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| − | This page is to record the current status of discussion
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| − | on the proposed Complexity Category of the Termination Competition.
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| − |
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| − | The first installation of this event is planned for November 1, 2008.
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| − |
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| − | (Discussion should take place on the termtools mailing list.)
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| − |
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| − | == Overview of the Event ==
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| − |
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| − | It is a challenging topic to automatically determine upper bounds on
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| − | the complexity of rewrite systems. By complexity of a TRS, we mean
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| − | the maximal length of derivations, where either no restrictions on the
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| − | initial terms are present ("derivational complexity") or only
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| − | constructor based terms are considered ("runtime complexity"). See
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| − | (Hirokawa, Moser, 2008) for further reading on the notion of runtime
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| − | complexity. Additionally one distinguishes between complexities
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| − | induced by full rewriting as opposed to complexities induced by
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| − | specific strategies, as for example innermost rewriting.
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| − | We propose four sub-categories:
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| − | # Derivational Complexity (DC),
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| − | # innermost Derivational Complexity (iDC),
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| − | # Runtime Complexity (RC), and
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| − | # innermost Runtime Complexity (iRC)
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| − |
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| − | == Syntax/Semantics for Input/Output ==
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| − |
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| − | As competition semantics, we propose to focus on <em>polynomial</em>
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| − | bounds.
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| − |
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| − | === Input Format ===
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| − | Problems will be given in the newly TPDB-format, cf.
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| − | [http://www.termination-portal.org/wiki/XTC_Format_Specification], where
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| − | the XML-element ''problem'' will have the type ''complexity'' given.
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| − | Further, depending on the category DC, iDC, RC and iRC, the attributes
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| − | ''strategy'' and ''startterm'' will be set to FULL/INNERMOST and full/constructor-based
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| − | respectively.
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| − | In particluar, this allows the upload of one single tool for all categories the authors want to participate in.
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| − |
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| − |
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| − | === Output Format ===
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| − | The output format is adapted so that additional
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| − | information on the asymptotic complexity is given for lower as well
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| − | as upper bounds. Hence the output written to the first line of STDOUT
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| − | shall be a complexity statement according to the following grammar:
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| − |
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| − | <pre>
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| − | S -> NO | MAYBE | YES( F, F) | YES( ?, F) | YES( F, ?)
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| − | F -> O(1) | O(n^Nat) | POLY
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| − | </pre>
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| − |
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| − | where "Nat" is a non-zero natural number and YES(F1, F2) means F2 is
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| − | upper bound and that F1 is a lower-bound. "O(n^k)" is the usual big-Oh
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| − | notation and "POLY" indicates an unspecified polynomial. Either of
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| − | the functions F1, F2 (but not both) may be replaced by ``don't know'',
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| − | indicated by ?. Any remaining output on STDOUT will be considered as
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| − | proof output and has to follow the normal rules for the competition.
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| − |
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| − | <em>Example</em>: Consider R= {a(a(x)) -> b(c(x)), b(b(x)) -> a(c(x)), c(c(x)) -> a(b(x))}. Within
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| − | the derivational complexity category a syntactically correct output would be "YES(O(n^2),POLY)".
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| − | (Whether this output would also indicate a correct tool, is another question.)
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| − |
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| − | == Scoring ==
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| − |
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| − | Currently we focus on (polynomial) <em>upper</em> bounds. As
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| − | the output format indicates, this restriction should be lifted
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| − | later, see below. In order to take into account the quality of the upper
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| − | bound provided by the different tools, we propose the following
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| − | scoring algorithm, where we suppose the number of competitors is x.
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| − |
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| − | Firstly, for each TRS the competing tools are ranked, where constant
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| − | complexity, i.e., output "YES(?,O(1))" is best and "MAYBE", "NO" or
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| − | time-out is worst.
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| − | As long as the output is of form "YES(?,O(n^k))" or "YES(?,POLY)" the
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| − | rank of the tool defines the number of points. More precisely the
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| − | best tool gets x+1 points, the second gets x points and so on. On the
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| − | other hand a negative output ("MAYBE", "NO" or time-out) gets 0
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| − | points.
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| − | If two or more tools would get the same rank, the rank of the
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| − | remaining tools is adapted in the usual way.
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| − |
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| − | Secondly, all resulting points for all considered systems are summed
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| − | up and the contestant with the highest number of points wins. If this
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| − | cannot establish a winner, the total number of wins is counted. If
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| − | this still doesn't produce a winner, we give up and provide two (or
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| − | more) winners.
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| − |
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| − | The maximal allowed CPU time is 60 seconds.
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| − |
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| − | == Problem selection ==
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| − |
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| − | We propose to run each complexity-subcategory
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| − | on all TRS and SRS families from the newly organised TPDB, after
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| − | the following selection function defined below has been applied.
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| − |
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| − | === Selection function ===
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| − |
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| − | In the following, we denote by ''select'' the function that relates
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| − | each family from the TPDB to the number of randomly chosen examples within this family as defined
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| − | for the termination competition.
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| − | The idea is to make ''select''
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| − | aware of different difficulties of proving complexity bounds. We do so by
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| − | # 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'':
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| − | #:* '''subcategories RC, iRC and iDC:''' we propose to partition each family into
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| − | #:*:(i) those upon which a polynomial bound could be shown automatically in last years competition (denoted by ''F_auto'' below) and
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| − | #:*:(ii) those where a polynomial bound could not be shown (''F_nonauto'').
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| − | #:* '''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)
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| − | # 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:
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| − | #: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''.
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| − | # From each partition ''F_i'' of a family ''F'', we randomly select ''select_comp(F,i)'' examples.
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| − |
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| − | == Test Cases ==
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| − | In the following test cases we restrict to full rewriting.
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| − | <em>
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| − | test cases - derivational complexity
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| − | </em>
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| − |
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| | <pre> | | <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))"
| + | This page is no longer maintained. |
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| − | 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(?,?)"
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| − | | |
| − | R= {+(s(x),+(y,z)) -> +(x,+(s(s(y)),z)), +(s(x),+(y,+(z,w))) -> +(x,+(z,+(y,w)))}, expected output "YES(?,?)"
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| | </pre> | | </pre> |
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| − | <em>test cases - runtime complexity </em>
| + | The entry point to the complexity category is here: [http://cl-informatik.uibk.ac.at/users/georg/cbr/competition/ complexity competition]. |
| − | <pre>
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| − | R = {a(b(x)) -> b(b(a(x)))}, expected output "YES(?,O(n^1))" or "YES(O(n^1),O(n^1))"
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| − | | |
| − | 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))"
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| − | | |
| − | 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))"
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| − | | |
| − | R= {f(0) -> c, f(s(x)) -> c(f(x),f(x))}, expected output "YES(?,?)"
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| − | </pre>
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| − | | |
| − | In the following test cases we restrict to innermost rewriting.
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| − | | |
| − | <em>test cases - derivational complexity </em>
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| − | <pre>
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| − | R = {f(x) -> c(x,x)}, expected output "YES(O(n^1),O(n^1))" or "YES(?,O(n^1))"
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| − | </pre>
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| − | | |
| − | <em>test cases - runtime complexity </em>
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| − | <pre>
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| − | 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))"
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| − | </pre>
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| − | | |
| − | == Wishlist ==
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| − | *
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| − | * assessment of lower bounds:<br>
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| − | In the future the tools should also be able to provide certificates on the
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| − | lower bound. This would imply to extend the grammar as follows
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| − | | |
| − | <pre>
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| − | F -> O(1) | O(n^Nat) | POLY | EXP | INF
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| − | </pre>
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| − | | |
| − | such that e.g. "YES(EXP,?)" indicated an exponential lower-bound,
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| − | or "YES(INF,INF)" indicated non-termination.
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| − | * as for the upper bound the lower bound certificate should be ranked and
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| − | both ranks could be compared lexicographically
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| − | | |
| − | == Questions ==
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| − | | |
| − | * JW would prefer the following output format as it is easier to parse:
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| − | | |
| − | F -> POLY(Nat) | POLY(?)
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| − | | |
| − | Here "POLY(k)" abbreviates "O(n^k)" and "POLY(?)" denotes an unspecified
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| − | polynomial.<br>
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| − | | |
| − | <em>resolved</em>
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| − | | |
| − | == Participation ==
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| − | | |
| − | === Requirements ===
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| − | In order to participate in the competition, the '''sources''' of your tool has to be '''publicly available'''.
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| − | | |
| − | === Participants ===
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| − | | |
| − | Insert your name here if you intend to participate:
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| − | | |
| − | ==== Competition 2008 ====
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| − | * Johannes Waldmann (Matchbox), but will need more time (December 2008)
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| − | * M. Avanzini, G. Moser, A. Schnabl (TCT)
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| − | * N. Hirokawa (Hydra), but might need more time
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| − | * M. Korp, C. Sternagel, H. Zankl (CaT)
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| − | | |
| − | ==== Competition 2009 ====
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| − | * M. Avanzini, G. Moser, A. Schnabl (TCT)
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