Functional Programming In Mathematica

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Functional Programming In Mathematica Talk, Egypt Mathematica Conference, 2010

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Functional Programming In Mathematica

  1. 1. Functional Programming in Mathematica An Introduction Hossam Karim ITWorx
  2. 2. 2 FunctionalProgramming-Egypt-2010.nb About Work for an Egyptian Software Services Company Design software for the Bioinformatics and Telecommunications industries Use Mathematica every day Implement prototypes and proof of concepts Design and implement solutions and algorithms Validate implementations' feasibility, performance and correctness Code, concepts and algorithms provided in this presentation are not related or affiliated by any means to my employer nor my clients. All the material in this presentations are samples not suitable for production, and are provided for the sole purpose of demonstration. |
  3. 3. FunctionalProgramming-Egypt-2010.nb 3 The Function Functions in Mathematics Function Definition A Function f is a mapping from the domain X to the co-domain Y and is defined by f : X Y The curve y 2 x4 x2 1 can be defined as function by the triple notation , , x, x4 x2 1 : x (1) where is the domain of real numbers and also the co-domain, alternatively, f: , x x4 x2 1 (2) A more common notation is f x x4 x2 1 (3) Plotting a Function Mathematica supports both function and curve plotting
  4. 4. 4 FunctionalProgramming-Egypt-2010.nb Function plot In[1]:= Plot x4 x2 1 , x4 x2 1 , x, 3, 3 , PlotRange 3, AspectRatio 1, PlotStyle Black, Thick , AxesLabel x, y y 3 2 1 Out[1]= x 3 2 1 1 2 3 1 2 3
  5. 5. FunctionalProgramming-Egypt-2010.nb 5 Curve Plot In[2]:= ContourPlot y2 x4 x2 1, x, 3, 3 , y, 3, 3 , Axes True, Frame False, ContourStyle Thick , AxesLabel x, y y 3 2 1 Out[2]= x 3 2 1 1 2 3 1 2 3 Functional Programming Languages Functional Programming Languages has always been the natural choice for implementing computer derived solutions for mathematical problems including Numerical Analysis, Combinatorics and Algorithms Haskell Implementation of the Quick Sort algorithm in Haskell. Demonstrates polymorphic types, pattern matching, list comprehension and immutability qsort :: Ord a => [a] -> [a] qsort [] = [] qsort (p:xs) = qsort lesser ++ [p] ++ qsort greater where lesser = [ y | y <- xs, y < p ] greater = [ y | y <- xs, y >= p ] Scala Implementation of the Quick Sort algorithm in Scala. Demonstrates polymorphic types, pattern matching, object-oriented support and partial functions application def qsort[T <% Ordered[T]](list:List[T]):List[T] = { list match { case Nil Nil case p::xs val (lesser,greater) = xs partition (_ <= p) qsort(lesser) ++ (p :: qsort(greater)) } }
  6. 6. 6 FunctionalProgramming-Egypt-2010.nb def qsort[T <% Ordered[T]](list:List[T]):List[T] = { list match { case Nil Nil case p::xs val (lesser,greater) = xs partition (_ <= p) qsort(lesser) ++ (p :: qsort(greater)) } } Mathematica Implementation of the Quick Sort algorithm in Mathematica. Demonstrates pattern matching, pattern guards and rule based programming In[3]:= ClearAll qsort ; qsort : ; qsort p_, xs___ : Cases xs , y_ ; y p , Cases xs , y_ ; y p . lesser_, greater_ qsort lesser , p, qsort greater Flatten In[6]:= qsort 0, 9, 1, 8, 3, 6, 2, 7, 5, 4 Out[6]= 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 |
  7. 7. FunctionalProgramming-Egypt-2010.nb 7 Functions in Mathematica Numeric Computations Numeric computations through function calls In[210]:= Plus 1, Exp Times I , Pi Out[210]= 0 Or through Mathematical notation Π In[8]:= 1 Out[8]= 0 Arbitrary precision In[9]:= N Π, 100 Out[9]= 3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068 Calculus Symbolic Calculus 1 In[10]:= Θ Sin Θ 1 Π Out[10]= 2 EllipticF Θ , 2 2 2 Accurate results In[11]:= Θ FullSimplify 1 Out[11]= Sin Θ Algorithms Optimized algorithms
  8. 8. 8 FunctionalProgramming-Egypt-2010.nb In[12]:= Timing Sort RandomInteger 1, 10 000 000 , 100 000 Short Out[12]//Short= 0.04, 9, 400, 418, 657, 719, 859, 947, 1057, 1061, 99 982 , 9 998 951, 9 998 953, 9 999 009, 9 999 077, 9 999 123, 9 999 227, 9 999 236, 9 999 314, 9 999 497 Algorithm analysis 2n In[13]:= T n . RSolve T n T 1, T 1 1 , T n , n 3 Log n Out[13]= 1 3 Log 2 Graphics Stunning graphics In[236]:= ParametricPlot3D Cos Φ Sin Θ , Sin Φ Sin Θ , Cos Θ , Φ, 0, 2 Π , Θ, 0, Π , PlotPoints 100, Mesh None, ColorFunction x, y, z, Φ, Θ Hue , ColorFunctionScaling False, Boxed False, Axes False Magnify , .5 & & Θ Φ Θ Φ Sin Θ Cos Θ , Sin Φ Cos Φ , Sin Θ Φ Cos Θ Φ , , , Partition , 3 & Grid Out[236]= |
  9. 9. FunctionalProgramming-Egypt-2010.nb 9 Define Your Own Function A Function as a Rule Functions can be defined as a delayed assignment rule In[15]:= f x_ : x3 x 1 Functions can accept multiple parameters In[16]:= f x_, a_, b_ : x3 ax b
  10. 10. 10 FunctionalProgramming-Egypt-2010.nb Calling a Function Default Form Default Form, notice the square brackets In[17]:= f 2 Out[17]= 7 Prefix Form Prefix Form, using the @ sign In[18]:= f 2 Out[18]= 7 Postfix Form Postfix Form, using the // sign In[19]:= 2Π Sin Out[19]= 0 Infix Form Infix Form, function surrounded by ~ sign In[20]:= 1, 2, 3 Join 4, 5, 6 Out[20]= 1, 2, 3, 4, 5, 6 Multiple Arguments Function call with multiple actual arguments
  11. 11. FunctionalProgramming-Egypt-2010.nb 11 In[223]:= Magnify ChemicalData "Ethanol", "MoleculePlot" , 1 Out[223]= |
  12. 12. 12 FunctionalProgramming-Egypt-2010.nb Domains and Patterns Domains as Patterns Domains can be specified as patterns In[22]:= ClearAll square ; square i_Integer : i2 2 square i_Real : Floor i 2 square i_Complex : Re i 2 square a_Symbol : a square SomeDataType a_ : a2 In[28]:= square 2.1 , square 2 , square 2 3 , square x , square SomeDataType x Out[28]= 4, 4, 4, x2 , x2 Patterns and Lists Expressive patterns on lists, notice how ordering is important One or more __ Zero or more ___ 2 3 In[29]:= ClearAll listOp ; listOp : listOp x_, y_ : y, x flip a tuple listOp x_, xs__ : xs drop the first element listOp l : __ ... : Reverse l match a list of lists In[34]:= listOp 1, 2, 3 , listOp 1, 2 , listOp 1, 2 , 3, 4 Out[34]= 2, 3 , 2, 1 , 3, 4 , 1, 2 |
  13. 13. FunctionalProgramming-Egypt-2010.nb 13 Guards on Patterns Guards on Domains Guards can be specified using the Pattern ? Predicate notation In[35]:= ClearAll collatz ; collatz n_Integer ? EvenQ : n 2 Matches only even integers collatz n_Integer : 3 n 1 Matches all integers In[38]:= collatz 3 , collatz 4 Out[38]= 10, 2 Arbitrary Conditions A condition can be specified on a pattern using Pattern /; Predicate notation Example : Bubble Sort By Dr Jon D Harrop, 2010 (Modified) Using pattern matching and conditions on patterns, bubble sort can then be defines as In[39]:= bubbleSort xs___, x_, y_, ys___ : bubbleSort xs, y, x, ys ; x y For example In[40]:= bubbleSort 3, 2, 1 Out[40]= bubbleSort 1, 2, 3 We can simply extract the sorted list using the rule In[211]:= bubbleSort 3, 2, 1 . _ sorted_ sorted Out[211]= 1, 2, 3 |
  14. 14. 14 FunctionalProgramming-Egypt-2010.nb Pure Functions Defining a Pure Function The Notation A function that squares its argument In[41]:= x x2 2 Out[41]= Function x, x Square function applied at 3 In[42]:= x x2 3 Out[42]= 9 A function with a list as its argument In[43]:= x, y x2 y2 3, 4 Out[43]= 5 The (# &) Notation The notation is programmatically equivalent to the notation Square function applied at 3 2 In[44]:= & 3 Out[44]= 9 A pure function with 2 arguments, notice the numbering after # In[45]:= 12 22 & 3, 4 Out[45]= 5
  15. 15. FunctionalProgramming-Egypt-2010.nb 15 Higher-Order Functions Map Map as a function call In[46]:= Map x x2 , a, b, c, d Out[46]= a2 , b2 , c2 , d2 Using the ( /@ ) Notation In[47]:= x x2 a, b, c, d Out[47]= a2 , b2 , c2 , d2 The mapped function can be composed of any type of expression In[48]:= Mean , Variance , PDF , x & NormalDistribution Μ, Σ , MaxwellDistribution Σ , GammaDistribution Α, Β x2 x Μ 2 2 x 2 Σ2 x2 2 Σ2 2 8 3 Π Σ2 Π Β x 1 Α Β Α 2 2 Out[48]= Μ, Σ , , 2 Σ, , , Α Β, Α Β , 2Π Σ Π Π Σ3 Gamma Α Or a composed function In[215]:= ChemicalData "Caffeine", Magnify , .5 & & "CHColorStructureDiagram", "CHStructureDiagram", "ColorStructureDiagram", "StructureDiagram", "MoleculePlot", "SpaceFillingMoleculePlot" Partition , 3 & Grid , Frame All & H H O H O H C C O H H H H H H C C C C N N N C N C N H H N C H C H C C C C N N O N O N N O N H C H H C H H H Out[215]= O N N N O N Or used inside a manipulation
  16. 16. 16 FunctionalProgramming-Egypt-2010.nb Or used inside a manipulation In[50]:= ClearAll tux, browsers ; tux, browsers , , , ; Manipulate Map ImageCompose tux, ImageResize , Scaled scale , horizontal, vertical &, browsers GraphicsRow, scale, 0.5, 1 , horizontal, 60, 200, 10 , vertical, 60, 200, 10 scale horizontal vertical Out[52]= Select In[53]:= Select 1, 3, 2, 5, 0 , n 0 n Out[53]= 3, 5 Fold In[54]:= Fold x, y x y, 0, a, b, c, d Out[54]= a b c d Folding to the left or to the right
  17. 17. FunctionalProgramming-Egypt-2010.nb 17 In[218]:= Manipulate Fold F 1, 2 &, x, Take a, b, c, d , step TreeForm , AspectRatio 1.2, PlotLabel "Fold Left" &, Fold F 2, 1 &, x, Take a, b, c, d , step TreeForm , AspectRatio 1.2, PlotLabel "Fold Right" & GraphicsRow Magnify , 1 &, step, 0, 4, 1 step Fold Left Fold Right F F Out[218]= F d d F F c c F F b b F x a a x Power Set In[56]:= Fold set, element set Append element & set , , Α, Β, Γ Out[56]= , Α , Β , Γ , Α, Β , Α, Γ , Β, Γ , Α, Β, Γ One more time In[57]:= FoldList set, element set Append element & set , , Α, Β, Γ Column , Α Out[57]= , Α , Β , Α, Β , Α , Β , Γ , Α, Β , Α, Γ , Β, Γ , Α, Β, Γ NestWhileList x In[58]:= NestWhileList x , 32, i i 1 2 Out[58]= 32, 16, 8, 4, 2, 1 Pascal Triangle Pascal triangle can be defined using binomials
  18. 18. 18 FunctionalProgramming-Egypt-2010.nb In[59]:= Table Binomial n, k , n, 0, 4 , k, 0, n Column , Center & 1 1, 1 Out[59]= 1, 2, 1 1, 3, 3, 1 1, 4, 6, 4, 1 This can defined using the pattern In[60]:= tuples x_ : tuples x_, y_, ys___ : x y Join tuples y, ys pascal h_ : NestWhileList 1 Join tuples Join 1 &, 1 , Length h & pascal 9 Column , Center & 1 1, 1 1, 2, 1 1, 3, 3, 1 Out[63]= 1, 4, 6, 4, 1 1, 5, 10, 10, 5, 1 1, 6, 15, 20, 15, 6, 1 1, 7, 21, 35, 35, 21, 7, 1 1, 8, 28, 56, 70, 56, 28, 8, 1 And nicely manipulated, In[64]:= ClearAll hexagon, render ; 2Πk 2Πk hexagon x_, y_ : Polygon Table Sin x, Cos y , k, 6 6 6 Length l render l_List : Graphics Hue , hexagon 0, 0 , Text Style , White, Bold, 10 & l Π GraphicsRow , ImageSize 50 Length , 30 , Spacings 2, 0 & Manipulate Graphics render & pascal h GraphicsColumn , Alignment Center, Spacings 0, 8 & Magnify , 1.5 &, h, 1, 5, 1 h 1 1 1 Out[67]= 1 2 1 1 3 3 1 1 4 6 4 1 3 n + 1 Problem
  19. 19. FunctionalProgramming-Egypt-2010.nb 19 3 n + 1 Problem n 2 EvenQ n In[68]:= collatz : n ∂ 3n 1 OddQ n NestWhileList collatz, 200, m m 1 Out[69]= 200, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 In[70]:= Manipulate MapIndexed Text Style 1, Blue, Italic, 45 2 &, NestWhileList collatz, x, m m 1 , x, 10, 100, 10 x Out[70]= 70 35 106 53 160 , , , , , 80 40 20 10 5 16 8 4 2 1 , , , , , , , , , |
  20. 20. 20 FunctionalProgramming-Egypt-2010.nb Example: Binary Search Tree Haskell Haskell approach to Algebraic Data Types and Pattern Matching data Tree a = Empty | Node (Tree a) a (Tree a) deriving(Show, Eq) insert :: Ord a => a -> Tree a -> Tree a insert n Empty = Node Empty n Empty insert n (Node left x right) | n < x = Node (insert n left) x right | otherwise = Node left x (insert n right) inorder :: Ord a => Tree a -> [a] inorder Empty = [] inorder (Node left x right) = inorder left ++ [x] ++ inorder right bst :: Ord a => [a] -> Tree a bst [] = Empty bst (x : xs) = foldr(insert) (insert x Empty) xs Mathematica The same algorithm in Mathematica syntax In[71]:= ClearAll tree, insert, inorder, bst ; tree nil node _, _, _ ; insert n_, nil : node nil, n, nil ; insert n_, node l_, x_, r_ ; n x : node insert n , l , x , r ; insert n_, node l_, x_, r_ : node l, x, insert n, r ; inorder nil : ; inorder node l_, x_, r_ : inorder l Join x Join inorder r ; bst : nil bst x_, xs___ : Fold insert 2, 1 & , insert x, nil , xs ; In[80]:= inorder bst 8, 3, 10, 1, 6, 9, 12, 4, 7, 13, 11 Out[80]= 1, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13 Visualizing a Binary Search Tree In order to be able to visualize a BST, we need to convert the tree of nodes into a {src dst, ...} representation. Using our structure, the binary search tree of the list {8, 3, 10, 1, 6} is In[81]:= bst 8, 3, 10, 1, 6 Out[81]= node node node nil, 1, nil , 3, node nil, 6, nil , 8, node nil, 10, nil We can then define
  21. 21. FunctionalProgramming-Egypt-2010.nb 21 In[82]:= ClearAll tuple ; tuple nil : tuple node nil, x_, nil : tuple node l : node _, a_, _ , x_, nil : tuple l Join x a tuple node nil, x_, r : node _, b_, _ : x b Join tuple r tuple node l : node _, a_, _ , x_, r : node _, b_, _ : tuple l Join x a, x b Join tuple r Then In[88]:= tuple bst 8, 3, 10, 1, 6, 9, 12, 4, 7, 13, 11 Out[88]= 3 1, 3 6, 6 4, 6 7, 8 3, 8 10, 10 9, 10 12, 12 11, 12 13 Plotting the BST In[89]:= TreePlot tuple bst 8, 3, 10, 1, 0, 4, 6, 5, 9, 12, 2, 7, 13, 11 , Automatic, 8, VertexLabeling True, DirectedEdges True, PlotLabel " 8,3,10,1,0,4,6,5,9,12,2,7,13,11 ", VertexRenderingFunction ps, v White, EdgeForm Black, Thick , Disk ps, .2 , Black, Text v, ps 8,3,10,1,0,4,6,5,9,12,2,7,13,11 8 3 10 Out[89]= 1 4 9 12 0 2 6 11 13 5 7 We can then visualize a random BST construction step by step
  22. 22. 22 FunctionalProgramming-Egypt-2010.nb In[214]:= With l RandomInteger 1, 100 , 10 DeleteDuplicates , Manipulate Text Style Framed l , 12, Bold, Black , TreePlot tuple bst Take l, step , Automatic, l 1 , VertexLabeling True, DirectedEdges True, VertexRenderingFunction ps, v White, EdgeForm Black, Thick , Disk ps, .2 , Black, Text v, ps , ImageSize 400, 300 , ImagePadding 1 , Text Style Framed inorder bst Take l, step , 12, Bold, Blue Column , Center &, step, 2, Length l , 1 step 91, 57, 43, 99, 92, 50, 11, 48, 89 91 57 99 Out[214]= 43 92 11 50 48 11, 43, 48, 50, 57, 91, 92, 99 |
  23. 23. FunctionalProgramming-Egypt-2010.nb 23 Pattern Matching and Transformation Matching Cases Match Cases In[91]:= Cases a2 , b3 , c4 , d5 , e6 , _2 Out[91]= a2 Match Cases with a predicate In[92]:= Cases a2 , b3 , c4 , d5 , e6 , _n_ ; EvenQ n Out[92]= a2 , c4 , e6 Match Cases with a predicate and a transformation rule In[93]:= Cases a2 , b3 , c4 , d5 , e6 , x_n_ ; OddQ n xn 1 Out[93]= b4 , d6 Pattern Matching and Rules Swap is simple In[94]:= a, b . x_, y_ y, x Out[94]= b, a Decompose an expression In[95]:= x Sin Θ y Cos Θ . a_ Sin Α_ b_ Cos Α_ a, b, Α Out[95]= x, y, Θ Match rules
  24. 24. 24 FunctionalProgramming-Egypt-2010.nb In[96]:= ClearAll g ; g Μ Α, Μ Β, Α i, Α j, Β a, Β b ; GraphPlot g, VertexLabeling True, AspectRatio 0.2 Magnify , 1.5 & i b Out[98]= Α Μ Β j a Use delayed rules In[99]:= find the children of the vertex Β in the Graph g . Β x_ x , _ & g Flatten Out[99]= a, b Example: Palindrome Generate a sequence of probable palindrome integers In[100]:= alg196 n_Integer : n IntegerDigits n Reverse FromDigits NestWhileList alg196, 77, 10 000 000 & Out[101]= 77, 154, 605, 1111, 2222, 4444, 8888, 17 776, 85 547, 160 105, 661 166, 1 322 332, 3 654 563, 7 309 126, 13 528 163 Recursively test if a sequence is a palindrome In[102]:= isPalindrome seq_List : seq . x_ True, x_, xs___, y_ x y && isPalindrome xs Test the sequence In[103]:= Select NestWhileList alg196, 77, 10 000 000 & , isPalindrome IntegerDigits & Out[103]= 77, 1111, 2222, 4444, 8888, 661 166, 3 654 563 Example: Run Length Encoding Perform run length encoding on a finite sequence By Frank Zizza, 1990 Use replace repeated (//.)
  25. 25. FunctionalProgramming-Egypt-2010.nb 25 In[104]:= runLengthEncoding l_List : Map , 1 &, l . head___, x_, n_ , x_, m_ , tail___ head, x, n m , tail In[105]:= runLengthEncoding a, a, a, b, b, c, c, c, c, a, a Out[105]= a, 3 , b, 2 , c, 4 , a, 2 How does the magic happen? First define the magical rule In[106]:= ClearAll rule ; rule head___, x_, n_ , x_, m_ , tail___ head, x, n m , tail ; Second, generate a list tuples of the form e1 , 1 , e2 , 1 , ..., en , 1 In[108]:= Map , 1 &, a, a, a, b, b, c, c, c Out[108]= a, 1 , a, 1 , a, 1 , b, 1 , b, 1 , c, 1 , c, 1 , c, 1 Keep applying the transformation until the input is exhausted In[109]:= . rule Out[109]= a, 2 , a, 1 , b, 1 , b, 1 , c, 1 , c, 1 , c, 1 |
  26. 26. 26 FunctionalProgramming-Egypt-2010.nb Example: Mathematica in Bioinformatics XML in Mathematica Mathematica supports a large variety of data formats, XML happens to be one of them In[225]:= xml Import " work presentation Mathematica Conference 2010 code xml graph.xml", "XML" Out[225]= XMLObject Document , XMLElement v, id root , XMLElement v, id a , XMLElement v, id a1, cost 1 , , XMLElement v, id a2, cost 2 , , XMLElement v, id a3, cost 3 , , XMLElement v, id b , XMLElement v, id b1, cost 4 , , XMLElement v, id b2, cost 5 , , XMLElement v, id b3, cost 6 , , XMLElement v, id c , XMLElement v, id c1, cost 7 , , XMLElement v, id c2, cost 8 , , XMLElement v, id c3, cost 9 , , The XML document represents a graph, each vertex v is represented as an element. Children of an element are connected to the parent, the hierarchy represents the edges. The following functions use Mathematica XML support to create a garph representation of the XML docu- ment and plot it In[226]:= ClearAll root, id, cost, rec ; root XMLObject _ _, r_, _ : r id XMLElement _, as_, ___ : "id" . as cost XMLElement _, as__ , ___ : "cost" . as, _ "0" rec e : XMLElement "v", _, cs : ___ : id e id , cost & cs Join rec cs Flatten , 1 & Recursively walk the element structure and create a Mathematica graph representation In[231]:= rec root xml Out[231]= root a, 0 , root b, 0 , root c, 0 , a a1, 1 , a a2, 2 , a a3, 3 , b b1, 4 , b b2, 5 , b b3, 6 , c c1, 7 , c c2, 8 , c c3, 9 Plot the extracted graph
  27. 27. FunctionalProgramming-Egypt-2010.nb 27 In[232]:= rec root xml GraphPlot , VertexLabeling True, EdgeLabeling True & Magnify , 1 & a1 c3 1 9 a2 c2 2 8 a c 0 0 3 root 7 Out[232]= a3 c1 0 b 4 6 b1 5 b3 b2 Sequence Alignment In[118]:= xml Import " work presentation Mathematica Conference 2010 code xml sequenceML.xml", "XML" ; In[119]:= ClearAll root, sequenceList, sequence, element ; root XMLObject _ _, r_, _ : r sequenceList XMLElement "sequenceML", _, seqs_ : sequence seqs sequence e : XMLElement "sequence", "seqID" id_ , cs_ : seqID id, name element "name", cs , description element "description", cs , aminoAcidSequence element "aminoAcidSequence", cs element name_, elements_ : Cases elements, XMLElement n_, , value_ ;n name value First In[124]:= root xml sequenceList First Out[124]= seqID gi 58374180 gb AAW72226.1 , name HA, description Influenza A virus A duck Shandong 093 2004 H5N1 , aminoAcidSequence MEEIVLLLAIVSLVKSDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKTHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEFINVPEWSYIVEKANPAND LCYPGDFNDYEELKHLLSRINHFEKIQIIPKSSWSDHEASSGVSSACPYNGKSSFFRNVVWLIKKNSSYPTIKRSYNNTNQEDLLILWGIHHPNDAAE QTKLYQNPTTYISVGTSTLNQRLVPKIATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPEYAYKIVKKGDSAIMKSELEYGNCNTKCQTPMGA INSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRNTPQRERRRKKRGLFGAIAGFIEGGWQGMVDGWYGYHHSNEQGSGYAADKESTQKAIDGVTNKV NSIIDKMNTQFEAVGREFNNLERRIENLNKKMEDGFLDVWTYNAELLVLMENERTLDFHDSNVKNLYDKVRLQLRDNAKELGNGCFEFYHKCDNECME SVKNGTYDYPRYSEEARLNREEISGVKLESMGTYQILSIYSTVASSLALAIMVAGLSLWMCSNGSLQCRICI
  28. 28. 28 FunctionalProgramming-Egypt-2010.nb In[125]:= first, last root xml sequenceList . x_, ___, y_ x, y Out[125]= seqID gi 58374180 gb AAW72226.1 , name HA, description Influenza A virus A duck Shandong 093 2004 H5N1 , aminoAcidSequence MEEIVLLLAIVSLVKSDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKTHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEFINVPEWSYIVEKANPA NDLCYPGDFNDYEELKHLLSRINHFEKIQIIPKSSWSDHEASSGVSSACPYNGKSSFFRNVVWLIKKNSSYPTIKRSYNNTNQEDLLILWGIHHPN DAAEQTKLYQNPTTYISVGTSTLNQRLVPKIATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPEYAYKIVKKGDSAIMKSELEYGNCNTKC QTPMGAINSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRNTPQRERRRKKRGLFGAIAGFIEGGWQGMVDGWYGYHHSNEQGSGYAADKESTQKA IDGVTNKVNSIIDKMNTQFEAVGREFNNLERRIENLNKKMEDGFLDVWTYNAELLVLMENERTLDFHDSNVKNLYDKVRLQLRDNAKELGNGCFEF YHKCDNECMESVKNGTYDYPRYSEEARLNREEISGVKLESMGTYQILSIYSTVASSLALAIMVAGLSLWMCSNGSLQCRICI , seqID gi 108671045 gb ABF93441.1 , name hemagglutinin, description Influenza A virus St Jude H5N1 influenza seed virus 163222 , aminoAcidSequence MEKIVLLLAIVSLVKSDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKTHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEFLNVPEWSYIVEKINPA NDLCYPGNFNDYEELKHLLSRINHFEKIQIIPKSSWSDHEASSGVSSACPYQGRSSFFRNVVWLIKKNNAYPTIKRSYNNTNQEDLLVLWGIHHPN DAAEQTRLYQNPTTYISVGTSTLNQRLVPKIATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPENAYKIVKKGDSTIMKSELEYGNCNTKC QTPIGAINSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRNSPQIETRGLFGAIAGFIEGGWQGMVDGWYGYHHSNEQGSGYAADKESTQKAIDGV TNKVNSIIDKMNTQFEAVGREFNNLERRIENLNKKMEDGFLDVWTYNAELLVLMENERTLDFHDSNVKNLYDKVRLQLRDNAKELGNGCFEFYHRC DNECMESVRNGTYDYPQYSEEARLKREEISGVKLESIGTYQILSIYSTVASSLALAIMVAGLSLWMCSNGSLQCRICI In[126]:= SequenceAlignment aminoAcidSequence . first, aminoAcidSequence . last Out[126]= ME, E, K , IVLLLAIVSLVKSDQICIGYHANNSTEQVDTIMEKNVTVTHAQDILEKTHNGKLCDLDGVKPLILRDCSVAGWLLGNPMCDEF, I, L , NVPEWSYIVEK, A, I , NPANDLCYPG, D, N , FNDYEELKHLLSRINHFEKIQIIPKSSWSDHEASSGVSSACPY, N, Q , G, K, R , SSFFRNVVWLIKKN, SS, NA , YPTIKRSYNNTNQEDLL, I, V , LWGIHHPNDAAEQT, K, R , LYQNPTTYISVGTSTLNQRLVPKIATRSKVNGQSGRMEFFWTILKPNDAINFESNGNFIAPE, Y, N , AYKIVKKGDS, A, T , IMKSELEYGNCNTKCQTP, M, I , GAINSSMPFHNIHPLTIGECPKYVKSNRLVLATGLRN, T, S , PQ, R, I , E, RRRKK, T , RGLFGAIAGFIEGGWQGMVDGWYGYHHSNEQGSGYAADKESTQKAIDGVTNKVNSIIDKMNTQFEAVGREFNNLERRIENLNKKMEDGFLDVWTYNAELLVLMEN ERTLDFHDSNVKNLYDKVRLQLRDNAKELGNGCFEFYH, K, R , CDNECMESV, K, R , NGTYDYP, R, Q , YSEEARL, N, K , REEISGVKLES, M, I , GTYQILSIYSTVASSLALAIMVAGLSLWMCSNGSLQCRICI |
  29. 29. FunctionalProgramming-Egypt-2010.nb 29 Example: Mathematica XQuery XQuery A fluent XML Query Language from W3C XQuery FLWOR Expression An XQuery expression is typically a for, let, where, order by and return construct for $x in doc("books.xml")/bookstore/book where $x/price>30 order by $x/title return $x/title XML in Mathematica Mathematica XML Support Import the XML document In[127]:= xml Import " work presentation Mathematica Conference 2010 xml books.xml", "XML" Out[127]= XMLObject Document XMLObject Declaration Version 1.0, Encoding ISO 8859 1 , XMLElement bookstore, , XMLElement book, category COOKING , XMLElement title, lang en , Everyday Italian , XMLElement author, , Giada De Laurentiis , XMLElement year, , 2005 , XMLElement price, , 30.00 , XMLElement book, category CHILDREN , XMLElement title, lang en , Harry Potter , XMLElement author, , J K. Rowling , XMLElement year, , 2005 , XMLElement price, , 29.99 , XMLElement book, category WEB , XMLElement title, lang en , XQuery Kick Start , XMLElement author, , James McGovern , XMLElement author, , Per Bothner , XMLElement author, , Kurt Cagle , XMLElement author, , James Linn , XMLElement author, , Vaidyanathan Nagarajan , XMLElement year, , 2003 , XMLElement price, , 49.99 , XMLElement book, category WEB , XMLElement title, lang en , Learning XML , XMLElement author, , Erik T. Ray , XMLElement year, , 2003 , XMLElement price, , 39.95 , XQuery like DSL in Mathematica Define an XQuery like Domain Specific Language (DSL) for XML processing using Mathematica's Functional approach The tiny language is a set of higher - order functions, each function returns a function that can act on the XML axis
  30. 30. 30 FunctionalProgramming-Egypt-2010.nb In[128]:= ClearAll doc, where, orderBy, e, att, attribute, data, return ; doc XMLObject Document _, root_, _ : root; where Select; orderBy Sort; e n_String : es Cases es, XMLElement n , _, _ , ; att XMLElement _ , rules_, _ , n_String : n . rules Join _ Φ ; attribute XMLElement _ , rules_, _ , n_String : n n . rules ; attribute n_String : es attribute , n & es; attribute n_String, pred_ : el pred att el, n ; data n_String : es Cases es, XMLElement n , _, d_ d, ; data n_String, pred_ : es Cases es, XMLElement n , _, d_ ; pred d , ; return es_, f_ : f es; Mathematica XQuery DSL in Action XPath Expressions All book titles In[140]:= doc xml e "book" data "title" Out[140]= Everyday Italian, Harry Potter, XQuery Kick Start, Learning XML All book authors In[141]:= doc xml e "book" data "author" Out[141]= Giada De Laurentiis, J K. Rowling, James McGovern, Per Bothner, Kurt Cagle, James Linn, Vaidyanathan Nagarajan, Erik T. Ray The return function Return a { {author..} title} tuple In[142]:= doc xml e "book" return bs data "author" . a_ a data "title" . t_ t & bs Out[142]= Giada De Laurentiis Everyday Italian, J K. Rowling Harry Potter, James McGovern, Per Bothner, Kurt Cagle, James Linn, Vaidyanathan Nagarajan XQuery Kick Start, Erik T. Ray Learning XML The where function All titles with price > 30 In[143]:= doc xml e "book" where b b data "price", ToExpression 30 & return data "title" Out[143]= XQuery Kick Start, Learning XML All titles in the COOKING category
  31. 31. FunctionalProgramming-Egypt-2010.nb 31 In[144]:= doc xml e "book" where b b attribute "category", "COOKING" & return data "title" Out[144]= Everyday Italian All titles in the WEB category with price > 40 In[145]:= doc xml e "book" where b b data "price", ToExpression 40 & && b attribute "category", "WEB" & return data "title" Out[145]= XQuery Kick Start The order by function Order by title in descending order In[146]:= doc xml e "book" where b b attribute "category", "WEB" & orderBy Order 1 data "title" , 2 data "title" 0 & return data "title" Out[146]= XQuery Kick Start, Learning XML All books in WEB category, ordered by title in ascending order, formatted as { title price } list In[147]:= doc xml e "book" where b b attribute "category", "WEB" & orderBy Order 1 data "title" , 2 data "title" 0 & return bs data "title" . t_ t data "price" . p_ p & bs Out[147]= Learning XML 39.95, XQuery Kick Start 49.99 |
  32. 32. 32 FunctionalProgramming-Egypt-2010.nb Example: SQL Establish a Database Connection In[148]:= HSQL memory db Needs "DatabaseLink`" bookstore OpenSQLConnection ; Create the Book table In[150]:= SQLDropTable bookstore, & SQLTableNames bookstore ; SQLCreateTable bookstore, SQLTable "BOOK" , SQLColumn "ID", "DataTypeName" "INTEGER" , SQLColumn "TITLE", "DataTypeName" "VARCHAR", "DataLength" 128 , SQLColumn "YEAR", "DataTypeName" "VARCHAR", "DataLength" 4 , SQLColumn "PRICE", "DataTypeName" "FLOAT" ; SQLTableNames bookstore Out[152]= BOOK Load books from XML In[153]:= ClearAll books ; books doc xml e "book" return bs data "title" . t_ t, data "year" . y_ y, data "price" . p_ p & bs Out[154]= Everyday Italian, 2005, 30.00 , Harry Potter, 2005, 29.99 , XQuery Kick Start, 2003, 49.99 , Learning XML, 2003, 39.95 Load books into the Database In[155]:= MapIndexed SQLInsert bookstore, "BOOK", "ID", "TITLE", "YEAR", "PRICE" , 2 Join 1 &, books ; Query the Database In[156]:= SQLSelect bookstore, "BOOK" TableForm Out[156]//TableForm= 1 Everyday Italian 2005 30. 2 Harry Potter 2005 29.99 3 XQuery Kick Start 2003 49.99 4 Learning XML 2003 39.95 Close the Database Connection
  33. 33. FunctionalProgramming-Egypt-2010.nb 33 Close the Database Connection In[157]:= CloseSQLConnection bookstore |
  34. 34. 34 FunctionalProgramming-Egypt-2010.nb Example: Geometric Transformation The RotationTransform Function Understanding the Function In[158]:= RotationTransform Θ Cos Θ Sin Θ 0 Out[158]= TransformationFunction Sin Θ Cos Θ 0 0 0 1 In[159]:= RotationTransform Θ x, y Out[159]= x Cos Θ y Sin Θ , y Cos Θ x Sin Θ Creating a Replacement Rule In[160]:= RotationTransform Θ x, y . a_, b_ x a, y b Out[160]= x x Cos Θ y Sin Θ , y y Cos Θ x Sin Θ Testing Our Rule In[161]:= y x2 . x x Cos Θ y Sin Θ , y y Cos Θ x Sin Θ 2 Out[161]= y Cos Θ x Sin Θ x Cos Θ y Sin Θ 2 In[162]:= y Cos Θ x Sin Θ x Cos Θ y Sin Θ .Θ 90 ° Out[162]= x y2
  35. 35. FunctionalProgramming-Egypt-2010.nb 35 In[163]:= ContourPlot y x2 , x y2 , x, 2 Π, 2 Π , y, 2 Π, 2 Π , Axes True, Frame False, ContourStyle Thick, Red , Thick, Blue 6 4 2 Out[163]= 6 4 2 2 4 6 2 4 6 Creating a Simple Rotate Function In[164]:= rotate eq_, Θ_ : eq . RotationTransform Θ x, y . a_, b_ x a, y b In[165]:= rotate y x2 , 90 ° , rotate y Sin x , 30 ° FullSimplify 1 Out[165]= x y2 , x 3 y 2 Sin 3 x y 2
  36. 36. 36 FunctionalProgramming-Egypt-2010.nb The Rotate Function in Action In[233]:= Show ContourPlot rotate y Sin x , 1 ° Evaluate, x, 2 Π, 2 Π , y, 2 Π, 2 Π , Frame False, Exclusions rotate y Sin x , 1 ° Evaluate, x2 y2 25 , 1 ContourStyle Thickness 0.004 , Hue Π & Range 0, 360, 10 Magnify , 1 & Out[233]= |
  37. 37. FunctionalProgramming-Egypt-2010.nb 37 Example: Tree Chains Graph Algebraic Data Type Graph Structure and supporting functions In[167]:= ClearAll Graph, graph, Vertex, vertex, Leaf, leaf, Branch, tail, branch, succ, dft, concatMap, chains, toRules, vrf ; Vertex Vertex d$ : _ ; vertex d_ : Vertex d ; value Vertex : d$; Graph Graph rep$ : _Vertex , _Vertex ... ... ; graph rep : _Vertex , _Vertex ... ... : Graph rep ; graph rep : _Vertex _Vertex ... ... : Graph . x_ y : __ x, y & rep ; rep Graph : rep$; Leaf Leaf v$ : Vertex ; leaf v : Vertex : Leaf v ; vertex Leaf : v$; Branch Branch v$ : Vertex, tail$ : __ ; branch v : Vertex, : leaf v ; branch v : Vertex, l : ___ : Branch v, l ; vertex Branch : v$; tail Branch : tail$; succ g : Graph, v : Vertex : Cases rep g, u : Vertex, adj_ ; u v adj Flatten; dft g : Graph, v : Vertex : branch v, dft g, & succ g, v ; concatMap f_, l : __ : Fold Join, , Flatten f & l ; chains g : Graph : chains dft g, rep g 1 1 ; chains l : Leaf : vertex l ; chains b : Branch : concatMap vertex b, &, chains & tail b Flatten , 1 &; toRules g : Graph : . x_, y : __ x & y & rep g Flatten; toRules ch : _Vertex ... : ch . x_ , x_, y_, xs___ x y Join toRules y, xs ; toRules chs : _Vertex ... ... : toRules & chs; vrf ps, v White, EdgeForm Black , Disk ps, .3 , Black, Text value v, ps ; Graph Instance In[193]:= Dynamic g ; g graph vertex Μ vertex Α, Β, Γ , vertex Α vertex Α1 , Α2 , Α3 , vertex Β vertex Β1 , Β2 , Β3 , vertex Γ vertex Γ1 , Γ2 , Γ3 ;
  38. 38. 38 FunctionalProgramming-Egypt-2010.nb Graph Plot In[219]:= LayeredGraphPlot g toRules, VertexRenderingFunction vrf Magnify , 1 & Dynamic Μ Out[219]= Α Β Γ Α1 Α2 Α3 Β1 Β2 Β3 Γ1 Γ2 Γ3 Chains and Chains Plot In[196]:= chains g Dynamic Out[196]= Vertex Μ , Vertex Α , Vertex Α1 , Vertex Μ , Vertex Α , Vertex Α2 , Vertex Μ , Vertex Α , Vertex Α3 , Vertex Μ , Vertex Β , Vertex Β1 , Vertex Μ , Vertex Β , Vertex Β2 , Vertex Μ , Vertex Β , Vertex Β3 , Vertex Μ , Vertex Γ , Vertex Γ1 , Vertex Μ , Vertex Γ , Vertex Γ2 , Vertex Μ , Vertex Γ , Vertex Γ3 In[220]:= GraphPlot , VertexRenderingFunction vrf Magnify , 1 & & toRules chains g Partition , 3 & TableForm Dynamic Μ Α Α1 Μ Α Α2 Out[220]= Μ Β Β1 Μ Β Β2 Μ Γ Γ1 Μ Γ Γ2 |
  39. 39. FunctionalProgramming-Egypt-2010.nb 39 Example: Sound The Sound of Mathematics In[198]:= n11 11, 2, 7, 9, 11, 2, 7, 5, 11, 2, 7, 4, 2, 2, 2 ; a11 n11 ; a12 12 & n11 ; a13 a12 Join a11 ; a14 a13 Join Reverse a12 Join n11 ; s1 Sound SoundNote , .1, "Flute" & a14 ; n21 2, 3, 7, 2, 3, 7, 2, 3, 9, 9, 7, 3 ; a21 n21 ; a22 12 & n21 . xs__ xs, xs ; a23 a22 Join a21 Join Reverse 12 & a21 Join Reverse a21 ; s2 Sound SoundNote , .1, "Piano" & a23 ; EmitSound s1 , s2 |

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