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Added figures to ONS

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Joshua Moerman 2018-10-01 18:40:45 +02:00
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biblio.bib
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@ -889,7 +889,7 @@
author = {Gian Luigi Ferrari and author = {Gian Luigi Ferrari and
Ugo Montanari and Ugo Montanari and
Emilio Tuosto}, Emilio Tuosto},
title = {Coalgebraic minimization of HD-automata for the Pi-calculus using title = {Coalgebraic minimization of {HD}-automata for the $\pi$-calculus using
polymorphic types}, polymorphic types},
journal = {Theor. Comput. Sci.}, journal = {Theor. Comput. Sci.},
volume = {331}, volume = {331},

247
content/ordered-nominal-sets.tex
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@ -379,19 +379,19 @@ Combining this, we find that the orbit is $\{(a,(a,b)) \mid a,b \in \Q, a < b\}$
We summarise our concrete representation in the following table. We summarise our concrete representation in the following table.
\in{Propositions}[thm:pres-nom], \in{}[eqimap_rep] and \in{}[orbstring] correspond to the three rows in the table. \in{Propositions}[thm:pres-nom], \in{}[eqimap_rep] and \in{}[orbstring] correspond to the three rows in the table.
\todo{Table} \placetable[here][tab:overview]{Overview of representation.}
% \starttabular}{l|p{5.7cm} \starttabulate[|l|p|]
% \emph{Object} & \emph{Representation} \\ \NC \emph{Object} \NC \emph{Representation} \NC\NR
% \hline \HL %----------------------------------------
% Single orbit $O$ & Natural number $n = \dim(O)$ \\ \NC Single orbit $O$ \NC Natural number $n = \dim(O)$ \NC\NR
% Nominal set $X = \bigcup_i O_i$ & Multiset of these numbers \\ \NC Nominal set $X = \bigcup_i O_i$ \NC Multiset of these numbers \NC\NR
% \hline \HL %----------------------------------------
% Map from single orbit $f \colon O \to Y$ & The orbit $f(O)$ and a bit string $F$ \\ \NC Map from single orbit $f \colon O \to Y$ \NC The orbit $f(O)$ and a bit string $F$ \NC\NR
% Equivariant map $f \colon X \to Y$ & Set of tuples $(O, F, f(O))$, one for each orbit \\ \NC Equivariant map $f \colon X \to Y$ \NC Set of tuples $(O, F, f(O))$, one for each orbit \NC\NR
% \hline \HL %----------------------------------------
% Orbit in a product $O \subseteq X \times Y$ & The corresponding orbits of $X$ and $Y$, and a string $P$ relating their supports \\ \NC Orbit in a product $O \subseteq X \times Y$ \NC The corresponding orbits of $X$ and $Y$, and a string $P$ relating their supports \NC\NR
% Product $X \times Y$ & Set of tuples $(P, O_X, O_Y)$, one for each orbit \\ \NC Product $X \times Y$ \NC Set of tuples $(P, O_X, O_Y)$, one for each orbit \NC\NR
% \stoptabular \stoptabulate
Notice that in the case of maps and products, the orbits are inductively represented using the concrete representation. Notice that in the case of maps and products, the orbits are inductively represented using the concrete representation.
As a base case we can represent single orbits by their dimension. As a base case we can represent single orbits by their dimension.
@ -404,7 +404,7 @@ As a base case we can represent single orbits by their dimension.
reference=sec:impl] reference=sec:impl]
The ideas outlined above have been implemented in the \cpp{} library \ONS{} The ideas outlined above have been implemented in the \cpp{} library \ONS{}
\footnote{\ONS{} can be found at \todo{https://github.com/davidv1992/ONS}.} \footnote{\ONS{} can be found at \href{https://github.com/davidv1992/ONS}.}
The library can represent orbit-finite nominal sets and their products, (disjoint) unions, and maps. The library can represent orbit-finite nominal sets and their products, (disjoint) unions, and maps.
A full description of the possibilities is given in the documentation included with \ONS{}. A full description of the possibilities is given in the documentation included with \ONS{}.
@ -463,20 +463,18 @@ Recall that $\Nsize(X)$ denotes the number of orbits of $X$.
We use $p$ and $f$ to denote functions implemented in whatever way the user wants, which we assume to take $O(1)$ time. We use $p$ and $f$ to denote functions implemented in whatever way the user wants, which we assume to take $O(1)$ time.
The software assumes these are equivariant, but this is not verified. The software assumes these are equivariant, but this is not verified.
\todo{Table} \placetable[here][tab:complexity]{Time complexity of operations on nominal sets.}
% \startcenter \starttabulate[|r|l|]
% \starttabular}{r|l \NC \emph{Operation} \VL \emph{Complexity} \NC\NR
% Operation & Complexity\\ \HL
% \hline \NC Test $x \in X$ \VL $O(\log \Nsize(X))$ \NC\NR
% Test $x \in X$ & $O(\log \Nsize(X))$\\ \NC Test $X\subseteq Y$ \VL $O(\min(\Nsize(X)+\Nsize(Y), \Nsize(X)\log \Nsize(Y)))$ \NC\NR
% Test $X\subseteq Y$ & $O(\min(\Nsize(X)+\Nsize(Y), \Nsize(X)\log \Nsize(Y)))$\\ \NC Calculate $X \cup Y$ \VL $O(\Nsize(X)+\Nsize(Y))$ \NC\NR
% Calculate $X \cup Y$ & $O(\Nsize(X)+\Nsize(Y))$\\ \NC Calculate $X \cap Y$ \VL $O(\Nsize(X)+\Nsize(Y))$ \NC\NR
% Calculate $X \cap Y$ & $O(\Nsize(X)+\Nsize(Y))$\\ \NC Calculate $\{x \in X \mid p(x)\}$ \VL $O(\Nsize(X))$ \NC\NR
% Calculate $\{x \in X \mid p(x)\}$ & $O(\Nsize(X))$\\ \NC Calculate $\{f(x) \mid x \in X\}$ \VL $O(\Nsize(X)\log \Nsize(X))$ \NC\NR
% Calculate $\{f(x) \mid x \in X\}$ & $O(\Nsize(X)\log \Nsize(X))$\\ \NC Calculate $X\times Y$ \VL $O(\Nsize(X\times Y)) \,\subseteq\, O(3^{\dim(X)+\dim(Y)}\Nsize(X)\Nsize(Y))$ \NC\NR
% Calculate $X\times Y$ & $O(\Nsize(X\times Y)) \,\subseteq\, O(3^{\dim(X)+\dim(Y)}\Nsize(X)\Nsize(Y))$\\ \stoptabulate
% \stoptabular
% \stopcenter
\stoptheorem \stoptheorem
\startproof \startproof
@ -535,27 +533,25 @@ Such a transition structure is made precise by the notion of nominal automata.
\startplacefigure \startplacefigure
[title={Example automaton that accepts the language $\Lint$.}, [title={Example automaton that accepts the language $\Lint$.},
reference=fig:lint-minimal] reference=fig:lint-minimal]
\todo{Plaatje} \starttikzpicture[node distance=3.5cm]
% \starttikzpicture \node (S 0) [state] {$q_0$};
% \node (S 0) at (0,0) [state, minimum size=1.5cm] {$q_0$}; \node (S 1) [state, right of=S 0] {$q_1(a)$};
% \node (S 1) at (3,0) [state, minimum size=1.5cm] {$q_1(a)$}; \node (S 2) [state, accepting, right of=S 1] {$q_2(a,b)$};
% \node (S 2) at (7,0) [state, minimum size=1.5cm, accepting] {$q_2(a,b)$}; \node (S 3) [state, right of=S 2] {$q_3(a,b)$};
% \node (S 3) at (11,0) [state, minimum size=1.5cm] {$q_3(a,b)$}; \node (S 4) [state, below=1.5cm of S 2] {$q_4$};
% \node (S 4) at (7,-2.5) [state, minimum size=1.5cm] {$q_4$};
% \node (S 5) at (11,-2.5) [state,minimum size=1.5cm,draw=none] {};
% \path \path[->]
% (S 0) edge node[above] {$a$} (S 1) (S 0) edge node[above] {$a$} (S 1)
% (S 1) edge node[above] {$b > a$} (S 2) (S 1) edge node[above] {$b > a$} (S 2)
% (S 1) edge[bend right] node[below left] {$b \le a$} (S 4) (S 1) edge[bend right] node[below left] {$b \le a$} (S 4)
% (S 2) edge[bend left=10] node[above,align=center] {$a<c<b$\\$a\leftarrow c$} (S 3) (S 2) edge[bend left=10] node[above,align=center] {$a<c<b$\\$a \leftarrow c$} (S 3)
% (S 3) edge[bend left=10] node[below,align=center] {$a<c<b$\\$b\leftarrow c$} (S 2) (S 3) edge[bend left=10] node[below,align=center] {$a<c<b$\\$b \leftarrow c$} (S 2)
% (S 2) edge[bend right=15] node[left] {$c\le a$} (S 4) (S 2) edge[bend right=10] node[left] {$c \le a$} (S 4)
% (S 2) edge[bend left=15] node[right] {$c\ge b$} (S 4) (S 2) edge[bend left=10] node[right] {$c \ge b$} (S 4)
% (S 3) edge[bend left] node[above left] {$c\le a$} (S 4) (S 3) edge[bend left] node[above left] {$c \le a$} (S 4)
% (S 3) edge[bend left=60] node[below right] {$c\ge b$} (S 4); (S 3) edge[bend left=60] node[below right] {$c \ge b$} (S 4)
% \draw[-{Latex},rounded corners=5] (S 4.south west) -- ([yshift=-.5cm]S 4.south west) -- node[midway,below]{$a$} ([yshift=-.5cm]S 4.south east) -- (S 4.south east); (S 4) edge[loop below] node [below] {$a$} (S 4);
% \stoptikzpicture \stoptikzpicture
\stopplacefigure \stopplacefigure
\startdefinition \startdefinition
@ -636,22 +632,27 @@ We say an automaton is \emph{minimal} if its set of states has the least number
Minimal deterministic automata are unique up to isomorphism.} Minimal deterministic automata are unique up to isomorphism.}
We generalise Moore's minimisation algorithm to nominal DFAs (\in{Algorithm}[alg:moore]) and analyse its time complexity using the bounds from \in{Section}[sec:impl]. We generalise Moore's minimisation algorithm to nominal DFAs (\in{Algorithm}[alg:moore]) and analyse its time complexity using the bounds from \in{Section}[sec:impl].
\todo{algoritme} \startplacealgorithm
% \startalgorithm [title={Moore's minimisation algorithm for nominal DFAs.},
% \caption{Moore's minimisation algorithm for nominal DFAs}\label{alg:moore} reference=alg:moore]
% \startalgorithmic[1] \startalgorithmic
% \Require{ Nominal automaton $(S,A,F,\delta)$.} \REQUIRE{Nominal automaton $M = (S,A,F,\delta)$}
% \State{$i \leftarrow 0$, ${\equiv_{-1}} \leftarrow S\times S$, ${\equiv_{0}} \leftarrow F\times F \cup (S\backslash F)\times (S\backslash F)$} \ENSURE{Minimal nominal automaton equivalent to $M$}
% \While{${\equiv_i} \ne {\equiv_{i-1}}$} \startlinenumbering
% \State{${\equiv_{i+1}} = \{(q_1, q_2) \mid (q_1, q_2) \in {\equiv_i} \wedge \forall a\in A, (\delta(q_1,a), \delta(q_2,a))\in {\equiv_i} \}$} \STATE $i \gets 0$
% \State{$i \leftarrow i+1$} \STATE ${\equiv_{-1}} \gets S \times S$
% \EndWhile \STATE ${\equiv_{0}} \gets F \times F \cup (S\backslash F)\times (S\backslash F)$
% \State{$E\leftarrow S/_{\equiv_i}$} \WHILE{${\equiv_i} \ne {\equiv_{i-1}}$\startline[line:loop]}
% \State{$F_E \leftarrow \{e\in E \mid \forall s\in e, s\in F\}$} \STATE ${\equiv_{i+1}} \gets \{ (q_1, q_2) \mid (q_1, q_2) \in {\equiv_i} \wedge \forall a \in A, (\delta(q_1, a), \delta(q_2, a)) \in {\equiv_i} \}$\someline[line:main-work]
% \State{Let $\delta_E$ be the map such that, if $s\in e$ and $\delta(s,a)\in e'$, then $\delta_E(e,a) = e'$.} \STATE $i \gets i+1$
% \State{\Return{$(E,A,F_E,\delta_E)$.}} \ENDWHILE\stopline[line:loop]
% \stopalgorithmic \STATE $E \gets S/_{\equiv_i}$
% \stopalgorithm \STATE $F_E \gets \{e \in E \mid \forall s \in e, s \in F\}$
\STATE Let $\delta_E$ be the map such that, if $s \in e$ and $\delta(s, a) \in e'$, then $\delta_E(e, a) = e'$
\RETURN $(E,A,F_E,\delta_E)$
\stoplinenumbering
\stopalgorithmic
\stopplacealgorithm
\starttheorem[reference=thm:moore] \starttheorem[reference=thm:moore]
The runtime complexity of Moore's algorithm on nominal deterministic automata is $O(3^{5k} k \Nsize(S)^3 \Nsize(A))$, where $k=\dim(S\cup A)$. The runtime complexity of Moore's algorithm on nominal deterministic automata is $O(3^{5k} k \Nsize(S)^3 \Nsize(A))$, where $k=\dim(S\cup A)$.
@ -659,8 +660,8 @@ The runtime complexity of Moore's algorithm on nominal deterministic automata is
\startproof \startproof
This is shown by counting operations, using the complexity results of set operations stated in \in{Theorem}[thmcomplexity]. This is shown by counting operations, using the complexity results of set operations stated in \in{Theorem}[thmcomplexity].
We first focus on the while loop on \todo{lines 2 through 5}. We first focus on the while loop on \inline[line:loop].
The runtime of an iteration of the loop is determined by \todo{line 3}, as this is the most expensive step. The runtime of an iteration of the loop is determined by \inline[line:main-work], as this is the most expensive step.
Since the dimensions of $S$ and $A$ are at most $k$, computing $S \times S \times A$ takes $O(\Nsize(S)^2 \Nsize(A) 3^{5k})$. Since the dimensions of $S$ and $A$ are at most $k$, computing $S \times S \times A$ takes $O(\Nsize(S)^2 \Nsize(A) 3^{5k})$.
Filtering $S \times S$ using that then takes $O(\Nsize(S)^2 3^{2k})$. Filtering $S \times S$ using that then takes $O(\Nsize(S)^2 3^{2k})$.
The time to compute $S \times S \times A$ dominates, hence each iteration of the loop takes $O(\Nsize(S)^2 \Nsize(A) 3^{5k})$. The time to compute $S \times S \times A$ dominates, hence each iteration of the loop takes $O(\Nsize(S)^2 \Nsize(A) 3^{5k})$.
@ -702,38 +703,33 @@ The output of these programs was manually checked to see if the minimisation was
The results (shown in \in{Table}[minimize_results]) for random automata show a clear advantage for \ONS{}, which is capable of running all supplied testcases in less than one second. The results (shown in \in{Table}[minimize_results]) for random automata show a clear advantage for \ONS{}, which is capable of running all supplied testcases in less than one second.
This in contrast to both \LOIS{} and \NLambda{}, which take more than $2$ hours on the largest random automata. This in contrast to both \LOIS{} and \NLambda{}, which take more than $2$ hours on the largest random automata.
\todo{Table} \placetable[here][minimize_results]
% \starttable {Running times for \in{Algorithm}[alg:moore] implemented in the three libraries.
% \centering $N(S)$ is the size of the input and $N(S^{\text{min}})$ the size of the minimal automaton.
% \starttabular}{l|l|l|r|r|r|r For \ONS{}, the time used to generate the automaton is reported separately (in grey).}
% Type & $\Nsize(S)$ & $\Nsize(S^\text{min})$ & \multicolumn{2}{l|}{\ONS{}} & \NLambda{} & \LOIS{} \\ \starttabulate[|l|l|l|r|r|r|r|][distance=none]
% & & & & \color{black!70} Gen. & &\\ \NC Type \NC $\Nsize(S)$ \NC $\Nsize(S^{\text{min}})$ \VL \ONS{} \NC \color[darkgray]{(Gen.)} \VL \NLambda{} \VL \LOIS{} \NC\NR
% \hline \HL
% rand$_{5,1}$ (x10) & $5$ & n/a & $0.02$s & \color{black!70} n/a & $0.82$s & $3.14$s \\ \NC rand$_{5,1}$ (x10) \NC $5$ \NC n/a \VL $0.02$s \NC \color[darkgray]{n/a} \VL $0.82$s \VL $3.14$s \NC\NR
% rand$_{10,1}$ (x10) & $10$ & n/a & $0.03$s & \color{black!70} n/a & $17.03$s & $1$m $32$s \\ \NC rand$_{10,1}$ (x10) \NC $10$ \NC n/a \VL $0.03$s \NC \color[darkgray]{n/a} \VL $17.03$s \VL $1$m $32$s \NC\NR
% rand$_{10,2}$ (x10) & $10$ & n/a & $0.09$s & \color{black!70} n/a & $35$m $14$s & $> 60$m \\ \NC rand$_{10,2}$ (x10) \NC $10$ \NC n/a \VL $0.09$s \NC \color[darkgray]{n/a} \VL $35$m $14$s \VL $> 60$m \NC\NR
% rand$_{15,1}$ (x10) & $15$ & n/a & $0.04$s & \color{black!70} n/a & $1$m $27$s & $10$m $20$s \\ \NC rand$_{15,1}$ (x10) \NC $15$ \NC n/a \VL $0.04$s \NC \color[darkgray]{n/a} \VL $1$m $27$s \VL $10$m $20$s \NC\NR
% rand$_{15,2}$ (x10) & $15$ & n/a & $0.11$s & \color{black!70} n/a & $55$m $46$s & $> 60$m \\ \NC rand$_{15,2}$ (x10) \NC $15$ \NC n/a \VL $0.11$s \NC \color[darkgray]{n/a} \VL $55$m $46$s \VL $> 60$m \NC\NR
% rand$_{15,3}$ (x10) & $15$ & n/a & $0.46$s & \color{black!70} n/a & $> 60$m & $> 60$m \\ \NC rand$_{15,3}$ (x10) \NC $15$ \NC n/a \VL $0.46$s \NC \color[darkgray]{n/a} \VL $> 60$m \VL $> 60$m \NC\NR
% \hline \HL
% FIFO($2$) & $13$ & $6$ & $0.01$s & \color{black!70} $0.01$s & $1.37$s & $0.24$s \\ \NC FIFO($2$) \NC $13$ \NC $6$ \VL $0.01$s \NC \color[darkgray]{$0.01$s} \VL $1.37$s \VL $0.24$s \NC\NR
% FIFO($3$) & $65$ & $19$ & $0.38$s & \color{black!70} $0.09$s & $11.59$s & $2.4$s \\ \NC FIFO($3$) \NC $65$ \NC $19$ \VL $0.38$s \NC \color[darkgray]{$0.09$s} \VL $11.59$s \VL $2.4$s \NC\NR
% FIFO($4$) & $440$ & $94$ & $39.11$s & \color{black!70} $1.60$s & $1$m $16$s & $14.95$s \\ \NC FIFO($4$) \NC $440$ \NC $94$ \VL $39.11$s \NC \color[darkgray]{$1.60$s} \VL $1$m $16$s \VL $14.95$s \NC\NR
% FIFO($5$) & $3686$ & $635$ & $> 60$m & \color{black!70} $39.78$s & $6$m $42$s & $1$m $11$s \\ \NC FIFO($5$) \NC $3686$ \NC $635$ \VL $> 60$m \NC \color[darkgray]{$39.78$s} \VL $6$m $42$s \VL $1$m $11$s \NC\NR
% \hline \HL
% $ww(2)$ & $8$ & $8$ & $0.00$s & \color{black!70} $0.00$s & $0.14$s & $0.03$s \\ \NC $ww(2)$ \NC $8$ \NC $8$ \VL $0.00$s \NC \color[darkgray]{$0.00$s} \VL $0.14$s \VL $0.03$s \NC\NR
% $ww(3)$ & $24$ & $24$ & $0.19$s & \color{black!70} $0.02$s & $0.88$s & $0.16$s \\ \NC $ww(3)$ \NC $24$ \NC $24$ \VL $0.19$s \NC \color[darkgray]{$0.02$s} \VL $0.88$s \VL $0.16$s \NC\NR
% $ww(4)$ & $112$ & $112$ & $26.44$s & \color{black!70} $0.25$s & $3.41$s & $0.61$s \\ \NC $ww(4)$ \NC $112$ \NC $112$ \VL $26.44$s \NC \color[darkgray]{$0.25$s} \VL $3.41$s \VL $0.61$s \NC\NR
% $ww(5)$ & $728$ & $728$ & $> 60$m & \color{black!70} $6.37$s & $10.54$s & $1.80$s \\ \NC $ww(5)$ \NC $728$ \NC $728$ \VL $> 60$m \NC \color[darkgray]{$6.37$s} \VL $10.54$s \VL $1.80$s \NC\NR
% \hline \HL
% $\Lmax$ & $5$ & $3$ & $0.00$s & \color{black!70} $0.00$s & $2.06$s & $0.03$s \\ \NC $\Lmax$ \NC $5$ \NC $3$ \VL $0.00$s \NC \color[darkgray]{$0.00$s} \VL $2.06$s \VL $0.03$s \NC\NR
% $\Lint$ & $5$ & $5$ & $0.00$s & \color{black!70} $0.00$s & $1.55$s & $0.03$s \NC $\Lint$ \NC $5$ \NC $5$ \VL $0.00$s \NC \color[darkgray]{$0.00$s} \VL $1.55$s \VL $0.03$s \NC\NR
% \stoptabular \stoptabulate
% \caption{Running times for \in{Algorithm}[alg:moore] implemented in the three libraries.
% $N(S)$ is the size of the input and $N(S^\text{min})$ the size of the minimal automaton.
% For \ONS{}, the time used to generate the automaton is reported separately (in grey).}
% \label{minimize_results}
% \stoptable
The results for structured automata show a clear effect of the extra structure. The results for structured automata show a clear effect of the extra structure.
Both \NLambda{} and \LOIS{} remain capable of minimising the automata in reasonable amounts of time for larger sizes. Both \NLambda{} and \LOIS{} remain capable of minimising the automata in reasonable amounts of time for larger sizes.
@ -762,7 +758,7 @@ In particular, it learns a nominal DFA (over an infinite alphabet) using only fi
We implement \nLStar{} in the presented library and compare it to its previous implementation in \NLambda{}. We implement \nLStar{} in the presented library and compare it to its previous implementation in \NLambda{}.
The algorithm is not polynomial, unlike the minimisation algorithm described above. The algorithm is not polynomial, unlike the minimisation algorithm described above.
However, the authors conjecture that there is a polynomial algorithm. However, the authors conjecture that there is a polynomial algorithm.
\footnote{See \todo{joshuamoerman.nl/papers/2017/17popl-learning-nominal-automata.html} for a sketch of the polynomial algorithm.} \footnote{See \href{joshuamoerman.nl/papers/2017/17popl-learning-nominal-automata.html} for a sketch of the polynomial algorithm.}
For the correctness, termination, and comparison with other learning algorithms see \in{Chapter}[chap:learning-nominal-automata]. For the correctness, termination, and comparison with other learning algorithms see \in{Chapter}[chap:learning-nominal-automata].
@ -771,7 +767,7 @@ For the correctness, termination, and comparison with other learning algorithms
Both implementations in \NLambda{} and \ONS{} are direct implementations of the pseudocode for \nLStar{} with no further optimisations. Both implementations in \NLambda{} and \ONS{} are direct implementations of the pseudocode for \nLStar{} with no further optimisations.
The authors of \LOIS{} implemented \nLStar{} in their library as well. The authors of \LOIS{} implemented \nLStar{} in their library as well.
\footnote{Can be found on \todo{github.com/eryxcc/lois/blob/master/tests/learning.cpp}.} \footnote{Can be found on \href{github.com/eryxcc/lois/blob/master/tests/learning.cpp}.}
They reported similar performance as the implementation in \NLambda{} (private communication). They reported similar performance as the implementation in \NLambda{} (private communication).
Hence we focus our comparison on \NLambda{} and \ONS{}. Hence we focus our comparison on \NLambda{} and \ONS{}.
We use the variant of \nLStar{} where counterexamples are added as columns instead of prefixes. We use the variant of \nLStar{} where counterexamples are added as columns instead of prefixes.
@ -804,33 +800,30 @@ For languages which are equivariant for the equality symmetry, the \NLambda{} im
This is expected as the automata themselves have fewer orbits. This is expected as the automata themselves have fewer orbits.
It is interesting to see that these languages can be learned more efficiently by choosing the right symmetry. It is interesting to see that these languages can be learned more efficiently by choosing the right symmetry.
\todo{Table} \placetable[here][learning_results]
% \starttable {Running times and number of membership queries for the \nLStar{} algorithm.
% \centering For \NLambda{} we used two version: \NLambda{}$^{\mathit{ord}}$ uses the total order symmetry \NLambda{}$^{\mathit{eq}}$ uses the equality symmetry.}
% \starttabular}{l l l|r r|r r|r r \starttabulate[|l|l|l|r|r|r|r|r|r|][distance=none]
% & & & \multicolumn{2}{l|}{\ONS{}} & \multicolumn{2}{l|}{\NLambda{}$^{\mathit{ord}}$} & \multicolumn{2}{l}{\NLambda{}$^{\mathit{eq}}$} \\ \NC \NC \NC \VL \NC \ONS{} \VL \NC \NLambda{}$^{\mathit{ord}}$ \VL \NC \NLambda{}$^{\mathit{eq}}$ \NC\NR
% Model & $\Nsize(S)$ & $\dim(S)$ & time & MQs & time & MQs & time& MQs \\ \NC Model \NC $\Nsize(S)$ \NC $\dim(S)$ \VL time \NC MQs \VL time \NC MQs \VL time \NC MQs \NC\NR
% \hline \HL
% rand$_{5,1}$ & $4$ & $1$ & $2$m $7$s & $2321$ & $39$m $51$s & $1243$ & & \\ \NC rand$_{5,1}$ \NC $4$ \NC $1$ \VL $2$m $7$s \NC $2321$ \VL $39$m $51$s \NC $1243$ \VL \NC \NC\NR
% rand$_{5,1}$ & $5$ & $1$ & $0.12$s & $404$ & $40$m $34$s & $435$ & & \\ \NC rand$_{5,1}$ \NC $5$ \NC $1$ \VL $0.12$s \NC $404$ \VL $40$m $34$s \NC $435$ \VL \NC \NC\NR
% rand$_{5,1}$ & $3$ & $0$ & $0.86$s & $499$ & $30$m $19$s & $422$ & & \\ \NC rand$_{5,1}$ \NC $3$ \NC $0$ \VL $0.86$s \NC $499$ \VL $30$m $19$s \NC $422$ \VL \NC \NC\NR
% rand$_{5,1}$ & $5$ & $1$ & $> 60$m & n/a & $> 60$m & n/a & & \\ \NC rand$_{5,1}$ \NC $5$ \NC $1$ \VL $> 60$m \NC n/a \VL $> 60$m \NC n/a \VL \NC \NC\NR
% rand$_{5,1}$ & $4$ & $1$ & $0.08$s & $387$ & $34$m $57$s & $387$ & & \\ \NC rand$_{5,1}$ \NC $4$ \NC $1$ \VL $0.08$s \NC $387$ \VL $34$m $57$s \NC $387$ \VL \NC \NC\NR
% \hline \HL
% FIFO$(1)$ & $3$ & $1$ & $0.04$s & $119$ & $3.17$s & $119$ & $1.76$s & $51$ \\ \NC FIFO$(1)$ \NC $3$ \NC $1$ \VL $0.04$s \NC $119$ \VL $3.17$s \NC $119$ \VL $1.76$s \NC $51$ \NC\NR
% FIFO$(2)$ & $6$ & $2$ & $1.73$s & $2655$ & $6$m $32$s & $3818$ & $40.00$s & $434$ \\ \NC FIFO$(2)$ \NC $6$ \NC $2$ \VL $1.73$s \NC $2655$ \VL $6$m $32$s \NC $3818$ \VL $40.00$s \NC $434$ \NC\NR
% FIFO$(3)$ & $19$ & $3$ & $46$m $34$s & $298400$ & $> 60$m & n/a & $34$m $7$s & $8151$ \\ \NC FIFO$(3)$ \NC $19$ \NC $3$ \VL $46$m $34$s\NC $298400$ \VL $> 60$m \NC n/a \VL $34$m $7$s \NC $8151$ \NC\NR
% \hline \HL
% $ww(1)$ & $4$ & $1$ & $0.42$s & $134$ & $2.49$s & $77$ & $1.47$s & $30$ \\ \NC $ww(1)$ \NC $4$ \NC $1$ \VL $0.42$s \NC $134$ \VL $2.49$s \NC $77$ \VL $1.47$s \NC $30$ \NC\NR
% $ww(2)$ & $8$ & $2$ & $4$m $26$s & $3671$ & $3$m $48$s & $2140$ & $30.58$s & $237$ \\ \NC $ww(2)$ \NC $8$ \NC $2$ \VL $4$m $26$s \NC $3671$ \VL $3$m $48$s \NC $2140$ \VL $30.58$s \NC $237$ \NC\NR
% $ww(3)$ & $24$ & $3$ & $> 60$m & n/a & $> 60$m & n/a & $> 60$m & n/a \\ \NC $ww(3)$ \NC $24$ \NC $3$ \VL $> 60$m \NC n/a \VL $> 60$m \NC n/a \VL $> 60$m \NC n/a \NC\NR
% \hline \HL
% $\Lmax$ & $3$ & $1$ & $0.01$s & $54$ & $3.58$s & $54$ & & \\ \NC $\Lmax$ \NC $3$ \NC $1$ \VL $0.01$s \NC $54$ \VL $3.58$s \NC $54$ \VL \NC \NC\NR
% $\Lint$ & $5$ & $2$ & $0.59$s & $478$ & $1$m $23$s & $478$ & & \NC $\Lint$ \NC $5$ \NC $2$ \VL $0.59$s \NC $478$ \VL $1$m $23$s \NC $478$ \VL \NC \NC\NR
% \stoptabular \stoptabulate
% \caption{Running times and number of membership queries for the \nLStar{} algorithm. For \NLambda{} we used two version: \NLambda{}$^{\mathit{ord}}$ uses the total order symmetry \NLambda{}$^{\mathit{eq}}$ uses the equality symmetry.}
% \label{learning_results}
% \stoptable
\stopsubsubsection \stopsubsubsection

4
environment.tex
View file

@ -63,6 +63,10 @@
\setupalgorithmic[before={\startframedtext[width=0.85\textwidth,offset=none,frame=off]},after={\stopframedtext}] % To enable linenumbers in a float \setupalgorithmic[before={\startframedtext[width=0.85\textwidth,offset=none,frame=off]},after={\stopframedtext}] % To enable linenumbers in a float
\def\href#1{\useURL[#1][{#1}]{\tt\goto{\url[#1]}[url(#1)]}}
% Debugging % Debugging
%\enabletrackers[references.references, visualizers.justification] %\enabletrackers[references.references, visualizers.justification]
%\showmakeup %\showmakeup