Execute a Markov algorithm

You are encouraged to solve this task according to the task description, using any language you may know.


Task

Create an interpreter for a Markov Algorithm.

Rules have the syntax:

<ruleset> ::= ((<comment> | <rule>) <newline>+)*
<comment> ::= # {<any character>}
<rule> ::= <pattern> <whitespace> -> <whitespace> [.] <replacement>
<whitespace> ::= (<tab> | <space>) [<whitespace>]

There is one rule per line.

If there is a   .   (period)   present before the   <replacement>,   then this is a terminating rule in which case the interpreter must halt execution.

A ruleset consists of a sequence of rules, with optional comments.


 Rulesets 

Use the following tests on entries:



Ruleset 1

# This rules file is extracted from Wikipedia:
# http://en.wikipedia.org/wiki/Markov_Algorithm
A -> apple
B -> bag
S -> shop
T -> the
the shop -> my brother
a never used -> .terminating rule

Sample text of:

  I bought a B of As from T S. 
Should generate the output:

  I bought a bag of apples from my brother. 


Ruleset 2

A test of the terminating rule

# Slightly modified from the rules on Wikipedia
A -> apple
B -> bag
S -> .shop
T -> the
the shop -> my brother
a never used -> .terminating rule
Sample text of:

 I bought a B of As from T S.
Should generate:

 I bought a bag of apples from T shop.


Ruleset 3

This tests for correct substitution order and may trap simple regexp based replacement routines if special regexp characters are not escaped.

# BNF Syntax testing rules
A -> apple
WWWW -> with
Bgage -> ->.*
B -> bag
->.* -> money
W -> WW
S -> .shop
T -> the
the shop -> my brother
a never used -> .terminating rule

Sample text of:

 I bought a B of As W my Bgage from T S.
Should generate:

 I bought a bag of apples with my money from T shop.


Ruleset 4

This tests for correct order of scanning of rules, and may trap replacement routines that scan in the wrong order.   It implements a general unary multiplication engine.   (Note that the input expression must be placed within underscores in this implementation.)

### Unary Multiplication Engine, for testing Markov Algorithm implementations
### By Donal Fellows.
# Unary addition engine
_+1 -> _1+
1+1 -> 11+
# Pass for converting from the splitting of multiplication into ordinary
# addition
1! -> !1
,! -> !+
_! -> _
# Unary multiplication by duplicating left side, right side times
1*1 -> x,@y
1x -> xX
X, -> 1,1
X1 -> 1X
_x -> _X
,x -> ,X
y1 -> 1y
y_ -> _
# Next phase of applying
1@1 -> x,@y
1@_ -> @_
,@_ -> !_
++ -> +
# Termination cleanup for addition
_1 -> 1
1+_ -> 1
_+_ -> 

Sample text of:

  _1111*11111_ 
should generate the output:

  11111111111111111111 


Ruleset 5

A simple Turing machine,
implementing a three-state busy beaver.

The tape consists of 0s and 1s,   the states are A, B, C and H (for Halt), and the head position is indicated by writing the state letter before the character where the head is.
All parts of the initial tape the machine operates on have to be given in the input.

Besides demonstrating that the Markov algorithm is Turing-complete, it also made me catch a bug in the C++ implementation which wasn't caught by the first four rulesets.

# Turing machine: three-state busy beaver
#
# state A, symbol 0 => write 1, move right, new state B
A0 -> 1B
# state A, symbol 1 => write 1, move left, new state C
0A1 -> C01
1A1 -> C11
# state B, symbol 0 => write 1, move left, new state A
0B0 -> A01
1B0 -> A11
# state B, symbol 1 => write 1, move right, new state B
B1 -> 1B
# state C, symbol 0 => write 1, move left, new state B
0C0 -> B01
1C0 -> B11
# state C, symbol 1 => write 1, move left, halt
0C1 -> H01
1C1 -> H11

This ruleset should turn

  000000A000000 
into

  00011H1111000 


-- utility method to escape punctuation
function normalize(str)
    local result = str:gsub("(%p)", "%%%1")
    -- print(result)
    return result
end
 
-- utility method to split string into lines
function get_lines(str)
    local t = {}
    for line in str:gmatch("([^\n\r]*)[\n\r]*") do
        table.insert(t, line)
    end
    return t
end
 
local markov = {}
local MARKOV_RULE_PATTERN = "(.+)%s%-%>%s(%.?)(.*)"
 
function markov.rule(pattern,replacement,terminating)
    return {
        pattern = pattern,
        replacement = replacement,
        terminating = (terminating == ".")
    }, normalize(pattern)
end
 
function markov.make_rules(sample)
    local lines = get_lines(sample)
    local rules = {}
    local finders = {}
    for i,line in ipairs(lines) do
        if not line:find("^#") then
        s,e,pat,term,rep = line:find(MARKOV_RULE_PATTERN)
        if s then
            r, p = markov.rule(pat,rep,term)
            rules[p] = r
            table.insert(finders, p)
        end
        end
    end
    return {
        rules = rules,
        finders = finders
    }
end
 
function markov.execute(state, sample_input)
 
local rules, finders = state.rules, state.finders
local found = false -- did we find any rule?
local terminate = false
 
repeat
found = false
 
for i,v in ipairs(finders) do
    local found_now = false -- did we find this rule?
    if sample_input:find(v) then 
        found = true
        found_now = true
    end
    sample_input = sample_input:gsub(v, rules[v].replacement, 1)
    -- handle terminating rules
    if found_now then 
        if rules[v].terminating then terminate = true end
        break
    end
end
 
until not found or terminate
 
return sample_input
end
------------------------------------------
------------------------------------------
 
local grammar1 = [[
# This rules file is extracted from Wikipedia:
# http://en.wikipedia.org/wiki/Markov_Algorithm
A -> apple
B -> bag
S -> shop
T -> the
the shop -> my brother
a never used -> .terminating rule
]]
 
local grammar2 = [[
# Slightly modified from the rules on Wikipedia
A -> apple
B -> bag
S -> .shop
T -> the
the shop -> my brother
a never used -> .terminating rule
]]
 
local grammar3 = [[
# BNF Syntax testing rules
A -> apple
WWWW -> with
Bgage -> ->.*
B -> bag
->.* -> money
W -> WW
S -> .shop
T -> the
the shop -> my brother
a never used -> .terminating rule
]]
 
local grammar4 = [[
### Unary Multiplication Engine, for testing Markov Algorithm implementations
### By Donal Fellows.
# Unary addition engine
_+1 -> _1+
1+1 -> 11+
# Pass for converting from the splitting of multiplication into ordinary
# addition
1! -> !1
,! -> !+
_! -> _
# Unary multiplication by duplicating left side, right side times
1*1 -> x,@y
1x -> xX
X, -> 1,1
X1 -> 1X
_x -> _X
,x -> ,X
y1 -> 1y
y_ -> _
# Next phase of applying
1@1 -> x,@y
1@_ -> @_
,@_ -> !_
++ -> +
# Termination cleanup for addition
_1 -> 1
1+_ -> 1
_+_ -> 
]]
 
local grammar5 = [[
# Turing machine: three-state busy beaver
#
# state A, symbol 0 => write 1, move right, new state B
A0 -> 1B
# state A, symbol 1 => write 1, move left, new state C
0A1 -> C01
1A1 -> C11
# state B, symbol 0 => write 1, move left, new state A
0B0 -> A01
1B0 -> A11
# state B, symbol 1 => write 1, move right, new state B
B1 -> 1B
# state C, symbol 0 => write 1, move left, new state B
0C0 -> B01
1C0 -> B11
# state C, symbol 1 => write 1, move left, halt
0C1 -> H01
1C1 -> H11
]]
 
local text1 = "I bought a B of As from T S."
local text2 = "I bought a B of As W my Bgage from T S."
local text3 = '_1111*11111_'
local text4 = '000000A000000'
 
------------------------------------------
------------------------------------------
 
function do_markov(rules, input, output)
    local m = markov.make_rules(rules)
    input = markov.execute(m, input)
    assert(input == output)
    print(input)
end
 
do_markov(grammar1, text1, 'I bought a bag of apples from my brother.')
do_markov(grammar2, text1, 'I bought a bag of apples from T shop.')
-- stretch goals
do_markov(grammar3, text2, 'I bought a bag of apples with my money from T shop.')
do_markov(grammar4, text3, '11111111111111111111')
do_markov(grammar5, text4, '00011H1111000')