κ with x ⊑α f(x), we will denote the element y con- structed above byfα(x). We have shown above that whenx⊑α f(x), thenfα(x)satisfies the three properties of Lemma 5.2.
Remark 5.3 According to Lemma 5.2 fα(x) =α f(fα(x))and fα(x)is a⊑α+1-least element of [fα(x)]α. Thus fα(x)⊑α+1f(fα(x)).
Theorem 5.4 Let (L,{⊑α}α<κ) be a stratified complete lattice. Suppose that f : L → L is α-monotonic for each ordinal α < κ. Then f has a least pre-fixed point with respect to the partial order⊑, which is also the least fixed point of f.
Proof. Let us define for each ordinal α < κ, xα = fα(yα), where y0 =⊥, yα = xβ when α = β+1and yα =⊔
{xβ :β <α}when α is a limit ordinal.
We are going to prove simultaneously by transfinite induction on α that:
• yα ⊑α f(yα)for all α <κ.
• xγ =γ xα whenever γ<α <κ.
• xα is the⊑-least element of[xα]α for all α <κ.
• xα =α f(xα)for all α <κ.
When α = 0 the first claim holds by Lemma 4.2. The second one is trivial. We apply Lemma 5.2 for the case α = 0 and x = y0, thus x0 = f0(y0) is the ⊑-least element of [x0]0and x0=0f(x0).
Suppose now that α = β+1. By the induction hypothesis, we have that yβ ⊑β f(yβ). We apply Lemma 5.2 for the case α =β and x=yβ and by Remark 5.3 we have xβ⊑β+1f(xβ), so that yα ⊑α f(yα). By applying Lemma 5.2 for the case α = α and x = yα we have xβ = yα ⊑α fα(yα) = xα, so that by Property 1, xβ =γ xα for all γ < α. If γ <α, then γ ≤ β, so by the induction hypothesis xγ =γ xβ for all γ ≤ β. It follows xγ =γ xα whenever γ < α.
Finally, by Lemma 5.2, xα is the⊑-least element of[xα]α and xα =α f(xα).
Suppose now that α > 0 is a limit ordinal and our claims hold for all ordinals less than α.
By Lemma 4.5 we have that xβ =β yα for all β < α and that yα is the ⊑-least element of (yα]α. Since f is β-monotonic, f(xβ) =β f(yα) for all β < α and by the induction hypothesis xβ =β f(xβ)for all β <α, so that f(yα)∈(yα]α. Thus, by Lemma 3.9, yα ⊑α f(yα). By applying Lemma 5.2 for the case α =α and x=yαwe have yα ⊑α fα(yα) = xα, so that by Property 1,
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yα =β xα for all β < α, hence xβ =β xα whenever β <α. Also, xα is the⊑-least element of [xα]α and xα =α f(xα).
Let us define x∞ =⊔
{xα : α < κ}. By Lemma 4.5, x∞ =α xα for all α < κ. Now since f is α-monotonic, also f(x∞) =α f(xα) =α xα for all α < κ. Thus, by Property 2, f(x∞) = x∞. It remains to show that x∞is the least pre-fixed point of f with respect to⊑.
Suppose that f(z) ⊑ z. We want to prove by induction that for all α < κ, either xγ ⊏γ z for some γ <α, or xγ ⊑γ z for all γ≤α. It then follows that x∞⊑z.
When α=0then by Lemma 4.1 and Lemma 4.2 it holds that f(z)⊑0z and⊥⊑0 z; thus, by Lemma 5.2, x0⊑0 z.
Suppose now that α > 0. If xγ ⊏γ z for some γ < α, then we are done. So suppose that this is not the case, ie., xγ =γ z for all γ <α.
Suppose that α = β+1. Then xβ =β z and thus z ∈ [xβ]β. Since xβ is⊑-least in [xβ]β, we have xβ ⊑ z and thus, by Lemma 3.9, xβ ⊑α z. We conclude that xβ ⊑α f(xβ) ⊑α f(z), ie., xβ ⊑α f(z). If f(z)⊏γ z for some γ <α, then by xβ ⊑α f(z)⊏γ z we have xβ⊏γ z, contradict- ing xβ =β z. Thus, since f(z)⊑z and f(z) =γ z for all γ<α, we must have f(z)⊑α z. Since xα =fα(xβ)and xβ ⊑α z, we conclude by Lemma 5.2 that xα ⊑α z.
Suppose that α > 0is a limit ordinal. Then, as shown above, yα =γ xγ for all γ <α. Since also xγ =γ z for all γ <α, we have z∈ (yα]α. But yα is the ⊑-least and a⊑α-least element of (yα]α, so that yα ⊑α z and thus yα ⊑α f(yα) ⊑α f(z); therefore yα ⊑α f(z). Suppose that f(z) ⊏γ z for some γ < α. Then yα ⊑α f(z) ⊏γ z and thus yα ⊏γ z, contradicting yα ⊑α z.
Thus, f(z) =γ z for all γ < α. Since f(z) ⊑ z and f(z) =γ z for all γ < α, we have f(z) ⊑α z.
Since yα ⊑α z and f(z)⊑α z, by Lemma 5.2 we have that xα ⊑α z. □
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6. THE CASE OF MONOTONIC FUNCTIONS
The above theorem has as a special case the well-known Knaster-Tarski fixed point theo- rem [8] when restricted to the case of monotonic functions.
Let (L,≤) be a complete lattice. Consider the case κ = 2, take ⊑0 to be equal to the ≤ relation and⊑1to be the equality relation overL. Properties 1 and 2 clearly hold. Moreover,
⊑ is equal to the ≤ relation, so (L,⊑) is indeed a complete lattice. Finally, (x]0 = L and (x]1={x}, so that the least upper bound ofXsatisfies the Property 4. Thus(L,{⊑α}α<2)is
a stratified complete lattice.
Let f : L→ L be a ≤-preserving function. It follows that f is 0 and 1-monotonic. According to Theorem 5.4,fhas a least fixed point.
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7. CONCLUSION
I have presented a simplified proof of a fixpoint theorem for (potentially) non-monotonic functions over stratified complete lattices. Moreover, Theorem 5.4 appears to have appli- cations in other classes of logic programs. One such case is extensional higher-order logic programming, which enhances classical logic programming with higher-order predicates.
This theorem has been used in order to obtain a minimum model semantics for extensional higher-order logic programming extended with negation [9]. Apart from logic programming, it would be interesting to investigate other applications of the derived theorem.
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[1] J.W. Lloyd. Foundations of Logic Programming. Springer-Verlag, 1987.
[2] P. Rondogiannis and W.W. Wadge. Minimum Model Semantics for Logic Programs with Negation-as- Failure. ACM Transactions on Computational Logic, 6(2):441–467, 2005.
[3] A. van Gelder, K. A. Ross and J. S. Schlipf. The Well-Founded Semantics for General Logic Programs.
Journal of the ACM, 38(3):620–650, 1991.
[4] Z. Ésik and P. Rondogiannis. A fixed point theorem for non-monotonic functions. Theor. Comput. Sci., 574:18–38, 2015.
[5] M. Fitting. Fixpoint Semantics for Logic Programming: A Survey. Theoretical Computer Science, 278(1- 2):25–51, 2002.
[6] T.C. Przymusinski. Every Logic Program has a Natural Stratification and an Iterated Least Fixed Point Model. Proceedings of the Eighth ACM Symposium on Principles of Database Systems (PODS), 11–21, 1989.
[7] B. A. Davey and H. A. Priestley. Introduction to Lattices and Order (2nd ed.). Cambridge University Press, 2002.
[8] A. Tarski. A Lattice-theoretical Fixpoint Theorem and its Applications. Pacific Journal of Mathematics, 5(2): 285–309, 1955.
[9] A. Charalambidis, Z. Ésik and P. Rondogiannis. Minimum Model Semantics for Extensional Higher-Order Logic Programming with Negation. (submitted), 2014.
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