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1. Extending Tiny Programming Language Principles Lecture 27 Prepared by Manuel E. Bermúdez, Ph.D. Associate Professor University of Florida

2. Tiny’s Denotational Semantics in RPAL let EQ x y = Istruthvalue x & Istruthvalue y -> (x & y) or (not x & not y) | Isstring x & Isstring y or Isinteger x & Isinteger y -> x eq y | false in let COMP f g x = let R = f x in R @EQ 'error' -> 'error' | g R in

3. Tiny’s Denotational Semantics in RPAL let PIPE x f = x @EQ 'error' -> 'error' | (f x) in let Return v s = (v,s) in let Check Dom (v,s) = Dom eq 'Num' -> Isinteger v -> (v,s) | 'error' | Dom eq 'Bool' -> Istruthvalue v -> (v,s) | 'error' | 'error' in

4. Tiny’s Denotational Semantics in RPAL let Dummy s = s in let Cond F1 F2 (v,s) = s @PIPE (v -> F1 | F2) in let Replace m i v x = x @EQ i -> v | m x in let Head i = i 1 in let Tail T = rtail T (Order T) where rec rtail T N = N eq 1 -> nil | (rtail T (N-1) aug (T N)) in

5. Tiny’s Denotational Semantics in RPAL let rec EE E (m,i,o) = Isinteger E -> Return E (m,i,o) | Isstring E -> ( E eq 'true' -> Return true (m,i,o) | E eq 'false' -> Return false (m,i,o) | E eq 'read' -> Null i -> 'error' | (Head i,(m,Tail i,o)) | (let R = m E in R @EQ 'undef' -> 'error' | (R,(m,i,o)) ) )

6. Tiny’s Denotational Semantics in RPAL | Istuple E -> ( (E 1) @EQ 'not' -> (m,i,o) @PIPE EE(E 2) @PIPE (Check 'Bool') @PIPE (fn(v,s).(not v,s))

7. Tiny’s Denotational Semantics in RPAL | (E 1) @EQ '<=' -> (m,i,o) @PIPE EE(E 2) @PIPE (Check 'Num') @PIPE (fn(v1,s1). s1 @PIPE EE(E 3) @PIPE (Check 'Num') @PIPE (fn(v2,s2).(v1 le v2,s2)) )

8. Tiny’s Denotational Semantics in RPAL | (E 1) @EQ '+' -> (m,i,o) @PIPE EE(E 2) @PIPE (Check 'Num') @PIPE (fn(v1,s1). S1 @PIPE EE(E 3) @PIPE (Check 'Num') @PIPE (fn(v2,s2).(v1 + v2,s2)) ) | 'error' // not 'not', '<=', '+' ) | 'error' // not a tuple in

9. Tiny’s Denotational Semantics in RPAL let rec CC C s = not (Istuple C) -> 'error' |(C 1) @EQ ':=' -> s @PIPE EE (C 3) @PIPE (fn(v,s). (Replace (s 1) (C 2) v,s 2,s 3)) |(C 1) @EQ 'print' -> s @PIPE EE (C 2) @PIPE (fn(v,s). (s 1,s 2,s 3 aug v))

10. Tiny’s Denotational Semantics in RPAL | (C 1) @EQ 'if' -> s @PIPE EE (C 2) @PIPE (Check 'Bool') @PIPE (Cond (CC(C 3)) (CC(C 4))) | (C 1) @EQ 'while' -> s @PIPE EE (C 2) @PIPE (Check 'Bool') @PIPE Cond (CC(';',C 3,C)) Dummy

11. Tiny’s Denotational Semantics in RPAL |(C 1) @EQ ';' -> s @PIPE CC (C 2) @PIPE CC (C 3) | 'error' // not ':=', 'if', ... in let PP P = not (Istuple P) -> (fn i. 'error') | not ((P 1) @EQ 'program') -> (fn i. 'error') | ((fn i. CC (P 2) ((fn i.'undef'),i,nil) //start state! ) @COMP (fn s.(s 3)) ) in

12. Tiny’s Denotational Semantics in RPAL Print ( PP ('program', // test program (';', (':=', 'x',3), ('print', 'x') ) ) (nil aug 3) // the input ) Whew ! Now, RUN IT !!

13. Tiny’s Denotational Semantics in RPAL • Executable semantic specification of Tiny. • Add a parser, and voilà ... Tiny is implemented ! • Could even write the parser in RPAL ...  • Inefficient, but who cares ... • 'error' (and others) should probably be '<error>', so we allow those as variable names in Tiny. • Subject to change: • Alter order of evaluation of operands. • Allow comparison of booleans.

14. Extending Tiny • First, add more comparison operators, and lots of arithmetic operators (easy). Example: EE[<- E1 E2>] = EE[E1] o (Check Num) o (λ(v1,s1). s1 => EE[E2] => (Check Num) => (λ(v2,s2).(v1 - v2,s2) )

15. Extending Tiny • Let’s add the '=' comparison operator. Allow for Num and Bool. This allows type mixing ! EE[<= E1 E2>] = EE[E1] o (Check Num) o (λ(v1,s1). s1 => EE[E2] => (Check Num) => (λ(v2,s2).(v1 eq v2,s2) )

16. Add Conditional Expression Need a new auxiliary function: Econd. ECond: (State → Val xState) → (State → Val xState) → (Val x State) → (Val x State) Econd = λEF1. λEF2. λ(v,s). s => (v → EF1 | EF2) EE[<cond E1 E2 E3>] = EE[E1] o (Check Bool) o (Econd EE[E2] EE[E3])

17. Add prefix auto-increment operator EE[<++ I>] = | EE[I] o (Check Num) (λ(v,(m,i,o)). v eq → error | (v+1, (Replace m I (v+1)), i, o) ) For postfix (n++), change this to v !

18. Adding the one-armed ‘if’ to Tiny CC[<if E C>] = EE[E] o (Check Bool) o (Cond CC[C] Dummy) Of course, for most of these, the syntactic domains need to be updated.

19. Adding a ‘repeat’ statement to Tiny CC[<repeat C E>] = CC[C] o EE[E] o (Check Bool) o (Cond Dummy (CC[<repeat C E>])) or better yet, CC[<repeat C E>] = CC[C] o CC[<while <not E> C>]

20. Adding a read statement to Tiny CC[<read I>] = | λ(m,i,o). Null i → error | (Replace m I (Head i), Tail i, o) Would need to remove the ‘read’ expression.

21. Adding the Pascal ‘for’ loop to Tiny CC[<for I F L C>] = EE[L] o (Check Num) o (λ(l,s). S => EE[F] => (Check Num) => (λ(f,(m,i,o)). (Replace m I f, i, o) => CC[<while < ≤ I l> <; C <:= I <+ I 1>>> ] ) ) o (λ(m,i,o). (Replace m I ,i, o)) Yuck. Can’t enforce lots of rules.

22. Adding the C ‘for’ loop to Tiny CC[<for F E I C>] = CC[F] o CC[<while E <; C I>>] or CC[<; F <while E <; C I>>] Remarkably simple, eh ? Of course, Tiny has no continue statement to get in the way. We assume default values have been added for any missing parts, e.g. true for E.

23. Adding a ‘case’ statement to Tiny CC[<case E CC1 ... CCn >] = EE[E] o (Check Num) o C_CC[CC1] ... C_CC[CCn] o (λ(v,s).s) Need a new syntactic domain, for case-clauses: C_C = <c_c n C> Also, a new semantic function to process them: C_CC: C_C → (Val x State) → (Val x State)

24. Adding a ‘case’ statement to Tiny To process one case clause: C_CC[<c_c n C>] = λ(v,s). v eq  → (v,s) | v ne n → (v,s) | ( , s => CC[C]) Aborts all subsequent case clauses. To process them all, change this to v !

25. Remarks on Denotational Semantics • Exercise: implement these in RPAL ! (see ‘medium’ on website) • Can this be done for “real” programming languages ? Yes, but ... • We now have three formalisms for specifying the semantics of programming languages: • Operational (RPAL) • Attribute grammars (Tiny) • Denotational (Tiny)

26. Remarks on Semantic Specifications • Remember, parsing was *easy*  • Reason: one formalism (CFG’s) good for everyone: • Language user. • Language implementer. • Language designer. • Not so in the world of semantics.

27. Remarks on Semantic Specifications User Designer Implementer Operational Denotational Attribute Grammar E – Excellent, G – Good F – Fair, T - Terrible

28. Extending Tiny Programming Language Principles Lecture 27 Prepared by Manuel E. Bermúdez, Ph.D. Associate Professor University of Florida