Saturday, June 4, 2016

Markov Chains with Erlang

My previous posts about getting started with epgsql helped me get familiar so I could store my Makov chain data in a database to help Markov-lang learn how to talk.

Today we’re going to discuss how we generate the Markov chain data that will go into that database. I’m going to forgo the OTP portion of this application. Though the full version can be found on my github.

For those unfamiliar, Markov chains describe how likely you are to transition from one state to the next. Read about them here and here. Or you can just Google it.

I wanted to model text as a chain. This chain represents the transition from a prefix of two words to a one word suffix.

this is a string, this is a string, this is a ending

start of sentence -> this
start of sentence this -> is
this is -> a
is a -> string,
a string, -> this
string, this -> is
this is -> a
is a -> string,
a string, -> this
string, this -> is
this is -> a
is a -> ending
a ending -> end of sentence

To start this we will represent start of sentence as two spaces [" ", " "]. We will tokenize the string. Then we will represent end of sentence as a special string ["<<< undefined>>>"]. (Please note the space in <<< undefined>>> is there because of the markdown).

The string:

"This is a string, this is a string, this is a ending"

Turns into:

[" ", " ", "This", "is", "a", "string", " this", "is", "a", "string", " this", "is", "a", "ending", "<<< undefined>>>"]

We will write a splitString function to take our incoming string, remove any unprintable chars, and return the list of tokens.

%% markov.erl
splitString(String) ->
Tokens = filterPrintableAscii(string:tokens(String, " ")),
[" ", " "] ++ Tokens ++ ["<<<undefined>>>"].

filterPrintableAscii(TokenList) ->
lists:map(fun(Token) ->
re:replace(Token, "[^ -~]", "", [global, {return, list}])
end, TokenList).

We pad it with two blank space to represent the start of a sentence. Then we append "<<< undefined>>>" so that we know when we’ve reached the end of a sentence.

Now that we have a list of the words, we need to turn this into the chains we described above. In order to represent these I created a chain record:

%% records.hrl
-record(chain, {prefix="", suffix=""})

Now lets turn our list of tokens into a table of chains.

%% markov.erl
genTable(Tokens) ->
genTable(Tokens, tl(Tokens), tl(tl(Tokens)));
genTable(_, _, [], Acc) ->
genTable(L1, L2, L3, Acc) ->
Prefix = string:join([hd(L1), hd(L2)], " "),
Suffix = hd(L3),
Chain = factories:chainFactory(Prefix, Suffix),
genTable(tl(L1), tl(L2), tl(L3), Acc ++ [Chain]).

We will discuss the call to reduceTable shortly. genTable/1 takes in a list of tokens and then calls itself with the list, the tail of the list and the tail of the tail of the list. We keep accumulating chains until we run out of suffixes.

When we reach tl(L3) == [] the accumulator acc looks like:

1 [
2 {chain, " ", "this"},
3 {chain, " this", "is"},
4 {chain, "this is", "a"},
5 {chain, "is a", "string,"},
6 {chain, "a string,", "this"},
7 {chain, "string, this", "is"},
8 {chain, "this is", "a"},
9 {chain, "is a", "string,"},
10 {chain, "a string,", "this"},
11 {chain, "string, this", "is"},
12 {chain, "this is", "a"},
13 {chain, "is a", "ending"},
14 {chain, "a ending", "<<<undefined>>>"}
15 ]

You’ll notice that some chains such as lines 4 and 8 have the same prefix and suffix. Also, you’ll notice chains such as lines 5, 9, and 13 all share the same prefix, but line 13 has a different suffix. In order to weight chains properly I came up with 2 more records.

%% records.hrl
-record(reducedChain, {prefix="", suffixes=[]}).
-record(suffix, {word="", count=0}).

The reducedChain record represents a prefix and a list of suffixes. The suffixes in the list are of the suffix record type which keeps track of the suffix and how many times it has appeared after the prefix for the reducedChain.

The call to reduceTable, told you we would get to this, takes the list of chains and turns it into a list of reducedChains.

%% markov.erl
reduceTable(Table) ->
reduceTable(Table, []).
reduceTable([], Acc) ->
reduceTable(Table, Acc) ->
Chain = hd(Table),
Prefix = Chain#chain.prefix,
ChainsSamePrefix = gatherChainsWithPrefix(Prefix, Table),
Suffixes = genSuffixes(ChainsSamePrefix),
reduceTable(Table -- ChainsSamePrefix, Acc ++ [factories:reducedChainFactory(Prefix, Suffixes)]).

Our first step is to get all the chains with the same prefix:

hasPrefix(Prefix, #chain{prefix=Prefix, suffix=_}) ->
hasPrefix(_,_) ->
gatherChainsWithPrefix(Prefix, Table) ->
lists:filter(fun(Chain) ->
hasPrefix(Prefix, Chain)
end, Table).

If we gather the chains for the prefix "is a" we would get the following output:

1 [
2 {chain, "is a", "string,"},
3 {chain, "is a", "string,"},
4 {chain, "is a", "ending"}
5 ]

The next step is to create the suffixes for our reducedChain.

hasSuffix(Suffix, #chain{prefix=_, suffix=Suffix}) ->
hasSuffix(_,_) ->

gatherChainsWithSuffix(Suffix, Table) ->
lists:filter(fun(Chain) ->
hasSuffix(Suffix, Chain)
end, Table).

countSameSuffix(Suffix, Chains) ->
lists:foldl(fun(Chain, NumSame) ->
case hasSuffix(Suffix, Chain) of
true -> NumSame + 1;
false -> NumSame
end, 0, Chains).

genSuffixes([], Acc) ->
genSuffixes(Chains, Acc) ->
Chain = hd(Chains),
Suffix = Chain#chain.suffix,
SameSuffixes = gatherChainsWithSuffix(Suffix, Chains),
NumSameSuffix = countSameSuffix(Suffix, Chains),
genSuffixes(Chains -- SameSuffixes, Acc ++ [factories:suffixFactory(Suffix, NumSameSuffix)]).

We would get the following list of suffixes:

{suffix, "string,", 2},
{siffix, "ending", 1}

Now we have everything we need to create a reducedChain. We do this for all items and we end up with the following reducedTable:

{reducedChain, " ", [{suffix, "this", 1}],
{reducedChain, " this", [{suffix, "is", 1}],
{reducedChain, "is a", [{suffix, "string,", 2}, {suffix, "ending", 1}],
{reducedChain, "this is", [{suffix, "a", 3}],

Now we have generated out weighted chains and can use these to generate sentences. I will go over this in my next blog post when I cover how we attach the database.

A version of Markov-lang that can generate sentences without the database exists at this commit. Just feed the output of genTable into genSentence

Written with StackEdit.

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