Description of background ideas, and the module itself.

Transcription

1 CO3008 Semantic of Programming Language 1 Chapter 1 Decription of background idea, and the module itelf. Review ome mathematic.

2 CO3008 Semantic of Programming Language 2 Overview: Background Introduction to MC308 What i a Language? What i a Programming Language? What i Syntax? What i Semantic?

3 CO3008 Semantic of Programming Language 3 Two kind of language Natural language: Some Anwer Recognized method of communicating thought and feeling: peech, hand ignal, ending gift... Formal language: A rigourouly defined ytem to convey meaning or information. We do not have a precie definition of language. Try looking up language in, ay, the Cambridge Encyclopaedia of Language.

4 CO3008 Semantic of Programming Language 4 Programming Language are formal language ued to communicate with a computer. Programming language may be low level. They give direct intruction to the computer (machine code). Programming language may be high level. The intruction given to the computer are indirect, but much cloer to general concept undertood by the uer (Java, C++,... ).

5 CO3008 Semantic of Programming Language 5 Syntax refer to particular arrangement of word and letter eg David hit the ball or if t 2 then H = Off. A grammar i a et of rule which can be ued to pecify how yntax i created. Example can be een in automata theory, or programming manual. Theorie of yntax and grammar can be developed idea are ued in compiler contruction.

6 CO3008 Semantic of Programming Language 6 Semantic i the tudy of meaning. In particular, yntax can be given meaning. The word run can mean execution of a computer program, pread of ink on paper,... Programming language yntax can be given a emantic. We need thi to write program.

7 CO3008 Semantic of Programming Language 7 Semantic decription are often informal. Conider while (expreion) command ; adapted from Kernighan and Ritchie 1978/1988, p 224: The command i executed repeatedly o long a the value of the expreion remain unequal to 0; the expreion mut have arithmetic or pointer type. The tet, including all ide effect from the expreion, occur before each execution of the command. We want to be more precie, more uccinct.

8 P CO3008 Semantic of Programming Language 8 High Level view of MC308 For variou language we hall define yntax for program P and type σ ; define type aignment P :: σ ; define operational emantic looking like P P or P V define algorithm to check that P :: σ; and compile P to a lit of machine intruction P.

9 CO3008 Semantic of Programming Language 9 Overview: Dicrete Mathematic We briefly review Logic Set Relation Function

10 CO3008 Semantic of Programming Language 10 Logic If P and Q are propoition, we can form new propoition a follow: P implie Q (ometime written P... ee the note. for all x, P (ometime written x P); Q or P Q); We hall often prove propoition of the form x X P x P where x i a propoition depending on x, and X i a given et. Eg n 2n 1i odd

11 B, interection A A and b CO3008 Semantic of Programming Language 11 Set We aume a et i undertood. A or B or... often ued to denote et. Write a Afor element of. If a i not an element of A, we write a A. Union A known. B, hould already be The carteian product of A and B i a et given by A B def a b a B

12 b CO3008 Semantic of Programming Language 12 Relation A relation R between et A and B i a ubet R A B. A binary relation R on A i a relation between A and A. If R A intead of B i a relation, it i convenient to write a R b a b R. R i reflexive iff for all a A we have a R a; R i tranitive iff for all a implie a R c; c A, a R b and b R c For example.

13 CO3008 Semantic of Programming Language 13 Function You hould know what a (total) function f :A You hould know what a partial function f :A B i. B i. Recall undefinedne and application notation, compoition, and domain of definition.

14 CO3008 Semantic of Programming Language 14 Chapter 2 Define abtract yntax tree a bit like pare tree. Explain inductive definition. Explain Rule Induction.

15 CO3008 Semantic of Programming Language 15 Overview: Abtract Syntax Outline the idea of concrete yntax (eg program a acii file) and abtract yntax (the pare tree of program).

16 CO3008 Semantic of Programming Language 16 Abtract and Concrete Syntax The text tring if true then 2 ele 3 i concrete yntax. A compiler will recognize a conditional expreion (an if-then-ele ) and three data, namely the Boolean and the two number. The three data, together with the knowledge that the tring denote a conditional, make up the emantic content of the expreion.

17 2 T 2 CO3008 Semantic of Programming Language 17 We can capture thi emantic content a a tree T 2 3 which can be denoted by the formal notation T 3 and informally by the ugared notation 3 Lexer and parer tranform text program into pare tree, ometime referred to a abtract yntax.

18 B 0 CO3008 Semantic of Programming Language 18 Here i another example of ugared tree notation l 0 l l It ha the form E1 E2 where, for example, B i l. The abtract yntax tree i l l l Think of the conditional a a contructor which act on three argument (ubprogram) to contruct a new program.

19 T CO3008 Semantic of Programming Language 19 In CO3008 we need to give precie definition of abtract yntax tree. An example: Let C l 1 l 2 l 3 c1 c 2 are label for tree node. We can pecify a et of finite be a et of contructor, which tree built from thi et by a grammar of the form T :: l1 l 2 l 3 c 1 T T c 2 T T

20 CO3008 Semantic of Programming Language 20 You need to undertand the definition of node leaf root contructor (a label for any node) children (of non-leaf node) ubtree We alo talk about ubprogram, ubexpreion outermot contructor ( = root label).

21 CO3008 Semantic of Programming Language 21 Overview: Inductively Defined et Specify inductively defined et; program, type etc will be defined thi way. BNF grammar are a form of inductive definition; abtract yntax tree were defined inductively. Define Rule Induction; propertie of program will be proved uing thi. It i important.

22 A CO3008 Semantic of Programming Language 22 Example Inductive Definition Let Var be a et of propoitional variable. Then the et Prpn of propoition of propoitional logic i inductively defined by the rule P P Var φ φ ψ ψ φ ψ φ ψ φ φ ψ φ ψ φ Each propoition i created by a deduction...

23 CO3008 Semantic of Programming Language 23 Two More Example A et R of rule for defining the et E number i R 6 R1 R2 0 R1 where E iff there i a deduction of 6. e e 2 R2 of even Suppoe that Σ i any et, which we think of a an alphabet. Each element l of Σ i letter. We inductively define the et Σ of word over the alphabet Σ by l l Σ 1 w ww w 2

24 R CO3008 Semantic of Programming Language 24 Some Notation for Rule A rule R i a pair H c where H i any finite et. Note that H might be i a bae rule., in which cae we ay that R c R If H i non-empty (ay H h 1 hk where 1 k) we ay R i an inductive rule. h 1 h 2 hk c

25 c CO3008 Semantic of Programming Language 25 that Inductively Defined Set Given a et of rule, a deduction i a finite tree uch each leaf node label c occur a a bae rule R for any non-leaf node label c, if H i the et of children of c then R i an inductive rule. H c The et I inductively defined by R conit of thoe element e which have a deduction with root node e.

26 R2 u3 u3 CO3008 Semantic of Programming Language 26 An Abtract Example Let R be the et of rule R1 R 3 R4 where R 1 u1 R2 u3 R3 u 1 u3 u4 R 4 u 1 u 4 u5 R5 u 2 u3 u6 Then a deduction for u 5 i given by u 1 u 3 u 1 u 3 u 1 u 3 u 4 u 5 u 1 u 3 u 4 u 5 The inductively defined et i I u 1 u 4 u5

27 CO3008 Semantic of Programming Language 27 Rule Induction Let I be inductively defined by a et of rule R. Suppoe we wih to how the truth of i I φ i To do thi, it i enough to how for every bae rule b R that φ b hold; and for every inductive rule h 1 whenever h i I, φ c hk R prove that h 1 and φ h 2 and and φ h k implie φ c We call φ out the h j inductive hypothee. We refer to carrying tak a verifying property cloure.

28 T L T2 N T2 CO3008 Semantic of Programming Language 28 Example Conider the et of tree T defined inductively by n n T1 T 1 T 2 T2 Let L number of T be the number of leave in T, and N -node of T. We prove T T L T N T 1 be the where the function L N:T are defined recurively by L n 1 and L T1 T2 T 1 L N n 0 and N T1 T2 T 1 N 1

29 CO3008 Semantic of Programming Language 29 Chapter 3 Decribe the program (yntax) of a imple imperative language called IMP. Review and motivate type. Give a type ytem to IMP. Decribe compile time type checking and type inference.

30 CO3008 Semantic of Programming Language 30 Overview: Syntax for IMP Program Decribe the baic building block for program. Specify the program expreion. Comment on ome grammatical convention.

31 Syntax for 1 1 CO3008 Semantic of Programming Language 31 Program Expreion for IMP built out of element of the et def def 0 T F Loc ICt BCt IOpr BOpr def def def def def l 1 n b l 2 n b (** NB **)

32 BOpr aign cond CO3008 Semantic of Programming Language 32 The et of expreion contructor i pecified by Loc ICt kip equence while The program expreion are given by P :: ccontant l iop P P P l memory location integer operator P aignment P P P while loop P equencing

33 CO3008 Semantic of Programming Language 33 We adopt ome abbreviation (known a yntactic ugar): We write P iop P for iop P P ; l : P for P l ; P ; P for P P ;... Bracketing convention: Arithmetic operator group to the left. Thu P 1 op P 2 op P 3 abbreviate Sequencing aociate to the right. P1 op P2 op P 3

34 CO3008 Semantic of Programming Language 34 Overview: Type for IMP Program Explain what a type i. Motivate the ue for type. Explain ome terminology. Define IMP type checking (compilation check). Define IMP type inference.

35 CO3008 Semantic of Programming Language 35 Defining and Motivating Type Type in a programming language are collection of object ( et ), with collection of operation acting on thee object. The type int conit of the collection of integer, together with operation uch a might be pecified a,, and o on. The action of int int bool

36 CO3008 Semantic of Programming Language 36 Statically typed language carry out type checking at compile-time. Need ome explicit type information. Ue of type Expreion organized to reduce program error. Polymorphim mean function can have many type. Thi allow code re-ue. Type tructure data, uing ADT and module.

37 CO3008 Semantic of Programming Language 37 Run time error trapped error execution halt immediately. An untrapped error execution doe not necearily halt. An example i acceing data pat the end of an array, which one can do in C! A language i afe if all yntactically legal program do not yield certain run-time error. JAVA wa claimed to be afe, but in 1997 thi wa hown not to be the cae. Proof ue MC 308 method!

38 CO3008 Semantic of Programming Language 38 Technical Definition If P can be aigned a type σ we write P :: σ and call the tatement a type aignment. Type afety i the property that if P :: σ then certain kind of error can not occur at P run-time. Given P and σ, type checking validate P :: σ. Given P, type inference i the proce of trying to find σ for which P :: σ the proce can fail.

39 CO3008 Semantic of Programming Language 39 Type for IMP The type of the language grammar σ :: are given by the A location environment L i a finite et of (location, type) pair, with type being jut or: L l1 :: l n :: l n 1 : : l m :: Given L, then any P whoe location all appear in L can (ometime) be aigned a type; we write P :: σ to indicate thi.

40 CO3008 Semantic of Programming Language 40 n :: [any n ] :: INT T :: :: TRUE F :: l :: l :: L P1 :: P2 :: P1 bop P2 :: [ bop BOpr] :: BOP :: l :: σ P :: σ l : P :: P1 :: P2 :: P3 :: P1 :: P2 :: P1 P3 :: P2 P1 P2 ::

41 CO3008 Semantic of Programming Language 41 Example: Deduction of a Type Aignment l :: l l 5 :: 5 :: 5 D2 l : 1 l : l l : l 1 ; l D3 1 ; l D4 : l l :: : l l ::

42 CO3008 Semantic of Programming Language 42 Type Inference Given L and P, there i an algorithm which will infer if P can be aigned a type. If uch a type exit we ay P i typable. The algorithm will ucceed and will output the type. If not, the algorithm fail. In a real language, uch type inference i often performed by the compiler. Given L and P, we define a function Φ which given P a input will either return a type for P, or will FAIL.

43 T P1 bop P2 P2 CO3008 Semantic of Programming Language 43 Φ Φ l τ FAIL if l :: τ L otherwie and τ or Φ if Φ FAIL otherwie P1 and Φ

44 P or CO3008 Semantic of Programming Language 44 Φ l : P if Φ l and τ Φ FAIL otherwie τ Φ P1 P2 if Φ FAIL otherwie P1 and Φ P2

45 CO3008 Semantic of Programming Language 45 Chapter 4 Explain how IMP programme execute an operational emantic. Show that the type of a program doe not change on execution. Show that a program alway give the ame anwer when run IMP i determinitic. Typed program don t yield certain error.

46 CO3008 Semantic of Programming Language 46 Overview: Tranition Semantic Motivate and define tranition emantic a method for tating preciely how a program execute. Give ome example.

47 CO3008 Semantic of Programming Language 47 State A tate i a partial function Loc. For example l1 4 l 2 T l 3 21 There i a tate denoted by l c : Loc which i the partial function l c l def c l if l l otherwie We ay that tate i updated at l by c.

48 CO3008 Semantic of Programming Language 48 Tranition Semantic Conider the following tranition, which model one tep in a program execution l :2 5 l 8 l :7 l 8 l 8 l 7 The element of Exp configuration. State will be known a Wehall inductively define a binary relation. We call it tranition relation, and any intance of a relationhip in i called a tranition tep.

49 P1 P2 P1 P2 P1 op P P2 op P P1 P2 CO3008 Semantic of Programming Language 49 l l provided that l i defined OP1 n op P1 n op P2 OP2 n1 op n2 n1 op n2 OP3 l : P1 l :P2 ASS1 l : c l c ASS2 LOC

50 P T P P1 ; P P1 CO3008 Semantic of Programming Language P2 ; P P2 2 2 P SEQ1 ; P SEQ2 P1 P2 P P P1 P2 COND1 P1 P2 P1 COND2 P1 P2 P1 P2 ; P1 P2 LOOP

51 2 ; l : 2 ; l : CO3008 Semantic of Programming Language 51 Example of Tranition A deduction (for any P): l l : : l : 2 l l 1 1 ; P l 2 ; l : l ASS2 1 ; l : l l 2 1 ; P SEQ1 l 2 SEQ1

52 CO3008 Semantic of Programming Language 52 Q i l 0 Q where Q i l : l 2 ; l : l 1. Q l 1 l 0 l 0 Q ; Q l 1 l Q ; Q l 1 l 0 T Q ; Q l 1 l 0 l : 2 ; l : l 1 ; Q l 1 l 0 ; l : l 1 ; Q l 1 l 2 l : l1 ; Q l 1 l 2

53 CO3008 Semantic of Programming Language 53 Overview: Propertie of the Semantic Program type do not change on execution. IMP i determinitic the final reult of a program run i unique; and in fact the tage of the run are unique.

54 or CO3008 Semantic of Programming Language 54 Type Preervation Given L, i enible for L if for all l :: σ in L (l) i defined (all location initialized), and l:: σ (the type of data tored in a location matche the type of the location). Take L and enible 1. Then atifie Let P 1 :: σ. Then for any P1 1 P2 2 we have P 2 :: σ. Further, if σ i either, then 1 2.

55 P1 :: σ implie P2 : : σ CO3008 Semantic of Programming Language 55 Proving Type Preervation P1 1 P2 2 σ We have to check property cloure for each of the rule defining. We look at a couple of example. (Property Cloure for LOC) We have to how that l :: σ implie l :: σ for any σ. Thi i immediate a i enible.

56 P1 :: σ implie P2 : : σ CO3008 Semantic of Programming Language 56 (Property Cloure for OP2) The induction hypothei i σ IH P1 n op P 1 P2 n op P 2 OP2 We have to prove σ n op P 1 :: σ implie n op P 2 :: σ C

57 The operational emantic of CO3008 Semantic of Programming Language 57 IMP i Determinitic i determinitic: If P P and P P then P P and

58 CO3008 Semantic of Programming Language 58 Proof of Determinim We can prove thi reult by Rule Induction. We how P P X x P X x implie X P and x

59 z y P2 and y CO3008 Semantic of Programming Language 59 We conider property cloure for P1 l : P1 P2 l : P2 ASS1 The inductive hypothei IH i Y P1 Y y implie Y We need to prove the concluion C Z l: P1 Z z implie Z l: P2 and z

60 CO3008 Semantic of Programming Language 60 Overview: IMP i Type Safe We decribe ome pecial program; we decribe ome pecial kind of tranition, and ue the idea to how IMP i type afe.

61 CO3008 Semantic of Programming Language 61 Different Kind of Tranition We define V :: c. V configuration are called terminal. They indicate proper termination of program run. Any configuration and there i no P P for which i tuck if P i non-terminal P P. WARNING: Note that any terminal configuration ha no tranition.

62 CO3008 Semantic of Programming Language 62 Given any configuration P there i a unique equence of tranition P P1 1 P2 2 An infinite tranition equence take the form P P1 1 P2 2 Pi i where no configuration Pi i i terminal or tuck.

63 CO3008 Semantic of Programming Language 63 A P finite tranition equence for a configuration take the form P P1 1 P2 2 Pm m m 1 If Pm m i either tuck or terminal we call the tranition equence complete. Make up lot of example of thee idea!!

64 CO3008 Semantic of Programming Language 64 Some Reult about IMP Type Safety Let be enible for L. Then if P :: σ i any type aignment, P i not tuck. If alo P P, then i alo enible. If P P, then might be terminal). Thu P cannot be tuck (but i type afe. Thi follow from the two reult above why?

65 or CO3008 Semantic of Programming Language 65 We prove P :: σ P i not tuck by Rule Induction on type aignment. (Property Cloure for :: IOP) The inductive hypothee are that neither are tuck, where P1 : : P1 P 2 and P2 ::. We have to prove that P1 iop P2 i not tuck, where P 1 iop P 2 ::. Let work thi on the board...

66 CO3008 Semantic of Programming Language 66 We prove, for a given L, P P σ (P::σ and enible) implie enible by rule induction for. We check property cloure for l : c l c ASS2 Suppoe i enible, and l : c :: σ. We need to verify that l c i enible, that i All location in L are in the domain of definition of l c. l :: τ in L we have l c l :: τ.

67 CO3008 Semantic of Programming Language 67 Overview: Evaluation Relation We decribe a emantic which tell u immediately the final reult of a program run. We how how thi connect with tranition.

68 T ; l : and CO3008 Semantic of Programming Language 68 An Evaluation Relation Conider the following evaluation relationhip l : 4 1 l T l 5 The idea i Starting program final reult We decribe an operational emantic which ha aertion which look like P n P 1 2

69 P1 ; P2 P1 P P1 P1 bop P2 provided l CO3008 Semantic of Programming Language 69 l l domain of n1 P2 n1 bop n2 n2 OP2 l :P l n n ASS1 l :P l P b b ASS P SEQ LOC

70 P F P1 P1 P CO3008 Semantic of Programming Language 70 1 F 1 P2 P1 1 P COND2 1 T 1 P2 1 2 P1 P2 2 3 P1 P2 1 3 P1 P2 LOOP2

71 CO3008 Semantic of Programming Language 71 We derive deduction for Example Evaluation and l 1 l : l 1 l 1 l 0

72 CO3008 Semantic of Programming Language 72 A Mutual Correctne Proof For any configuration V, P P V and terminal configuration iff P V where denote reflexive, tranitive cloure of.

73 CO3008 Semantic of Programming Language 73 We break the proof into three part: Prove Induction. P V implie P V by Rule Prove by Rule Induction for that P P V implie P V Ue previou reult to deduce P V implie P V

74 H1 H2 H3 P2 1 2 P1 P V P V CO3008 Semantic of Programming Language 74 We hall prove by Rule Induction that We need to prove that P1 1 P2 2 T 1 3 P1 P2 1 3 C

75 LOOP T H1 H2 H3 Q P2 ; Q Q CO3008 Semantic of Programming Language 75 Let u write Q for P1 P2. Then 1 P1 P2 ; Q 1 P2 ; Q 1 & COND1 1 COND2 ; Q 2 & SEQ1 2 SEQ2 3 which prove (C).

76 LOOP COND1 CO3008 Semantic of Programming Language 76 We hall prove by Rule Induction for that P P V P V implie P V Let u jut conider property cloure for the rule any V and uppoe that. Pick P1 P2 ; Q V 1 We need to how that Q V 2 But (1) can hold only if it ha been deduced either from or COND2. In either cae V mut be.

77 CO3008 Semantic of Programming Language 77 Chapter 5 Decribe the CSS machine, which execute compiled IMP program. Show how to compile IMP program to CSS intruction equence. Give ome example execution.

78 CO3008 Semantic of Programming Language 78 Motivating the CSS Machine An operational emantic give a ueful model of we eek a more direct, computational method for evaluating configuration. If P e V, how do we effectively compute V from P? The tranition relation i not quite right. It i eay for human to ee that but etablihing thi involve a deduction tree...

79 CO3008 Semantic of Programming Language 79 We eek a way of taking a program P, and mechanically producing the value V : P P 0 P1 P2 Pn V Mechanically produce can be made precie uing a relation P Pdefined by a et of rule in which there are no hypothee. Such rule are called re-write: n m m n Etablihing P deduction tree: P will not require the contruction of a

80 CO3008 Semantic of Programming Language 80 P 0 P1 P2 P3 P4 V Rewrite Rule (Abtract Machine) deduction tree deduction tree deduction tree P 0 P1 P 2 V Tranition Semantic

81 CO3008 Semantic of Programming Language 81 An Example Let l 6. Execute 10 l on the CSS machine. Firt, compile the program. 10 l FETCH l : PUSH 10 : OP Then FETCH l : PUSH 10 : OP PUSH 10 : OP 6 OP 10 : 6 4

82 FETCH OP BR CO3008 Semantic of Programming Language 82 Defining the CSS Machine A CSS code C i a lit : in ::PUSH c l op SKIP STO l C C LOOP C C C :: in in : C The object in are CSS intruction. A tack σ i produced by the grammar σ :: c c : σ

83 σ σ1 σ2 CO3008 Semantic of Programming Language 83 A CSS configuration i a triple C. A CSS tranition take the form C1 1 C2 2 Defined inductively by a et of rule, each rule having the form C1 σ1 1 C2 σ2 2 We call a binary relation (uch a) which i inductively defined by rule with no hypothee a re-write relation. R

84 C2 : LOOP l CO3008 Semantic of Programming Language 84 PUSH c : C σ C c : σ FETCH l : C σ C l : σ OP op : C n 1 : n 2 : σ C n1 op n2 : σ SKIP : C σ C σ STO l : C c : σ C σ c BR C1 C2 : C F : σ C2 : C σ LOOP C1 C2 : C σ C 1 : BR C 1 C2 SKIP : C σ

85 P CO3008 Semantic of Programming Language 85 c def PUSH c l def FETCH l l : P P1 op P2 P2 def def def SKIP : P1 P : STO l : OP op P1 P2 P1 ; P2 def def P1 : P2 P P1 P2 P1 P2 def LOOP P1 P2 : BR

86 CO3008 Semantic of Programming Language 86 An Example Execution Execute l : 2 ; l the program. : 5l on the CSS machine. Firt, compile l : 2 ; l : 5l PUSH 2 : STO l : FETCH l : PUSH 5 : OP : STO l Then PUSH 2 : STO l : FETCH l : PUSH 5 : OP : STO l l 2 l 10

87 CO3008 Semantic of Programming Language 87 Chapter 6 Motivate a language in which we can write higher order function. Decribe it type. Decribe it expreion yntax. Outline a type aignment ytem. Explain how to write imple program.

88 CO3008 Semantic of Programming Language 88 Overview: Motivating and Defining FUN Give a broad outline of FUN. Define it yntax and type ytem. Explain ome technical convention and definition.

89 CO3008 Semantic of Programming Language 89 ct :: Int ct = 76 Example of FUN Declaration f :: Int -> Int f x = x g :: Int -> Int -> Int g x y = x+y h :: Int -> Int -> Int -> Int h x y z = x+y+z

90 CO3008 Semantic of Programming Language 90 empty_lit :: [Int] empty_lit = nil l1 :: [Int] l1 = 5:(6:(8:(4:(nil)))) l2 :: [Int] l2 = 5:6:8:4:nil h :: Int h = hd (5:6:8:4:nil)

91 CO3008 Semantic of Programming Language 91 p :: (Int,Int) ft :: (Int,Int) -> Int length :: [Bool] -> Int map :: (Int -> Bool) -> [Int] -> [Bool] p = (3,4) ft (x,y) = x length l = if elit(l) then 0 ele (1 + length t) map f l = if elit(l) then nil ele (f h) : (map f t)

92 σ σ2 σn σ2 CO3008 Semantic of Programming Language 92 FUN Type The type of e are σ :: σ σ σ σ We hall write Type for the et of type. We hall write σ 1 σ 3 σ for σ 1 σ2 σ3 σn σ Thu for example σ 1 σ 3 mean σ 1 σ2 σ3.

93 CO3008 Semantic of Programming Language 93 FUN Expreion E :: xvariable contant identifier function identifier E 1 E 2 E 1 : E 2 Bracketing convention apply... E E E firt projection function application tail of lit con for lit Boolean tet for empty lit

94 x x y z CO3008 Semantic of Programming Language 94 Subtitution (for next chapter) E The variable x occur in the expreion x op 3 op x. If E and E 1 E1 En x 1 En are expreion, then xn denote the expreion E with Ei imultaneouly replacing x i for each 1 Eg i n. u y 6 2 z u x 2 x 6

95 CO3008 Semantic of Programming Language 95 Overview: FUN Type Sytem Show how to declare the type of variable and identifier; an identifier i (the name of) a contant or function. Define a type aignment ytem. Give ome example. Verify that FUN i monomorphic each program ha a unique type.

96 CO3008 Semantic of Programming Language 96 Context When we write a FUN program, we hall declare the type of variable, for example x :: y : : z : : A context take the form Γ x1 :: σ1 xn : : σn Thu a context pecifie type declaration for variable. The variable mut be ditinct.

97 CO3008 Semantic of Programming Language 97 Environment When we write a FUN program, we want to declare the type of contant and function. A imple example of an identifier environment i :: :: and another i :: and another i :: :: ::

98 CO3008 Semantic of Programming Language 98 σ 1 An identifier type look like σ2 σ 3 number and σ i NOT a function type. σk σ where k i a natural If k If k 0 then the identifier i called a contant. 0 then the identifier i called a function. An identifier environment look like I 1 :: ι 1 m : : ιm

99 y : : z : : CO3008 Semantic of Programming Language 99 Example Type Aignment With the previou identifier environment x :: x : y : z : :: We have T 2 : 2 : 6 : ::

100 CO3008 Semantic of Programming Language 100 E1 : : E1 iop E2 : : E2 : : n : : Inductively Defining Type Aignment Start with an identifier environment and a context. Then Γ x :: σ where x :: σ Γ :: VAR Γ :: INT Γ Γ Γ :: OP1

101 σ σ CO3008 Semantic of Programming Language 101 Γ E 1 :: σ 2 σ1 Γ E 2 :: σ 2 :: AP Γ E1 E2 :: σ1 Γ Γ E :: E :: σ 1 σ1 σ2 :: FST Γ :: ι where :: ι I :: IDR Γ σ : : :: NIL Γ E1 :: σ Γ E 2 :: Γ E1 : E2 :: σ :: CONS

102 CO3008 Semantic of Programming Language 102 FUN i Monomorphic Given I, Γ and E, if there i a type σ for which Γ then uch a type i unique. E :: σ, We verify Γ E :: σ 1 σ2 Γ E :: σ 2 implie σ 1 σ2 uing Rule Induction. We check property cloure for the rule HD:

103 σ σ E E CO3008 Semantic of Programming Language 103 The inductive hypothei i σ2 Γ E :: σ2 implie σ2 where Γ E ::. Γ Γ E :: E σ :: σ :: HD We wih to prove that σ2 Γ :: σ 2 implie σ σ2 where Γ :: σ.

104 CO3008 Semantic of Programming Language 104 Overview: Function Declaration and Program Show how to code up function. Define what make up a FUN program. Give ome example.

105 CO3008 Semantic of Programming Language 105 Introducing Function Declaration To declare can write x x x. To declare x x 1 1 x x 1 And to declare that denote T we write T. In e, can pecify E x E xy E

106 v CO3008 Semantic of Programming Language 106 An Example Declaration Let I 1 :: 2 : : 3 : : an example of an identifier declaration dec I i. Then 1 l y l 2 y 2x x x 3 T 4 u v w u w

107 σ2 σk CO3008 Semantic of Programming Language 107 Defining Declaration Let I 1 :: ι 1 m : : ιm where for example ι j σ1 σ 3 σ j j 1 m Then an identifier declaration dec I conit of. j x 1 xk E j. for each j 1 m

108 y : : :: :: CO3008 Semantic of Programming Language 108 An Example Program Let I ::. Then an identifier declaration dec I i x y x 7 y 10 An example of a program i dec I in 81. Note that 8 :: and that x :: y : : x 7 and

109 CO3008 Semantic of Programming Language 109 Program A program expreion P i any expreion containing no variable. A program in e i a judgement of the form dec I in P where P :: σ and the declaration in dec I atify. x 1 :: σ 1 xk : : σk E j :: σ j.

110 CO3008 Semantic of Programming Language 110 Example Program x x 1 1 x x 1 in 4 1 xyz 2 x x 1 x1 2 x 1 y x z 1 in 2 4 l code to ort lin 3 : 6 : 2 : 8 :

111 CO3008 Semantic of Programming Language 111 Chapter 7 Explain call-by-value (eager) and call-by-need (lazy) function calling method. Give FUN an eager and lazy evaluation tyle operational emantic. Prove propertie uch a determinim. Extend the language to give local declaration.

112 CO3008 Semantic of Programming Language 112 Overview: Program and Value Look at the notion of evaluation order. Define value, which are the reult of eager program execution. Define an eager evaluation emantic: P e V. Give ome example.

113 4 CO3008 Semantic of Programming Language 113 Evaluation Order The operational emantic of e ay when a program P evaluate to a value V. It i like the IMP evaluation emantic. Write thi in general a P e V, and example are 3 10 e 17 and 2 : e 2

114 Let CO3008 Semantic of Programming Language 114 xy x y. We would expect e 26. We could evaluate 2 3 to get value 6 yielding 6 4 5, then evaluate 4 5 to get value 20 yielding 620. We then call the function to get 6 20, which evaluate to 26. Thi i call-by-value or eager evaluation. Or the function could be called firt yielding and then we continue to get and 6 i called call-by-name or lazy evaluation. The order of evaluation i different. 20 and 26. Thi

115 σ2 σk CO3008 Semantic of Programming Language 115 Defining and Explaining (Eager) Value Let dec I be a identifier declaration, with typical typing :: σ1 σ 3 σ A value expreion i any expreion V produced by V :: c σ V V V V : V where V abbreviate V 1 V 2 Vl 1Vl and 0 lk, and k i the. CARE!!! maximum number of input taken by Note that contant are not value. Note alo that l i trictly le than k, and that if k 1then denote V.

116 :: CO3008 Semantic of Programming Language 116 A value i any value expreion for which dec I i a valid e program. in V Suppoe that and P e 2 5 and P 3 and that P1 e 7 with P i not value. Then e 2 P V P V P 1 2 P P P 1 P 2 P 3 14 Of coure P1 P2 P3 e 14.

117 P V 2 P 2 V 2 P V 2 CO3008 Semantic of Programming Language 117 V e V e VAL P 1 e m P 2 e n P 1 op P e 2 m op n e OP P 1 P1 e T P 2 P2 e V P3 e V e COND1 P 1 P1 e F P 3 P2 e V P3 e V e COND2 P 1 e V 1 P 2 e V 2 e PAIR P 1 e V 1 P e V 1 e FST P e V 1 e SND e V 1 e V 2

118 Vk j CO3008 Semantic of Programming Language 118 P 1 e V P 2 e V 2 V V 2 e V where either P 1 or P 2 i not a value P 1 P 2 e V e AP E V1 V1 x 1 V k xk e V e V x E declared in deci e FID E e V e V E declared in deci e CID

119 P P P P P CO3008 Semantic of Programming Language 119 P e e σ σ e NIL P e V : V e V e HD P e V : V e V e TL P 1 P 1 : P 2 e V P 2 e V e V : V e CONS P e σ e T e ELIST1 P e V : V e F e ELIST2

120 CO3008 Semantic of Programming Language 120 Suppoe that dec I i Example of Evaluation x x 2 3 e VAL 3 e 3 VAL e 3 CID 3 e 6 e VAL 3 x 2 3 x e 6 2 e VAL 2 e 6 OP FID AP

121 CO3008 Semantic of Programming Language 121 e VAL SND e D 4 VAL VAL e 4 e 2 2 OP 4 e e 8 FID e 8 AP

122 :: y CO3008 Semantic of Programming Language 122 Let where xyz x z 2 and 23 are (program and) value i a program, but not a value Note that i ugar for 23 i ugar for and that V In the Definition of value, k and l , and in 23 we have

123 y CO3008 Semantic of Programming Language 123 We can prove that e 10 where xyz x z a follow: 23 e 23 e VAL e e 10 e 5 e 1 T e AP

124 y CO3008 Semantic of Programming Language 124 where T i the tree 2 e 2 3 x e 3 e y z e 10 x e 10 z e 10 e 5 e FID

125 CO3008 Semantic of Programming Language 125 Overview: FUN Propertie of Eager Evaluation Explain and define determinim. Explain and define ubject reduction, that i, preervation of type during program execution.

126 CO3008 Semantic of Programming Language 126 Propertie of FUN The evaluation relation for More preciely, for all P, V 1 and V 2, if e i determinitic. P e V 1 and P e V 2 then V 1 V2. (Thu e i a partial function.) Evaluating a program dec I type. More preciely, in P doe not alter it P :: σ and P e V implie V :: σ for any P, V, σ and I. The conervation of type during program evaluation i called ubject reduction.

127 CO3008 Semantic of Programming Language 127 Proving Determinim To prove determinim, we prove by Rule Induction that P e V 1 V2 P e V 2 implie V 1 V2 See the board...

128 CO3008 Semantic of Programming Language 128 Proving Subject Reduction We prove by Rule Induction that given dec I in P P e V σ P :: σ implie V :: σ The tricky rule i E V1 V 1 Vk j x 1 Vk xk e V e V x E declared in deci e FID Suppoe that we need to prove V1 we jut need to prove Vk :: σ where σ i any type. Then V :: σ. By the induction hypothei, E V1 Vk j x 1 xk :: σ.

129 CO3008 Semantic of Programming Language 129 Overview: Program and (Lazy) Value Define value, which are the reult of program execution. Define a lazy evaluation emantic: P l V. Give ome example.

130 P CO3008 Semantic of Programming Language 130 Defining and Explaining Value Let dec I be a identifier declaration, with typical typing :: σ 1 σ2 σ 3 A value expreion i any expreion V produced by the grammar σk σ V :: c σ P P P : P where P abbreviate P 1 P 2 Pl 1 Pl and 0 i the maximum number of input taken by l k, and k. A value i any value expreion for which dec I l i a valid program. in V

131 P1 P CO3008 Semantic of Programming Language 131 P l P 1 P 2 P 1 l V l V lfst P 1 l P PP 2 l V where either P 1 or P 2 i not a value P 1 P 2 l V l AP E P1 P k x 1 P k xk l V l V x E declared in deci l FID P 1 l P 2 : P 3 P 2 P 1 l V l V lhd

132 :: CO3008 Semantic of Programming Language 132 Example of Evaluation Let I be there i a program dec I 1 l 3., and dec I be in 1 xx : x 2. Then. We prove that 1 : l 1 : l 1 : l VAL l FID T1 1 l 1 2 : T2 1 l 3

133 CO3008 Semantic of Programming Language 133 T2 1 l l 3 l 2 T1 1 2 : l 1 l : : l VAL

134 CO3008 Semantic of Programming Language 134 Let x 1 x in thi programme to a value V 3 0. We try to evaluate 3 0 l P1 P2 3 0 R l V P 1 l V R l FST for which we mut have P 1 and R are both intance of l VAL. 3, P2 0, V 3 and R

135 CO3008 Semantic of Programming Language 135 Overview: Locality Explain unnamed function and local definition. Decribe free and bound variable. Extend the yntax of FUN, and it operational emantic. Give ome example.

136 x x CO3008 Semantic of Programming Language 136 Motivating Function and Locality We can define unnamed function. The expreion x 2 i a program whoe intended meaning i the function which add 2. But it i not (necearily) named by an identifer. x 2 4 will evaluate to 4 2 (and thu to 6).

137 y CO3008 Semantic of Programming Language 137 If xx 2then x x 2would be interchangeable. i the name of the function. and The yntax example x 5 x E1 x y E2 give local declaration. For x. We explain local with the next example: x 7 x x 5 x x

138 We call CO3008 Semantic of Programming Language 138 Syntax and Type Aignment E :: x E x E E We call body of x E. x E a function abtraction. We call E the x E1 E2 a local declaration. Γ E 1 :: σ Γ Γ x E1 E 2 E1 x E2 :: σ :: σ LET Γ Γ x : : σ E :: τ x E : : σ τ ABS

139 E y x x y y y x CO3008 Semantic of Programming Language 139 Convention and Example x E mean x x E1 E2 mean x E1 E2 Thu yy2 = y 2 y x y x 2 = x y 2 x 4 y T x x 4 y T x

140 E2 E2 CO3008 Semantic of Programming Language 140 Motivating Free and Bound Variable Write F def emantic xe 1. Given any expreion E 2, in a tranition F E 2 E1 x Thu if E 1 i x y, then F E 2 x y x def E2 y and the intended meaning of F xx y i the function with add y. E x y ought to be the function which add x. But in fact E x y clearly the expreion xi x x, which i the function which double an integer input!

141 x CO3008 Semantic of Programming Language 141 We ay that the ubtituted x fall in the cope of the coping x. The expreion x x yand x x ycan be regarded a the ame. We ay that x and x are bound, and y i free. Note that x y x y x x x We re-name the bound variable x in x x ya a new variable x o that when x i ubtituted for y it doe not become bound.

142 v E CO3008 Semantic of Programming Language 142 Definition of Free and Bound Variable The yntax tree for look like thi coping variable v E cope In v E1 uch a v a coping variable. E2, the cope of v i E2. We alo call

143 x E CO3008 Semantic of Programming Language 143 Suppoe x doe occur in E. Each occurrence of x (in E) i either free or bound (but not both!!). We ay that an occurrence of x i bound if and only if the occurrence of x i in a ubexpreion of the form or x E1 E2 where the occurrence i in E2. Thu an occurrence of x in E i bound jut in cae the occurrence i a coping variable; the occurrence occur within the cope of a coping occurrence of x.

144 CO3008 Semantic of Programming Language 144 If there i an occurrence of x in uch E or E 2 then we ometime ay that thi bound occurrence of x ha been captured by the coping x. An occurrence of x in E i free iff the occurrence of x i not bound.

145 CO3008 Semantic of Programming Language 145 Subtitution Example x x y 2 y x x 2 x x y x y x x x x y 4 x z 7 u v z x y 4 x u v 7 x y 4 x z 7 u y z x y 4 x u y 7 x z 4 x x z 7 x y z x x y 4 x x x y 7 u u u 7 7 u u 7 u 7

146 E2 CO3008 Semantic of Programming Language 146 Extending the Eager Semantic P 1 e x E P2 P 1 P 2 e V e V E V x e V e AA E 1 e V 1 E 2 x E1 V1 x e V e V e LET

147 CO3008 Semantic of Programming Language 147 An Example e VAL e 3 3?? x 2 x x 2 3 : x x 2 VAL 3 x x x 23 e 5 : 3 : e e 5 e 5 VAL e 5 OP AA CONS HD

148 CO3008 Semantic of Programming Language 148 Chapter 8 Give overview of polymorphim. Introduce type variable into e. Give example of type aignment. Explain local polymorphim. Explain the polymorphic type inference algorithm.

149 CO3008 Semantic of Programming Language 149 Overview: Simple Type Deduction with Variable Explain different kind of polymorphim. Give example of type aignment deduction.

150 CO3008 Semantic of Programming Language 150 Varietie of Type Sytem A language i trongly typed if every legal expreion ha at leat one type. A trongly typed language i monomorphic if every legal expreion ha a unique type (for example Pacal). A trongly typed language i polymorphic if a legal expreion can have everal type (for example Standard ML and Hakell and Java).

151 CO3008 Semantic of Programming Language 151 Overloading: The ame ymbol i ued to denote (finitely many) function, implemented by different algorithm. Parametric: One expreion belong to a family of tructurally related type. The expreion encode one algorithm which work at each type in the family. An example i lit orting. Implicit: Thi i a particular form of parametric polymorphim, and we meet it later on.

152 The et Type of type of σ σ CO3008 Semantic of Programming Language 152 PFUN Type Sytem pecified by the grammar i inductively σ :: X σ σ σ σ Each type i a finite tree. Two type are equal if the tree are identical. Example on the board. We hall write TV appearing in σ. for the et of type variable The rule for deriving type aignment are a before. expreion are from the extended language.

153 T : CO3008 Semantic of Programming Language 153 Example of Type Aignment Deduction Prove that ::. T :: TRUE T : :: :: NIL CONS

154 x CO3008 Semantic of Programming Language 154 Show that Γ 0 : x :: for any context Γ. We produce a deduction tree: note that the expreion i a function, o the final rule ued in the deduction mut be ABS, where E : x, and σ 0 τ. Γ x : : 0 :: Γ x : : INT x Γ 0 : x Γ x : : 0 : x :: :: x :: VAR CONS ABS

155 Show that Γ i not typable in σ σ CO3008 Semantic of Programming Language 155 y : 3 in any context Γ. Working backward we have: Γ y :: σ VAR Γ y : 3 :: Γ y : 3 3 :: :: σ INT CONS HD Looking at the rule INT (which mut be ued to type 3) we mut have cannot be typable. σ, a contradiction. So the expreion

156 f X X X X X f X X f X T T T CO3008 Semantic of Programming Language 156 Show that f :: Y Y. f :: Y f :: Y VAR f :: f :: f f :: Y Y :: Y f Y Y :: NIL AP :: Y D PAIR :: Y ABS

157 f CO3008 Semantic of Programming Language 157 Show that f f y y i not typable for any context of the form y :: τ. (Note that y i the only free variable). We uppoe, for a contradiction, that the expreion i typeable. Let u call thi type σ 1, ay. We have: y :: τ f :: σ 2 f :: σ 3 σ 1 VAR y :: τ VAR f :: σ 2 y :: σ 3 AP y :: τ y :: τ y :: τ f y :: σ 2 f f y f :: σ 2 f y :: σ 1 ABS σ1d y :: σ 1 AP where D i y :: τ y :: σ 2 VAR

158 x X; any type σ CO3008 Semantic of Programming Language 158 Motivating Type Subtitution 6 T ha no type. 1 :: σ hold only for σ. However, x :: σ σ hold for any type σ. In, of all the type that can be aigned to an expreion, there i a mot general one: all other type are intance of it. We call thi the principal type. The principal type of x x i X obtained by ubtituting σ for X. σ i

159 X CO3008 Semantic of Programming Language 159 Type Subtitution Define S def U Y. Let σ def X Y Z. Then S σ U Z S a type ubtitution if it i a (poibly empty) finite et of (type-variable,type) pair in which all the type-variable are ditinct. We will write a typical S in the form X 1 σ1 Xn σn We write the empty type ubtitution a.

160 CO3008 Semantic of Programming Language 160 If τ i any type, we hall write S τ to denote the type τ in which any occurrence of X i i changed to σ i. Thu X 1 σ1 Xn σn τ def τ σ1 σn X1 Xn We will define equality of type ubtitution in a imilar way to function equality, namely S S iff τ S τ S τ

161 V CO3008 Semantic of Programming Language 161 Given ubtitution S 1 and S 2 we define the effect of the ubtitution S 1 S 2 by etting S 1 S 2 Warning!! A type ubtitution i a et of (type-variable,type) pair. What et i S 1 S 2? τ def S1 S 2 τ. to be If S def X1 σ σ1 X1 Xn σ1 σn Xn σn and alo then we define S V V to be.

162 CO3008 Semantic of Programming Language 162 σ generalie σ for which if there exit a type ubtitution S σs σ and ay that σ i an intance of σ. In, if P :: σ, the type σ aigned to the expreion P i principal if σ generalie any other type which can be aigned to P. The principal type of x x i X X. Note that the principal type i unique up to a conitent renaming of variable. Another principal type for x x i V V.

163 σ Γ y : : Y y : : X Z Y X Y X Y Y y : : S Y U X CO3008 Semantic of Programming Language 163 Type Subtitution Example Define S def Γ def x :: X U Z. Then X Y and Y. Let σ def Z S Z and S x :: S X Z x :: U Z Note that Z generalie for S Z where S def

164 X X X X Z Y X CO3008 Semantic of Programming Language 164 It follow from the definition that X. The definition of compoition of type ubtitution doe not decribe S 1 S 2 a an explicit et of pair. Conider X. The compoition i Y Y X Z Now conider compoition i Y X. The Y

165 Y X Z X Y X Y CO3008 Semantic of Programming Language 165 X Z Y X Y X Z X U Y X Z Y U A an exercie, try to write down a formula for Y1 τ1 Ym τm X1 σ1 Xn σn

166 CO3008 Semantic of Programming Language 166 Local Polymorphim in The LET rule permit different occurrence of x in E 2 to have different implicit type in a local declaration x E1 E2. Thu, E 1 can be ued polymorphically in the body E 2. Thi idea i bet explained by example...

167 x x X X x X X X x X x x T T : : CO3008 Semantic of Programming Language 167 and D1 x :: x :: x :: VAR x ABS T :: TRUE AP and D2 x :: x :: x X x :: VAR x ABS D1 D2 x :: :: :: PAIR NIL AP

168 x x x x x X CO3008 Semantic of Programming Language 168 x :: Y x : : Y x :: Y VAR Y f ABS x f T f T T x f f X X D1 D2 :: x x f :: :: LET

169 X ha implicit type X X CO3008 Semantic of Programming Language 169 In the above deduction of f xx f T f :: occurrence of f labelled (2) ha implicit type occurrence of f labelled 3 X. The principal type of xx i Y Y The implicit type of f are ubtitution intance of thi principal type, (2) with S Y (3) and S Y

170 CO3008 Semantic of Programming Language 170 Can Function Abtraction Yield Implicit Poly m? It i only poible for bound variable to poe polymorphic intance. ha one other variable binding operation, that found in function abtraction x E. Can uch bound variable have polymorphic intance within the cope of x abtraction? The anwer i in fact no. An example illutrate thi.

171 f f σ8 CO3008 Semantic of Programming Language 171 f T f i not typable (in the empty context). VAR NIL D f :: σ 2 f :: σ 7 σ 5 f :: σ 2 f f :: σ 2 :: σ5 :: σ7 AP f :: σ 2 f T f T f f :: σ 3 :: σ 1 σ2 σ4 σ5 σ3 ABS where D i f :: σ 2 f :: σ 6 f :: σ 2 σ 4 VAR f T :: σ4 f :: σ 2 T :: σ 6 TRUE AP

172 X E E CO3008 Semantic of Programming Language 172 A Type Inference Algorithm The type and expreion are now jut given by σ :: σ σ E :: n E iop E x E x E E S σ and τ are unifiable if we can find S for which Sτ. We call S a unifier. σ S i a mot general unifer if, given another unifer S there exit T for which S TS.,

173 CO3008 Semantic of Programming Language 173 X X X MGU σ σ here σ i any type MGU X Y Y here X and Y are ditinct here σ i either or a function type MGU X σ FAIL σ if X TV otherwie σ here σ i either or a function type MGU σ X σ if X TV FAIL otherwie σ

174 CO3008 Semantic of Programming Language 174 MGU σ1 σ2 τ1 τ2 S2 S1 where σ i τi any type S 1 def MGU S 2 def MGU S 1 FAIL otherwie σ1 τ1 S1 σ2 τ 2 MGU σ τ FAIL here σ τ any type MGU σ τ FAIL here σ τ any type

175 σ1 σn CO3008 Semantic of Programming Language 175 A typing for the judgement x 1 :: σ 1 xn : : σn E i a pair S τ for which x 1 :: S xn : : S E :: τ Such a typing i aid to be principal if given any other S there i ome T for which STSand τ Tτ. τ There i a type inference function Φ which given any input of the form will either return a principal typing, or FAIL if there i none. To define Φ we need more notation.

176 σ1 σn Γ σ1 σn CO3008 Semantic of Programming Language 176 Given a context Γ abuing notation) TV x1 :: σ1 Γ for the et xn : : σn let u write (by TV TV We hall alo write S to mean x 1 :: S xn : : S and we define S def.

177 Γ Γ CO3008 Semantic of Programming Language 177 Φ x1 :: σ1 xn :: σn xi Φ x1 :: σ1 i xn :: σn y FAIL xi y Φ n Φ E1 iop E2 S4 S3 S2 S1 where S1 S2 S3 S4 τ2 τ1 Γ Φ MGU Φ MGU τ1 τ2 S4 τ2 E1 S2 S1 Γ E2 σi

178 Γ Γ S τ Γ Γ CO3008 Semantic of Programming Language 178 Φ x E S V where V S TV V Φ Γ x:v τ E Φ E1 E2 S3 V S2 S1 where S1 S2 S3 V τ2 τ1 Γ Φ Φ MGU TV S2 S2 τ1 S1 τ1 τ2 E1 E2 V or TV τ2 V S3

179 CO3008 Semantic of Programming Language 179 Φ Γ x E1 E2 S 2 S 1 τ2 where S 1 τ1 Φ Γ E1 S 2 τ2 Φ S 1 Γ E 2 E1 x

180 x x V V CO3008 Semantic of Programming Language 180 Example We claimed that the principal type of have x i X X. We Φ x S V S τ where S τ Φ x :: V x V Thu Φ x x V V V. So, up to a renaming of type variable, the principal type i V V. V

181 f V X V U CO3008 Semantic of Programming Language 181 We calculate Φ x :: X f x. Thi i S V S τ where S τ Φ x :: X f :: V f x A3 U A 2 A 1 A3 where A1 τ1 Φ x :: X f :: V f and A2 τ2 Φ x :: X f :: V x

182 τ f X X U and o CO3008 Semantic of Programming Language 182 and A3 MGU V X X U V U U Therefore S V U Φ x :: X f x U U V X V X