Operational Semantics

1
Operational Semantics

• References Kurtz (Ch. 8.4, 8.5, 8.6)
• Many ways to define o.s. of a language L
• Define an interpreter for L
• Define a compiler for L, plus an interpreter for
  the assembly language used
• Specify how the state changes as various commands
  execute
• We will now consider the third approach

2
Oper. Sem. (contd)

• Will define a relation ltc, ?gt ? ?' to mean
  execution of c starting in ? terminates in ?'
• E.g. ltskip , ?gt ? ?
• Also need to define ltae, ?gt ? n if value of ae in
  ? is n
• And define ltbe, ?gt ? bv if value of be in ? is
  bv

3
Oper. Sem. (contd)

• What if ae or be are not well defined in ??
• Then there will be no n such that ltae, ?gt ?
  nSimilarly for be.
• What if c doesn't terminate or aborts (when
  started in ?)?
• Then there will be no ?' such that ltc, ?gt ? ?'
• What if ae (or be) have side-effects?
• Will have to consider ltae, ?gt ? ltn, ?'gt

4
Oper. Sem. of Arith. Exps.

• a n X a0 a1 a0 ? a1 a0 ? a1
• ltn, ?gt ? n
• ltX, ?gt ? ?(X)
• lt a0, ?gt ? n0 , lt a1, ?gt ? n1 ----------------
  --------------------------
• lt a0 a1 , ?gt ? nwhere n is the sum of n0
  and n1
• lt a0, ?gt ? n0 , lt a1, ?gt ? n1 ----------------
  --------------------------
• lt a0 ? a1 , ?gt ? nwhere n is the product of
  n0 and n1 Etc.
• Using these, we can derive, for e.g., ltXY, ?gt ?
  10 if ?(X) is 4 and ?(Y) is 6

Look like axiomatic inf. rules but are not. No
assertions here
5
Oper. Sem. of Bool. Exps.

• b true false a0 a1 a0 ? a1 ?b b0
  ? b1 b0 ? b1
• lttrue, ?gt ? true
• lt a0, ?gt ? n0 , lt a1, ?gt ? n1 ----------------
  ------------------------- if n0 equal to n1
• lt a0 a1 , ?gt ? true
• lt a0, ?gt ? n0 , lt a1, ?gt ? n1 ----------------
  ------------------------- if n0 not equal to
  n1
• lt a0 a1 , ?gt ? false
• Similarly for a0 ? a1

ltfalse, ?gt ? false
6
Oper. Sem. of Bool. Exps. (contd)

• lt b, ?gt ? true ------------------------
• lt ?b, ?gt ? false
• ltb0, ?gt ? t0 , lt b1, ?gt ? t1 -------------------
  ------------------ t true if t0 and t0 are
  both
• lt b0 ? b1 , ?gt ? t true and false
  otherwise
• ltb0, ?gt ? false ltb0, ?gt ? true, ltb1, ?gt ?
  false -------------------------------- --------
  ---------------------------------
• lt b0 ? b1 , ?gt ? false lt b0 ? b1 , ?gt ?
  false
• ltb0, ?gt ? true, ltb1, ?gt ? true
  -----------------------------------------
• lt b0 ? b1 , ?gt ? true

lt b, ?gt ? false ------------------------ lt ?b,
?gt ? true
b0 ? b1 is similar
7
Oper. Sem. of Commands

• Notation ?X/m is the same as ? except value of
  X is m i.e. ?X/m(Y) m if Y ? X and
  ?(Y) otherwise
• ?X/m is also written ?X?m
• ltskip, ?gt ? ?
• lt a, ?gt ? m ------------------------------
• lt X a, ?gt ? ?X/m
• lt c0, ?gt ? ?', ltc1, ?'gt ? ?'' ------------------
  ---------------------
• lt c0 c1 , ?gt ? ?''

Need to define sem. of read and write
8
Oper. Sem. of Commands

• lt b, ?gt ? true, lt c0, ?gt ? ?' ------------------
  -----------------------
• lt if b then c0 else c1, ?gt ? ?'
• lt b, ?gt ? false, lt c1, ?gt ? ?' -----------------
  ------------------------
• lt if b then c0 else c1, ?gt ? ?'
• lt b, ?gt ? false ------------------------------
• lt while b do c, ?gt ? ?
• lt b, ?gt ? true, lt c, ?gt ? ?', ltwhile b do c, ?'gt
  ? ?'' ------------------------------------------
  -----------------------------
• lt while b do c, ?gt ? ?''

This semantics is coarse or big-step details of
individual steps are lost. Kurtz/Slonnegar focus
on small-step sem.
9
Small-step O.S.

• Example ltX 5 Y1, ?gt ? ltY1, ?X/5 gt ?
  ?X/5Y/1
• lt c, ?gt ? ltc', ?'gt -----------------------------
  ----
• lt c c1 , ?gt ? ltc' c1 ,?'gt
• K/S destroy small step OS with the following
  rule ltc, ?gt ? ?', ltc1 ,?'gt ? ?'' --------------
  ---------------------
• ltc c1 , ?gt ? ?''This removes distinction with
  big-step o.s., so makes no sense
• Problem What is a single step? Consider
  evaluation of right side of X5 as a step?

lt c, ?gt ? ?'----------------------------- lt c
c1 , ?gt ? ltc1 ,?'gt
10
Denotational Semantics

• References Kurtz (Ch. 9, 10(?)) Pagan (Ch. 4.2)
• Idea The d.s. of each construct is the function
  computed by it
• Important Compositionality Semantics of a
  construct should be obtained by appropriately
  composing the semantics of its components (and
  should not depend on the details of the
  components)

11
Notation Sem. of Expressions

• Notation C?c? denotes the den. sem. of c
• C?c?(?) ?' means executing c starting in ?
  will lead to ?'
• E?ae?(?) n means value of ae in state ? is n
• B?be?(?) bv means value of be in state ? is
  bv
• "?...?" means whatever is enclosed in ?? is a
  syntactic entity
• B?? and E?? are easy to define
• E?n?(?) n
• E?X?(?) ?(X)
• E?ae1 ae2?(?) E?ae1?(?) E?ae2?(?) but!
  "" vs. ""
• ...
• B?true?(?) true B?false?(?) false
  "true" vs. "true"
• B?ae1ae2?(?) E?ae1?(?) E?ae2?(?) ""
  vs. ""

12
Semantics of Commands

• C?skip?(?) ?
• C?x ae?(?) ? x ? E?ae?(?) (or ?
  x/E?ae?(?)
• C?write ae?(?) ? out ? ?(out)E?ae?(?)
• C?read x?(?) ? in ? tail(?(in)), x ?
  head(?(in))
• C?c1 c2?(?) C?c2? (C?c1?(?))
• C?if be then c1 else c2?(?)
• C?c1?(?) if B?be?(?) true
• C?c2?(?) if B?be?(?) false
• C?while be do c?(?)
• ? if B?be?(?) false
• C?while be do c?(C?c?(?)) if B?be?(?)
  true

13
Comparison with Oper. Sem.

• Den. sem. looks like oper. sem. But
• No such thing as small-step den. sem.
• The den. sem. of the loop has a subtle and
  important problem

14
Nested Blocks

• Consider nested blocks (but no procedures/function
  s)
• begin int x x 0
• begin int x x 1 end
• write x
• end
• should output 0, not 1
• But when we reach end of block, we can't just go
  back to the state we had prior to entering the
  block
• begin int x int y x 0 y 0
• begin int x x 1 y 1 end
• write x write y
• end
• should output 0, 1 not 0, 0

15
Semantics of Nested Blocks

• Solution Split the state, ?, into two
  components
• ? (environment) maps each program variable to
  its current address or location
• ? (store) maps each address to the value at
  that location
• D?d?(?) ?' sem. of a decl. d maps the env. ?
  to a new ?'
• C?c?(?, ?) ?' sem. of a command gives a new
  store (same env)
• C?skip?(?, ?) ?
• C?x 5?(?, ?) ??x ? 5
• D?int x?(?) ?x ? new(?)
• C?begin ds cs end?(?, ?) C?cs?(D?ds?(?), ?)
• C?c1 c2?(?, ?) C?c2?(?, C?c1?(?, ?))
• This restores, at start of c2, env. to what it
  was at start of c1

16
Semantics of Nested Blocks (contd)

• Work out semantics of begin int x int y x
  0 y 0
• begin int x x 1 y 1 end
• write x write y
• end
• You would not want to implement using this
  approach
• What if sem. of decl. depended on the store?

17
Den. Sem. of Loops (revisited)

• Our definition of den. sem. of while loops
• C?while be do c?(?)
• ? if B?be?(?) false
• C?while be do c?(C?c?(?)) if B?be?(?)
  true
• Let ww be "while true do skip"
• C?ww?(?) ? if B?true?(?) false
• C?ww?(C?skip?(?)) if B?true?(?) true
• Reduces to
• C?ww?(?) C?ww?( ? ) (???)
• Any function would satisfy that "definition"!