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A Haskell Implementation of Turing Machines
Lim Shao En Zhang Licheng
Computer Science Computability Theory
A Haskell Implementation of Turing Machines
A Haskell Implementation of Turing Machines Lim Shao En Zhang - - PowerPoint PPT Presentation
A Haskell Implementation of Turing Machines Lim Shao En Zhang - - PowerPoint PPT Presentation
A Haskell Implementation of Turing Machines Lim Shao En Zhang Licheng Computer Science Computability Theory A Haskell Implementation of Turing Machines Computability Theory Turing Machine Recursion theory Lambda-calculus Post
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Computability Theory
A Haskell Implementation of Turing Machines
- Turing Machine
- Recursion theory
- Lambda-calculus
- Post system
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A Haskell Implementation of Turing Machines
Aim and Hypothesis
Bridging the theory – practice gap
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Aim and Hypothesis
Computability Theory
A Haskell Implementation of Turing Machines
Abstract Descriptions Computer programs
Our Aim
Turing Machines Haskell Codes
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A Haskell Implementation of Turing Machines
Turing Machine
Tape (Infinite Length)
Cells
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A Haskell Implementation of Turing Machines
Turing Machine
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Symbols (0,1 or Blank)
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A Haskell Implementation of Turing Machines
Turing Machine
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q1
Head
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A Haskell Implementation of Turing Machines
Turing Machine
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q1 q2 q3 q4 q5
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State Read and Write Symbols
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A Haskell Implementation of Turing Machines
Turing Machine -- -- Transitions
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A Haskell Implementation of Turing Machines
Haskell
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A Haskell Implementation of Turing Machines
Haskell
- Typed , Functional Programming
Language
- Typed - Data types in haskell are built up from
the basic data types: Int, Bool, (a,b) and X->Y
- Functional - functions can be passed as data
types, data is passed recursively from function to function
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A Haskell Implementation of Turing Machines
Methods
5 Major Stages
Learn Haskell 1 2 Create ‘Types’ 3 Construct ‘UTM’ Prototype 4 Test - Functions 5 Refine & Debug
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A Haskell Implementation of Turing Machines
Results and Discussion
Translating knowledge of Turing Machines into Haskell
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Typ ypes Construction
type pe Dir Dir = Int Int
- - {-1,0,1}
type pe Sta State te = Int Int
- - state numbers
- - start at q(0);
- - halts at q(-1)
type pe Sym Sym = Int Int
- - {0,1,2} cell symbols;
- - 2 represents Blank
type pe Hdp Hdp = Int Int
- - head position
type pe Tap Tape = [Sym Sym]
- - strings on tape
type pe Tmac mac = Stat tate
- >
Sym Sym
- >
(Stat State,S ,Sym,Di m,Dir)
- - generic TM
- - (a set of transitions)
Results and Discussion:
Translating knowledge of Turing Machines into Haskell
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Ma Main Program ram
- f UT
f UTM proto totyp type
tu turr rr :: :: Tmac Tmac -> Tap Tape -> Ta Tape pe turr m t = fst (auxt m 0 (t,0)) au auxt xt :: :: Tmac Tmac -> Sta State te -> (Tap Tape, e,Hdp Hdp) -> (Tap Tape,H e,Hdp dp) auxt m (-1) (t,i) = (t,i) auxt m q (t,i) = let t2 = edit (i,mysnd(m q (t!!i)),t) in auxt m (myfst(m q (t!!i))) (t2,i + mythd(m q (t!!i))) ed edit it :: :: (Hdp Hdp,S ,Sym, ym,Ta Tape pe) -> Tape Tape edit (i,x,t) = (take i t) ++ [x] ++ (drop (i+1) t) ++ [2] myfst :: (State,Sym,Dir) -> State myfst (q,x,d) = q mysnd :: (State,Sym,Dir) -> Sym mysnd (q,x,d) = x mythd :: (State,Sym,Dir) -> Dir mythd (q,x,d) = d
Results and Discussion:
Translating knowledge of Turing Machines into Haskell
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Test – Function: Addition
add add :: :: Tmac mac add 0 2 = (1,2,1) add 1 1 = (1,1,1) add 1 0 = (2,1,1) add 2 1 = (2,1,1) add 2 2 = (3,2,-1) add 3 1 = (4,2,-1) add 4 1 = (4,1,-1) add 4 2 = (-1,2,0)
3 + 2 = 5
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Test – Function: Multiplication
mult lt :: :: Tmac Tmac mult 0 0 = (7,0,1) mult 0 2 = (0,2,1) mult 1 1 = (2,2,1) mult 1 0 = (6,0,-1) mult 2 0 = (3,0,1) mult 2 1 = (2,1,1) mult 3 1 = (3,1,1) mult 3 2 = (4,1,-1) mult 4 0 = (5,0,-1) mult 4 1 = (4,1,-1) mult 5 1 = (5,1,-1) mult 5 2 = (1,2,1) mult 6 2 = (6,1,-1) mult 6 0 = (9,0,-1) mult 7 0 = (10,2,1) mult 7 1 = (8,2,1) mult 8 0 = (1,0,1) mult 8 1 = (8,1,1) mult 9 1 = (9,1,-1) mult 9 2 = (7,2,1)
- mult 10 1 = (10,2,1)
mult 10 0 = (11,2,-1) mult 11 2 = (11,2,-1) mult 11 0 = (-1,2,0)
3 x 2 = 6
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UTM Prototype
Results and Discussion:
Translating knowledge of Turing Machines into Haskell
Successful !
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A Haskell Implementation of Turing Machines
Conclusion
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State-transition table / diagram Haskell Codes Prototype of UTM Theory-Practice Gap
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A Haskell Implementation of Turing Machines
Future Work
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More Functions (e.g. Logarithm, Factorial) Visualise operation of Turing Machine Multi-tape Turing Machine
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