Formal Semantics of Programming Language s

Formal Semantics of Programming Languages Topic 4: Denotational Semantics 虞慧群 yhq@ecust.edu.cn

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[a0a1] = {(, 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[a0a1] = {(, true) |   & A[a0]  A[a1] }{(, false) |   & A[a0]A[a1] } • B[b] = {(, T t) |  , (, t) B[b]} • B[b0b1] = {(, t0 Tt1) |  , (, t0) B[b0], (, t1) B[b1] } • B[b0b1] = {(, 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> ’  ( , ’)  Cc • Proof • P(c, , ’)  ( , ’)  Cc • Rule induction

Equivalence of Semantics • Theorem 5.7 For all c  Com • Cc = {( , ’) | <c, > ’ } • Proof • , ’: (, ’)  Cc <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 XP • 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 XP • 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: DE between cpos D and E is monotonic if • d,d’ D.d  d’ f(d)  f(d’) • A function: DE between cpos D and E is continuous if for all chainsd0 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: xf(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 (1) (2)

Formal Semantics of Programming Language s