I. Declarative Programming Techniques (Ch 3) A. Why declarative programming? When is declarative programming useful? Why is it useful? ------------------------------------------ WHEN IS DECLARATIVE PROGRAMMING USEFUL? WHY IS IT USEFUL? ------------------------------------------ Why is declarative programming deterministic? What do we mean by an expression being deterministic? What is referential transparency? ------------------------------------------ REFERENTIAL TRANSPARENCY def: A model is referentially transparent ------------------------------------------ B. How What technique can be used to simplify declarative program structures? How can we tell if a program is declarative? ------------------------------------------ WHEN IS A PROGRAM DECLARATIVE? ------------------------------------------ Why does the declarative model lead to declarative programs? II. Declarative Programming Techniques (3.2-3.5) A. iterative computation (3.2) What is an iterative computation? 1. general schema (3.2.1, 3.3.3, 3.4.3) ------------------------------------------ ITERATIVE COMPUTATION (3.2) Is this iterative? fun {SumFromTo I J} if I > J then 0 else I + {SumFromTo I+1 J} end end ------------------------------------------ How to make it iterative? ------------------------------------------ FOR YOU TO DO Make the following iterative: fun {Product Ls} case Ls of E|Es then E*{Product Es} else 1 end end ------------------------------------------ 2. When to use Iteration ------------------------------------------ WHEN TO USE ITERATION 0. When need efficiency 1. When the data doesn't maintain "place" 2. When need to return directly to caller ------------------------------------------ what are some examples of this that we have seen? B. Data-driven recursion (3.4) 1. Programming tips ------------------------------------------ TIPS FOR SAVING TIME WHEN DOING RECURSIVE PROBLEMS 1. Write out your own examples if needed a. base case(s) b. related simpler example 2. Write out the type. Assume the inputs match this type. 3. Use an outline that matches the grammar for the input data type. 4. Fill in the outline by generalizing the examples a. What's returned in the base case(s)? b. How do we get the answer from - recursive call's answer, and - other information? 5. Debug bottom up from subexpressions, replacing recursive calls with expected results. 6. One function per nonterminal. Don't mix 2 recursions in a function. ------------------------------------------ 2. Type notation (3.4.1) ------------------------------------------ TYPE NOTATION (GRAMMARS) ::= red | blue | green ::= zero | succ( ) > ::= nil | '|' > Equivalently: > ::= nil | '|'(1: 2: >) > ::= leaf | tree( key: value: left: > right: > ) Examples in these grammars: ------------------------------------------ How does the type definition resemble a grammar? 3. Natural Numbers ------------------------------------------ RECURSIVE NATURAL NUMBERS ::= zero | succ ( ) Examples: 0 represented by zero 1 represented by succ(zero) 2 represented by succ(succ(zero)) 3 represented by succ(succ(succ(zero))) ------------------------------------------ ------------------------------------------ FOLLOWING THE GRAMMAR FOR RECURSIVE FUNCTIONS OVER NATURAL NUMBERS ::= zero | succ ( ) Outline that follows this grammar: fun {F N} case N of zero then ... % basis [] succ (P) then ...{F P}... % inductive case end end ------------------------------------------ In what sense does that look like the grammar? ------------------------------------------ EXAMPLE DEVELOPMENT Write Plus, which adds two What are some examples? What's the type? What's the outline? How do we fill it in? ------------------------------------------ How does the structure of the program resemble the type definition? ------------------------------------------ FOR YOU TO DO Write Mult that multiplies two s. Write Equals that tells if two s are equal, without using Oz's == on arguments. ------------------------------------------ 4. Working with lists (3.4.2.6) ------------------------------------------ RECURSION OVER FLAT LISTS > ::= nil | T '|' > Write Add1Each: {Add1Each nil} == nil {Add1Each 1|3|5|nil} == 2|4|6|nil {Add1Each 3|5|nil} == 4|6|nil ------------------------------------------ ------------------------------------------ FOR YOU TO DO Write DeleteFirst: > T}: >> such that {DeleteFirst Ls Sought} returns a new list that is like Ls, but without the first occurrence of Sought in Ls (if any). {Test {DeleteFirst nil b} '==' nil } {Test {DeleteFirst a|b|c|b|nil b} '==' a|c|b|nil } ------------------------------------------ 5. structure of data determines structure of code a. non-empty lists ------------------------------------------ GENERALIZING HOW TO WRITE RECURSIONS ::= sing(T) | cons(T ) Write: HeadNEL: }: T> TailNEL: }: > Write AppendNEL: }: > such that {Test {AppendNEL sing(3) sing(4)} '==' cons(3 sing(4))} {Test {AppendNEL cons(5 sing(3)) sing(4)} '==' cons(5 cons(3 sing(4)))} {Test {AppendNEL cons(7 cons(5 sing(3))) cons(8 sing(1))} '==' cons(7 cons(5 cons(3 cons(8 sing(1)))))} ------------------------------------------ ------------------------------------------ FOR YOU TO DO Using Max: }: > Write MaxNEL: }: T> such that {MaxNEL sing(3)} == 3 {MaxNEL cons(5 sing(3))} == 5 {MaxNEL cons(3 cons(5 sing(3)))} == 5 ------------------------------------------ b. Multiple nonterminals ------------------------------------------ MULTIPLE NONTERMINALS Recall our last tip: 6. One function per nonterminal. Don't mix 2 recursions in a function. Example: A computer manufacturer (HAL) sells single computers and "grids" ::= single(name: diskSizes: >) | grid(name: elements: >) ::= > ::= > ::= nil | T '|' > Examples: single(name: "xz400" diskSizes: [2 4 8]) grid(name: "small" elements: [single(name: "xz400" diskSizes: [2 4 8])]) grid(name: "maxFlow" elements: [single(name: "xz400" diskSizes: [2 4 8]) single(name: "9000" diskSizes: [8 16]) grid(name: "small" elements: [single(name: "xz400" diskSizes: [2 4 8])])]) ------------------------------------------ ------------------------------------------ EXAMPLE PROBLEM Write "SuperSize", which (a) puts "super-" in front of each name, (b) doubles all disk sizes. Examples: {Test {SuperSize single(name: "xz400" diskSizes: [2 4 8])} '==' single(name: "super-xz400" diskSizes: [4 8 16])} {Test {SuperSize grid(name: "small" elements: [single(name: "xz400" diskSizes: [2 4 8])])} '==' grid(name: "super-small" elements: [single(name: "super-xz400" diskSizes: [4 8 16])])} {Test {SuperSize grid(name: "maxFlow" elements: [single(name: "xz400" diskSizes: [2 4 8]) single(name: "9000" diskSizes: [8 16]) grid(name: "small" elements: [single(name: "xz400" diskSizes: [2 4 8])])])} '==' grid(name: "super-maxFlow" elements: [single(name: "super-xz400" diskSizes: [4 8 16]) single(name: "super-9000" diskSizes: [16 32]) grid(name: "super-small" elements: [single(name: "super-xz400" diskSizes: [4 8 16])])])} What's the type? What's the outline? How do we fill it in? ------------------------------------------ c. More programming language like grammars ------------------------------------------ RECURSION OVER GRAMMARS ::= boolLit( ) | intLit( ) | subExp( ) | equalExp( ) | ifExp( ) Write the following Eval: }: > such that {Eval subExp(intLit(5) intLit(4))} == intLit(1) {Eval equalExp(subExp(intLit(5) intLit(4)) intLit(1))} == boolLit(true) ------------------------------------------ What are the base cases? Where should there be a recursion? Examples for each recursive case? 6. Difference Lists (3.4.4) a. Basics of Difference Lists ------------------------------------------ DIFFERENCE LISTS (3.4.4) Grammar: > ::= > # > Idea L1 # L2 represents list L1 minus elements of L2 invariant: L2 is a tail of L1. Example: (1|2|3|X) # X means (1|2|3|nil) Main advantage: Lists of form (a|b|...|X) # X can be appended in constant time Example: To append (1|2|3|X) # X and (4|5|Y) # Y bind X to (4|5|Y) to get (1|2|3|4|5|Y) # Y ------------------------------------------ Why not just use (1|2|3|X) instead of (1|2|3|X) # X ? ------------------------------------------ FOR YOU TO DO Write AppendD: }: > Examples: {AppendD (2|3|X)#X (3|5|Y)#Y} == (2|3|3|5|Y)#Y {AppendD X#X (3|5|Y)#Y} == (3|5|Y)#Y ------------------------------------------ What are the limitations of difference lists? b. Applications (skip) c. Queues and Performance (3.4.5) What's efficiency issue in implementing FIFO queues in the declarative model? What's the difference between ephemeral and persistent data? How could we get amortized constant time queues? How could we get constant time queues with dataflow variables? Can we delete elements from a queue that aren't present? d. Trees (3.4.6-7) e. Parsing (3.4.8) C. Time and space efficiency (3.5) (skip, put on supplement) What's the recommended general approach for calculating resource usage? 1. Time (3.5.1) ------------------------------------------ EXECUTION TIMES OF KERNEL INSTRUCTIONS S T(S) skip k X = Y k0*max(size(X),size(Y)) X = myrec(f1:X1 k1 + k2*n ... fn:Xn) X = proc k3' + k3*|FV(S) - {X1,...,Xn}| {$ X1 ... Xn} S end S1 S2 T(S1) + T(S2) local X in S end k4 + T(S) if X then S1 k5 + max(T(S1),T(S2)) else S2 end case X of r(f1:X1...fn:Xn) k7 + max(k6*n+T(S1), T(S2)) then S1 else S2 end {F Y1 ... Yn} T_F(size_F( I_F({Y1,...,Yn}))) where I_F is the subset of used arguments and size_F is a measure function and T_F is a function specific to F ------------------------------------------ Why is the time needed for skip constant? Can unification be done constant time? What's the time to do closure formation? How can pattern matching in case be constant time? 2. Memory usage (3.5.2) What needs to be measured for space? Which is more important? ------------------------------------------ MEMORY CONSUMPTION OF KERNEL INSTRUCTIONS S M(S), in words skip 0 X = Y 0 X = V memsize(V) X = proc k1 {$ X1 ... Xn} + n S end + k2*|FV(S) - {X1,...,Xn}| S1 S2 max(M(S1),M(S2)) local X in S end 1 + M(S) if X then S1 max(M(S1),M(S2)) else S2 end case X of max(n+M(S1), M(S2)) r(f1:X1 ... fn:Xn) then S1 else S2 end {F Y1 ... Yn} M_F(size_F( I_F({Y1,...,Yn}))) where I_F is the subset of used arguments and size_F is a measure function and M_F is a function specific to F ------------------------------------------ Does the value always need to be completely created in X=V? What's the size of a closure? 3. Amortized complexity (3.5.3) What's amortized complexity? What are the techniques used to compute it? 4. Does performance still matter? (3.5.4) III. Procedural Abstraction (Higher-Order Programming, 3.6) ------------------------------------------ def: the *order* of a function F is 1+(maximum order of F's arguments), where the order of a non-function argument is 0. def: A function is *higher-order* if its order is greater than 1. Examples: Add1: Map: }:S>}: > ------------------------------------------ A. Basic Operations (3.6.1) What are the 4 basic operations of higher-order programming? ------------------------------------------ 4 KINDS OF HIGHER-ORDER PROGRAMMING ------------------------------------------ 1. procedural abstraction How can you make a statement S into a procedure? Suppose we want two variants of a procedure that are similar, but differ a bit? 2. genericity a. for iteration (3.2.4) ------------------------------------------ ABSTRACTION OF ITERATION (3.2.4) Consider fun {SumFromToIter JnR} J#R=JnR in if I > J then R else {SumFromToIter J-1#R+J} end end fun {SqrtIter Guess} if {GoodEnough Guess} then Guess else {SqrtIter {Improve Guess}} end end What do they have in common? How do they differ? Can we make the differences arguments? ------------------------------------------ b. for lists ------------------------------------------ ABSTRACTING A COMMON PATTERN fun {Sum Ls} case Ls of E|Es then E+{Sum Es} else 0 end end fun {Product Ls} case Ls of E|Es then E*{Product Es} else 1 end end ------------------------------------------ What are the parts specific to computing the sum? the product? How can you write Sum and Product using FoldR? ------------------------------------------ FOLDR AS A LIST HOMOMORPHISM {FoldR 1|(2|(3|nil)) Number.'-' 0} == 1-(2-(3-0)) In general: {FoldR '|'(X1'|'(X2'|'(...'|'(Xn nil)...))) == {F X1 {F X2 ... {F Xn U} ...}}} ------------------------------------------ ------------------------------------------ FOR YOU TO DO Using FoldR: S}: S> Append: }: > and "andthen" write: Concat: >}: > {Concat nil} == nil {Concat [[4 5 6] nil} == [4 5 6] {Concat [[1] nil [2 3] [4 5 6] nil]} == [1 2 3 4 5 6] All: }: Boolean> {All [true true false]} == false {All [true true]} == true {All nil} == true ------------------------------------------ c. for trees ------------------------------------------ FOR YOU TO DO > ::= leaf | tree( key: value: T left: > right: > ) Generalize: fun {Preorder T} case T of leaf then nil [] tree(key: L value: V left: Left right: Right) then (L#V) | {Append {Preorder Left} {Preorder Right}} end end fun {IncTree T} case T of leaf then leaf [] tree(key: L value: V left: Left right: Right) then tree(key: L value: 1+V left: {IncTree Left} right: {IncTree Right}) end end WRITE FOLDTREE AS A GENERALIZATION ------------------------------------------ How would you use the result to write Preorder and IncTree? 3. instantiation (currying) (3.6.6) ------------------------------------------ INSTANTIATION or RULES THAT PRODUCE RULES Aka: factories, generators, curried functions Examples: fun {MakeSort Comparison} fun {$ Ls} {Sort Ls Comparison} end end Use of MakeSort: SortGT = {MakeSort fun {$ X Y} X > Y end} {SortGT List1} {SortGT List2} ... ------------------------------------------ How could we do this for the tree example? How would you use the that to write Preorder and IncTree? ------------------------------------------ AN EXAMPLE: COMPOSE Write the function Compose such that: {{Compose Head Tail} [a b c]} == {Head {Tail [a b c]}} == {Head [b c]} == b {{Compose Not IsEmpty} nil} == {Not {IsEmpty nil}} == false (The examples assume that: fun {Head L} X|_=L in X end fun {Tail L} _|Xs=L in Xs end fun {IsEmpty L} L == nil end ) How to write this: ------------------------------------------ ------------------------------------------ SUMMARY OF STEPS FOR MAKING A HIGHER-ORDER FUNCTION 1. Starting from examples, name the roles 2. Generalize the result expression, using role names, write it down 3. Wrap fun declarations around it, corresponding to the arguments ------------------------------------------ ------------------------------------------ FOR YOU TO DO Write a procedure Twice Twice: }: > Examples: {{Twice Not} true} == {Not {Not true}} == true {{Twice fun {$ N} N+1 end} 3} == {{fun {$ N} N+1 end} {{fun {$ N} N+1 end} 3}} == 5 ------------------------------------------ 4. embedding ------------------------------------------ EMBEDDING Putting closures in data is useful for: - explicit lazy evaluation - modules = records of operations - components, which return modules - manipulating actions as data (e.g., in testing) ------------------------------------------ ------------------------------------------ EXAMPLE: INFINITE SEQUENCES Write the following functions to implement Repeat: }: > Generate: : }: > Nth: }: > Add: }: > For example: declare Ones = {Repeat 1} Halves = {Generate fun {$ N} 1.0/{IntToFloat {Pow 2 N}} end} {Show {Nth Ones 33}} {Show [{Nth Halves 0} {Nth Halves 1} {Nth Halves 2} {Nth Halves 3} {Nth Halves 30}]} shows: 1 [1.0 0.5 0.25 0.125 9.3132e~010] ------------------------------------------ How can we represent something infinite in a computer? B. Loop abstractions (3.6.2-3) 1. FoldL and other loops over lists (3.6.2) Does it do any good in the declarative model to run an action {A I} for each integer I in 1 to 10? ------------------------------------------ FOLDL {FoldL X1|X2|...|Xn|nil F U} == {F ... {F {F U X1} X2} ... Xn} {FoldL nil F U} == U fun {FoldL L F U} case L of nil then U [] X|L2 then {FoldL L2 F {F U X}} end end ------------------------------------------ Is FoldL iterative? 2. linguistic support for loops (3.6.3) What's the difference between a declarative and imperative loop? ------------------------------------------ FOR LOOPS IN OZ Simple counting: for I in A .. B do {Stmt I} end With step S: for I in A .. B; S do {Stmt I} end For lists: for E in L do {Stmt E} end With pattern match: for foo(X Y) in L do {Stmt X Y} end FORM WITH COLLECT: USEFUL IN DECLARATIVE MODEL Collection of results: for Elem in Ls collect: Results do {Results {F Elem}} end for foo(X _) in L collect: Xs do {Xs X} done ------------------------------------------ How would you write a function that returns all the numbers in a list LoN that are strictly greater than N? a. for loop semantics (skip) ------------------------------------------ FOR LOOP DESUGARING for in collect: do end ==> local in local Body = proc {$ } end in {ForLoop Body} end end where Body is fresh and ForLoop is defined by: % ForLoop: > % }> % }>}: % >> fun {ForLoop Exp ?Name Body} local Z Res = Z#Z % the automatically generated proc Name = proc {$ X} local _#ResEnd = Res in ResEnd = X|_ end end % processing each list element proc {Loop Ls} case Ls of H|T then {Body H} {Loop T} else skip end end in {Loop Exp} % convert the difference list to a list local ResFront#ResEnd = Res in ResEnd = nil ResFront end end end ------------------------------------------ What variables can be used in ? Why does Body have to be a proc? IV. Abstract data types (3.7) A. definition and examples ------------------------------------------ ABSTRACT DATA TYPES (3.7) def: A *data type* or *type* is a set of values with a set of operations on them. Examples: Integers with +, -, *, div, ... Bool with Not, And, Or def: An *abstract data type* or ADT is a data type ------------------------------------------ How is an ADT different from a data type specification? ------------------------------------------ 1 INCH = 1 MILE? ^^^^^^^^^^^^^) ( Abstraction ) ( 1/2 ) ( 3/4 ) (uuuuuuuuuuuuu) ^ | | Abstraction Function | | |------------------| | [1 2] [6 8] | |------------------| ------------------------------------------ 1. stacks (3.7.1) ------------------------------------------ DECLARATIVE STACK (3.7.1) Signature: NewStack: > Push: T}: > Pop: T}: > IsEmpty: }: > ------------------------------------------ Which are observers? Which construct new stacks? ------------------------------------------ EXAMPLES USING THESE declare fun {StackSize Stk} if {IsEmpty Stk} then 0 else 1+{StackSize {Pop Stk _}} end end fun {StackElements Stk} if {IsEmpty Stk} then nil else local E Rest in Rest={Pop Stk E} E|{StackElements Rest} end end end fun {StackEqual S1 S2} {StackElements S1} == {StackElements S2} end ------------------------------------------ ------------------------------------------ Specification forall E:T, ST: : {IsEmpty {NewStack}} == true {IsEmpty {Push ST E}} == false {StackEqual {Pop {Push ST E} _} ST} local Y in _={Pop {Push ST E} Y} Y end == E try _={Pop {NewStack} _} false catch _ then true end == true ------------------------------------------ What's going on with Pop? What do these last two equations mean? What's an implementation of Stack that satisfies these specifications? What's another one? How do we specify that this is persistent? 2. A non-free type, Sample ------------------------------------------ HIDDEN DATA WITH AN INVARIANT Signature: NewSample: }: > Average: }: > Median: }: > Specification for all I, J, K, L: , F: : try {Average {NewSample I J K}} catch X then F end == if 0 =< I andthen I < J andthen J < K then {IntToFloat I+J+K}/3.0 else F end try {Median {NewSample I J K}} catch X then L end == if 0 =< I andthen I < J andthen J < K then J else L end ------------------------------------------ Suppose Inc is a value of type , what can we say about {Average Inc} and {Median Inc}? Using only the operations on a value Inc of type , can one figure out what the 3 arguments to the constructor are? ------------------------------------------ Q: What are two possible implementations of this ADT? ------------------------------------------ Can someone figure out what the 3 arguments to the constructor are if they know how the implementation works? 3. dictionaries (3.7.1) (skip) ------------------------------------------ DICTIONARIES (3.7.1) Signature: NewDictionary: > Put: }: > CondGet: }: > Domain: }: >> where ::= | Specification: {Domain {NewDictionary}} = nil {Domain {Put D F V} = F|{Domain D} {CondGet {NewDictionary} F V} = V {CondGet {Put D F V} F V2} = V {CondGet {Put D F V} F2 V2} = {CondGet D F2 V2} ------------------------------------------ What would be an implementation? B. security (3.7.4) 1. problems What kinds of problems could happen if an ADT is implemented as a collection of functions? ------------------------------------------ SECURITY OF ADTS Want to prevent: alteration = discovery = impersonation = -- James Morris, 1973 Encapsulation = all code for an ADT is in one place ------------------------------------------ Why is alteration a bad thing? Why is discovery a bad thing? Why is impersonation a bad thing? Are the ADT implementations we've seen so far secure? What's the connection between encapsulation and maintenance? What's the connection between encapsulation and security? What's an open program? Why is that a problem for security? 2. solutions (3.7.4-5) What are general approaches to securing your cell phone? a. names (3.7.5) ------------------------------------------ NAMES TO PROTECT VALUES (3.7.5) Signature of Value type: NewName: > == : }: > Specification: ({NewName} == {NewName}) = false ------------------------------------------ Is NewName referentially transparent? Is this possible in the declarative model? ------------------------------------------ WRAPPING AND UNWRAPPING proc {NewWrapper ?Wrap ?Unwrap} Key={NewName} in fun {Wrap X} {Chunk.new w(Key:X)} end fun {Unwrap W} try W.Key catch _ then raise error(unwrap(W)) end end end end ------------------------------------------ ------------------------------------------ SAMPLE AS A SECURE ADT declare local Wrap Unwrap in {NewWrapper Wrap Unwrap} fun {NewSample I J K} if 0 =< I andthen I < J andthen J < K then {Wrap sample(average: {IntToFloat I+J+K}/3.0 median: J)} else raise badArguments(I J K) end end end fun {Average Sample} sample(average: AV ...) = {Unwrap Sample} in AV end fun {Median Sample} sample(median: MV ...) = {Unwrap Sample} in MV end end ------------------------------------------ How does this prevent discovery? Impersonation? Can you write Stack using NewWrapper? b. protecting undetermined variables Does wrapping protect undetermined variables? ------------------------------------------ READ-ONLY VIEW OF VARIABLE (FUTURES) !!X means X, which can only be read from ------------------------------------------ What does read only mean in the declarative model? How to implement this? 3. capabilities (3.7.7) ------------------------------------------ CAPABILITIES (3.7.7) def: a *capability* is an unforgeable entity with an associated set of rights ------------------------------------------ How could we represent rights? How can we implement the principle of least authority using capabilities? V. Nondeclarative needs (3.8) Are nondeclarative operations needed to interface with the world? What is a module? A. text input/output with files (3.8.1) ------------------------------------------ FILE MODULE declare [File] = {Module.link ['File.ozf']} L = {File.readList "foo.txt"} D = {WordFreq L} {File.writeOpen 'word.freq'} for X in {Domain D} do {File.write {Get D X} # ' occurrences of word ' # X # '\n'} end ------------------------------------------ ------------------------------------------ VIRTUAL STRINGS - Data that specifies a string - Avoids actual concatenation Examples: 'a virtual string' 'a' # ' virtual' # " string" ['a' ' virtual ' ' string'] ------------------------------------------ Why is it useful to avoid explicit concatenation? B. Text input/output with a graphical user interface (3.8.2) C. Stateless data I/O with files VI. Program Design and Modules (3.9) What do the book's authors consider the distinction between small and large programs? A. design methodology (3.9.1) (skip) ------------------------------------------ METHODOLOGY FOR DEVELOPING SMALL APPLICATIONS - informal specification (requirements) - examples, including boundary conditions - exploration of programming fragments and basic operations (bottom up exploration) - structure and coding (some structure from modules) - testing and reasoning to check correctness and efficiency - judging quality (correctness, efficiency, maintainability extensibility, simplicity) ------------------------------------------ B. example program requirements (3.9.2) (skip) C. Software components (3.9.3) 1. modules and functors ------------------------------------------ MODULES Modules have: - an interface, represented by a record - an implementation client | v interface(op1: Op1, ... opn: OpN) | | | module | | implementation | | | |-------------------------------| ------------------------------------------ what is the implementation? can the interface record see the implementation? how is the implementation hidden from clients? What is this like in C, C++, and Java? ------------------------------------------ FUNCTORS A functor is a function that: - takes module interfaces as arguments - creates a new module - returns the new module's interface Syntax (Table 3.8): ::= ... | functor [ import { }+ ] [ export { }+ ] define { }+ [ in ] end ::= [ at ] | ( { } + ) ::= [ : ] ::= | ::= [ : ] ::= ... | functor [ $ ] [ import { }+ ] [ export { }+ ] define { }+ [ in ] end ------------------------------------------ What is this like in C, C++, and Java? can we also make a functor into an expression? what's the unit of compilation in Mozart? How are modules linked into a program? What's the module environment? ------------------------------------------ EXAMPLE %% file Combinators.oz % Example of how to write a module in Oz functor $ import % This section isn't needed for this example System(showInfo show) export s: S k: K define % The S combinator fun {S F} fun {$ G} fun {$ X} {{F X} {G X}} end end end % The K (constant) combinator fun {K C} fun {$ X} C end end end ------------------------------------------ 2. compilation 3. linking ------------------------------------------ % LINKING MODULES local [Combinators] = {Module.link ['Combinators.ozf']} S = Combinators.s K = Combinators.k in % In this part we use the abbriviations defined above {StartTesting 'Combinators'} {Test {{Combinators.k 7} 6} '==' 7} local I = {{S K} K} in {Test {I 3} '==' 3} end end ------------------------------------------ What is Module.link returning? 4. libraries