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COMP3258: Functional Programming

创建日期：2024-07-04 15:29:40

所属分类：计算机computer science

标签： COMP3258

Functional Programming

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1. Expression Trees

This time our expression trees are about lists instead of integers. And we are going to define an evaluation function which returns a value of Maybe [Int].

In Haskell, Maybe is a type constructor used to represent “a value or nothing”. A value of type Maybe a is either Just a (just a value of type a) or Nothing, as Maybe has 2 data constructors by the following definition (which is included in Prelude library).

Maybe describes values with a context of possible failure attached. For example, if we expect a function returns a integer, but it might have error, we can set its return type to be Maybe Int, and let it returns Nothing when error happens. When there is no error, the function can return Some 3 for example if 3 is the calculated result.

If the input type of your function is Maybe a, you need to take care of the 2 possible forms of the argument. The following example shows how to use pattern matching to define a function f that converts Nothing to -1.

Let’s start from defining operations.

We support 3 operations on lists. Add and Sub can be understood as vector addition and substraction. And Div works in a similar way. Essentially, an operation takes 2 lists with the same lengths. For each element from one list, add it to, or substract it from, or divide it by the element from in the same position from the other list. Specially, if the 2 operands are both empty lists, the result should be empty list.

Problem 1. (10 pts.) Implement a function applyOp :: Op -> Maybe [Int] -> Maybe [Int] -> Maybe [Int] that calculate the result when applying an operation.

The function should return Nothing when: Any of its input is Nothing, or its input lists are in different lengths, or dividing by zero happens.

Expected running results:

*Main> applyOp Add (Just [1,2,3]) (Just [1,1,1]) Just [2,3,4]

*Main> applyOp Sub (Just [1,2,3]) (Just [1,1,1]) Just [0,1,2]

*Main> applyOp Div (Just [2,3,4]) (Just [2,2,2]) Just [1,1,2]

*Main> applyOp Div (Just []) (Just []) Just []

*Main> applyOp Div (Just [2,3,4]) (Just [2,0,2]) Nothing

*Main> applyOp Add (Just [1,2]) (Just [1,1,1]) Nothing

*Main> applyOp Sub (Just [1,2,3]) Nothing Nothing

Our Expr data type is defined for expression trees on lists.

Problem 2. (5 pts.) Implement a function eval :: Env -> Expr -> Maybe [Int] to evaluate expression trees. eval should return Nothing if the lengths of 2 lists for one operation is different, or any divisor is 0, or a variable cannot be found in the environment.

Expected running results:

eval [] (Bin Add (Val [1,2,3]) (Val [1,1,1])) Just [2,3,4]

eval [(“x”,[1..3])] (Bin Add (Var “x”) (Val [1,1,1])) Just [2,3,4]

eval [] $ Bin Add (Bin Add (Val [1]) (Val [2])) (Bin Div (Val [3]) (Val [2])) Just [4]

*Main> eval [(“x1”,[1..10]),(“x2”,[2..11])] (Bin Add (Var “x1”) (Var “x2”)) Just [3,5,7,9,11,13,15,17,19,21]

2. Parsing Expressions

Then let’s write a parser for those expression trees. You may want to review Tutorial 4 and Lecture 7 when doing this section. And don’t forget you can use functions defined in Parsing.h.

Here is the grammer of our expressions:

expr := term op_term

op_term := (′+′ | ′−′) term op_term | ϵ term := factor op_factor

op_factor := ′/′ factor op_factor | ϵ factor := ′(′ expr ′)′ | list | identif ier list := ′[′ (integer some_ints | ϵ) ′]′

some_ints := ′,′ interger some_ints| ϵ

You can assume the identifiers start with a lower case letter, and may contain any alphabetic or numeric characters after the first one, without any spaces in it.

Notice:

Use the token function in hs to remove leading and trailing spaces.
Your parser should reflect the left-associativity of the See the last example in question 4.
Integers can be

Problem 3. (10 pts.) Implement a function pList :: Parser Expr for parsing lists based on the last 2 rules in our grammer above.

Expected running results:

*Main> parse pList “[1,2,3]” [(Val [1,2,3],””)]

*Main> parse pList “[]” [(Val [],””)]

*Main> parse pList ” [ 1 , 2 , 3 ]” [(Val [1,2,3],””)]

Problem 4. (20 pts.) Implement a function pExpr :: Parser Expr for parsing Exprs.

Expected running results:

*Main> parse pExpr “[1,2,3]+[1,1,1]” [(Bin Add (Val [1,2,3]) (Val [1,1,1]),””)]

*Main> parse pExpr “[1,2,3]-[1,1,1]” [(Bin Sub (Val [1,2,3]) (Val [1,1,1]),””)]

*Main> parse pExpr “[] / []” [(Bin Div (Val []) (Val []),””)]

*Main> parse pExpr ” [10] – [3] – [2] – [1]”

[(Bin Sub (Bin Sub (Bin Sub (Val [10]) (Val [3])) (Val [2])) (Val [1]),””)]

Now we are going to implement a general function that works for any Parser. Instead of parse, this function runs a parser and returns the parsed thing, but it fails when the parser does not parse the whole string. Thurs, for example, it can tell if a given string is grammacially incorrect.

data Parser a = P (String -> [(a,String)])

In the data type definition of Parser in Parsing.h, we are using empty list to represent failure of a parsing process. In our function, we would like to convert it to Nothing.

Problem 5. (5 pts.) Implement a function runParser :: Parser a -> String -> Maybe a. runParser runs a given parser once to parse the full string and returns the parsed result.

Notice:

Return Nothing when the parser fails
Return Nothing when the parser only consumes part of the input

Expected running results:

*Main> runParser (char ‘a’) “a” Just ‘a’

*Main> runParser (char ‘a’) “aa” Nothing

*Main> runParser pList “[]” Just (Val [])

*Main> runParser pList “[]]” Nothing

*Main> runParser pExpr “x+[1]” Just (Bin Add (Var “x”) (Val [1]))

*Main> runParser pExpr “x++” Nothing

3. Compilation

Instead of defining eval directly, we can give expressions meaning by compiling expres- sions onto a simple stack machine.

Consider that the stack machine supports the following instructions:

data Instr = IVal [Int] | IBin Op | IVar String

deriving (Eq, Show)

Instructions contains integer literal IVal, variable Var, and binary operations IBin. The evaluation of instructions are based on a simple stack, which is modeled as a list of list of integers:

type Stack = [[Int]]

Initially the stack would be empty, when implementing instruction, the state of the stack changes. Instruction IVal xs will push xs into the stack, and IVar x will push the value of variable x (if it is in the environment) into stack. IBin op will pop two values from the stack, and apply the binary operator to them, and then push the result into the stack.

type Prog = [Instr]

A program is a list of instructions. Implementing a program is implementing instructions in it sequentially.

Before compiling, we can do static checking on an expression to catch some possible errors eariler. It is static in contrast with dynamtic checking because we do not need to run the program or calculate the result of expressions.

Problem 6. (5 pts.) Implement a function check :: Expr -> Env -> Bool that when any binary operations in the input expression takes 2 lists with different lengths, or when any variable cannot be found in the enviroment.

Expected running results:

*Main> check (Val [1..10]) [] True

*Main> check (Bin Add (Val [1,2]) (Val [0,1])) [] True

*Main> check (Bin Div (Val [1]) (Val [0])) [] True

*Main> check (Bin Add (Val [1]) (Bin Sub (Val [2]) (Val [1]))) [] True

*Main> check (Bin Add (Var “x”) (Bin Sub (Val [2]) (Val [1]))) [(“x”,[1,2])] False

Problem 7. (10 pts.) Implement a function compile :: Expr -> Env -> Maybe Prog that compiles an expression into a program that can be run in the stack machine after checking the expression with given environment using check. It returns Nothing when checking fails.

Expected running results:

*Main> compile (Val [1..10]) [] Just [IVal [1,2,3,4,5,6,7,8,9,10]]

*Main> compile (Bin Add (Val [1,2]) (Val [0,1])) []

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MAT136H1 GY 6183 COMP3170 AGRI90089 PHYSICS 1002 ECON10003 IEOR E4004 ACS61012 MATH 40550 MKTG2113 TELE4653 ENSC1004 ECE 232E MK726 MTRX2700 SHEET 3 CSCA08 COMP30027 Math 2XX3 statistic MMAN3200 BISM7213 ECMM109 ECON 2022 FNCE 20005 GR5067 CSCB09 ME 3017 CS 5100 PHYS 2903 COMS 4771 COMP222 COMP 1046 CSC148H1 ISYS2038 FINS2615 OT5206 COMP2048 A6 MECH5770M MA826/20 CMPT 726 STA457 INFS6071 INFS588 integral ECON 216 IRDR 0017 MA323 PA 15213 PSYC2001 ENV222 Statistics ACCT5930 ACCT12 LAND2121 ME 5405 COMM1180 TELE3113 A7 ECE 568 ACCT2019 INFS1200 ENGG1340 CS 1400 ASST3 FINM7409 SWEN30006 INFO2222 FIN20150 SOSS1000 CSSE7231 FINS5513 COMS3200 CECN 702 ECON4060H MN3515 HW2 PSYA02H3 ECON 2112 COMP90051 CS 1652 MA30059 AF1605 MKTG7501 ECO-6003B COM4521 CTM ECON3350 COMP 3804 ELEC9762 BISM 7255 ST22 MATH20014 MNE3122 COMP343 MATH5165 ELE7029 ENGR30002 SEEM3500 ECON7350 RISK5002 FINM7405 COMP90059 MOS 3320B STAT 431 BISM7221 MATH3014 MTH783P EBA 35302 ACS133 BUS2206 IE5600 ECOS 3997 EDST2003 FIN 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ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 101H1 IB9520 FINM 3008 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