scholarly journals Equivalence of two Fixed-Point Semantics for Definitional Higher-Order Logic Programs

2015 ◽  
Vol 191 ◽  
pp. 18-32 ◽  
Author(s):  
Angelos Charalambidis ◽  
Panos Rondogiannis ◽  
Ioanna Symeonidou
Keyword(s):  
Fixed Point ◽  
Higher Order ◽  
Order Logic ◽  
Logic Programs ◽  
Higher Order Logic ◽  
Fixed Point Semantics

scholarly journals Equivalence of two fixed-point semantics for definitional higher-order logic programs

2017 ◽  
Vol 668 ◽  
pp. 27-42 ◽  
Author(s):  
Angelos Charalambidis ◽  
Panos Rondogiannis ◽  
Ioanna Symeonidou
Keyword(s):  
Fixed Point ◽  
Higher Order ◽  
Order Logic ◽  
Logic Programs ◽  
Higher Order Logic ◽  
Fixed Point Semantics

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

2017 ◽  
Vol 17 (5-6) ◽  
pp. 974-991
Author(s):  
PANOS RONDOGIANNIS ◽  
IOANNA SYMEONIDOU
Keyword(s):  
Infinite Number ◽  
Higher Order ◽  
Order Logic ◽  
Stable Model ◽  
Logic Programs ◽  
Higher Order Logic ◽  
Truth Values ◽  
Positive Side ◽  
Stable Models

AbstractM. 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.

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

2018 ◽  
Vol 18 (3-4) ◽  
pp. 421-437
Author(s):  
ANGELOS CHARALAMBIDIS ◽  
PANOS RONDOGIANNIS ◽  
IOANNA SYMEONIDOU
Keyword(s):  
Logic Program ◽  
Higher Order ◽  
Order Logic ◽  
Logic Programs ◽  
Higher Order Logic ◽  
Source Language ◽  
Monotonic Functions ◽  
Consistent Approximation ◽  
Extensional Model ◽  
Type Hierarchy

AbstractWe 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 ofconsistent approximation fixpoint theory(Deneckeret 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 aminimalone. 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 thewell-founded semantics for higher-order logic programs.

scholarly journals Learning higher-order logic programs

2019 ◽  
Vol 109 (7) ◽  
pp. 1289-1322 ◽  
Author(s):  
Andrew Cropper ◽  
Rolf Morel ◽  
Stephen Muggleton
Keyword(s):  
Inductive Logic Programming ◽  
Inductive Logic ◽  
Higher Order ◽  
Order Logic ◽  
Sample Complexity ◽  
Logic Programs ◽  
Higher Order Logic ◽  
First Order ◽  
Hypothesis Space ◽  
Theoretical Results

AbstractA key feature of inductive logic programming is its ability to learn first-order programs, which are intrinsically more expressive than propositional programs. In this paper, we introduce techniques to learn higher-order programs. Specifically, we extend meta-interpretive learning (MIL) to support learning higher-order programs by allowing for higher-order definitions to be used as background knowledge. Our theoretical results show that learning higher-order programs, rather than first-order programs, can reduce the textual complexity required to express programs, which in turn reduces the size of the hypothesis space and sample complexity. We implement our idea in two new MIL systems: the Prolog system $$\text {Metagol}_{ho}$$ Metagol ho and the ASP system $$\text {HEXMIL}_{ho}$$ HEXMIL ho . Both systems support learning higher-order programs and higher-order predicate invention, such as inventing functions for and conditions for . We conduct experiments on four domains (robot strategies, chess playing, list transformations, and string decryption) that compare learning first-order and higher-order programs. Our experimental results support our theoretical claims and show that, compared to learning first-order programs, learning higher-order programs can significantly improve predictive accuracies and reduce learning times.

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

2018 ◽  
Author(s):  
Panos Rondogiannis ◽  
Ioanna Symeonidou
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Logic Programs ◽  
Higher Order Logic ◽  
Positive Side

In this paper we examine the problem of providing a purely extensional three-valued semantics for higher-order logic programs with negation. We demonstrate that a technique that was proposed by M. Bezem for providing extensional semantics to positive higher-order logic programs, fails when applied to higher-order logic programs with negation. On the positive side, we demonstrate that for stratified higher-order logic programs, extensionality is indeed achieved by the technique. We analyze the reasons of the failure of extensionality in the general case, arguing that a three-valued setting can not 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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