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The entry point to the complexity category is here: [http://cl-informatik.uibk.ac.at/users/georg/cbr/competition/ complexity competition].
 
The entry point to the complexity category is here: [http://cl-informatik.uibk.ac.at/users/georg/cbr/competition/ complexity competition].
 
== Overview of the Event ==
 
 
It is a  challenging topic to automatically determine  upper bounds on
 
the complexity  of rewrite systems.  By  complexity of a  TRS, we mean
 
the maximal length of derivations, where either no restrictions on the
 
initial  terms  are  present  ("derivational  complexity")  or  only
 
constructor  based terms are  considered ("runtime  complexity").  See
 
(Hirokawa, Moser, 2008)  for further reading on the  notion of runtime
 
complexity.  Additionally  one  distinguishes  between  complexities
 
induced  by  full rewriting  as  opposed  to  complexities induced  by
 
specific strategies, as for example innermost rewriting.
 
We  propose four sub-categories:
 
# Derivational Complexity (DC),
 
# innermost Derivational Complexity (iDC),
 
# Runtime Complexity (RC), and
 
# innermost Runtime Complexity (iRC)
 
 
== Syntax/Semantics for Input/Output ==
 
 
As  competition  semantics,  we  propose  to  focus  on <em>polynomial</em>
 
bounds.
 
 
=== Input Format ===
 
Problems will be given in the newly TPDB-format, cf.
 
[http://www.termination-portal.org/wiki/XTC_Format_Specification], where
 
the XML-element ''problem'' will have the type ''complexity'' given.
 
Further, depending on the category DC, iDC, RC and iRC, the attributes
 
''strategy'' and ''startterm'' will be set to FULL/INNERMOST and full/constructor-based
 
respectively. 
 
In particluar, this allows the upload of one single tool for all categories the authors want to participate in.
 
 
 
=== Output Format ===
 
The output  format is  adapted so  that additional
 
information on the  asymptotic complexity is given for  lower as well
 
as upper bounds.  Hence the output written to the first line of STDOUT
 
shall be a complexity statement according to the following grammar:
 
 
<pre>
 
S -> NO | MAYBE | YES( F, F) | YES( ?, F) | YES( F, ?)
 
F -> O(1) | O(n^Nat) | POLY
 
</pre>
 
 
where "Nat" is  a non-zero natural number and YES(F1,  F2) means F2 is
 
upper bound and that F1 is a lower-bound. "O(n^k)" is the usual big-Oh
 
notation and  "POLY" indicates  an unspecified polynomial.  Either of
 
the functions F1, F2 (but not both) may be replaced by ``don't know'',
 
indicated by ?.  Any remaining  output on STDOUT will be considered as
 
proof output and has to follow the normal rules for the competition.
 
 
<em>Example</em>: Consider R= {a(a(x)) -> b(c(x)), b(b(x)) -> a(c(x)), c(c(x)) -> a(b(x))}. Within
 
the derivational complexity category a syntactically correct output would be "YES(O(n^2),POLY)".
 
(Whether this output would also indicate a correct tool, is another question.)
 
 
== Scoring ==
 
 
Currently we focus on (polynomial) <em>upper</em> bounds.  As
 
the output format indicates, this restriction should be lifted
 
later, see below.  In order to take  into account the quality of the upper
 
bound  provided  by the  different  tools,  we  propose the  following
 
scoring algorithm, where we suppose the number of competitors is x.
 
 
Firstly, for each  TRS the competing tools are  ranked, where constant
 
complexity, i.e., output "YES(?,O(1))" is best and "MAYBE", "NO" or
 
time-out is worst.
 
As long as the output  is of form "YES(?,O(n^k))" or "YES(?,POLY)" the
 
rank of  the tool  defines the number  of points.  More  precisely the
 
best tool gets x+1 points, the second gets x points and so on.  On the
 
other  hand a  negative  output  ("MAYBE", "NO"  or  time-out) gets  0
 
points.
 
If  two or  more  tools  would get  the  same rank,  the  rank of  the
 
remaining tools is adapted in the usual way.
 
 
Secondly, all  resulting points for all considered  systems are summed
 
up and the contestant with the  highest number of points wins. If this
 
cannot establish  a winner, the total  number of wins  is counted.  If
 
this still  doesn't produce a winner,  we give up and  provide two (or
 
more) winners.
 
 
The maximal allowed CPU time is 60 seconds.
 
 
== Problem selection ==
 
 
We propose to run subcategories DC and iDC
 
on all TRS and SRS families from the newly organised TPDB, after
 
the selection function defined below has been applied.
 
For categories RC and iRC, we propose to run the competition on all TRS families
 
after application of the selection function stated below:
 
 
=== Selection function ===
 
 
In the following, we denote by ''select'' the function that relates
 
each family from the TPDB to the number of randomly chosen examples within this family as defined
 
for the termination competition. 
 
The idea is to make ''select''
 
aware of different difficulties of proving complexity bounds. We do so by
 
# partitioning each family ''F'' into ''n'' different sets ''F = F_1 \cup ... \cup F_n'', where the sets ''F_i'' may be seen as collections of TRSs similar in difficulty. For this years competition we propose following partitioning of a family ''F'':
 
#:* '''subcategories RC, iRC and iDC:''' we propose to partition each family into
 
#:*:(i) those upon which a polynomial bound could be shown automatically in last years competition (denoted by ''F_auto'' below) and
 
#:*:(ii) those where a polynomial bound could not be shown (''F_nonauto'').
 
#:* '''subcategory DC:''' as above, but we split (ii) into duplicating TRS (''F_duplicating'') and non-duplicating TRSs (note that any TRS from (i) is non-duplicating)
 
# In accordance to the above described partitioning, we define a probability distribution ''p'' on ''F'' such that ''p(F_1) + ... p(F_n) = 1''. For this year's competition we propose the following distribution:
 
#:for all subcategories and families ''F'', we propose ''p(F_auto) = 0.4'' and ''p(F_nonauto) = 0.6'' (For the category DC, we additionally set ''p(F_duplicating) = 0.0''). That is, we want to consider 40% examples that could be solved automatically in last years competition, and 60% of examples that could not be solved automatically. Additionally for DC we want to exclude duplicating TRS as those admit exponential derivational complexity. Based on the probability distribution ''p'' we define the extended selection function ''select_comp(F,i) = min(|F_i|, p(i) * select(F))''. Here ''|F_i|'' denotes the size of ''F_i''.
 
# From each partition ''F_i'' of a family ''F'', we randomly select ''select_comp(F,i)'' examples.
 
 
== Test Cases ==
 
In the following test cases we restrict to full rewriting.
 
<em>
 
test cases - derivational complexity
 
</em>
 
 
<pre>
 
R = {a(b(x)) -> b(a(x))}, expected output "YES(?,O(n^2))" or "YES(O(n^1),O(n^2))" or "YES(O(n^2),O(n^2))"
 
 
R= {a(a(x)) -> b(c(x)), b(b(x)) -> a(c(x)), c(c(x)) -> a(b(x))}, expected output "YES(O(n^2),?)" or "YES(?,?)"
 
 
R= {+(s(x),+(y,z)) -> +(x,+(s(s(y)),z)), +(s(x),+(y,+(z,w))) -> +(x,+(z,+(y,w)))}, expected output "YES(?,?)"
 
</pre>
 
 
<em>test cases - runtime complexity </em>
 
<pre>
 
R = {a(b(x)) -> b(b(a(x)))}, expected output "YES(?,O(n^1))" or "YES(O(n^1),O(n^1))"
 
 
R = {plus(0,y) -> y, plus(s(x),y) -> s(plus(x,y)), mul(0,y) -> 0, mul(s(x),y) -> plus(mul(x,y),y)}, expected output "YES(?,O(n^2))" or "YES(O(n^1),O(n^2))" or "YES(O(n^2),O(n^2))"
 
 
R = {f(x,0) -> s(0), f(s(x),s(y)) -> s(f(x,y)), g(0,x) -> g(f(x,x),x)}, expected output "YES(?,O(n^1))" or "YES(O(n^1),O(n^1))"
 
 
R= {f(0) -> c, f(s(x)) -> c(f(x),f(x))}, expected output "YES(?,?)"
 
</pre>
 
 
In the following test cases we restrict to innermost rewriting.
 
 
<em>test cases - derivational complexity </em>
 
<pre>
 
R = {f(x) -> c(x,x)}, expected output "YES(O(n^1),O(n^1))" or "YES(?,O(n^1))"
 
</pre>
 
 
<em>test cases - runtime complexity </em>
 
<pre>
 
R= {f(x) -> c(x,x), g(0) -> 0, g(s(x)) -> f(g(x))}, expected output "YES(O(n^1),O(n^1))" or "YES(?,O(n^1))"
 
</pre>
 
 
 
== Participation ==
 
 
=== Requirements ===
 
In order to participate in the competition, the '''sources''' of your tool have to be '''publicly available'''.
 
 
=== Participants ===
 
 
Insert your name here if you intend to participate:
 
 
==== Competition 2009 ====
 
* M. Avanzini, G. Moser, A. Schnabl ([http://cl-informatik.uibk.ac.at/software/tct TCT])
 
* J. Waldmann ([[Tools:Matchbox]]) (derivational complexity for full rewriting)
 
 
==== Competition 2008 ====
 
* M. Avanzini, G. Moser, A. Schnabl (TCT)
 
* M. Korp, C. Sternagel, H. Zankl (CaT)
 
 
== Complexity proof techniques ==
 
 
A list of the complexity proof techniques applied by the participants of the competition can be found at [[Complexity_Techniques]]
 
 
== Discussion ==
 
 
=== lower bounds ===
 
 
In the future the tools should also be able to provide certificates on the
 
lower bound. This would imply to extend the grammar as follows
 
 
<pre>
 
F -> O(1) | O(n^Nat) | POLY | EXP | INF
 
</pre>
 
 
such that e.g. "YES(EXP,?)" indicated an exponential lower-bound,
 
or "YES(INF,INF)" indicated non-termination.
 
 
(JW: I don't like the looks of an answer starting "YES" and indicating non-termination. See "BOUNDS" proposal below.)
 
 
Scoring (proposals):
 
 
* as for the upper bound the lower bound certificate should be ranked and both ranks could be compared lexicographically (with the upper bound as the primary criterion)
 
 
* JW prefers: don't define some artificial total order on the bounds. The natural partial ordering is given by the inclusion relation on the sets of functions that are described by the bounds. This inclusion can be computed from
 
<PRE>
 
(low1, up1) "is better than" (low2, up2)  iff  low1 >= low2 and up1 <= up2
 
</PRE>Then for each problem, answer A gets awarded k points if A is strictly better than k of the answers, where "no answer" counts as BOUNDS(LIN,INF), and "strictly better = better and not equal".
 
This would imply that if all answers are identical, then no-one gets a point.
 
Perhaps we want to add one virtual prover that always says"BOUNDS(LIN,INF)" -
 
so anyone who gives a better answer, gets at least one point.
 
 
=== Concrete syntax ===
 
 
* JW would prefer the following output format as it is easier to parse:
 
 
F -> POLY(Nat) | POLY(?)
 
 
Here "POLY(k)" abbreviates "O(n^k)" and "POLY(?)" denotes an unspecified
 
polynomial.<br>
 
 
<em>resolved</em>
 
 
* JW: I'm not giving up ... one more reason against the O(n^k) syntax: 3. it cannot be used for lower bounds, as we would need Omega instead of Oh. (The other two reasons are: 2. needlessly complicated, and 1. n is an undefined variable)
 
 
* proposal to replace YES/NO/MAYBE by BOUNDS: http://dev.aspsimon.org/bugzilla/show_bug.cgi?id=85#c4
 
 
LN: I'd like to support this notation. But I think "?" for an unknown bound is unnecessary. It can always be replaced by POLY(0) for the lower
 
bound and INF for the upper bound. [[User:Noschinski|Noschinski]] 13:26, 13 February 2010 (UTC)
 
 
[[Category:Categories]]
 

Latest revision as of 12:05, 8 June 2012

This page is no longer maintained. 

The entry point to the complexity category is here: complexity competition.