The right representation is the key to successful reasoning
What is the ‘ right representation ’ of a problem?
; A representation that allows the most time/space efficient computation of its solution(s)
Tends to be a representation which causes the associated search space to be small
Where do different representations come from?
; How a problem is represented depends on
(a) choice of language
(b) choice of expression
Both influence the complexity of reasoning
Choice of language not always under the control of the user
Ullrich Hustadt, University of Liverpool (U.Hustadt@csc.liv.ac.uk; http://www.csc.liv.ac.uk/˜ullrich) – p.1

The right representation is the key to successful reasoning
A lot is known about the influence of language choice
(1) The description logic ALC ( A , ¬ C , C u D , ∀ R .
C ) can be translated to first-order logic FOL
Satisfiability of ALC is PSPACE complete, while the satisfiability problem of FOL is undecidable
But reasoning in FOL about a translated ALC problem has exactly as difficult as reasoning about it in ALC
Ullrich Hustadt, University of Liverpool (U.Hustadt@csc.liv.ac.uk; http://www.csc.liv.ac.uk/˜ullrich) – p.2

The right representation is the key to successful reasoning
A lot is known about the influence of language choice
(2) Quantified propositional logic QPL ( p , ¬ ϕ , ϕ ∧ ψ , ∀ p ϕ ) and basic modal logic K ( p , ¬ ϕ , ϕ ∧ ψ , 2 ϕ ) are both PSPACE complete, so any problem expressible in QPL can be expressed in K using about the same space
But reasoning about these problems in K seems to be much more difficult than in QPL
Ullrich Hustadt, University of Liverpool (U.Hustadt@csc.liv.ac.uk; http://www.csc.liv.ac.uk/˜ullrich) – p.3

The right representation is the key to successful reasoning
A lot is known about the influence of language choice
(3) The first-order logic of time FLT ( P ( t ), t ≤ t 0 , ¬ ϕ , ϕ ∧ ψ , ∀ t ϕ ) is expressivly equivalent to propositional temporal logic PTL ( p , ¬ ϕ , ϕ ∧ ψ , ϕ S ψ , ϕ U ψ ), but reasoning in FLT is much more difficult than reasoning in
PTL
On the other hand, expressing problems in PTL requires much more space
How difficult is it to reason in FLT about a translated PTL problem?
Ullrich Hustadt, University of Liverpool (U.Hustadt@csc.liv.ac.uk; http://www.csc.liv.ac.uk/˜ullrich) – p.4

The right representation is the key to successful reasoning
A lot is known about the influence of language choice
. . . but the conclusions we should draw from that are not clear
I
FLT vs PTL is an example of the standard way we use
‘language’ in Computer Science
(not a particularly successful one, though)
I
Both ALC vs FOL and K vs QPL seem to indicate that complexity theory isn’t the right tool for a fine-grained analysis of reasoning in various logics
(This seems mainly because reasoning methods for a logic operate ‘uniformly’ on all expressions of that logic)
Ullrich Hustadt, University of Liverpool (U.Hustadt@csc.liv.ac.uk; http://www.csc.liv.ac.uk/˜ullrich) – p.5

The right representation is the key to successful reasoning
Much less is known about the influence of expression choice
I
Switching from one language to another always implies a change of expression
This can
– open new and more efficient ways of reasoning , or
– it doesn’t make any difference ( ALC to FOL), or
– it makes matters worse (QPL to K)
So far, we have a few results in this direction, but nothing general
I
The same is true for changing the way we express a problem while keeping the language constant
But we have even less of a clue of what is going on
Ullrich Hustadt, University of Liverpool (U.Hustadt@csc.liv.ac.uk; http://www.csc.liv.ac.uk/˜ullrich) – p.6

The right representation is the key to successful reasoning
How do we human beings find good representations ?
; Humans simply try lots of things and are good at interpreting feedback
How can we model that on a computer?
; Representation modification and generation methods can be implemented; resulting representations then evaluated; success rate low, but brute force might win
How do we know whether a particular formulation of a problem is really a representation of the problem at hand ?
; Empirically. If the solution of the problem representation is correct, then the representation is correct.
Ullrich Hustadt, University of Liverpool (U.Hustadt@csc.liv.ac.uk; http://www.csc.liv.ac.uk/˜ullrich) – p.7

The right representation is the key to successful reasoning
Don’t give up on reasoning!
; In the area of non-classical logics, work on reasoning systems is still in its infancy
Existing reasoning systems are primitive compared to those for propositional or predicate logic (despite some claims to the contrary)
; One goal of current development is robustness with respect to problem representation
This is also important for the usability of such systems
Ullrich Hustadt, University of Liverpool (U.Hustadt@csc.liv.ac.uk; http://www.csc.liv.ac.uk/˜ullrich) – p.8