A SIMPLE IMPERATIVE LANGUAGE THE STORE FUNCTION NON-TERMINATING COMMANDS

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1 A SIMPLE IMPERATIVE LANGUAGE Eventually we will preent the emantic of a full-blown language, with declaration, type and looping. However, there are many complication, o we will build up lowly. Our firt verion will be language like the example ued in the yntax ection, but without a while loop. It ha only one type (natural number) and imple arithmetic. However, a a half-way houe to the full language, we will add a primitive command whoe ole job i to tand in for a non-terminating loop (whoe emantic we will examine later). Thi will be called diverge. The full yntax i: P Progam ommand E Expreion B Boolean-expr I Identifier N Numeral P :: :: ; if B then if B then ele I : E diverge E :: E + E I N B :: E E ~B THE STORE FUNTION Unlike the calculator, thi language need an unbounded tore, which can repreent the binding of identifier to value (which are jut natural number). Jut a in operational emantic, we will call on the tore function to do thi job. The tore will be a emantic domain baed on the function that map identifier to number: Store Id Nat where Id i the (primitive) domain of identifier, and Nat i the uual domain of natural number. We will alo need the Boolean domain, a before. The tore will be defined more accurately then we did in operational emantic a a emantic algebra: Domain Store Id Nat Operation newtore: Store newtore λi.zero acce: Id Store Nat acce λi.λ.(i) update: Id Nat Store Sore update λ i.λ n.λ[ i ] n The newtore operation i a contant function that alway return zero given any identifier a argument. acce imply ue the function that i paed to return the value bound to the identifier argument. update return a new function updated the new mapping of identifier to value. Thee are all preented a lambda function o that the order of the argument can be hown. That i why the functionality of each function i hown in curried form. NON-TERMINATING OMMANDS The tore will clearly be the main emantic argument in the evaluation of a command: : ommand Store Store

2 Since every one of the command form will bottom out in an aignment, the tore will change when a command i executed. However, what happen if we execute diverge? We don t want the program to continue after that point, becaue we want the emantic to be like that of an infinite loop. The olution i called lifting a domain. We will add a pecial value to the tore domain called bottom, written a. It will repreent a tore that i locked, unchangeable and inacceible. In general, a lifted domain i one which ha bottom added to it. i.e. for any domain A, A A {}. Thi change the behavior of function that map lifted domain: For a function with functionality A B, we can define a trict function by: λ xe. : A B ( λ xe. ) ( ) [ ] λ xe. a x\ a e, for all a In other word, if a trict function receive bottom, it immediately return bottom; in all other cae it imply doe it normal job. A non-trict function doe not check for bottom, but imply carrie on: λ xe. : A B ( ) [ ] λ x. e a x\ a e, for all a A a imple example, let apply firt a trict function, and then a non-trict function to an argument that i itelf an application. ( ) ( λ x. zero) ( λ y. one) ( λ xzero. ) uing the rule for trictne twice. On the other hand, ( ) ( λ x. zero) ( λ y. one) zero We do not need to reduce the argument, becaue the function being applied i a contant function that alway return zero regardle of the argument. We thu have a method of applying a trict function that i bet done by reducing the argument before applying the function. Thi i the applicative method of expreion reduction. We can make thi even plainer by uing the let form: (let x e in e ) i an abbreviation for ( λ x. e ) e So we will ue the let form to put trict function application into a more readable form. Reviiting the valuation function for command, it will actually operate on a lifted tore o that we can ue trict function to effectively prevent any further execution of command. : ommand Store Store SEMANTI ALGEBRAS We will not repeat the emantic algebra for the Boolean and the natural number. The Store domain ha been defined above. The only new domain i one for identifier. We will ditinguih between the emantic domain Id and the et Identifier, but they are baically identical, and there are no operation, it i jut a et of contant object with name x, y, z etc. VALUATION FUNTIONS We will proceed bottom-up again to leave the mot complex for lat. Mot of thee are the ame a we have een already in the calculator.

3 N: Numeral Nat There i not much to ay here. The et B: Boolean-expr Store Tr E+ E λ. E E ~B λ. not ( BB) B E equal E B i primitive. Again, thi i traightforward, except that we are expreing the right-hand ide a a lambda function with one argument, a Store. E : Expreion Store Tr + I λ. acce I N λ. NN E E E λ. E E plu E E E E The firt argument to acce i the identifier that correpond to the yntax I. We ignify thi by writing it in the funny quare bracket. The valuation function for command are: : ommand Store Store ; λ. ( ) B λ. B B λ. B I : E λ. update I ( EE) λ. if then B if then ele B diverge The firt four are reminicent of the operational rule, but intead of being indirect rule of inference (connected by an implication) thee function very directly given the denotation for each form. The function are all trict becaue if at any point any of them get a improper tore, i.e. bottom, then it immediately return bottom and refue to look at anything ele. However, if, an if-then i being executed and i handed a good tore (not bottom), then even when the then command return bottom, if the tet i fale the tore returned i the original good one. The aignment function reduce to a call to update to actually return a changed tore. The intereting addition i the function for diverge. Thi guarantee to return an improper tore (bottom) whatever it i given. It need not be trict, but i o for conitency purpoe. We finih off with the function for program. Here we alo add ome intreret by having the ability to pa a ingle number to a program, and to return a ingle number. Thi will erve a primitive input-output. Since there are no read or write command, we mut aign the input value to a variable in the emantic, and retrieve the final reult from a variable. We will chooe A for the input and Z for the output. Thee are arbitrary choice, not preent in the yntax, but preent only in the emantic. P: Program P Nat Nat n ( update n newtore) let in accez λ.let A in Note that the return domain i lifted ince the reult of executing the program may be bottom, repreenting a failed execution at ome tage (i.e. a diverge command). The ue of the let form aure u that bottom will be handled correctly.

4 AN EXAMPLE DERIVATION Firt, a derivation of a program with an input value will be examined. The program i: Z : ; if A 0 then diverge; Z : 3 The derivation tart a: Z : ; A 0 ; Z : 3( two) let updatea two newtore in if then diverge accez P if then diverge let Z : ; A 0 ; Z : 3 in Let updatea two newtore A two ( λ i. zero). So in the above expreion, Working on Z :, we have Z : ( λ. updatez( E ) ) updatez( N) Z one Thi i the tore for the ret of the program. So A 0 then diverge; Z : 3 ( λ. Z : 3 ( if A 0 then diverge) ) Z : 3 ( if A 0 then diverge ) Z : 3 ( λ. BA 0 diverge ) Z : 3( BA 0 diverge ) if ( ) ( λ. ) diverge, o we will get non-termination if the tet i true. However A 0 ( λ. A equal 0) EA equal E0 acce A equal zero (( Z A ) )A B E E one two newtore equal zero two equal zero fale So, diverge i not executed. The lat command i thu executed in the (unchanged) tore : Z Z : 3 3 three

5 The denotation for the whole program i then Z3 (( Z three) )Z acce three which i a imple number, a we expected. However, if the input value i zero, we get zero newtore, and the conditional i 4 A B A 0 diverge true diverge4 4 diverge 4 ( λ. ) So the tore for Z : 3 i improper, and then Z : 3 ( λ. updatez( E 3) ), becaue of trictne So the update i never carried out. The final denotation i let in acce Z, directly from the definition of let OMPILED DENOTATIONS 4 We can do more by paing the program a general input n, intead of an actual number. Of coure, now we cannot reduce the denotation to a number; it mut remain a a conditional. The denotation i: Z λ n.let update A n newtore in ( ) ( ) Zthree let update Z one in let let acce A equal zero λ. in in update acce In thi expreion, all the yntax i gone (except for identifier), but we are left with a function, which, if applied to two, will produce the denotation three, a before, and bottom if applied to zero. Thi reemble compiling, and hint that thee denotation can be ued to tudy how language may be compiled for a particular virtual machine (or a real machine). Here the virtual machine i one that can evaluate lambda expreion by ubtitution. One final tep i to attempt to implify thi rather large expreion. If we apply function to proper tore, when we know them, and alo ue a tranformation which i eay to prove by extenionality of function: ( ) [ ] let e e in e i the ame a e e \ e 3 3 then we can reduce the expreion to λ n. n equal zero three

6 which i intuitively correct. Here we have optimized all identifier away, leaving only number. PROVING PROGRAM EQUIVALENE Since the method produce tatic denotation, we can compare program by comparing their denotation. So, X : 0; Y : X +, and Y : ; X : 0 hould be equivalent in that they end up with the ame the two program value in the tore. We can prove thi by deriving the denotation in each cae, and then uing the principle of extenionality of function. The derivation of the firt program yield a denotation of Y one X zero for any tore, and the econd give a denotation of X Y zero one for the ame tore. There i no way to alter the firt tore into the econd, or vice vera by any of the rule we have. We mut prove that thee two tore are the ame function omehow. We do it by uing the fact that if x i any argument and f x g x, then f and g are the ame function. So if we can how that our two tore produce the ame reult for all argument, then we have hown they are the ame tore. learly, there are two pecial cae: X,and Y. ( X ), and ( X) ( Y ), and ( Y) zero zero one one The lat cae i the anything ele cae: I I, and I ( I ) for I which i not X nor Y. Hence the equivalence i proved. ( ) ( ) ( )