The Intricacies of Three-Valued Extensional Semantics for Higher-Order Logic Programs

Rondogiannis, Panos; Symeonidou, Ioanna

doi:10.24963/ijcai.2018/750

“…One can see that similar arguments hold for the technique of (Charalambidis et al 2014). Therefore, if we seek an extensional three-valued semantics for higher-order logic programs with negation, we need to follow an approach that is radically different from both (Charalambidis et al 2014) and (Rondogiannis and Symeonidou 2017).…”

Section: Introductionmentioning

confidence: 91%

“…An intriguing and difficult question regarding logic programming, is whether it can be extended to a higher-order setting without sacrificing its semantic simplicity and clarity. Research results in this direction (Wadge 1991;Bezem 1999;Charalambidis et al 2013;Rondogiannis and Symeonidou 2016;Rondogiannis and Symeonidou 2017) strongly suggest that it is possible to design higher-order logic programming languages that have powerful expressive capabilities, and which, at the same time, retain all the desirable semantic properties of classical first-order logic programming. In particular, it has been shown that higher-order logic programming can be given an extensional semantics, namely one in which program predicates denote sets.…”

Section: Introductionmentioning

confidence: 99%

“…Section 6 develops the well-founded semantics of higher-order logic programs with negation, based on an extension of consistent approximation fixpoint theory. Section 7 compares the present work with that of (Charalambidis et al 2014;Rondogiannis and Symeonidou 2017), and concludes by identifying some promising research directions. The proofs of most results of the paper are given in the supplementary material corresponding to this paper at the TPLP archives.…”

Section: Introductionmentioning

confidence: 99%

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Approximation Fixpoint Theory and the Well-Founded Semantics of Higher-Order Logic Programs

Charalambidis¹,

Rondogiannis²,

Symeonidou³

2018

Theory and Practice of Logic Programming

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We define a novel, extensional, three-valued semantics for higher-order logic programs with negation. The new semantics is based on interpreting the types of the source language as three-valued Fitting-monotonic functions at all levels of the type hierarchy. We prove that there exists a bijection between such Fitting-monotonic functions and pairs of two-valued-result functions where the first member of the pair is monotone-antimonotone and the second member is antimonotone-monotone. By deriving an extension of consistent approximation fixpoint theory (Denecker et al. 2004) and utilizing the above bijection, we define an iterative procedure that produces for any given higher-order logic program a distinguished extensional model. We demonstrate that this model is actually a minimal one. Moreover, we prove that our construction generalizes the familiar well-founded semantics for classical logic programs, making in this way our proposal an appealing formulation for capturing the well-founded semantics for higher-order logic programs.A. Charalambidis et al.The above line of research started many years ago by W. W. Wadge (Wadge 1991) who considered positive higher-order logic programs (i.e., programs without negation in clause bodies). Wadge argued that if such a program obeys some simple and natural syntactic rules, then it has a unique minimum Herbrand model. It is well-known that the minimum model property is a cornerstone of the theory of first-order logic programming (van Emden and Kowalski 1976). In this respect, Wadge's result suggested that it might be possible to extend all the elegant theory of classical logic programming to the higher-order case. The results in (Wadge 1991) were obtained using standard techniques from denotational semantics involving continuous interpretations and Kleene's least fixpoint theorem. A few years after Wadge's initial result, M. Bezem came to similar conclusions (Bezem 1999) but from a different direction. In particular, Bezem demonstrated that by using a fixpoint construction on the ground instantiation of the source higher-order program, one can obtain a model of the original program that satisfies an extensionality condition defined in (Bezem 1999). Despite their different philosophies, Wadge's and Bezem's approaches have recently been shown (Charalambidis et al. 2017) to have close connections. Apart from the above results, recent work (Charalambidis et al. 2013) has also shown that we can define a sound and complete proof procedure for positive higher-order logic programs, which generalizes classical SLD-resolution. In other words, the central results for positive first-order logic programs, generalize to the higher-order case.A natural question that arises is whether one can still obtain an extensional semantics if negation is added to programs. Surprisingly, this question proved harder to resolve. The first result in this direction was reported in (Charalambidis et al. 2014), where it was demonstrated that every higher-order logic program with negation has a minimum extens...

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“…Research results in this direction (Wadge 1991;Bezem 1999;Charalambidis et al 2013;Rondogiannis and Symeonidou 2016;Rondogiannis and Symeonidou 2017) strongly suggest that it is possible to design higher-order logic programming languages that have powerful expressive capabilities, and which, at the same time, retain all the desirable semantic properties of classical first-order logic programming. In particular, it has been shown that higher-order logic programming can be given an extensional semantics, namely one in which program predicates denote sets.…”

Section: Introductionmentioning

confidence: 99%

“…Section 6 develops the well-founded semantics of higher-order logic programs with negation, based on an extension of consistent approximation fixpoint theory. Section 7 compares the present work with that of (Charalambidis et al 2014;Rondogiannis and Symeonidou 2017), and concludes by identifying some promising research directions. The proofs of most results of the paper are given in the appendices.…”

Section: Introductionmentioning

confidence: 99%

Approximation Fixpoint Theory and the Well-Founded Semantics of Higher-Order Logic Programs

Charalambidis¹,

Rondogiannis²,

Symeonidou³

2018

EasyChair Preprints

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We define a novel, extensional, three-valued semantics for higher-order logic programs with negation. The new semantics is based on interpreting the types of the source language as three-valued Fitting-monotonic functions at all levels of the type hierarchy. We prove that there exists a bijection between such Fitting-monotonic functions and pairs of two-valued-result functions where the first member of the pair is monotone-antimonotone and the second member is antimonotonemonotone. By deriving an extension of consistent approximation fixpoint theory (Denecker et al. 2004) and utilizing the above bijection, we define an iterative procedure that produces for any given higher-order logic program a distinguished extensional model. We demonstrate that this model is actually a minimal one. Moreover, we prove that our construction generalizes the familiar well-founded semantics for classical logic programs, making in this way our proposal an appealing formulation for capturing the well-founded semantics for higher-order logic programs.

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The intricacies of three-valued extensional semantics for higher-order logic programs

Rondogiannis¹,

Symeonidou²

2017

Theory and Practice of Logic Programming

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M. Bezem defined an extensional semantics for positive higher-order logic programs. Recently, it was demonstrated by Rondogiannis and Symeonidou that Bezem's technique can be extended to higher-order logic programs with negation, retaining its extensional properties, provided that it is interpreted under a logic with an infinite number of truth values. Rondogiannis and Symeonidou also demonstrated that Bezem's technique, when extended under the stable model semantics, does not in general lead to extensional stable models. In this paper, we consider the problem of extending Bezem's technique under the well-founded semantics. We demonstrate that the well-founded extensionfailsto retain extensionality in the general case. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting cannot distinguish between certain predicates that appear to have a different behaviour inside a program context, but which happen to be identical as three-valued relations.

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The Intricacies of Three-Valued Extensional Semantics for Higher-Order Logic Programs

Cited by 3 publications

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Approximation Fixpoint Theory and the Well-Founded Semantics of Higher-Order Logic Programs

Approximation Fixpoint Theory and the Well-Founded Semantics of Higher-Order Logic Programs

Approximation Fixpoint Theory and the Well-Founded Semantics of Higher-Order Logic Programs

The intricacies of three-valued extensional semantics for higher-order logic programs

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