-Calculus

Kangwon National University

임현승

Programming Languages

These slides are based on the slides by Prof. Sungwoo Park at POSTECH.

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What is a core of functional languages?

fun x -> e

e1 e2

0, 1, 2, ..., +, -, ...

true, false, if e then e else e

patterns

datatypes

exceptions

structures

functors

let f x = evariables

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Core of functional languages

fun x -> ee1 e2

x

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-calculus

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Outline• Brief history• Abstract syntax of the -calculus• Operational semantics of the l-calculus• Substitutions• Programming in the -calculus

Liebniz’s Ideal

Gottfried Wilhelm von Leibniz

(1646-1716)

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1) Create a “universal language” in which all possible problems can be stated.

2) Find a decision method to solve all the problem stated in the universal language.

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Can one solve all problems formulated in the universal language?

Need a formalization of the notion of“decidable” or “computable”

Two Models of Computation

• -Calculus (Church 1936)

• Turing machine (Turing 1937)

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-Calculus (1936)

Alonzo Church

(1903-1995)

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• Minimalist

• Three expression types:

x [variables]

x.e [anonymous function]

e1 e2 [function application]

• Foundations of functional languages: Lisp, ML, Haskell, etc.

Turing Machine (1937)

Alan Turing

(1912-1954)

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• A hypothetical device that manipulates symbols on a strip of tape according to a table of rules

• The first accepted definition of a general-purpose computer

• Foundations of imperative languages: Java, C/C++, C#, Fortran, Pascal, assembler languages, etc.

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-calculus is Turing-complete

fun x -> ee1 e2

x

Turing machine-calculus

=

-Everywhere• Not only in FPLs…• C++ 11• C#• Java 8• Javascript• PHP• Python• Scala• Apple Swift

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Outline• Brief history V• Abstract syntax of the -calculus• Operational semantics of the l-calculus• Substitutions• Programming in the -calculus

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Syntax for a Programming Language

Concrete syntax• program =

string of characters• specifies rules for parsing.

– operator precedence– associativity– keywords, ...

1 + 2 * 31 + (2 * 3)1 + (2 * (3))

Abstract syntax• abstracts away from details of parsing.• focuses on the high-level structure of programs.• suitable for studying the semantics

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• x– variable– z, s, t, f, arg, accum, ...

• x. e– -abstraction– x = formal argument, e = body– fun x -> e

• e1 e2

– application– left-associative (as in OCaml):

• e1 e2 e3 = (e1 e2 ) e3

• e1 e2 e3 e1 (e2 e3 )

Abstract Syntax of the -Calculus

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Examples

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Outline• Brief history V• Abstract syntax of the -calculus V• Operational semantics of the l-calculus• Substitutions• Programming in the -calculus

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Semantics of Languages• Answers "what is the meaning of a given program?"

– ML has a formal semantics.– What about C?

• Three styles– denotational semantics– axiomatic semantics– operational semantics

• The 1990s saw the renaissance of operational semantics.

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Operational Semantics• Specifies how to transform a program into a value

via a sequence of operations

Program ValueP2

operation operation operationPnoperation...

let rec fac = function 1 -> 1 | n -> n * fac (n - 1)in fac 4

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Operational Semantics of -Calculus

• Specifies how to transform an expression into a value via a sequence of reductions

Expr ValueE2

reduction reduction reductionEnreduction...

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Values and Reductions

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Reductions

redex = reducible expression

: -reduction

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_____ = Redex

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-Reduction Not Unique

So we need a reduction strategy.

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Call-by-name Call-by-value

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Call-by-name Call-by-value

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Call-by-name Call-by-value• Used in Haskell• Lazy or non-strict

functional languages• The implementation uses

call-by-need.

• Superb!

• Used in OCaml• Eager or strict

functional languages

• Superb!

(fn x => 0) <some horrible computation>

(fn x => 0) <non-terminating computation>

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Outline• Brief history V• Abstract syntax of the -calculus V• Operational semantics of the l-calculus V• Substitutions• Programming in the -calculus

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Values and Reductions

redex = reducible expression

: -reduction

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Call-by-name Call-by-value

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[e' / x] e• Informally

"substitute e' for every occurrence of x in e."

• Examples

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Easy Cases First

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Two Remaining Cases

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• First (stupid) attempt

• Second attempt

• But wait:

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• Names of bound variables do not matter.

• Hence

– for a fresh variable y,

Bound Variables

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One Remaining Case

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A Naive Attempt

• An anomaly:

something for y

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Free Variables• Variables that are bound nowhere

FV(e) = set of free variables in e

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Free Variables Remain Free• From the point of view of an outside observer,

a free variable remains free until it is explicitly

replaced.

outside observer

? variable capture

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Capture-Avoiding Substitution

• What happens if– the free variable y is captured and becomes a

bound variable. – To an outside observer, it suddenly disappears!

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Substitution Completed

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• We have to rename bound variables as necessary.

Capture-Avoiding Substitution in Action

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-conversion• Renaming bound variables when necessary• Okay because the names of bound variables do not

matter.• Examples

Formalization of -Conversion

• Variable swapping

replace all occurrences of in by and all occurrences of in by .

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-conversion and Substitution

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Last case of substitution

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Outline• Brief history V• Abstract syntax of the -calculus V• Operational semantics of the l-calculus V• Substitutions V• Programming in the -calculus

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• A boolean value– "Give me two options and I will choose one for

you!"

• Syntactic sugar

Booleans

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Examples• Under the call-by-name strategy,

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Logical Operators

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Natural Numbers• A natural number n

– has the capability to repeat a given process n

times.– "Give me a function f and I will return

f n = f o f ... f o f "

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Church Numerals

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Addition• Key observation:

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Multiplication• Key observation:

• Alternatively

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Recursive Functions in -Calculus• Plan of attack

1. Assume a recursive function construct

2. Rewrite it as an expression in the -calculusi.e., show that it is syntactic sugar.

fun f x. e

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• Plan of attack

1. Assume a recursive function construct

2. Rewrite it as an expression in the -calculusi.e., show that it is syntactic sugar.

Example: Factorial

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fac to FAC

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FAC produces a new function from f

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Now we only need to find a fixed point of:

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Fixed Point• Fixed point of function f

V such that V = f (V)

• Ex.

f (x) = 2 - x

fixed point of f = 1

1 = f (1)

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Magic Revealed• Fixed point combinator / Y combinator

(call-by-value)

• fix F returns a fixed point of F.

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Now we only need to find a fixed point of:

Now we only need to compute:

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Answer