Mapping Inferences: Constraint Propagation and Diamond Satisfaction
Rosella Gennari

Abstract:
This thesis is concerned with knowledge representation, and
efficient automated reasoning on the chosen representation.

Part I treats constraint propagation algorithms. These constitute 
a broad class of widely used, efficient, non-deterministic algorithms
for constraint satisfaction problems. Such problems are introduced in
the first chapter of the thesis; e.g., the map colourability problem,
N-SAT, Allen temporal reasoning problems, scheduling.

Our aim in Part I is purely theoretical: a general theory, based
on function iterations, underpinning those algorithms; this unifying 
framework allows us to verify, explain, compare and differentiate
constraint propagation algorithms through functions and their iterations.

For instance, this theory can answer the following sort of questions
in terms of properties of functions or their iterations.

  1. Do constraint propagation algorithms follow a common strategy?
     If so, how can it be ``characterised'' in general?
  2. How do those algorithms \emph{differentiate}?
     That is: which function properties are responsible for the diverse 
     strategies and efficiency?
  3. What guarantees their termination?
  4. What properties of functions guarantee the absence of backtracking?

Soft constraints are more expressive than ``standard'' constrains: the
former allow the user a certain form of uncertainty. Yet, the developed 
theory of function iterations is sufficiently general and expressive to
represent and analyse soft constraint propagation algorithms too.

In Part II of the thesis, relations are still under focus: in fact, this 
part deals with modal logics, introduced in Chapter 7.  More precisely, in
this part of the thesis we propose two main ways to tackle the following 
question:

  how can the satisfiability of modal formulas be determined in an 
  efficient manner?

In Chapter 8 we answer the question by embedding modal logics into
first-order logics: we refine the standard translation
from modal to first-order logic, and show how this refinement
improves the performances of resolution-based theorem provers on the
resulting fragment of first-order logic.

Then, in Chapter 9  the same semantic intuitions underlying the
refinement of the standard translation are at the basis of a constraint
solver for propositional satisfiability; these decides the satisfiability of
modal formulas ``layer by layer'', using constraint based algorithms as 
explained in the first part of the thesis.

This chapter opens an interesting research topic:

      how efficient are constraint propagation and satisfaction 
      algorithms for modal reasoning?

Part III summarises the contents of this thesis, and concludes it by
discussing some other open questions.