Operational Semantics Class notes for a lecture given by Mooly Sagiv Tel Aviv University 24/5/2007 By Roy Ganor and Uri Juhasz

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1 Operational emantic Page Operational emantic Cla note for a lecture given by Mooly agiv Tel Aviv Univerity 4/5/7 By Roy Ganor and Uri Juhaz Reference emantic with Application, H. Nielon and F. Nielon, Chapter. hp:// Introduction emantic i the tudy of meaning of language. Formal emantic for programming language i the tudy of formalization of the practice of computer programming. A computer language conit of a (formal) yntax decribing the actual tructure of program and the emantic which decribe the meaning of program. Many tool exit for programming language (compiler, interpreter, verification tool, ), the bai on which thee tool can cooperate i the formal emantic of the language. Formal emantic When deigning a programming language, formal emantic give an (hopefully) unambiguou definition of what a program wrien in the language hould do. Thi definition ha everal ue: People learning the language can undertand the ubtletie of it ue The model over which the emantic i defined (the emantic domain) can indicate what the requirement are for implementing the language (a a compiler/interpreter/ ) Global propertie of any program wrien in the language, and any tate occurring in uch a program, can be undertood from the formal emantic Implementer of tool for the language (parer, compiler, interpreter, debugger etc) have a formal reference for their tool and a formal definition of it correctne/completene Program wrien in the language can be verified formally againt a formal pecification (or at leat a definition for their correctne exit) different program in the language can be proved formally a equivalent/non-equivalent From a computer readable verion of the emantic, an interpreter can be automatically generated full compiler generation i not (yet) feaible emantic Domain Formal emantic give rule for tranlation from one domain (uually the program abtract yntax) to another formally defined domain, the domain include the tate pace and the pace of function/relation over the tate: Operational emantic decribe the effect of each tatement on the tate uually giving the pot-tate a a function of the pre-tate.

2 Operational emantic Page Axiomatic emantic uually define a tranlation into logic formulae in ome well defined logic language the formulae decribe, for each tatement, the relation between the pre-tate and the pot-tate of the executing the tatement. Thi i uually given a a predicate tranformer: a formula i true at the pot-tate if ome yntactic tranformation of it i true at the pre-tate or the other way around. Logic model are ometime given over which thee formulae are interpreted the effect of a program in thee model i the interpretation of the formulae uually a relation. Denotational emantic define a tranlation into ome (partial) function pace uually defined in et/category theory. The meaning of a program i a (partial) function/relation in that pace. Uually emantic are given in a mathematical, machine independent way. Mot formal emantic are tailored for language in which the abtract yntax of a program i a tree hence a well-formed program can be een a a tree of tatement, with imple tatement (e.g. aignment) in leave and compound tatement (e.g. while loop) in node. There are everal flavour for formal emantic: Operational emantic: o Give the effect of each tatement a an operation or et of operation on ome abtract machine o The effect i given for imple tatement (leave). For compound tatement (node) a rule i given to combine the effect of it underlying tatement o Major type are Large tep emantic (Natural emantic) and mall tep emantic (tructural emantic) o Mot cloely related to interpreter and abtract interpretation o Example: Aignment: x : = c, x c ( ) Meaning after the aignment x:=c the value of the variable x would be c., '' '', while e do ' While:, while e do ' Denotational emantic: o Give the effect of each tatement a an equation decribing a relation between the input tate and the output tate hence give a tranlation of the program into ome relation in a wellformalized mathematical domain (e.g. et theory) o The effect of a whole program i a olution of the et of emantic equation which are uually complicated (e.g. for loop uually fix point are ued) o Example: Aignment: D x: = c ( ) x c

3 Operational emantic Page 3 Meaning that the aignment x:=c i tranlated into the function that take a tate and return with the value of x replaced by the value of the expreion c in the. While: ( ( )) while do ( ) ( ) = D e lfp F x ite a, x, id D Where lfp i the leat fix point of an operator (Here F i an operator over emantic function a function over relation over tate) and ite i the if-then-ele operator Axiomatic emantic: o Give the effect of each tatement a a pair of predicate for the pre and pot-tate defined over a logical language, along with deduction rule for compound tatement o A program atifie a property at a given point iff thi property can be proved from the axiom and deduction rule of the emantic o Can be ued to convert program correctne into a et of verification condition which can be proved with any automatic/manual theorem proving too o Example: Aignment: { P[ x/ c } x: = c{ P} (P i an arbitrary formula and [a/b denote textual ubtitution of free occurrence of a with b) the meaning i that we can prove that P hold after the aignment if P[x/c held before the aignment for example we can prove: y > 5 x: = y x> 5 { } { } While: { e I} { I} {} I while e do I{ I e } Here I i the loop invariant. Thi roughly mean: if we can prove that the loop body() preerve the invariant auming e hold at the pre-tate, then we can prove that at the end of the loop the invariant hold and e doen t. Uually, if everal emantic are given for the ame language, we would like to prove ome form of an equivalence theorem: tating that all emantic decribe the ame language. Abtract Interpretation and emantic Abtract interpretation i cloely related to operational emantic a it i the execution of abtraction of the program hence the exact emantic mut be defined.

4 Operational emantic Page 4 The concrete emantic (uually operational emantic) i defined over ome concrete domain (the tate pace e.g. for a program with variable thi would be the et of partial function from variable Id to value) In abtract interpretation we define ome abtract domain, define a function mapping the concrete domain to the abtract domain (the abtraction function which mut atify certain requirement) and then tranlate the concrete emantic, uing the abtraction function, to the abtract emantic we get an abtract tranformer for each concrete tranformer in the concrete emantic. Abtract repreentation Abtract tranformer Abtract repreentation concretization et of tate (in emantic domain) Concrete tranformer (emantic) abtraction et of tate (in emantic domain) The accuracy of thi tranlation ha direct effect on the accuracy of the analyi. For example, if our concrete domain i partial function from variable id to value, and the abtract domain i the ign of variable (plu the unknown ign), then: The concrete emantic for addition:, x: = y+ z [ x ( y) + ( z) The abtract emantic would be: # # # # #, x: = y+ z x ( y) + ( z) Where: N+ # N=N P+ # P=P For all other cae x+ # x=? (unknown ign) Abtract interpretation then execute program uing only the abtract emantic (tranformer) the tranlation enure that any concrete execution i covered by ome abtract execution, o if a property hold for all abtract execution it would hold for all concrete execution (after concretization of the property).

5 Operational emantic Page 5 The While Programming Language We define a imple programming language in order to demontrate defining formal emantic to a language for which we know the intuitive meaning of tatement. The abtract yntax of the language i: ::= kip x := a ; if b then ele while b do Where: a i an integer expreion over variable b i a Boolean expreion over variable x i a variable id. emantic Domain for While: The yntactic categorie of while are: Variable: Var Aexp: Arithmetic Expreion Bexp: Boolean Expreion tatement: tatement (a defined in the abtract yntax) The emantic categorie of while are: Number: N : are the natural number Boolean: T : Boolean truth value {,ff} tate: tate : Variable Value tate lookup : v the value of variable v in (function application) v c (v i in Var and c in Aexp) give: tate update : [ o v [ cv=c o v [ cu=u (when u v)

6 Operational emantic Page 6 emantic of Expreion For mot kind of program language emantic, the emantic of ide-effect-free expreion i independent of the emantic of tatement. Expreion appear in pecific tate in the language for example in while we have integer expreion in aignment and Boolean expreion in if and while tatement. Expreion are interpreted over tate and not over pair of tate. The ymbol ued for interpretation of arithmetic expreion i A x where x i a term in the abtract yntax for arithmetic expreion and i a tate. emantic of integer expreion for While: A x A n = n = x A e = A e + = + * = A e e A e A e A e e A e A e emantic of Boolean expreion for While: B true = B fale = ff B x = x = = = B e e if A e A e then ele ff B e e = if B e = and B e = then ele ff A can be een, the emantic for all expreion i compoitional (in the abtract yntax tree for an expreion, the meaning of each expreion depend olely on the meaning of it direct decendant). Thi property allow proving propertie inductively on expreion for example, if we want to how that an arithmetic expreion with only even number and no variable evaluate to an even number, we would how thi for leave (n,x), and for node (-e,e+e,e*e), auming it for direct decendant. Natural Operational emantic Natural operational emantic (large tep operational emantic) give, for each program, the effect of the program uually the pottate a a function of the pretate and the effect of other command. Notation: : a tate, ' : The program, when run on input, terminate in the tate

7 Operational emantic Page 7 The econd tatement i partial the program may alo terminate in other tate than or not terminate at all. The exact meaning of the program i the et of pair of tate.t. (, '):, '. { } The emantic rule let u generate the et of true tatement, ', where all other uch tatement are fale by definition. The abtract yntax of the language i given a a context free grammar hence all program can be een a tree (a derivation tree of the abtract yntax) and the emantic can be given in a compoitional way. Compoitional emantic are very ueful becaue we can give inductive proof for many program/language propertie. Each rule i given a (here a rule for if when the condition hold):, ' if N b, ' B b = if then ele Where: if N i the rule name (for later reference), ' are the premie (may be everal) if b then ele, ' i the conequence B b = i a condition If there are no premie and no condition we only write the conequence (an axiom cheme) The meaning of thi rule i that if the premie hold and the condition hold (the condition uually refer only the pretate) then the conequence hold a well. The rule are compoitional hence tatement that appear in the premie mut be ubtatement (yntactically) of the tatement in the conequence (here i a ubtatement of if b then ele ). There would uually be at leat one rule per each ubtatement of the premie (otherwie ome ubtree may be redundant)

8 Operational emantic Page 8 The operational emantic of while: imple tatement: kip kip, kip: [ N x:=a, x A a Compound tatement: equential compoition :, '','' ' [ comp N ;, ' Aignment: [ a N Conditional:, ' if N if b then ele, ' B b, ' ff if N if b then ele, ' B b Loop:, '' while b do,'' ' while b do, ' = =ff while N B b ff while N =ff while b do, B b The proof of a tatement, ' i a derivation of, ' from the rule thi form a tree of rule intance, where at each node there i a rule, at each leaf there i an axiom (a premie free rule) and for each node, there i exactly one decendant per premie of the rule, with it conequence equal to that premie the condition mut hold for all premie. The root of the tree i the tatement to be proved. (Thee tree mut be finite) In order to prove a tatement, ', we have to find a derivation tree for it: tart with the root (the tatement) At each tage, for each leaf n in the tree, for each premie p of the rule in the leaf for which there i no correponding decendant, find a rule where p i the conequence and add it a a decendant to n. Terminate when there are no more premie to atify (all leave are imple tatement). When chooing a rule to apply, we may have everal option (everal rule with the ame conequence) in order to prove our tatement we have to chooe the correct option, in order to prove it wrong ( cannot be reached from through ) we have to recurively check each option. When uing a rule, we may have everal poible intance of the ame rule (for example: in the rule comp N, doe not appear in the conequence therefore we can ue any ). To prove our tatement we have to chooe the =

9 Operational emantic Page 9 correct one (for example, for each and each there i a unique.t., '', therefore there i exactly one intance of the rule that can be applied). In while the choice of rule i determinitic (alway at mot one option), and the choice of premie for true tatement i unique (becaue the emantic i determinitic). Proof of if x=5 then x:=x- ele x:=-x, [ x 5 [ x 4 5 [ 5 B x= x = [ [ if x=5 then x:=x- ele x:=-x, x 5 x 4 if N [ [ [ a N x:=x-, x 5 x 4. Proving program propertie - equivalence: Two program and are equivalent iff forall, :, ' iff, '. We will how a proof, uing the natural emantic, of the equivalence of the following two program cheme (parametric in and b): : if b then ; while b do ele kip : while b do Auming, ' we ll how, ' We have two cae: If B b =ff then: ff Uing if N (the only poible derivation) we get, kip, ' - hence =' ff For, uing while N we get =' If B b =then: ' iff

10 Operational emantic Page Uing if N (the only poible derivation) we get, ; while b do, '. Now uing [ comp N we get, ''.t., '' and while b do,'' '. tarting from, ', uing while N we get, exit ''.t.., '' and while b do,'' '. Hence, ' iff, ' The other direction i hown imilarly ' iff ' iff there exit an ' iff there Proving language propertie determinim: We will how a proof for determinim of the emantic we gave for while. Determinim i a language property every program wrien in the language i determinitic. Formally, determinim mean (in our notation): For a program, i determinitic iff for every,','' :, ' and, '' imply '=''. We can prove thi property by induction on the derivation tree: We have to how it for the leave (axiom) and for node auming their ubnode are determinitic. The proof for while: imple tatement: kip: kip, ' and kip, '' iff ='and =''iff'='' Aignment: x:=a, ' x:=a, Hence x:=a, iff '= x A a '' iff ''= x A a ' and x:=a, '' imply '= x A a = '' Compound tatement: equential compoition :, ''',''' ' In the derivation ''' i unique (by induction for ;, ' ), o if ;, ' and ;, '' we have to have the exact ame ''' in both derivation and becaue,''' ' and,''' '' imply '='' (induction hypothei for ) we get ;, ' and ;, '' imply '=''

11 Operational emantic Page Other compound tatement are proved imilarly. Determinim will hold for ANY program in the language which i a trong theorem. The emantic Function: The emantic function i the tranlation of a program to a partial function (or relation for non-determinitic language) over tate n : tatement ( tatey tate) The emantic function decribe mathematically what the program doe hence it can be ued to compare different emantic. Example: [ n kip =id [ x y x [ y, ( )! if x> n [ y:=; while x> do (y:=y*x;x:=x-) = if x<= Example for incorrect rule: Many programming language (e.g. PACAL) have a repeat contruct in addition to the while contruct. If we try to formalize repeat we might end up with:, '' repeat until b,'' ' ff repeat N B b repeat until b, ', ' = repeat until b, ' B b repeat N =ff However thee rule do not give the intended meaning a the condition b i evaluated before the execution of rather than after it. o, for example, the tatement: repeat x:=x+ until x<= When executed with the initial tate x= will give the pottate x= while the actual program will not terminate. The correct rule would be: ff, '' repeat until b,'' ' repeat N B b ''=ff repeat until b, ', ' '= repeat until b, ' B b repeat N

12 Operational emantic Page tructural Operational emantic Introduction A we will ee hortly, natural emantic i ometime not enough for comprehenive analyi, tructural Operational emantic (O) i ued for decribing a more detailed cae like emantic for pointer, multi-threading and more. The main difference between natural emantic and O i that the laer emphaize the individual tep, meaning we have the ability to relate any portion of a tatement. In O we write the tranition relation a, γ, thi hould be conidered a the firt tep of executing the tatement on tate lead to γ. Therefore we can write two poibilitie for γ :. γ = ', ' : the execution of from tate i not completed, and the remaining computation i expreed by the intermediate configuration ', ' (which need to be performed ubequently).. γ = ' : the execution of from tate ha terminated with a final tate In cae the reult of, i not available we ay that γ i a tuck configuration and there i no ubequent tranition. Although O emphaize to the intermediate execution, the meaning of a program P on an input tate i the et of final tate (alo tuck configuration) that can be executed in arbitrary finite tep. Example emantic Going back to the While programming language example, the firt two axiom [ a and [ kip have not changed at all becaue the aignment and kip tatement are fully executed in one tep. [ a x : = a, x Α a [ [ [ kip kip, The two rule [ comp and [ comp for a tatement ; expre that the execution tart with the firt tep of from. Then there are two poible outcome derived from the two poibilitie for tranition relation:. [ comp - The execution of ha not been completed we have to complete it before embarking on the execution of. In thi cae the firt tep of, i an intermediate configuration ', ' then the next configuration i ;, '. ', ', ' [ comp ;, ';, '. [ comp - The execution of ha been completed we can tart on the execution of. In thi cae the reult of execution from i a final tate then the next configuration i, '

13 Operational emantic Page 3 [ comp ;,, ', ' The firt tep of the condition tatement tart with teting b, then branching according to the outcome: [ if if b then ele,, if Β[ b = ff [ if if b then ele,, if Β[ b = ff The firt tep of the while tatement i to unfold it one level, that i to rewrite it a a condition. In the next execution tep a tet i performed and the execution reume. Note that the [ while rule break the compoitional tructure of the emantic (the reaon i that the rule define the emantic in term of itelf), hence tructural induction doen t work here fine. [ while while b do, if b then ( ; while b do ) ele kip, Derivation equence A derivation equence of a tatement tarting in tate i either:. a finite equence: γ, γ,. γ,.., γ k of configuration atifying: γ =, γ γ + for i < k, k i i i either a terminal configuration or tuck configuration γ k. an infinite equence γ γ,,... of configuration atifying: γ, =, γ γ γ + for i i i More notation:. For a terminal or a tuck configuration γ i we write: i γ γ i - indicate that there are i tep in the execution from γ to γ i γ * γ - indicate that there i a finite number of tep from i γ to γ i. we contruct a derivation tree for each tep in the equence (ee example below) Example:. ( z : = x; x : = y) ; y : = z auming x = 5 and y = 7 The derivation equence can be contructed a follow: z : = x; x : = y ; y : = z, ( ) x : = y; y : = z, [ z 5 y : = z,( [ z 5 [ ) x 7 (( [ z 5 [ ) x 7[ ) y 5

14 Operational emantic Page 4 We can, for example contruct a derivation tree for the firt tep, which i z = x; x : = y ; y : = z, x : = y; y : = z, z 5 ( ) [ : And the derivation tree: ( z = x), [ z 5 : ( z = x; x : = y) ; y : = z, x : = y; y : = z, [ z 5 : ( z = x; x : = y) ; y : = z, x : = y; y : = z, [ z 5 :. The firt tep of the factorial program, auming x = 3 : The firt derivation tep i: y : = ; while (x = ) do ( y : = y * x; x : = x ), ( ) ( while (x = ) do ( y : = y * x; x : = x ) ), [ y We ued the axiom [ a and rule [ comp a preented in the next derivation tree: ( y : = ), [ y ( y : = ; while (x = ) do ( y : = y * x; x : = x ) ), ( while (x = ) do ( y : = y * x; x : = x ) ), [ y * Full derivation equence along with the derivation tree are preented in the book page tarting and Terminating a Program: Given a tatement and a tate :. It i alway poible to find at leae one derivation equence that tart. The reaon i that we apply axiom and rule forever or until a terminal or tuck.. The execution terminate or loop: a. terminate there i a finite derivation equence tarting with, b. loop there i an infinite derivation equence tarting with, 3. The execution terminate uccefully if, * ' for ome tate, where i a final tate and not a tuck configuration. 4. The execution alway terminate if it terminate on all tate.

15 Operational emantic Page 5 5. The execution alway loop if it loop on all tate. Propertie of O:. Two tatement and are emantically equivalent if for all tate both of the following are true: * * a., γ, γ (γ i either tuck or terminal, the length of the two equence may be different) b. There i infinite derivation equence tarting in, There i infinite derivation equence tarting in,. Determinitic - if for all choice of,, γ and γ ' the following i true Lemma: ;,, γ and, γ ' imply γ = γ ' k '' there exit ' and k k Ν.t. k k, ' and, ' ' ' where k = k + k. Proof: For O it i often ueful to conduct a proof by induction on the length of the derivation equence, the induction ha two tep:. The bai: Prove that the property hold for all derivation equence of length.. The inductive tep: Aume that the property hold for all derivation equence of length at mot k and how that it hold for a derivation equence of length k+. inpecting one of the following: a. the tructure of the yntactic element b. the derivation tree validating the firt tranition of the derivation equence. k o our induction will be on k - the length of the derivation equence ;, ' '.. For baic tep k= the reult hold vacuouly.. For the induction tep we aume that the lemma hold for k k and we hall prove it for k +, k + k ;, '', can be rewrien a ;, γ ' ' for ome configuration γ. Regarding ;, γ there are two rule that can be ued: [ comp and [ comp. a) if we ued [ comp - ;, ';, ' and taking the econd part k of the initial equence we have ' ;, ' ' ' and the induction hypothei can be applied here (ince thi i horter that the one we tarted with), hence there exit ' and k, k Ν.t k ', ' and k k +, '' where k + k = k. o we conclude that, k and, '' where k + ) + k = k., ( +

16 Operational emantic Page 6 b) if we ued [ comp - then we have, ' and (from thatγ i k, ' ) we get, ' '' o we can chooe k = and k =k for our proof. The emantic Function The meaning of a tatement i a partial function (not every element of the domain ha to be aociated with an element of the co-domain) from tate to tate: O : tm ( tate tate) uch that: O [ ' = undef if, * otherwie ' An Equivalent Reult of N and O For every tatement of While we have [ [ n = O.. Thi theorem derive that if the execution of from tate terminate in one emantic then it alo terminate in the other and the reulting tate will be equal (the ame can be derived for loop). the full proof of thi theorem i preented in book page the ummary of the proof i given here: a) Prove by induction on the hape of derivation tree that for each derivation tree in the N there i a correponding finite derivation equence in the O. b) Prove by induction on the length of derivation equence that for each finite derivation equence in the O emantic there i a correponding derivation tree in the N. Extenion to While We preent here five different extenion to While programming language, to illutrate the power and weakne of the two approache to operational emantic. Every time we add a new component to the baic language we how how to modify the emantic. We could alo add them one by one. The extenion are:. abort tatement. Non determinim 3. Parallelim 4. Local Variable and Procedure abort tatement yntax: ::= x := a kip ; if b then ele while b do abort Meaning: abort top the execution of the complete program. abort a program i modeled by enuring that the configuration of the form abort, i tuck. Therefore the extended language i till defined in the original N and O

17 Operational emantic Page 7 emantic axiom and rule. Not eing a new rule to the new tatement reult in a tuck configuration when calling it from a program. emantically Equivalent and Different to kip and while true do kip From the O point of view the three tatement are different.. kip i not emantically equivalent to abort following the firt condition of equivalency - ince kip, ' but there i no tate uch that abort, '. while true do kip i not emantically equivalent to abort following the econd condition of equivalency - there i infinite derivation equence tarting while true do kip, but there i not tarting abort,. From the N point of view thing are bit different:. kip i (till) not emantically equivalent to abort following the condition of equivalency - ince kip, but there i no tate uch that abort, '. (while true do kip) i emantically equivalent to abort ince the condition i true (hold vacuouly there i no uch tate), we are alway concerned with execution that terminate properly. Concluion:. The natural emantic cannot ditinguih between looping and abnormal termination (unle the tate are modified). In the tructural operational emantic looping reflected by infinite derivation and abnormal termination i reflected by tuck configuration Non-Determinim yntax: ::= x := a kip ; if b then ele while b do or Meaning: we can non-determinitically chooe to execute either one tatement or another. Another example of non-determinim (in other language) i allocating new memory or uing random data. ince we have two poible final tate (depend who i executed), we will have two rule in our operational emantic (and ubequently we will have two tree). From the N point of view we extend with two rule, ' [ orn or ' [ or N, ' or ' From the O point of view we extend with two rule [ or or, [ or O O or,

18 Operational emantic Page 8 Important note: a non-determinitic tatement may or may not be equivalent under N and O: (while true do kip) or x := Under N we will have only one derivation tree, rather than two under O (one infinite repreenting the while loop and one finite repreenting the aignment). x := or (x := ; x := x+) equivalent under the two emantic. Concluion:. In the natural emantic non-determinim will uppre looping if poible (mnemonic). In the tructural operational emantic non-determinim doe not uppre termination configuration 3. Comparing the N and O we ee that the laer can chooe the wrong branch of the or tatement wherea the firt alway chooe the right branch. Parallelim yntax: ::= x := a kip ; if b then ele while b do par Meaning: for a tatement par both tatement have to be executed but the execution can be interleaved (like multi-proceing i) the order i not et by the order of the tatement. Example: x := par (x := ; x := x+), can be executed a one of the following:. x := x := x := x+. x := x := x+ x := 3. x := x := x := x+ From the O point of view we write four extra rule:, ', ' [ par par, ' par, ' [ par [ par [ par 3 4,, ' par,, par, ' par,, ' ', ', ' par ', ' The firt two rule take account of the cae where we begin by executing the firt tep of tatement. if the execution of i not fully completed we modify the configuration o a to remember how far we have reached. Otherwie only ha to be executed and we update the configuration accordingly. The lat two rule are imilar for the cae where we begin by executing the firt tep of. From the N point of view we can not write extra rule in order to expre the parallelim ince we conider the execution of a tatement a an atomic entity that cannot be plit into maller piece. If we have tried to tart defining ome rule we could ee thi will not do becaue the rule only expre that either i executed before or vice vera. Concluion:

19 Operational emantic Page 9. In the natural emantic immediate contituent i an atomic entity o we cannot expre interleaving of computation.. In the tructural operational emantic we concentrate on mall tep o interleaving of computation can be eaily expreed Local Variable and procedure (p. 6-68) yntax: ::= x := a kip ; if b then ele while b do begin D v D p end call p D v := var x:= a; D v D p := proc p i ; D p * - empty declaration Meaning: We extend the language with block containing declaration of variable and procedure. The idea that the variable declared inide a block tatement are local to it. In order to add thee feature to our language we need to add two new concept:. Variable and procedure environment (model the cope) thi will allow u to write a tranition function for variable and procedure declaration with a notation of D which pecifie the relationhip between initial and final tate. We generalize the ubtitution operation on tate and write ' [ X for the tate that i a except for variable in the et X where it i a pecified by. Formally, x if x X ' [ X = ' x if x X. Location and tore (model the tack) in order to handle imple and recurive procedure call we introduce the location of the environment. o rather than having a ingle mapping from variable to value we have plit it into two mapping env v and to and the idea i that = to o env v - to determine the variable value we oberve both the location and the relative value of the variable. Giving thee two new concept we can write and repreent new rule for our extended language, for example we hall ee how to repreent a block in the natural emantic: Dv, D ',, ' '' [ block N begin Dv end, '' [ DV ( Dv ) And variable declaration rule will be: Dv, [ x A[ a D ' [ varn var x : = a; D, ' [ none ε, N D v Example uage: begin var y := ; ( x:= ; begin var x := ; y := x + end; x := y + x) end D

20 Operational emantic Page Note that there i a different uage with x a a local and global variable. From the N point of view we have one tranition for each of the yntactic categorie. The operation of begin / end will enure that local variable are retored to their previou value when the block i let. Concluion:. The natural emantic can remember local tate. Need to introduce tack or heap into tate of the tructural emantic Tranition ytem Tranition ytem are ued to decribe a more low-level emantic on program that ha relation over the tate. The advantage i that we focu on the lowet tructure of a program. More propertie: It i a private cae of O. We include the program counter (pc) in the et of tate. The meaning of a program i a relation τ X and the execution i finite number of tate. ummary. O i powerful enough to decribe imperative program, we can define the et of trace or repreent program counter implicitly or even handle goto.. Natural operational emantic i an abtraction 3. Different emantic may be ued to jutify different behavior 4. Thinking in concrete emantic i eential for a compiler writer