Logic and AI Planning: Default Logic - Department of Computer and ...
Logic and AI Planning: Default Logic - Department of Computer and ...
Logic and AI Planning: Default Logic - Department of Computer and ...
Create successful ePaper yourself
Turn your PDF publications into a flip-book with our unique Google optimized e-Paper software.
<strong>Logic</strong> <strong>and</strong> <strong>AI</strong> <strong>Planning</strong>: <strong>Default</strong> <strong>Logic</strong><br />
Peep Küngas<br />
peep@idi.ntnu.no<br />
<strong>Department</strong> <strong>of</strong> <strong>Computer</strong> <strong>and</strong> Information Science<br />
Norwegian University <strong>of</strong> Science <strong>and</strong> Technology<br />
http://www.idi.ntnu.no/emner/tdt14<br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.1/10
<strong>Default</strong> <strong>Logic</strong> (DL)<br />
One <strong>of</strong> the oldest <strong>and</strong> most studied nonmonotonic<br />
logic<br />
An extension <strong>of</strong> classical logic with default rules,<br />
which are nonmonotonic<br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.2/10
¥<br />
¢¤£<br />
¡<br />
¢<br />
¨© £ <br />
¨<br />
¦§<br />
<br />
<br />
<br />
<br />
¨ £<br />
<br />
<br />
¦<br />
¦<br />
¨ <br />
<br />
<br />
<br />
<br />
£<br />
<br />
<br />
Definition <strong>of</strong> DL (1)<br />
A default theory is a pair<br />
, where<br />
is a set <strong>of</strong> first-order formulae representing facts<br />
is a set <strong>of</strong> defaults in form<br />
whereas<br />
formulae<br />
,<br />
<strong>and</strong><br />
are classical<br />
The default has the following intuitive meaning—if<br />
derivable <strong>and</strong>, for all<br />
then derive<br />
.<br />
,<br />
is<br />
is not derivable,<br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.3/10
¦<br />
<br />
<br />
<br />
¨ £<br />
<br />
<br />
<br />
<br />
<br />
<br />
<br />
Definition <strong>of</strong> DL (2)<br />
The formula is called the prerequisite<br />
is a consistency condition or<br />
justification<br />
is called the consequent <strong>of</strong> the default<br />
For a default<br />
pre(<br />
), just(<br />
) <strong>and</strong> cons(<br />
) are used to<br />
denote respectively the prerequisite, the set <strong>of</strong> justifications<br />
<strong>and</strong> the consequent <strong>of</strong><br />
.<br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.4/10
¥<br />
)<br />
(<br />
) <br />
(<br />
¥<br />
¥<br />
A <strong>Default</strong> Theory (1)<br />
Here we describe another variant <strong>of</strong> the Yale Shooting<br />
domain. New states are managed through situation<br />
calculus.<br />
# £ $&%<br />
¡<br />
<br />
"!<br />
<br />
<br />
<br />
'(<br />
<br />
$.%<br />
¥¥<br />
,!1<br />
¥ £<br />
¡+<br />
<br />
<br />
<br />
) ¥0/<br />
$.%<br />
¡+<br />
¡*<br />
<br />
<br />
¡ <br />
¡*<br />
,!-<br />
¥ £<br />
'(<br />
<br />
)<br />
A causal law:<br />
¡2<br />
"3<br />
<br />
<br />
<br />
)£<br />
¡<br />
<br />
!<br />
<br />
<br />
<br />
#£<br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.5/10
)<br />
(<br />
¥¥<br />
,<br />
) £ 6 ) <br />
A <strong>Default</strong> Theory (2)<br />
¥ £ <br />
¥<br />
¥ £ <br />
4 ¡ <br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.6/10<br />
¡<br />
$7 <br />
¡54<br />
<br />
¡*<br />
<br />
<br />
¡2<br />
3<br />
<br />
<br />
<br />
An action
¥<br />
¡8 £<br />
$.%<br />
<br />
§<br />
¥<br />
¡8 £<br />
$.%<br />
¥<br />
¡8 £<br />
$.%<br />
¥¥<br />
,<br />
¡8 £<br />
6 ) <br />
¥¥<br />
,<br />
¡8 £<br />
6 ) <br />
¥¥<br />
,<br />
¡8 £<br />
6 ) <br />
¥¥<br />
,<br />
¡8 £<br />
6 ) <br />
A <strong>Default</strong> Theory (3)<br />
St<strong>and</strong>ard part <strong>of</strong> all domains are presented here.<br />
A fuent is either true or false in the initial situation<br />
<br />
<br />
<br />
§<br />
$.%<br />
¥<br />
<br />
¡8 £<br />
<br />
<br />
<br />
<br />
<br />
“Commonsense law <strong>of</strong> inertia”—things do not change<br />
unless they are made to<br />
¡ ( £ <br />
<br />
<br />
¡8 £ <br />
¥ §<br />
<br />
<br />
¡ ( £ <br />
<br />
<br />
¡ ( £ <br />
<br />
<br />
<br />
¡8 £ <br />
¥ §<br />
<br />
<br />
<br />
¡ ( £ <br />
<br />
<br />
<br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.7/10
¥<br />
)<br />
(<br />
)<br />
(<br />
,<br />
Applying the Theory<br />
The initial state<br />
<br />
!<br />
¡+<br />
<br />
# *<br />
¥0/<br />
*<br />
¡+<br />
<br />
,!1<br />
A consequence <strong>of</strong> the default theory:<br />
,!-<br />
,!-<br />
¥ £<br />
¡+ <br />
¡<br />
$7 <br />
,!1<br />
¥ £<br />
¡+ <br />
¡<br />
$7 <br />
# £ 6 ) <br />
¡<br />
6 ) <br />
,<br />
<br />
!<br />
<br />
<br />
<br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.8/10
Complexity<br />
the first order default logic is even not<br />
semi-decidable<br />
for the propositional default logic PSPACE<br />
procedures exist<br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.9/10
Homework<br />
Present a comparison <strong>of</strong> all approached covered<br />
during this course<br />
Consider syntax, semantics, expressivity, usability<br />
<strong>and</strong> computational complexity<br />
Summarise the comparison in a table<br />
TDT14 - <strong>Logic</strong> <strong>and</strong> <strong>Planning</strong> – p.10/10
Change language