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>=stealth', -% node distance=2.5cm, % specifies the minimum distance between two nodes. Change if necessary. -%% every state/.style={thick, fill=gray!10}, % sets the properties for each ’state’ node -% initial text=$ $, % sets the text that appears on the start arrow -% ] -% \node[state, initial] (00) {$q_{0,0}$}; -% \node[state, right of=00] (10) {$q_{1,0}$}; -% \node[state, right of=10] (20) {$q_{2,0}$}; -% \node[accepting, state, right of=20] (30) {$q_{3,0}$}; -% \node[accepting, state, right of=30] (40) {$q_{4,0}$}; -% -% \node[state, above of=00, shift={(-2cm,0cm)}] (01) {$q_{0,1}$}; -% \node[state, right of=01] (11) {$q_{1,1}$}; -% \node[state, right of=11] (21) {$q_{2,1}$}; -% \node[state, right of=21] (31) {$q_{3,1}$}; -% \node[accepting, state, right of=31] (41) {$q_{4,1}$}; -% -% \draw [->] (00) edge[below] node{$\texttt{N}$} (10); -% \draw [->] (10) edge[below] node{$\texttt{+}$} (20); -% \draw [->] (20) edge[below] node{$\texttt{*}$} (30); -% \draw [->] (30) edge[below] node{$\texttt{N}$} (40); -% -% \draw [->] (01) edge[below] node{$\texttt{N}$} (11); -% \draw [->] (11) edge[below] node[shift={(-0.2cm,0cm)}]{$\texttt{+}$} (21); -% \draw [->] (21) edge[below] node[shift={(-0.2cm,0cm)}]{$\texttt{*}$} (31); -% \draw [->] (31) edge[below] node[shift={(-0.2cm,0cm)}]{$\texttt{N}$} (41); -% -% \draw [->] (00) edge[left] node{\tiny{$[\neq \texttt{N}]$}} (11); -% \draw [->] (10) edge[left] node{\tiny{$[\neq \texttt{+}]$}} (21); -% \draw [->] (20) edge[left] node{\tiny{$[\neq \texttt{*}]$}} (31); -% \draw [->] (30) edge[left] node{\tiny{$[\neq \texttt{N}]$}} (41); -% -% \draw [->] (00) edge[bend left=10, left] node{\tiny{$[\neq \texttt{N}]$}} (01); -% \draw [->] (10) edge[bend left=10, left] node{\tiny{$[\neq \texttt{+}]$}} (11); -% \draw [->] (20) edge[bend left=10, left] node{\tiny{$[\neq \texttt{*}]$}} (21); -% \draw [->] (30) edge[bend left=10, left] node{\tiny{$[\neq \texttt{N}]$}} (31); -% \draw [->] (40) edge[bend left=10, left] node{\tiny{$[.^{\ast}]$}} (41); -% -% -% \draw [->, blue] (00) edge[bend right=11,below] node[shift={(0.5cm,0.9cm)}]{$\texttt{+}$} (21); -% \draw [->, blue] (10) edge[bend right=11,below] node[shift={(0.5cm,0.9cm)}]{$\texttt{*}$} (31); -% \draw [->, blue] (20) edge[bend right=11,below] node[shift={(0.5cm,0.9cm)}]{$\texttt{N}$} (41); -% \end{tikzpicture} -% } -% \caption{Levenshtein automaton recognizing every single edit of a string.}\label{fig:lev_automaton} -%\end{figure} +We will now give an intuitive overview of our approach, then proceed to formalize it in the following sections. By way of illustration, suppose we have string: $\texttt{( ) )}$ and want to find all repairs with a single edit. There is a nondeterministic finite automaton, called the Levenshtein automaton, recognizing every single string that can be formed by inserting, substituting or deleting a parenthesis. We adapt this to remove some unnecessary edges, which does not affect its semantics. + +\begin{figure}[h!] + \resizebox{0.45\textwidth}{!}{ + \begin{tikzpicture}[ +%->, % makes the edges directed + >=stealth', + node distance=2.5cm, % specifies the minimum distance between two nodes. Change if necessary. +% every state/.style={thick, fill=gray!10}, % sets the properties for each ’state’ node + initial text=$ $, % sets the text that appears on the start arrow + ] + \node[state, initial] (00) {$q_{0,0}$}; + \node[state, right of=00] (10) {$q_{1,0}$}; + \node[accepting, state, right of=10] (20) {$q_{2,0}$}; + \node[accepting, state, right of=20] (30) {$q_{3,0}$}; + + \node[state, above of=00, shift={(-2cm,0cm)}] (01) {$q_{0,1}$}; + \node[state, right of=01] (11) {$q_{1,1}$}; + \node[state, right of=11] (21) {$q_{2,1}$}; + \node[accepting, state, right of=21] (31) {$q_{3,1}$}; + + \draw [->] (00) edge[below] node{$\texttt{(}$} (10); + \draw [->] (10) edge[below] node{$\texttt{)}$} (20); + \draw [->] (20) edge[below] node{$\texttt{)}$} (30); + + \draw [->] (01) edge[below] node{$\texttt{(}$} (11); + \draw [->] (11) edge[below] node[shift={(-0.2cm,0cm)}]{$\texttt{)}$} (21); + \draw [->] (21) edge[below] node[shift={(-0.2cm,0cm)}]{$\texttt{)}$} (31); + + \draw [->] (00) edge[bend left=10] node[shift={(-0.15cm,0cm)}]{\tiny{$\texttt{(}$}} (11); + \draw [->] (10) edge[bend left=10] node[shift={(-0.15cm,0cm)}]{\tiny{$\texttt{(}$}} (21); + \draw [->] (20) edge[bend left=10] node[shift={(-0.15cm,0cm)}]{\tiny{$\texttt{(}$}} (31); + + \draw [->] (00) edge[bend left=10, left] node[shift={(-0.1cm,0cm)}]{\tiny{$\texttt{(}$}} (01); + \draw [->] (10) edge[bend left=10, left] node[shift={(-0.1cm,0cm)}]{\tiny{$\texttt{(}$}} (11); + \draw [->] (20) edge[bend left=10, left] node[shift={(-0.1cm,0cm)}]{\tiny{$\texttt{(}$}} (21); + + \draw [->] (00) edge[bend right=10, right] node{\tiny{$\texttt{)}$}} (11); + \draw [->] (10) edge[bend right=10, right] node{\tiny{$\texttt{)}$}} (21); + \draw [->] (20) edge[bend right=10, right] node{\tiny{$\texttt{)}$}} (31); + + \draw [->] (00) edge[bend right=10, right] node{\tiny{$\texttt{)}$}} (01); + \draw [->] (10) edge[bend right=10, right] node{\tiny{$\texttt{)}$}} (11); + \draw [->] (20) edge[bend right=10, right] node{\tiny{$\texttt{)}$}} (21); + + \draw [->] (30) edge[bend left=10, left] node[shift={(-0.1cm,0cm)}]{\tiny{$\texttt{(}$}} (31); + \draw [->] (30) edge[bend right=10, right] node{\tiny{$\texttt{)}$}} (31); + + \draw [->, blue] (00) edge[bend right=11,below] node[shift={(0.4cm,0.9cm)}]{$\texttt{)}$} (21); + \draw [->, blue] (10) edge[bend right=11,below] node[shift={(0.4cm,0.9cm)}]{$\texttt{)}$} (31); + \node[align=center, yshift=2em, xshift=-1cm] (title) at (current bounding box.north) {Original Levenshtein Automaton}; + \end{tikzpicture} + } + \resizebox{0.515\textwidth}{!}{ + \begin{tikzpicture}[ +%->, % makes the edges directed + >=stealth', + node distance=2.5cm, % specifies the minimum distance between two nodes. Change if necessary. +% every state/.style={thick, fill=gray!10}, % sets the properties for each ’state’ node + initial text=$ $, % sets the text that appears on the start arrow + ] + \draw[blue,->] (-4cm,1.2cm) -- (-3cm,1.2cm); + + \node[state, initial] (00) {$q_{0,0}$}; + \node[state, right of=00] (10) {$q_{1,0}$}; + \node[accepting, state, right of=10] (20) {$q_{2,0}$}; + \node[accepting, state, right of=20] (30) {$q_{3,0}$}; + + \node[state, above of=00, shift={(-2cm,0cm)}] (01) {$q_{0,1}$}; + \node[state, right of=01] (11) {$q_{1,1}$}; + \node[state, right of=11] (21) {$q_{2,1}$}; + \node[accepting, state, right of=21] (31) {$q_{3,1}$}; + + \draw [->] (00) edge[below] node{$\texttt{(}$} (10); + \draw [->] (10) edge[below] node{$\texttt{)}$} (20); + \draw [->] (20) edge[below] node{$\texttt{)}$} (30); + + \draw [->] (01) edge[below] node{$\texttt{(}$} (11); + \draw [->] (11) edge[below] node[shift={(-0.2cm,0cm)}]{$\texttt{)}$} (21); + \draw [->] (21) edge[below] node[shift={(-0.2cm,0cm)}]{$\texttt{)}$} (31); + + \draw [->] (00) edge[left] node{\tiny{$[\neq \texttt{(}]$}} (11); + \draw [->] (10) edge[left] node{\tiny{$[\neq \texttt{)}]$}} (21); + \draw [->] (20) edge[left] node{\tiny{$[\neq \texttt{)}]$}} (31); + + \draw [->] (00) edge[bend left=10, left] node{\tiny{$[\neq \texttt{(}]$}} (01); + \draw [->] (10) edge[bend left=10, left] node{\tiny{$[\neq \texttt{)}]$}} (11); + \draw [->] (20) edge[bend left=10, left] node{\tiny{$[\neq \texttt{)}]$}} (21); + \draw [->] (30) edge[bend left=10, left] node{\tiny{$[=.]$}} (31); + + + \draw [->, blue] (00) edge[bend right=11,below] node[shift={(0.4cm,0.9cm)}]{$\texttt{)}$} (21); + \draw [->, blue] (10) edge[bend right=11,below] node[shift={(0.4cm,0.9cm)}]{$\texttt{)}$} (31); + \node[align=center, yshift=2em, xshift=-0.4cm] (title) at (current bounding box.north) {Nominal Levenshtein Automaton (ours)}; + \end{tikzpicture} + } + \caption{Automaton recognizing every single 1-edit patch. We nominalize the original automata.}\label{fig:lev_automaton} +\end{figure} + +Let us consider a very simple grammar: $G = \{S \rightarrow ( ) \mid ( S ) \mid S S\}$. For convenience, $G$ can be reduced into an equivalent normal form, $G'$, by refactoring the right hand side of each production to be in either binary or unary form: $G'= \{S \rightarrow L R, S \rightarrow L I, S \rightarrow S S, I \rightarrow S R, L \rightarrow (, R \rightarrow )\}$. We will now proceed to stitch together the automaton and grammar to form a new grammar, $G_\cap$, recognizing every string in the intersection of their languages. + +This stitching process is known in the literature as the Bar-Hillel (BH) construction, and applies to any context-free grammar and nondeterministic finite automaton. It takes the following approach: for every initial state in the automaton (here there is just one, $q_{00}$) and every final state, ($q_{20}, q_{30}, q_{31}$), there will be a production $S\rightarrow q_{00}Sq_{xy}$. We call this rule $\downarrow$, and leave it unchanged. + +Likewise, the BH construction also states for every production $A\rightarrow a$ and arc $q \overset{a}{\rightarrow} r$ in the automaton, there will be a production, $qAr\rightarrow a$ in the intersection grammar. We call this rule $\uparrow$. Here is where our approach, the Levenshtein Bar-Hillel (LBH) construction, begins to diverge: we will only add the unit production if it matches the arc predicate. We call this rule $\hat\uparrow$. So far, the intersection grammar $G_\cap$ will contain the following productions:\vspace{-8pt} + +\begin{table}[H] + \begin{tabular}{c|cccccc} + $\downarrow$ & $\hat\uparrow$ & & & & \\\hline + $S \rightarrow q_{00}Sq_{20}$ & $q_{00}Rq_{01} \rightarrow \texttt{)}$ & $q_{10}Lq_{11} \rightarrow \texttt{(}$ & $q_{20}Rq_{21} \rightarrow \texttt{)}$ & $q_{01}Lq_{11} \rightarrow \texttt{(}$ \\ + $S \rightarrow q_{00}Sq_{30}$ & $q_{00}Rq_{11} \rightarrow \texttt{)}$ & $q_{10}Lq_{21} \rightarrow \texttt{(}$ & $q_{20}Rq_{31} \rightarrow \texttt{)}$ & $q_{11}Rq_{21} \rightarrow \texttt{)}$ \\ + $S \rightarrow q_{00}Sq_{31}$ & $q_{00}Lq_{10} \rightarrow \texttt{(}$ & $q_{10}Rq_{20} \rightarrow \texttt{)}$ & $q_{20}Lq_{30} \rightarrow \texttt{(}$ & $q_{21}Rq_{31} \rightarrow \texttt{)}$ \\ + & $q_{30}Lq_{31} \rightarrow \texttt{(}$ & $q_{10}Rq_{31} \rightarrow \texttt{)}$ & $q_{30}Rq_{31} \rightarrow \texttt{)}$ & $q_{00}Rq_{21} \rightarrow \texttt{)}$ + \end{tabular} +\end{table}\vspace{-8pt} + +Now, the rule we call $\Join$, and the most expensive of them all, states that for every $w \rightarrow xy$ in $G'$ and every state triplet $\langle p, q, r\rangle$, we will have $pwr \rightarrow (pxq)(qxr)$ in $G_\cap'$. Effectively, this is constructing a grammar that recognizes every combination of NFA states consistent with every nonterminal in every production. But most of these combinations are simply impossible. For example, we note that $q_{31}Sq_{00}$ is an impossible nonterminal, as there is no path from $q_{31}$ to $q_{00}$ in the automaton. Likewise, the terminal $q_{i, j}Iq_{i+1, j}$ is clearly impossible, as the nonterminal $I$ requires at least three tokens, and the only path between $q_{i, j}$ and $q_{i+1, j}$ has length 1. So we can safely ignore any productions containing these impossible nonterminals generated by the original BH $\Join$ rule. + +We will present a more precise heuristic for eliminating impossible productions from Bar-Hillel intersections, based on a sound overapproximation to the Parikh image. Nominalizing the automaton and refining the $\Join$ rule are the basic idea behind the Levenshtein Bar-Hillel construction, allowing us to scale up this technique to handle real-world programming languages and code snippets. \clearpage\section{Problem statement} diff --git a/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/BarHillelTest.kt b/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/BarHillelTest.kt index 10b5044b..440f2802 100644 --- a/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/BarHillelTest.kt +++ b/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/BarHillelTest.kt @@ -454,17 +454,4 @@ class BarHillelTest { allTriplesMinusOverwritten.forEach { println(it) } println("Found ${allTriplesMinusOverwritten.size} non-overwritten triples.") } - - /* - ./gradlew jvmTest --tests "ai.hypergraph.kaliningraph.parsing.BarHillelTest.testDeadSimple" - */ - @Test - fun testDeadSimple() { - val prompt = "N + % N" - val ds = Grammars.deadSimple - - assertFalse("+ N" in ds.language) - assertFalse(prompt in ds.language) - assertTrue("N + N" in ds.language) - } } \ No newline at end of file diff --git a/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/Grammars.kt b/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/Grammars.kt index 56bc3cdb..18deb9ec 100644 --- a/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/Grammars.kt +++ b/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/Grammars.kt @@ -25,7 +25,15 @@ object Grammars { S -> X | Y | Z """.parseCFG().noNonterminalStubs - val deadSimple = """S -> N | S + S | S % S""".parseCFG().noNonterminalStubs + val deadSimple = """S -> ( ) | ( S )""".parseCFG().noEpsilonOrNonterminalStubs + val dsNorm = """ + START -> START START + START -> A B + START -> A C + A -> ( + B -> ) + C -> START B + """.parseCFG().noEpsilonOrNonterminalStubs val ocamlCFG = """ S -> X diff --git a/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/SeqValiantTest.kt b/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/SeqValiantTest.kt index 2d0afc14..9f90b8cc 100644 --- a/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/SeqValiantTest.kt +++ b/src/commonTest/kotlin/ai/hypergraph/kaliningraph/parsing/SeqValiantTest.kt @@ -2,6 +2,7 @@ package ai.hypergraph.kaliningraph.parsing import Grammars.arith import Grammars.evalArith +import Grammars.seq2parsePythonVanillaCFG import Grammars.tinyC import Grammars.toyArith import ai.hypergraph.kaliningraph.* @@ -218,4 +219,12 @@ class SeqValiantTest { assertTrue("1 + 1 + 1 = 3".matches(cfg)) assertTrue("1 + 1 + 1 + 1 = 4".matches(cfg)) } + + /* + ./gradlew jvmTest --tests "ai.hypergraph.kaliningraph.parsing.SeqValiantTest.testBinaryMinimization" + */ + @Test + fun testBinaryMinimization() { + println(seq2parsePythonVanillaCFG.size) + } } \ No newline at end of file diff --git a/src/jvmTest/kotlin/ai/hypergraph/kaliningraph/repair/ProbabilisticLBH.kt b/src/jvmTest/kotlin/ai/hypergraph/kaliningraph/repair/ProbabilisticLBH.kt index 3353d0f8..1ce79467 100644 --- a/src/jvmTest/kotlin/ai/hypergraph/kaliningraph/repair/ProbabilisticLBH.kt +++ b/src/jvmTest/kotlin/ai/hypergraph/kaliningraph/repair/ProbabilisticLBH.kt @@ -2,6 +2,7 @@ package ai.hypergraph.kaliningraph.repair import Grammars import Grammars.shortS2PParikhMap +import ai.hypergraph.kaliningraph.automata.toDFA import ai.hypergraph.kaliningraph.parsing.* import ai.hypergraph.kaliningraph.tokenizeByWhitespace import ai.hypergraph.markovian.* @@ -433,6 +434,28 @@ class ProbabilisticLBH { } } } + + /* + ./gradlew jvmTest --tests "ai.hypergraph.kaliningraph.repair.ProbabilisticLBH.testDeadSimple" + */ + @Test + fun testDeadSimple() { + val prompt = "( ) )" + val ds = Grammars.dsNorm + val la = makeLevFSA(prompt.tokenizeByWhitespace(), 1) + val ig = ds.intersectLevFSA(la) + + println(ig.prettyPrint()) + + assertTrue("( )" in ds.language) + assertFalse(prompt in ds.language) + assertFalse("( ) )" in ds.language) + + val solutions = ig.toPTree().sampleStrWithoutReplacement().toSet() + solutions.forEach { println(it) } + + println(ig.toPTree().toDFA(true)!!.toDot()) + } } // NAME . NAME ( STRING , class = STRING ) . NAME ( STRING , NAME = NAME . NAME ( STRING ) ) NEWLINE