The purpose of the present paper is to give an overview of our joint work with Zoltán Ésik, namely the development of an abstract fixed point theory for a class of non-monotonic functions [4] and its use in providing a novel denotational semantics for a very broad extension of classical logic programming [1]. Our purpose is to give a high-level presentation of the main developments of these two works, that avoids as much as possible the underlying technical details, and which can be used as a mild introduction to the area

Charalambidis, Angelos

Rondogiannis, Panos

Charalambidis Angelos

Rondogiannis Panos

University of Szeged

Acta Cybernetica 23 (2017) 319–326.Overview of an Abstract Fixed Point Theory forNon-Monotonic Functions and its Applications toLogic ProgrammingAngelos Charalambidisa and Panos RondogiannisaAbstractThe purpose of the present paper is to give an overview of our joint workwith Zoltán Ésik, namely the development of an abstract fixed point theory fora class of non-monotonic functions [4] and its use in providing a novel denota-tional semantics for a very broad extension of classical logic programming [1].Our purpose is to give a high-level presentation of the main developmentsof these two works, that avoids as much as possible the underlying technicaldetails, and which can be used as a mild introduction to the area.Keywords: fixed point theory, higher-order logic programming, semanticsof logic programming1 IntroductionThe purpose of this paper is to present an overview of the authors’ joint work withZoltán Ésik. This work [4] concerned the development of an abstract fixed pointtheory for a class of functions that exhibit a type of “monotonicity in layers” butwhich are overall non-monotonic. Such functions prove to be quite common invarious investigations in logic programming and formal language theory, and maypotentially have other applications. We also describe our development [1], basedon the aforementioned abstract framework, of a novel denotational semantics for avery broad extension of classical logic programming. In the rest of this section weprovide a short description of the beginnings of our collaboration with Zoltán thatled to the above results.In 2005, the second author together with Bill Wadge proposed [5] the infinite-valued semantics for logic programs with negation. This particular work was some-what ad-hoc, namely the main results relied on techniques custom-tailored for logicprogramming. In 2013, the second author of the present paper, together with ZoltánaDepartment of Informatics and Telecommunications, National and Kapodistrian Universityof Athens, E-mail: {a.charalambidis,prondo}@di.uoa.grDOI: 10.14232/actacyb.23.1.2017.17320 Angelos Charalambidis and Panos RondogiannisÉsik started a collaboration supported by a “Greek-Hungarian Scientific Collabo-ration Program” with title “Extensions and Applications of Fixed Point Theoryfor Non-Monotonic Formalisms”. The purpose of the program was to create anabstract fixed point theory based on the infinite-valued approach, namely a theorythat would not only be applicable to logic programs but also to other non-monotonicformalisms. This abstract theory was successfully developed and is described in de-tail in [4]. As an application of these results, this abstract theory was used in [1] inorder to obtain the first extensional semantics for higher-order logic programs withnegation. Another application of the new theory to the area of non-monotonic for-mal grammars was proposed in [3]. Moreover, Zoltán himself further investigatedthe foundations and the properties of the infinite-valued approach [2], highlightingsome of its desirable characteristics. Unfortunately, the further joint developmentof the abstract infinite-valued approach to non-monotonic fixed point theory, wasabruptly interrupted by the untimely loss of Zoltán.In the next section we describe the basic concepts behind the abstract approachto non-monotonic fixed point theory. In Section 3 we describe the application ofthe theory to the class of higher-order logic programs with negation. The paperconcludes by giving pointers for future work.2 Non-Monotonic Fixed Point TheorySuppose that (L,≤) is a complete lattice in which the least upper bound operationis denoted by∨and the least element is denoted by ⊥. Let κ > 0 be a fixedordinal. We assume that for each ordinal α < κ, there exists a preordering vαon L. We denote with =α the equivalence relation determined by vα. We definex <α y iff x vα y but x =α y does not hold. Finally, we define <=⋃α<κ <αand let x v y iff x < y or x = y. Given an ordinal α < κ and x ∈ L, define(x]α = {y ∈ L : ∀β < α x =β y}. We require of our relations to satisfy thefollowing axioms:Axiom 1. For all ordinals α < β < κ, vβ is included in =α.Axiom 2.⋂α<κ =α is the identity relation on L.Axiom 3. For each x ∈ L, for every ordinal α < κ, and for any X ⊆ (x]αthere is some y ∈ (x]α such that:Overview of an Abstract Fixed Point Theory 321• X vα y, and• for all z ∈ (x]α, if X vα z then y vα z and y ≤ z.Axiom 4. If xj , yj ∈ L and xj vα yj for all j ∈ J then∨{xj : j ∈ J} vα∨{yj : j ∈ J}.The element y specified by the Axiom 3 above, can be shown to be unique andwe denote it by⊔αX.In the following, we will often talk about “models of the Axioms 1-4” (or simply“models”). More formally:Definition 1. A model of Axioms 1-4 or simply model consists of a completelattice (L,≤), an ordinal κ > 0 and a set of preorders vα for every α < κ, suchthat Axioms 1-4 are satisfied.Under the above axioms, the following theorem is established in [4]:Theorem 1. (L,v) is a complete lattice.The following definition will lead us to the main theorem of [4]:Definition 2. Suppose that L is a model and let α < κ. A function f : L → L iscalled α-monotonic if for all x, y ∈ L, if x vα y then f(x) vα f(y).The central fixed point theorem of [4] can now be stated:Theorem 2. Let L be a model. Suppose that f : L → L is α-monotonic for eachordinal α < κ. Then f has a least pre-fixed point with respect to the partial orderv, which is also the least fixed point of f .The article [4] contains many more results, but one could say that the abovetheorem is possibly the main technical achievement. Actually, the above theoremis also the main tool that we will need in the developments of the next section.3 Higher-Order Logic Programs with NegationIn this section we present the application of the non-monotonic fixed point theoryto the class of higher-order logic programs with negation. The approach presentednaturally extends the ideas behind the infinite-valued approach proposed in [5] intoa higher-order setting. The basic idea behind the approach in [5] is that in order toobtain minimum model semantics for higher-order logic programs with negation itis necessary to consider a multi-valued logic. We first present the syntax and thenthe semantics of our language.322 Angelos Charalambidis and Panos Rondogiannis3.1 SyntaxOur higher-order logic programming language is based on a simple type system thatsupports two base types: o, the boolean domain, and ι, the domain of individuals(data objects). The composite types are partitioned into three classes: functional(assigned to individual constants, individual variables and function symbols), pred-icate (assigned to predicate constants and variables) and argument (assigned toparameters of predicates).Definition 3. A type τ can either be functional, argument, or predicate, denotedas σ, π and ρ respectively and defined as:σ := ι | ι→ σπ := o | ρ→ πρ := ι | πDefinition 4. The set of expressions of our higher-order language is defined asfollows:1. Every predicate variable (respectively, predicate constant) of type π is an ex-pression of type π; every individual variable (respectively, individual constant)of type ι is an expression of type ι; the propositional constants false and trueare expressions of type o.2. If f is an n-ary function symbol and E1, . . . ,En are expressions of type ι, then(f E1 · · ·En) is an expression of type ι.3. If E1 is an expression of type ρ → π and E2 is an expression of type ρ, then(E1 E2) is an expression of type π.4. If V is an argument variable of type ρ and E is an expression of type π, then(λV.E) is an expression of type ρ→ π.5. If E1,E2 are expressions of type π, then (E1∧π E2) and (E1∨π E2) are ex-pressions of type π.6. If E is an expression of type o, then (∼E) is an expression of type o.7. If E1,E2 are expressions of type ι, then (E1 ≈ E2) is an expression of type o.8. If E is an expression of type o and V is a variable of type ρ then (∃ρVE) isan expression of type o.The notions of free and bound variables of an expression are defined as usual.An expression is called closed if it does not contain any free variables.A program clause is a clause p ←π E where p is a predicate constant of type πand E is a closed expression of type π. A program is a finite set of program clauses.Overview of an Abstract Fixed Point Theory 3233.2 SemanticsWe start by examining the semantics of types. The most crucial case is that of theboolean domain o. The boolean values range over a partially ordered set (V,≤) oftruth values. The number of truth values of V will be specified with respect to anordinal κ > 0. The set (V,≤) is the following:F0 < F1 < · · ·< Fα < · · ·< 0 < · · ·< Tα < · · ·< T1 < T0where α < κ. Intuitively, F0 and T0 are the classical False and True values and 0 isthe undefined value. The new values express different levels of truthness and falsity.The order of a truth value is defined as follows: order(Tα) = α, order(Fα) = α andorder(0) = +∞.We define the following preorderings vα on the set V for each α < κ:1. x vα x if order(x) < α;2. Fα vα x and x vα Tα if order(x) ≥ α;3. x vα y if order(x), order(y) > α.We then have the following result from [1]:Lemma 1. (V,≤) is a complete lattice and a model.Let us denote by [Am→ B] the set of functions from A to B that are α-monotonicfor all α < κ. Based on the above discussion, we can now state the semantics of allthe types of our language:Definition 5. Let D be a nonempty set. Then:• [[o]]D = V , and ≤o is the partial order of V ;• [[ι]]D = D, and ≤ι is the trivial partial order such that d ≤ι d, for all d ∈ D;• [[ιn → ι]]D = Dn → D. A partial order in this case will not be needed;• [[ι→ π]]D = D → [[π]]D, and ≤ι→π is the partial order defined as follows: forall f, g ∈ [[ι→ π]]D, f ≤ι→π g iff f(d) ≤π g(d) for all d ∈ D;• [[π1 → π2]]D = [[[π1]]Dm→ [[π2]]D], and ≤π1→π2 is the partial order defined asfollows: for all f, g ∈ [[π1 → π2]]D, f ≤π1→π2 g iff f(d) ≤π2 g(d) for alld ∈ [[π1]]D.Moreover, we have the following relations vα on our domains:• The relation vα on [[o]]D is the relation vα on V .• The relation vα on [[ρ → π]]D is defined as follows: f vα g iff f(d) vα g(d)for all d ∈ [[ρ]]D.324 Angelos Charalambidis and Panos RondogiannisThe following lemma can then be established following the results of [4]:Lemma 2. Let D be a non-empty set and π be a predicate type. Then, ([[π]]D,≤π)is a complete lattice and a model.For the rest of the section we focus on Herbrand interpretations and we assumefor a program P, D = UP where UP is the Herbrand universe and therefore wesimple write [[τ ]] instead of [[τ ]]UP . A Herbrand interpretation I for a program P isa function that maps a predicate of type π to an element of [[π]]. The set of all theinterpretation of P is denoted by IP. It follows directly from the results of [4] thatIP is a complete lattice and a model. A Herbrand state s is a function that assignsto each argument variable V of type ρ, of an element s(V) ∈ [[ρ]]UP .Let I be a Herbrand interpretation and s be a Herbrand state. The semanticsof expressions with respect to I and s, is defined as follows:1. [[false]]s(I) = F02. [[true]]s(I) = T03. [[c]]s(I) = I(c), for every individual constant c4. [[p]]s(I) = I(p), for every predicate constant p5. [[V]]s(I) = s(V), for every argument variable V6. [[(f E1 · · ·En)]]s(I) = I(f) [[E1]]s(I) · · · [[En]]s(I), for every n-ary function sym-bol f7. [[(E1E2)]]s(I) = [[E1]]s(I)([[E2]]s(I))8. [[(λV.E)]]s(I) = λd.[[E]]s[V/d](I), where d ranges over [[type(V)]]D9. [[(E1∨π E2)]]s(I) =∨π{[[E1]]s(I), [[E2]]s(I)}, where∨π is the lub function on[[π]]D10. [[(E1∧π E2)]]s(I) =∧π{[[E1]]s(I), [[E2]]s(I)}, where∧π is the glb function on[[π]]D11. [[(∼E)]]s(I) =Tα+1 if [[E]]s(I) = FαFα+1 if [[E]]s(I) = Tα0 if [[E]]s(I) = 012. [[(E1≈E2)]]s(I) ={T0, if [[E1]]s(I) = [[E2]]s(I)F0, otherwise13. [[(∃VE)]]s(I) =∨d∈[[type(V)]]D[[E]]s[V/d](I)Definition 6. Let P be a program and let M be a Herbrand interpretation of P.Then M will be called a model of P iff for all clauses p ←π E of P, it holds[[E]](M) ≤π M(p), where M(p) ∈ [[π]].Overview of an Abstract Fixed Point Theory 325We can now define the immediate consequence operator for our language:Definition 7. Let P be a program. The mapping TP : IP → IP is defined for everyp : π and for every I ∈ IP asTP(I)(p) =∨{[[E]](I) : (p←π E) ∈ P}As it turns out, TP enjoys the α-monotonicity property [1]:Lemma 3. For all α < κ, TP is α-monotonic.We now have all we need in order to apply the main Theorem of [4], getting thefollowing result [1]:Theorem 3 (Least Fixed Point Theorem). Let P be a program and let M be theset of all its Herbrand models. Then, TP has a least fixed point MP which is theleast model of P.4 ConclusionsWe have presented an overview of the abstract fixed point theory developed in [4]and its application [1] on a very broad class of logic programs, namely higher-orderlogic programs with negation. It is our belief that the framework of [4] can find otherinteresting applications, especially ones where non-monotonicity plays a prevailingrole. In particular, we believe that an area that has not yet been sufficientlyexplored is that of non-monotonic formal grammars. In [3] it was demonstrated thatthe semantics of Boolean grammars can be easily captured through an extensionof the framework of [4]. However, it is conceivable to have other non-monotonicextensions of formal grammars apart from the Boolean ones, such as for examplemacro-grammars with conjunction and negation in rule bodies. We believe thatthe results of [1] can be used as a yardstick in order to approach the semantics ofsuch grammar formalisms.References[1] Charalambidis, Angelos, Ésik, Zoltán, and Rondogiannis, Panos. Minimummodel semantics for extensional higher-order logic programming with negation.TPLP, 14(4-5):725–737, 2014.[2] Ésik, Zoltán. Equational properties of stratified least fixed points (extendedabstract). In de Paiva, Valeria, de Queiroz, Ruy J. G. B., Moss, Lawrence S.,Leivant, Daniel, and de Oliveira, Anjolina Grisi, editors, Logic, Language, In-formation, and Computation - 22nd International Workshop, WoLLIC 2015,Bloomington, IN, USA, July 20-23, 2015, Proceedings, volume 9160 of LectureNotes in Computer Science, pages 174–188. Springer, 2015.326 Angelos Charalambidis and Panos Rondogiannis[3] Ésik, Zoltán and Rondogiannis, Panos. Theorems on pre-fixed points of non-monotonic functions with applications in logic programming and formal gram-mars. In Kohlenbach, Ulrich, Barceló, Pablo, and de Queiroz, Ruy J. G. B.,editors, Logic, Language, Information, and Computation - 21st InternationalWorkshop, WoLLIC 2014, Valparáıso, Chile, September 1-4, 2014. Proceedings,volume 8652 of Lecture Notes in Computer Science, pages 166–180. Springer,2014.[4] Ésik, Zoltán and Rondogiannis, Panos. A fixed point theorem for non-monotonicfunctions. Theor. Comput. Sci., 574:18–38, 2015.[5] Rondogiannis, Panos and Wadge, William W. Minimum model semantics forlogic programs with negation-as-failure. ACM Trans. Comput. Log., 6(2):441–467, 2005.

Overview of an abstract fixed point theory for non-monotonic functions and its applications to logic programming

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Overview of an abstract fixed point theory for non-monotonic functions and its applications to logic programming

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