Formal Semantics of Programming Languages

虞慧群yhq@ecust.edu.cn

Topic 4: Denotational Semantics

Motivation

Operational semantics to IMP is syntax-oriented, which makes it difficult to compare two programs written in different programming languages.

A more abstract (syntax independent) level for the semantics alleviates this. Denotational semantics is such a semantics.

Outline

Denotational semantics of IMP Equivalence of Semantics Complete partial orders and continuous functions

Denotational semantics A: Aexp (N) B: Bexp (T) C: Com ( ) Defined by structural induction

Denotational semantics of Aexp

A: Aexp (N) A [n] = {(, n) | } A [X] = {(, (X)) | } A [a0+a1] = {(, n0+n1) | (, n0)A[a0]&(,n1)A[a1]}

A [a0-a1] = {(, n0-n1) | (, n0)A[a0]&(,n1)A[a1]}

A [a0a1] = {(, n0 n1) | (, n0)A[a0]&(,n1)A[a1]}

Lemma: A [a ] is a function

Equivalence of Semantics

Lemma 5.3: For all a Aexp A[a]={(, n) | < a, > n}

Proof P(a) A[a]={(, n) | < a, > n} Structural induction

Denotational semantics of Bexp

B: Bexp (T) B [true] = {(, true) | } B [false] = {(, false) | } B [a0=a1] = {(, true) | & A[a0]=A[a1] }

{(, false) | & A[a0] A[a1] }

B [a0a1] = {(, true) | & A[a0] A[a1] } {(, false) | & A[a0]A[a1] }

B [b] = {(, T t) | , (, t) B[b]}

B [b0b1] = {(, t0 Tt1) | , (, t0) B[b0], (, t1) B[b1] }

B [b0b1] = {(, t0 Tt1) | , (, t0) B[b0], (, t1) B[b1] }

Lemma: B[b] is a function

Equivalence of Semantics

Lemma 5.4: For all b Bexp B[b]={(, t) | < b, > t}

Proof P(b) B[b]={(, t) | < b, > n} Structural induction

Denotational semantics of Com C[skip]={(, )| } C [X:=a] = {(, [n/X]) | & n=A[a]} C [c0;c1] = C [c1] C [c0] C [if b then c0 else c1] =

{(, ’) | B[b]=true & (, ’) C[c0]} {(, ’) | B[b]=false & (, ’) C[c1]}

C [while b do c] = fix() where() = {(, ’) | B[b]=true & (, ’) C[c]} {(, ) | B[b]=false}

Operational Semantics of Commands

/<skip, > {<a, > n}/<X:=a, > [n/X]

{<c0, > ’’, <c1, ’’ > ’ }/<c0;c1, > ’

{<b, > true, <c0, > ’ }/<if b then c0 else c1, > ’{<b, > false, <c1, > ’ }/<if b then c0 else c1, > ’{<b, > false }/<while b do c , > {<b, > true, <c, > ’’, <while b do c, ’’ ’ >}/ <while b do c, > ’

Equivalence of Semantics

Lemma 5.6: For all c Com <, c> ’ ( , ’) Cc

Proof P(c, , ’) ( , ’) Cc Rule induction

Equivalence of Semantics

Theorem 5.7 For all c Com Cc = {( , ’) | <c, > ’ }

Proof , ’: (, ’) Cc <c, > ’ Structural++ induction

Partial Orders A partial order (p.o.) is a set P with a binary

relation reflexive p: p P. p p transitive: p, q, r P. p q & q r p r antisymmetric: p, q P. p q & q p p=q

For a partial order (P, ) and a subset XP p is an upper bound of X

q X. q p p is a least upper bound of X (denoted by U X) if

p is an upper bound of X For all upper bounds q of X: p q

d1 U d2 U … dn = U{d1, d2, …, dn}

Partial Orders For a partial order (P, ) and a subset XP

p is an lower bound of X q X. p q

p is a greatest lower bound of X (denoted by X) if p is a lower bound of X For all lower bounds q of X: q p

Examples (Pow(X), ) (Pow(X), ),

Complete Partial Orders Let (D, ) be a partial order

An -chain is an increasing chain d0 d1 …dn ..

D is a complete partial order (c.p.o) if every -chain has a least upper bound

D is a complete partial order with bottom if is a c.p.o. with a minimum element .

D is a complete lattice if every subset has both greatest and lower bounds

Example (Pow(X), )

Monotonic and Continuous Functions A function f: DE between cpos D and E is

monotonic if d,d’ D.d d’ f(d) f(d’)

A function: DE between cpos D and E is continuous if for all chains

d0 d1 …dn … in D: n f(dn) = f( n dn)

Scott’ Thesis

Any computable function is continuous preserve information order

More information as input gives more information as output

preserve limits The total information obtainable as output from an

infinite sequence of input elements with refinement information is the sum of total of all information obtainable from each input seperately

Fixed Points

Let f: D D be a continuous function on a cpo with in D.

A fixed point of f is an element d D such that f(d) =d

A pre fixed point of f is an element d D such that f(d) d

Thm: fix(f) = n fn()

Knaster-Tarski Theorem

Let f: L L be a monotonic function on a complete lattice L

The least fixed point lfp(f) exists

lfp(f) = {x L: xf(x)}

Summary

Denotational semantics provides a way to declare the meaning of programs in an abstract way Can handle side-effects Loops Recursion Gotos

Fixed point theory provides a declarative way to specify computations Many usages

Exercise 4

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