Mini Erlang

MINI ERLANG

ID1218 Lecture 03 2009-11-02

Christian [email protected]

Software and Computer SystemsSchool of Information and Communication

TechnologyKTH – Royal Institute of Technology

Stockholm, Sweden

mailto:[email protected]

Reminder & Overview

L03, 2009-11-02ID1218, Christian Schulte

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Roadmap: MiniErlang What to compute with

MiniErlang expressions and programs What are the results

MiniErlang Values What are the instructions

for compound value construction and function call

How are functions called parameters are passed by substitution considers only matching clauses clauses have patterns (we ignore guards)L03, 2009-11-02ID1218, Christian Schulte

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Evaluating Values

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ID1218, Christian Schulte

MiniErlang Values A MiniErlang value is an integer or a list

other values are similar In short notation

V := int | [] | [ V1 | V2 ] known as BNF notation: discussed later so: values are referred to by V (possibly

subscripted) can be: any integer, the empty list, a cons

consisting of two values V1 and V2


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MiniErlang Expressions A MiniErlang expression is a value, a

variable, or a function callE := int | [] | [ E1 |

E2 ] | X | F(E1,…, En)

expressions referred to by E variables referred to by X function names referred to by F


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MiniErlang Machine MiniErlang machine

Es ; Vs → Es’ ; Vs’transforms a pair (separated by ;) of

expression stack Es and value stack Vs

into a new pair of expression stack Es’ and value stack Vs’

Initial configuration: expression we want to evaluate on expression stack

Final configuration: single value as result on value stack


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Stacks We write stacks as

X1 … Xn Xr top of stack X1 n-th element Xn more elements Xr empty stack

Pushing X to stack Xr: X Xr Popping X from stack X Xr: Xr


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MiniErlang Execution Idea Simple case: an integer evaluates to

itself the result of an integer expression…

…is an integer value MiniErlang machine

i Er ; Vs → Er ; i Vs if the expression stack has the integer i as top

of stack… execution yields: the expression i is popped

from the expression stack and pushed on to the value stack

same for empty list L03, 2009-11-02ID1218, Christian Schulte

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MiniErlang Instruction Idea How to evaluate a list expression [ E1 |

E2 ] first evaluate E1 , to a value V1, … then evaluate E2 , to a value V2, … then construct a new value [ V1 | V2 ]

Use an instruction that says: build a list makes the assumption that values needed are

on the value stack execution will pop two values, push a new list

value when [ E1 | E2 ] is executed, E1 and E2 and the

instruction CONS are pushed on the expression stack


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Evaluating a List Expression Evaluate a list expression

[E1|E2]Er ; Vs→ E1E2CONSEr ; Vs

Execute a CONS instructionCONSEr ; V1V2Vs

→ Er ; [V2|V1]Vs


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Example We want to evaluate the expression[1|[]] (that is, just the list [1])

Start configuration of our machine [1|[]] ;

expression stack: [1|[]] empty value stack:

What should be the end configuration: ; [1|[]]

empty expression stack: result on value stack: [1|[]]


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Let’s Do It![1|[]] ;

→ …


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[E1|E2]Er ; Vs→ E1E2CONSEr ; Vs


→ 1 []CONS ; → …


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i Er ; Vs → Er ; i Vs


→ 1 []CONS ; → []CONS ; 1→ …


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i Er ; Vs → Er ; i Vs


→ 1 []CONS ; → []CONS ; 1→ CONS ; []1→ …


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CONSEr ; V1V2Vs→ Er ; [V2|V1]Vs


→ 1 []CONS ; → []CONS ; 1→ CONS ; []1→ ; [1|[]]


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Summary MiniErlang

values expressions

MiniErlang machine operates on expression and value stack evaluates topmost expression on expr stack executes topmost instruction on expr stack

Start state: single expr on expr stack Final state: single value on value stack


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Executing Functions

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ID1218, Christian Schulte

Roadmap How to evaluate arguments before

executing function… shuffle arguments on expression stack have a call instruction executing call instruction picks values from

value stack How to find the right clause

explain matching How to pass parameters

replace variables by substitutionL03, 2009-11-02ID1218, Christian Schulte

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Evaluating Function Call Evaluate call expression

F(E1, …, En)Er ; Vs→ E1…EnCALL(F/n)Er ; Vs


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MiniErlang Patterns Somehow: values + variables Or, crisply:

P := int | [] | [ P1 | P2 ] | X


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Pattern Matching Pattern matching takes a pattern P and a

value V and returns true, iff the value matches the pattern

If V matches P, a substitution is returned for each variable in the pattern P a matching

value Substitutions are applied to expressions

replacing variables by the respective values Details come later, just the big picture

nowL03, 2009-11-02ID1218, Christian Schulte

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Pattern Matching Examples [X|Xr] matches [2|[1|[]]]

substitution { X 2, Xr [1|[]]}

[X|[X|Xr]] matches [2|[2|[]]] substitution { X 2, Xr []}

[X|[X|Xr]] does not match [2|[1|[]]]

no substitution, of course


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Substitution Examples Application of

substitution { X 2, Xr [1|[]]} to expression X+len(Xr) yields yields 2+len([1|[]])

We refer to substitutions by s application to expression E by s(E)


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Reminder: Call Expression Evaluate call expression

F(E1, …, En)Er ; Vs→ E1…EnCALL(F/n)Er ; Vs


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Executing CALLCALL(F/n)Er ; V1…VnVs

→ s(E)Er ; Vs

F(P1, …, Pn) -> E is the first clause of F/n such that

the pattern [P1, …, Pn] matches……the list value [Vn, …, V1]

with substitution s


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Example Assume we want to evaluate

f([1|[]])where f/1 is defined by the single clause

f([X|Xr]) -> [X|f(Xr)].


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Let’s Do It!f([1|[]]) ;

→ [1|[]]CALL(f/1) ; → 1[]CONSCALL(f/1) ; → []CONSCALL(f/1) ; 1→ CONSCALL(f/1) ; []1→ CALL(f/1) ; [1|[]]→ [1|f([])] ; → …


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What Do We Ignore? Runtime errors

what if no clause matches Simplistic values No un-nesting No guards

simple: just check guards on values No case and if expressions

rewrite to additional functions


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An Important Fact… The expressions on the expression stack

must have an essential property…

hmmm… Look to the example again!


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Last Call Optimization MiniErlang has last call optimization

(LCO) built in remember what to do next on stack do not remember where to return to

What effect for recursive programs?


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MiniErlang Full Picture


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Making MiniErlang More Realistic Substitution replaces variables in clause

bodies by values values will be deconstructed and reconstructed

over and over again Add single rule that optimizes values

an expression that is a value can be directly moved from the expression to the value stack

subsumes the rules for integers and the empty list


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MiniErlang: Values, Expressions, Instructions MiniErlang values

V := int | [] | [ V1 | V2 ] MiniErlang expressions

E := int | [] | [ E1 | E2 ] | X | F(E1,…, En)

where X stands for a variable MiniErlang instructions

CONS, CALL


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MiniErlang: Expressions Evaluate values

V Er ; Vs → Er ; V Vs provided V is a value

Evaluate list expression[E1|E2]Er ; Vs→ E1E2CONSEr ; Vs

Evaluate function callF(E1, …, En)Er ; Vs→ E1…EnCALL(F/n)Er ; Vs


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MiniErlang: Instructions CONS instruction

CONSEr ; V1V2Vs→ Er ; [V2|V1]Vs

CALL instructionCALL(F/n)Er ; V1…VnVs

→ s(E)Er ; Vs F(P1, …, Pn) -> E first clause of F/n such that [P1, …, Pn] matches [Vn, …,V1] with substitution s


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MiniErlang Pattern Matching Patterns

P := int | [] | [ P1 | P2 ] | X match(P,V)

s:=try(P,V)if errors or XV1, XV2 s withV1≠V2 then no else s

wheretry(i,i) = try([],[]) = try([P1|P2],[V1|V2])= try(P1,V1)try(P2,V2)try(X,V) = {XV}otherwise = {error}


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Pattern Matching Example

match([A,A|[B|B]],[1,1,[]]) Uniform list notation

match([A|[A|[B|B]]],[1|[1|[[]|[]]]])

Evaluate trytry([A|[A|[B|B]]],[1|[1|[[]|[]]]])


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Evaluating trytry([A|[A|[B|B]]],[1|[1|[[]|[]]]]) =

try(A,1) try([A|[B|B]],[1|[[]|[]]]) ={A1} try(A,1) try([B|B],[[]|[]]) = {A1} {A1} try(B,[]) try(B,[]) ={A1} {B[]} {B[]} ={A1, B[]}

Matches with substitution: {A1, B[]}L03, 2009-11-02ID1218, Christian Schulte

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Pattern Matching Example

match([A,A|[B|B]],[1,2,[]]) Uniform list notation

match([A|[A|[B|B]]],[1|[2|[[]|[]]]])

Evaluate trytry([A|[A|[B|B]]],[1|[2|[[]|[]]]])


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Evaluating trytry([A|[A|[B|B]]],[1|[2|[[]|[]]]]) =

try(A,1) try([A|[B|B]],[2|[[]|[]]]) ={A1} try(A,2) try([B|B],[[]|[]]) = {A1} {A2} try(B,[]) try(B,[]) ={A1, A2} {B[]} {B[]} ={A1, A2, B[]}

Does not match! L03, 2009-11-02ID1218, Christian Schulte

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Substitution Defined over structure of expressions

s(i) = is([]) = []s([E1|E2]) = [s(E1)|s(E2)]s(F(E1, …, En)) = F(s(E1), …, s(En))s(X) = if XV s then V else X


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Matching and Substitutions If P matches V with substitution s

s=match(P,V)then

s(P)=V


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Substitution Assume s= {A1, B[]} s([A|[A|[B|B]]]) =

[s(A)|s([A|[B|B]])] =[1|[s(A)|s([B|B])]] =[1|[1|[s(B)|s(B)]]] =[1|[1|[[]|[]]]]


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Extending MiniErlang Built-in expressions, for example…

evaluate additionE1+E2Er ; Vs → E1E2ADDEr ; Vs

execute ADD instructionADDEr ; V1V2Vs → Er ; V2+V1Vs

Comma operator sequence of expressions replace substitution by environment lookup value for variable from environment


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What Did We Actually Do? Blueprint of MiniErlang implementation

operational semantics capable of explaining how calls are handelled stack machine

A real implementation would statically replace (compilation) call and

list construction to the respective instructions would replace substitution by environments

(registers) …of a stack machine is the JVM!


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What Can We Answer Faithful model for runtime

measure: number of function calls all remaining operations are constant wrt

function calls

Faithful model for memory MiniErlang does not use heap memory: value

stack stack space: number of entries on either stack space: size of entries on either stack


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Homework Hand execute app/2 in MiniErlang! Try all examples


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Documents

Mini Erlang