Under consideration for publication in Theory and Practice of Logic Programming                   1 
  A three-valued semantics for logic programmers 
                                          LEE NAISH 
                   Department of Computer Science and Software Engineering 
                                     University of Melbourne 
                                         Melbourne 3010 
                                             Australia 
                                    (e-mail: lee@www.book118.com.au) 
           submitted 15 June 2004; revised 30 November 2004; accepted 30 June 2005 
                                           Abstract 
This  paper  describes  a  simpler  way  for  programmers  to  reason  about  the  correctness  of 
their code. The study of semantics of logic programs has shown strong links between the 
model theoretic semantics (truth and falsity of atoms in the programmer’s interpretation 
of a program), procedural semantics (for example, SLD resolution) and ?xpoint semantics 
(which is useful for program analysis and alternative execution mechanisms). Most of this 
work  assumes  that  intended  interpretations  are  two-valued:  a  ground  atom  is  true  (and 
should succeed according to the procedural semantics) or false (and should not succeed). In 
reality, intended interpretations  are less  precise.  Programmers  consider  that  some atoms 
“should  not  occur”or  are  “ill-typed”or  “inadmissible”.  Programmers  don’t  know  and 
don’t care whether such atoms succeed. In this paper we propose a three-valued semantics 
for  (essentially)  pure  Prolog  programs  with  (ground)  negation  as  failure  which  re?ects 
this. The semantics of Fitting is similar but only associates the third truth value with non- 
termination. We provide tools to reason  about correctness of programs without the need 
for unnatural precision or undue restrictions on programming style. As well as theoretical 
results, we  provide a programmer-oriented  synopsis. This  work  has come out  of work  on 
declarative debugging, where it has been recognised that inadmissible calls are important. 
This paper has been accepted to appear in Theory and Practice of Logic Programming. 
KEYWORDS:  Models,         immediate    consequence    operator,  SLDNF  resolution,     negation, 
veri?cation, declarative debugging, inadmissibility 
                                       1  Introduction 
When programming, we often give too little attention to the meaning of our code. 
For these sins of sloth and pride (“I can’t be bothered rechecking everything —I’m sure I got it right”) we are forced to do penance, in the form of debugging. 
During the repetitive tedium we contemplate the relationships between our code, 
its behaviour and our desires: the domain of programming language semantics. Thus 
(in retrospect) it is natural for work on debugging to lead to work on semantics. 
All too often, work on semantics seems to bear little relation to any stage of the 
software life cycle. Here our aim is to provide theoretical support which can allow 
programmers to reason about the correctness of their code more easily. 
2                                     L. Naish 
A-three-valued-semantics-for-logic-programmers.pdf