Chapter 6. Logic and Action. 6.1 Actions in General

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1 Chpter 6 Logic nd Action Overview An ction is something tht tkes plce in the world, nd tht mkes difference to wht the world looks like. Thus, ctions re mps from sttes of the world to new sttes of the world. Actions cn be of vrious kinds. The ction of spilling coffee chnges the stte of your trousers. The ction of telling lie to your friend chnges your friend s stte of mind (nd mybe the stte of your soul). The ction of multiplying two numbers chnges the stte of certin registers in your computer. Despite the differences between these vrious kinds of ctions, we will see tht they cn ll be covered under the sme logicl umbrell. 6.1 Actions in Generl Sitting quietly, doing nothing, Spring comes, nd the grss grows by itself. From: Zenrin kushu, compiled by Eicho ( ) Action is chnge in the world. Chnge cn tke plce by itself (see the poem bove), or it cn involve n gent who cuses the chnge. You re n gent. Suppose you hve bd hbit nd you wnt to give it up. Then typiclly, you will go through vrious stges. At some point there is the ction stge: you do wht you hve to do to effect chnge. Following instructions for how to combine certin elementry culinry ctions (chopping n onion, firing up stove, stirring the contents of sucer) my mke you successful cook. Following instructions for how to combine communiction steps my mke you successful slesperson, or successful brrister. Lerning to combine elementry computtionl ctions in clever wys my mke you successful computer progrmmer. Actions cn often be chrcterized in terms of their results: stir in heted butter nd suté until soft, rinse until wter is cler. In this chpter you will lern how to use logic for 6-1

2 6-2 CHAPTER 6. LOGIC AND ACTION nlyzing the interply of ction nd sttic descriptions of the world before nd fter the ction. It turns out tht structured ctions cn be viewed s compositions of bsic ctions, with only few bsic composition recipes: conditionl execution, choice, sequence, nd repetition. In some cses it is lso possible to undo or reverse n ction. This gives further recipe: if you re editing file, you cn undo the lst delete word ction, but you cnnot undo the printing of your file. Conditionl or gurded execution ( remove from fire when cheese strts to melt ), sequence ( pour eggs in nd swirl; cook for bout three minutes; gently slide out of the pn ), nd repetition ( keep stirring until soft ) re wys in which cook combines his bsic ctions in prepring mel. But these re lso the strtegies for lwyer when plnning her defence ( only discuss the chrcter of the defendnt if the prosecution forces us, first convince the jury of the soundness of the libi, next cst doubt on the relibility of the witness for the prosecution ), or the bsic lyout strtegies for progrmmer in designing his code. In this chpter we will look t the logic of these wys of combining ctions. Action structure does not depend on the nture of the bsic ctions: it pplies to ctions in the world, such s prepring brekfst, clening dishes, or spilling coffee over your trousers. It lso pplies to communictive ctions, such s reding n English sentence nd updting one s stte of knowledge ccordingly, engging in converstion, sending n emil with cc s, telling your prtner secret. These ctions typiclly chnge the cognitive sttes of the gents involved. Finlly, it pplies to computtions, i.e., ctions performed by computers. Exmples re computing the fctoril function, computing squre roots, etc. Such ctions typiclly involve chnging the memory stte of mchine. Of course there re connections between these ctegories. A communictive ction will usully involve some computtion involving memory, nd the utternce of n impertive ( Shut the door! ) is communictive ction tht is directed towrds ction in the world. There is very generl wy to model ction nd chnge, wy tht we hve in fct seen lredy. The key is to view chnging world s set of situtions linked by lbeled rcs. In the context of epistemic logic we hve looked t specil cse of this, the cse where the rcs re epistemic ccessibility reltions: gent reltions tht re reflexive, symmetric, nd trnsitive. Here we drop this restriction. Consider n ction tht cn be performed in only one possible wy. Toggling switch for switching off your lrm clock is n exmple. This cn be pictured s trnsition from n initil sitution to new sitution: lrm on toggle lrm off Toggling the switch once more will put the lrm bck on:

3 6.1. ACTIONS IN GENERAL 6-3 toggle toggle lrm on lrm off lrm on Some ctions do not hve determinte effects. Asking your boss for promotion my get you promoted, but it my lso get you fired, so this ction cn be pictured like this: promoted employed sk for promotion fired Another exmple: opening window. This brings bout chnge in the world, s follows. open window The ction of window-opening chnges stte in which the window is closed into one in which it is open. This is more subtle thn toggling n lrm clock, for once the window is open different ction is needed to close it gin. Also, the ction of opening window cn only be pplied to closed windows, not to open ones. We sy: performing the ction hs precondition or presupposition. In fct, the public nnouncements from the previous chpter cn lso be viewed s (communictive) ctions covered by our generl frmework. A public nnouncement is n ction tht effects chnge in n informtion model. bc 0 : p bc 1 : p bc!p bc 0 : p

4 6-4 CHAPTER 6. LOGIC AND ACTION On the left is n epistemic sitution where p is in fct the cse (indicted by the grey shding), but b nd c cnnot distinguish between the two sttes of ffirs, for they do not know whether p. If in such sitution there is public nnouncement tht p is the cse, then the epistemic sitution chnges to wht is pictured on the right. In the new sitution, everyone knows tht p is the cse, nd everyone knows tht everyone knows, nd so on. In other words: p hs become common knowledge. Here is computtionl exmple. The sitution on the left in the picture below gives highly bstrct view of prt of the memory of computer, with the contents of three registers x, y nd z. The effect of the ssignment ction x := y on this sitution is tht the old contents of register x gets replced by the contents of register y. The result of the ction is the picture on the right. x 3 y 2 z 4 x := y x 2 y 2 z 4 The commnd to put the vlue of register y in register x mkes the contents of registers x nd y equl. The next exmple models trffic light tht cn turn from green to yellow to red nd gin to green. The trnsitions indicte which light is turned on (the light tht is currently on is switched off). The stte # is the stte with the green light on, the stte the stte with the yellow light on, nd the stte the stte with the red light on. yellow # green red These exmples illustrte tht it is possible to pproch wide vriety of kinds of ctions from unified perspective. In this chpter we will show tht this is not only possible, but lso fruitful. In fct, much of the resoning we do in everydy life is resoning bout chnge. If you reflect on n everydy life problem, one of the things you cn do is run through vrious scenrios in your mind, nd see how you would (re)ct if things turn out s you imgine. Amusing smples re in the Dutch Hndboek voor de Moderne Vrouw (The Modern Womn s Hndbook). See Here is smple question from Hndboek voor de Moderne Vrouw : I m longing for cosy Xms prty. Wht cn I do to mke our Xms event hppy nd joyful? Here is the recommendtion for how to reflect on this:

5 6.1. ACTIONS IN GENERAL 6-5 START guest your type? hostess become hostess? yes red tips pprecited? only by husbnd by no-one not relly only pssively pose s idel guest invite kids sk prticiption mke pizz Figure 6.1: Flow Digrm of Hppy Xms Procedure

6 6-6 CHAPTER 6. LOGIC AND ACTION Are you the type of guest or the type of hostess? If the nswer is guest : Would you like to become hostess? If the nswer is not relly then your best option is to profile s n idel guest nd hope for Xms prty invittion elsewhere. If the nswer is yes then here re some tips on how to become gret hostess:... If the nswer is hostess, then sk yourself: Are your efforts truly pprecited? If the nswer is Yes, but only by my own husbnd then probbly your kids re bored to deth. Invite friends with kids of the sme ge s yours. If the nswer is Yes, but nobody lifts finger to help out then Ask everyone to prepre one of the courses. If the nswer is No, I only gets mons nd sighs then put pizz in the microwve for your spouse nd kids nd get yourself invited by friends. Figure 6.1 gives so-clled flow digrm for the recommendtions from this exmple. Note tht the questions re put in boxes, tht the nswers re lbels of outgoing rrows of the boxes, nd tht the ctions re put in boxes. 6.2 Sequence, Choice, Repetition, Test In the logic of propositions, the nturl opertions re not, nd nd or. These opertions re used to mp truth vlues into other truth vlues. When we wnt to tlk bout ction, the repertoire of opertions gets extended. Wht re nturl things to do with ctions? When we wnt to tlk bout ction t very generl level, then we first hve to look t how ctions cn be structured. Let s ssume tht we hve set of bsic ctions. Cll these bsic ctions, b, c, nd so on. Right now we re not interested in the internl structure of bsic ctions. The ctions, b, c could be nything: ctions in the world, bsic cts of communiction, or bsic chnges in the memory stte of computer. Given such set of bsic ctions, we cn look t nturl wys to combine them. Sequence In the first plce we cn perform one ction fter nother: first et brekfst, then do the dishes. First execute ction, next execute ction b. First toggle switch. Then toggle it gin. Consider gin the lrm clock toggle ction. toggle toggle lrm on lrm off lrm on Writing the sequence of two ctions nd b s ; b, we get:

7 6.2. SEQUENCE, CHOICE, REPETITION, TEST 6-7 lrm on toggle; toggle lrm on Strting out from the sitution where the lrm is off, we would get: lrm off toggle; toggle lrm off Choice A complex ction cn lso consist of choice between simpler ctions: either drink te or drink coffee. Either mrry beggr or mrry millionnire. unmrried, poor -beggr mrried, poor unmrried, poor -millionnire mrried, rich mrried, poor unmrried, poor -beggr -millionnire mrried, rich Repetition Actions cn be repeted. The phrse lther, rinse, repet is used s joke t people who tke instructions too literlly: the stop condition until hir is clen is omitted. There is lso joke bout n dvertising executive who increses the sles of his client s shmpoo by dding the word repet to its instructions. If tken literlly, the compound ction lther, rinse, repet would look like this: lther ; rinse Repeted ctions usully hve stop condition: repet the lther rinse sequence until your hir is clen. This gives more sensible interprettion of the repetition instruction:

8 6-8 CHAPTER 6. LOGIC AND ACTION lther ; rinse hir clen? yes STOP no Looking t the picture, we see tht this procedure is mbiguous, for where do we strt? Here is one possibility: START lther ; rinse hir clen? yes STOP no And here is nother: START lther ; rinse hir clen? yes STOP no The difference between these two procedures is tht the first one strts with hir clen? check: if the nswer is yes, nothing hppens. The second procedure strts with lther; rinse sequence, no mtter the initil stte of your hir.

9 6.2. SEQUENCE, CHOICE, REPETITION, TEST 6-9 In mny progrmming lnguges, this sme distinction is mde by mens of choice between two different constructs for expressing condition controlled loops : while not hir clen do { lther; rinse } repet { lther ; rinse } until hir clen The first loop does not gurntee tht the lther ; rinse sequence gets performed t lest once; the second loop does. Test The condition in condition-controlled loop (the condition hir clen, for exmple) cn itself be viewed s n ction: test whether certin fct holds. A test to see whether some condition holds cn lso be viewed s bsic ction. Nottion for the ction tht tests condition ϕ is?ϕ. The question mrk turns formul (something tht cn be true or flse) into n ction (something tht cn succeed or fil). If we express tests s?ϕ, then we should specify the lnguge from which ϕ is tken. Depending on the context, this could be the lnguge of propositionl logic, the lnguge of predicte logic, the lnguge of epistemic logic, nd so on. Since we re tking n bstrct view, the bsic ctions cn be nything. Still, there re few cses of bsic ction tht re specil. The ction tht lwys succeeds is clled SKIP. The ction tht lwys fils is clled ABORT. If we hve tests, then clerly SKIP cn be expressed s? (the test tht lwys succeeds) nd ABORT s? (the test tht lwys fils). Using test, sequence nd choice we cn express the fmilir if then else from mny progrmming lnguges. if hir clen then skip else { lther ; rinse } This becomes choice between test for clen hir (if this test succeeds then nothing hppens) nd sequence consisting of test for not-clen-hir followed by lther nd rinse (if the hir is not clen then it is first lthered nd then rinsed).?hir clen {? hir clen ; lther ; rinse } The generl recipe for expressing if ϕ then α 1 else α 2 is given by:?ϕ; α 1? ϕ; α 2. Since exctly one of the two tests?ϕ nd? ϕ will succeed, exctly one of α 1 or α 2 will get executed. Using the opertion for turning formul into test, we cn first test for p nd next test for q by mens of?p;?q. Clerly, the order of testing does not mtter, so this is equivlent to?q;?p. And since the tests do not chnge the current stte, this cn lso be expressed s single test?(p q). Similrly, the choice between two tests?p nd?q cn be written s?p?q. Agin, this is equivlent to?q?p, nd it cn be turned into single test?(p q).

10 6-10 CHAPTER 6. LOGIC AND ACTION Converse Some ctions cn be undone by reversing them: the reverse of opening window is closing it. Other ctions re much hrder to undo: if you smsh piece of chin then it is sometimes hrd to mend it gin. So here we hve choice: do we ssume tht bsic ctions cn be undone? If we do, we need n opertion for this, for tking the converse of n ction. If, in some context, we ssume tht undoing n ction is generlly impossible we should omit the converse opertion in tht context. Exercise 6.1 Suppose ˇ is used for reversing bsic ctions. So ˇ is the converse of ction, nd bˇis the converse of ction b. Let ; b be the sequentil composition of nd b, i.e., the ction tht consists of first doing nd then doing b. Wht is the converse of ; b? 6.3 Viewing Actions s Reltions As n exercise in bstrction, we will now view ctions s binry reltions on set S of sttes. The intuition behind this is s follows. Suppose we re in some stte s in S. Then performing some ction will result in new stte tht is member of some set of new sttes {s 1,..., s n }. If this set is empty, this mens tht the ction hs borted in stte s. If the set hs single element s, this mens tht the ction is deterministic on stte s, nd if the set hs two or more elements, this mens tht ction is non-deterministic on stte s. The generl picture is: s 1 s 2 s s 3 s n Clerly, when we extend this picture to the whole set S, wht emerges is binry reltion on S, with n rrow from s to s (or equivlently, pir (s, s ) in the reltion) just in cse performing ction in stte s my hve s s result. Thus, we cn view binry reltions on S s the interprettions of bsic ction symbols. The set of ll pirs tken from S is clled S S, or S 2. A binry reltion on S is simply set of pirs tken from S, i.e., subset of S 2. Given this bstrct interprettion of bsic reltions, it mkes sense to sk wht corresponds to the opertions on ctions tht we encountered in Section 6.2. Let s consider them in turn.

11 6.3. VIEWING ACTIONS AS RELATIONS 6-11 Sequence Given tht ction symbol is interpreted s binry reltion R on S, nd tht ction symbol b is interpreted s binry reltion R b on S, wht should be the interprettion of the ction sequence ; b? Intuitively, one cn move from stte s to stte s just in cse there is some intermedite stte s 0 with the property tht gets you from s to s 0 nd b gets you from s 0 to s. This is well-known opertion on binry reltions, clled reltionl composition. If R nd R b re binry reltions on the sme set S, then R R b is the binry reltion on S given by: R R b = {(s, s ) there is some s 0 S : (s, s 0 ) R nd (s 0, s ) R b }. If bsic ction symbol is interpreted s reltion R, nd bsic ction symbol b is interpreted s reltion R b, then the sequence ction ; b is interpreted s R R b. Here is picture: s 11 s 12 s 1 s 13 s s 2 s 3 s 1m s n If the solid rrows interpret ction symbol nd the dshed rrows interpret ction symbol b, then the rrows consisting of solid prt followed by dshed prt interpret the sequence ; b. Choice Now suppose gin tht we re in stte s, nd tht performing ction will get us in one of the sttes in {s 1,..., s n }. And supposse tht in tht sme stte s, performing ction b will get us in one of the sttes in {s 1,..., s m}.

12 6-12 CHAPTER 6. LOGIC AND ACTION s 1 s 2 s 3 s n s s 1 s 2 s 3 s m Then performing ction b (the choice between nd b) in s will get you in one of the sttes in {s 1,..., s n } {s 1,..., s m}. More generlly, if ction symbol is interpreted s the reltion R, nd ction symbol b is interpreted s the reltion R b, then b will be interpreted s the reltion R R b (the union of the two reltions). Test A nottion tht is often used for the equlity reltion (or: identity reltion is I. The binry reltion I on S is by definition the set of pirs given by: I = {(s, s) s S}. A test?ϕ is interpreted s subset of the identity reltion, nmely s the following set of pirs: R?ϕ = {(s, s) s S, s = ϕ} From this we cn see tht test does not chnge the stte, but checks whether the stte stisfies condition. To see the result of combining test with nother ction:

13 6.4. OPERATIONS ON RELATIONS 6-13 s 1 s 2 s s 3 s n t 1 t 2 t t 3 t m The solid rrow interprets test?ϕ tht succeeds in stte s but fils in stte t. If the dshed rrows interpret bsic ction symbol, then, for instnce, (s, s 1 ) will be in the interprettion of?ϕ;, but (t, t 1 ) will not. Since is true in ny sitution, we hve tht? will get interpreted s I (the identity reltion on S). Therefore,? ; will lwys receive the sme interprettion s. Since is flse in ny sitution, we hve tht? will get interpreted s (the empty reltion on S). Therefore,? ; will lwys receive the sme interprettion s?. Before we hndle repetition, it is useful to switch to more gererl perspective. 6.4 Opertions on Reltions Reltions were introduced in Chpter 4 on predicte logic. In this chpter we view ctions s binry reltions on set S of situtions. Such binry reltion is subset of S S, the set of ll pirs (s, t) with s nd t tken from S. It mkes sense to develop the generl topic of opertions on binry reltions. Which opertions suggest themselves, nd wht re the corresponding opertions on ctions? In the first plce, there re the usul set-theoretic opertions. Binry reltions re sets of pirs, so tking unions, intersections nd complements mkes sense (lso see Appendix A). We hve lredy seen tht tking unions corresponds to choice between ctions. Exmple 6.2 The union of the reltions mother nd fther is the reltion prent. Exmple 6.3 The intersection of the reltions nd is the equlity reltion =.

14 6-14 CHAPTER 6. LOGIC AND ACTION In Section 6.3 we encountered the nottion I for the equlity (or: identity) reltion on set S. We hve seen tht tests get interpreted s subsets of I. We lso looked t composition of reltions. R 1 R 2 is the reltion tht performing n R 1 step followed by n R 2 step. To see tht order of composition mtters, consider the following exmple. Exmple 6.4 The reltionl composition of the reltions mother nd prent is the reltion grndmother, for x is grndmother of y mens tht there is z such tht x is mother of z, nd z is prent of y. The reltionl composition of the reltions prent nd mother is the reltion mternl grndprent, for x is mternl grndprent of y mens tht there is z such tht x is prent of z nd z is mother of y. Exercise 6.5 Wht is the reltionl composition of the reltions fther nd mother? Another importnt opertion is reltionl converse. The reltionl converse of binry reltion R, nottion Rˇ, is the reltion given by: Rˇ = {(y, x) S 2 (x, y) R}. Exmple 6.6 The reltionl converse of the prent reltion is the child reltion. Exercise 6.7 Wht is the reltionl converse of the reltion? The following lw describes the interply between composition nd converse: Converse of composition (R 1 R 2 )ˇ = R 2ˇ R 1ˇ. Exercise 6.8 Check from the definitions tht (R 1 R 2 )ˇ = R 2ˇ R 1ˇis vlid. There exists long list of logicl principles tht hold for binry reltions. To strt with, there re the usul Boolen principles tht hold for ll sets: Commuttivity R 1 R 2 = R 2 R 1, R 1 R 2 = R 2 R 1, Idempotence R R = R, R R = R. Lws of De Morgn R 1 R 2 = R 1 R 2, R 1 R 2 = R 1 R 2. Specificlly for reltionl composition we hve: Associtivity R 1 (R 2 R 3 ) = (R 1 R 2 ) R 3.

15 6.4. OPERATIONS ON RELATIONS 6-15 Distributivity R 1 (R 2 R3) = (R 1 R 2 ) (R 1 R 3 ) (R 1 R 2 ) R3) = (R 1 R 3 ) (R 2 R 3 ). There re lso mny principles tht seem plusible but tht re invlid. To see tht puttive principle is invlid one should look for counterexmple. Exmple 6.9 R R = R is invlid, for if R is the prent reltion, then the principle would stte tht grndprent equls prent, which is flse. Exercise 6.10 Show by mens of counterexmple tht R 1 (R 2 R 3 ) = (R 1 R 2 ) (R 1 R 3 ) is invlid. Exercise 6.11 Check from the definitions tht R 1 (R 2 R 3 ) = (R 1 R 2 ) (R 1 R 3 ) is vlid. Exercise 6.12 Check from the definition tht Rˇˇ = R is vlid. Exercise 6.13 Check from the definitions tht (R 1 R 2 )ˇ = R 1ˇ R 2ˇis vlid. Trnsitive Closure A reltion R is trnsitive if it holds tht if you cn get from x to y in two R-steps, then it is lso possible to get from x to y in single R-step (see pge 4-20 bove). This cn be redily expressed in terms of reltionl composition. R is trnsitive iff R R R. The trnsitive closure of reltion R is defined s the smllest trnsitive reltion S tht contins R. This mens: S is the trnsitive closure of R if (1) R S, (2) S S S, (3) if R T nd T T T then S T. Requirement (1) expresses tht R is contined in S, requirement (2) expresses tht S is trnsitive, nd requirement (3) expresses tht S is the smllest trnsitive reltion tht contins R: ny T tht stisfies the sme requirements must be t lest s lrge s S. The customry nottion for the trnsitive closure of R is R +. Here is n exmple. Exmple 6.14 The trnsitive closure of the prent reltion is the ncestor reltion. If x is prent of y then x is ncestor of y, so the prent reltion is contined in the ncestor reltion. If x is n ncestor of y nd y is n ncestor of z then surely x is n ncestor of z, so the ncestor reltion is trnsitive. Finlly, the ncestor reltion is the smllest trnsitive reltion tht contins the prent reltion.

16 6-16 CHAPTER 6. LOGIC AND ACTION You cn think of binry reltion R s recipe for tking R-steps. The recipe for tking double R-steps is now given by R R. The recipe for tking triple R-steps is given by R R R, nd so on. There is forml reson why the order of composition does not mtter: R 1 (R 2 R 3 ) denotes the sme reltion s (R 1 R 2 ) R 3. becuse of the bove-mentioned principle of ssocitivity. The n-fold composition of binry reltion R on S with itself cn be defined from R nd I (the identity reltion on S), by recursion (see Appendix, Section A.6), s follows: R 0 = I R n = R R n 1 for n > 0. Abbrevition for the n-fold composition of R is R n. This llows us to tlk bout tking specific number of R-steps. Notice tht R I = R. Thus, we get tht R 1 = R R 0 = R I = R. The trnsitive closure of reltion R cn be computed by mens of: R + = R R 2 R 3 This cn be expressed without the, s follows: R + = R n. n N,n>0 Thus, R + denotes the reltion of doing n rbitrry finite number of R-steps (t lest one). Closely relted to the trnsitive closure of R is the reflexive trnsitive closure of R. This is, by definition, the smllest reltion tht contins R nd tht is both reflexive nd trnsitive. The reflexive trnsitive closure of R cn be computed by: R = I R R 2 R 3 This cn be expressed without the, s follows: R = n N R n. Thus, R denotes the reltion of doing n rbitrry finite number of R-steps, including zero steps. Notice tht the following holds: R + = R R. Exercise 6.15 The following identity between reltions is not vlid: Explin why not by giving counter-exmple. (R S) = R S.

17 6.5. COMBINING PROPOSITIONAL LOGIC AND ACTIONS: PDL 6-17 Exercise 6.16 The following identity between reltions is not vlid: Explin why not by giving counter-exmple. (R S) = R S. For Loops In progrmming, repetition consisting of specified number of steps is clled for loop. Here is n exmple of loop for printing ten lines, in the progrmming lnguge Ruby: #!/usr/bin/ruby for i in puts "Vlue of locl vrible is #{i}" end If you hve system with Ruby instlled, you cn sve this s file nd execute it. While Loops, Repet Loops If R is the interprettion of ( doing once ), then R is the interprettion of doing n rbitrry finite number of times, nd R + is the interprettion of doing n rbitrry finite number of times but t lest once. These reltions cn be used to define the interprettion of while loops nd repet loops (the so-clled condition controlled loops), s follows. If is interpreted s R, then the condition-controlled loop while ϕ do is interpreted s: (R?ϕ R ) R? ϕ. First do number of steps consisting of?ϕ test followed by n ction, next check tht ϕ holds. Exercise 6.17 Supposing tht gets interpreted s the reltion R,?ϕ s R?ϕ nd? ϕ s R? ϕ, give reltionl interprettion for the condition controlled loop repet until ϕ. 6.5 Combining Propositionl Logic nd Actions: PDL The lnguge of propositionl logic over some set of bsic propositions P is given by: ϕ ::= p ϕ ϕ ϕ ϕ ϕ where p rnges over P. If we ssume tht set of bsic ction symbols A is given, then the lnguge of ctions tht we discussed in Sections 6.2 nd 6.3 bove cn be formlly defined s: α ::=?ϕ α; α α α α where rnges over A.

18 6-18 CHAPTER 6. LOGIC AND ACTION Note tht the test?ϕ in this definition refers to the definition of ϕ in the lnguge of propositionl logic. Thus, the lnguge of propositionl logic is embedded in the lnguge of ctions. Now here is new ide, for lso doing the converse: extend the lnguge of propositionl logic with construction tht describes the results of executing n ction α. If α is interpreted s binry reltion then in given stte s there my be severl sttes s for which (s, s ) is in the interprettion of α. Interpret α ϕ s follows: α ϕ is true in stte s if for some s with (s, s ) in the interprettion of α it holds tht ϕ is true in s. For instnce, if is the ction of sking for promotion, nd p is the proposition expressing tht one is promoted, then p expresses tht sking for promotion my result in ctully getting promoted. Another useful expression is [α]ϕ, with the following interprettion: [α]ϕ is true in stte s if for every s with (s, s ) in the interprettion of α it holds tht ϕ is true in s. For instnce, if gin expresses sking for promotion, nd p expresses tht one is promoted, then []p expresses tht, in the current stte, the ction of sking for promotion lwys results in getting promoted. Note tht p nd []p re not equivlent: think of sitution where sking for promotion my lso result in getting fired. In tht cse p my still hold, but []p does not hold. If one combines propositionl logic with ctions in this wy one gets bsic logic of chnge clled Propositionl Dynmic Logic or PDL. Here is the forml definition of the lnguge of PDL: Definition 6.18 (Lnguge of PDL propositionl dynmic logic) Let p rnge over the set of bsic propositions P, nd let rnge over set of bsic ctions A. Then the formuls ϕ nd ction sttements α of propositionl dynmic logic re given by: ϕ ::= p ϕ ϕ 1 ϕ 2 ϕ 1 ϕ 2 α ϕ [α]ϕ α ::=?ϕ α 1 ; α 2 α 1 α 2 α The definition does not hve or. But this does not mtter, for we cn introduce these opertors by mens of bbrevitions or shorthnds. is the formul tht is lwys true. From this, we cn define, s shorthnd for.

19 6.5. COMBINING PROPOSITIONAL LOGIC AND ACTIONS: PDL 6-19 Similrly, ϕ 1 ϕ 2 is shorthnd for ϕ 1 ϕ 2, ϕ 1 ϕ 2 is shorthnd for (ϕ 1 ϕ 2 ) (ϕ 2 ϕ 1 ). Propositionl dynmic logic bstrcts over the set of bsic ctions, in the sense tht bsic ctions cn be nything. In the lnguge of PDL they re toms. This mens tht the rnge of pplicbility of PDL is vst. The only thing tht mtters bout bsic ction is tht it is interpreted by some binry reltion on stte set. Propositionl dynmic logic hs two bsic syntctic ctegories: formuls nd ction sttements. Formuls re used for tlking bout sttes, ction sttements re used for clssifying trnsitions between sttes. The sme distinction between formuls nd ction sttements cn be found in ll impertive progrmming lnguges. The sttements of C or Jv or Ruby re the ction sttements. Bsic ctions in C re ssigning vlue to vrible. These re instructions to chnge the memory stte of the mchine. The so-clled Boolen expressions in C behve like formuls of propositionl logic. They pper s conditions or tests in conditionl expressions. Consider the following C sttement: if (y < z) x = y; else x = z; This is description of n ction. But the ingredient (y<z) is not sttement (description of n ction) but Boolen expression (description of stte) tht expresses test. Propositionl dynmic logic is n extension of propositionl logic with ction sttements, just like epistemic logic is n extension of propositionl logic with epistemic modlities. Let set of bsic propositions P be given. Then pproprite sttes will contin vlutions for these propositions. Let set of bsic ctions A be given. Then every bsic ction corresponds to binry reltion on the stte set. Together this gives lbeled trnsition system with vlutions on sttes s subsets from P nd lbels on rcs between sttes tken from A. Exercise 6.19 Suppose we lso wnt to introduce shorthnd α n, for sequence of n copies of ction sttement α. Show how this cn be defined by induction. (Hint: use α 0 :=? s the bse cse.) Let s get feel for the kind of things we cn express with PDL. For ny ction sttement α, α expresses tht the ction α hs t lest one successful execution. Similrly, [α] expresses tht the ction fils (cnnot be executed in the current stte).

20 6-20 CHAPTER 6. LOGIC AND ACTION The bsic ctions cn be nything, so let us focus on bsic ction tht is interpreted s the reltion R. Suppose we wnt to sy tht some execution of leds to p stte nd nother execution of leds to non-p stte. Then here is PDL formul for tht: p p. If this formul is true in stte s, then this mens tht R forks in tht stte: there re t lest two R rrows strting from s, one of them to stte s 1 stisfying p nd one of them to stte s 1 tht does not stisfy p. For the interprettion of P we need properties of sttes, for p is like one-plce predicte in predicte logic. If the bsic ctions re chnges in the world, such s spilling milk S or clening C, then [C; S]d expresses tht clening up followed by spilling milk lwys results in dirty stte, while [S; C] d expresses tht the occurrence of these events in the reverse order lwys results in clen stte. 6.6 Trnsition Systems In Section 6.7 we will define the semntics of PDL reltive to lbelled trnsition systems, or process grphs. Definition 6.20 (Lbelled trnsition system) Let P be set of bsic propositions nd A set of lbels for bsic ctions. Then lbelled trnsition system (or LTS) over toms P nd gents A is triple M = S, R, V where S is set of sttes, V : S P(P ) is vlution function, nd R = { S S A} is set of lbelled trnsitions, i.e., set of binry reltions on S, one for ech lbel. Another wy to look t lbelled trnsition system is s first order model predicte for lnguge with unry nd binry predictes. LTSs with designted node (clled the root node) re clled pointed LTSs or process grphs. The process of repetedly doing, followed by choice between b nd c cn be viewed s process grph, s follows: 0 b c 1

21 6.6. TRANSITION SYSTEMS 6-21 The root note 0 is indicted by. There re two sttes 0 nd 1. The process strt in stte 0 with the execution of ction. This gets us to stte 1, where there re two possible ctions b nd c, both of which get us bck to stte 0, nd there the process repets. This is n infinite process, just like n operting system of computer. Unless there is system crsh, the process goes on forever. Jumping out of process cn be done by creting n ction tht moves to n end stte. 0 b c 1 d 2 We cn think bout s proposition letter, nd then use PDL to tlk bout these process grphs. In stte 1 of the first model d is flse, in stte 1 of the second model d is true. This formul expresses tht d trnsition to stte is possible. In both models it is the cse in stte 0 tht fter ny number of sequences consisting of n step followed by either b or c step, further step is possible. This is expressed by the following PDL formul: [(; (b c)) ]. Exercise 6.21 Which of the following formuls re true in stte 0 of the two models given bove: (1) ; d. (2) [; d]. (3) []( b c ). (4) [] d. The following two pictures illustrte n importnt distinction:

22 6-22 CHAPTER 6. LOGIC AND ACTION b c b c In the picture on the left, it is possible to tke n ction from the root, nd next to mke choice between doing b or doing c. In the picture on the right, there re two wys of doing, one of them ends in stte where b is the only possible move nd the other one ending in stte where c is the only possible move. This difference cn be expressed in PDL formul, s follows. In the root stte of the picture on the left, []( b c ) is true, in the root stte of the picture on the right this formul is flse. Exercise 6.22 Find PDL formul tht cn distinguish between the root sttes of the following two process grphs: 0 0 b 1 1 b 2 The formul should be true in one grph, flse in the other.

23 6.7. SEMANTICS OF PDL 6-23 Exercise 6.23 Now consider the following two pictures of process grphs: 0 0 b b b 2 Is it still possible to find PDL formul tht is true in the root of one of the grphs nd flse in the root of the other? If your nswer is yes, then give such formul. If your nswer is no, then try to explin s clerly s you cn why you think this is impossible. 6.7 Semntics of PDL The formuls of PDL re interpreted in sttes of lbeled trnsition system (or: LTS, or: process grph), nd the ctions of PDL s binry reltions on the domin S of the LTS. We cn think of n LTS s given by its set of sttes S, its vlution V, nd its set of lbelled trnsitions R. We will give the interprettion of bsic ctions s. If n LTS M is given, we use S M to refer to its set of sttes, we use R M to indicte its set of lbelled trnsitions, nd we use V M for its vlution. Definition 6.24 (Semntics of PDL) Given is lbelled trnsition system M = S, V, R for P nd A. M, s = lwys M, s = p p V (s) M, s = ϕ M, s = ϕ M, s = ϕ ψ M, s = ϕ or M, s = ψ M, s = ϕ ψ M, s = ϕ nd M, s = ψ M, s = α ϕ for some t, (s, t) [α] M nd M, t = ϕ M, s = [α]ϕ for ll t with (s, t) [α] M it holds tht M, t = ϕ.

24 6-24 CHAPTER 6. LOGIC AND ACTION where the binry reltion [α] M interpreting the ction α in the model M is defined s [] M = M [?ϕ] M = {(s, s) S M S M M, s = ϕ} [α 1 ; α 2 ] M = [α 1 ] M [α 2 ] M [α 1 α 2 ] M = [α 1 ] M [α 2 ] M [α ] M = ([α] M ) Note tht the cluse for [α ] M uses the definition of reflexive trnsitive closure tht ws given on pge These cluses specify how formuls of PDL cn be used to mke ssertions bout PDL models. Exmple 6.25 The formul, when interpreted t some stte in PDL model, expresses tht tht stte hs successor in the reltion in tht model. A PDL formul ϕ is true in model if it holds t every stte in tht model, i.e., if [ϕ] M = S M. Exmple 6.26 Truth of the formul in model expresses tht is seril in tht model. (A binry reltion R is seril on domin S if it holds for ll s S tht there is some t S with srt.) A PDL formul ϕ is vlid if it holds for ll PDL models M tht ϕ is true in tht model, i.e., tht [ϕ] M = S M. Exercise 6.27 Show tht ; b b is n exmple of vlid formul. As ws note before,? is n opertion for mpping formuls to ction sttements. Action sttements of the form?ϕ re clled tests; they re interpreted s the identity reltion, restricted to the sttes stisfying the formul. Exercise 6.28 Let the following PDL model be given: b 1 : pq 2 : pq 3 : pq 4 : pq b

25 6.7. SEMANTICS OF PDL 6-25 Give the interprettions of?p, of?(p q), of ; b nd of b;. Exercise 6.29 Let the following PDL model be given: 1 : pq 2 : pq b 4 : pq b 3 : pq (1) List the sttes where the following formuls re true:. p b. b q c. [](p b q) (2) Give formul tht is true only t stte 4. (3) Give ll the elements of the reltions defined by the following ction expressions:. b; b b. b c. (4) Give PDL ction expression tht defines the reltion {(1, 3)} in the grph. (Hint: use one or more test ctions.) Let ˇ(converse) be n opertor on PDL progrms with the following interpre- Converse ttion: [αˇ] M = {(s, t) (t, s) [α] M }. Exercise 6.30 Show tht the following equlities hold: (α; β)ˇ = βˇ; αˇ (α β)ˇ = αˇ βˇ (α )ˇ = (αˇ) Exercise 6.31 Show how the equlities from the previous exercise, plus tomic converse ˇ, cn be used to define αˇ, for rbitrry α, by wy of bbrevition.

26 6-26 CHAPTER 6. LOGIC AND ACTION It follows from Exercises 6.30 nd 6.31 tht it is enough to dd converse to the PDL lnguge for tomic ctions only. To see tht dding converse in this wy increses expressive power, observe tht in root stte 0 in the following picture ˇ is true, while in root stte 2 in the picture ˇ is flse. On the ssumption tht 0 nd 2 hve the sme vlution, no PDL formul without converse cn distinguish the two sttes Axiomtistion The logic of PDL is xiomtised s follows. Axioms re ll propositionl tutologies, plus n xiom stting tht α behves s stndrd modl opertor, plus xioms describing the effects of the progrm opertors (we give box ([α])versions here, but every xiom hs n equivlent dimond ( α ) version), plus propositionl inference rule nd modl inference rule. The propositionl inference rule is the fmilir rule of Modus Ponens. (modus ponens) From ϕ 1 nd ϕ 1 ϕ 2, infer ϕ 2. The modl inference rule is the rule of modl generliztion (or: necessittion): (modl generlistion) From ϕ, infer [α]ϕ. Modl generliztion expresses tht theorems of the system hve to hold in every stte. Exmple 6.32 Tke the formul (ϕ ψ) ϕ. Becuse this is propositionl tutology, it is theorem of the system, so we hve (ϕ ψ) ϕ. And becuse it is theorem, it hs to hold everywhere, so we hve, for ny α: [α]((ϕ ψ) ϕ). Now let us turn to the xioms. The first xiom is the K xiom (fmilir from Chpter 5) tht expresses tht progrm modlities distribute over implictions: (K) [α](ϕ ψ) ([α]ϕ [α]ψ)

27 6.8. AXIOMATISATION 6-27 Exmple 6.33 As n exmple of how to ply with this, we derive the equivlent α version. By the K xiom, the following is theorem (just replce ψ by ψ everywhere in the xiom): [α](ϕ ψ) ([α]ϕ [α] ψ). From this, by the propositionl resoning principle of contrposition: From this, by propositionl resoning: ([α]ϕ [α] ψ) [α](ϕ ψ). [α]ϕ [α] ψ) [α](ϕ ψ). Now replce ll boxes by dimonds, using the bbrevition α ϕ for [α]ϕ: α ϕ α ψ) α (ϕ ψ). This cn be simplified by propositionl logic, nd we get: ( α ϕ α ψ) α (ϕ ψ). Exmple 6.34 This exmple is similr to Exmple 5.45 from Chpter 5. Above, we hve seen tht [α]((ϕ ψ) ϕ) is theorem. With the K xiom, we cn derive from this: [α](ϕ ψ) [α]ϕ. In similr wy, we cn derive: From these by propositionl resoning: [α](ϕ ψ) [α]ψ. [α](ϕ ψ) ([α]ϕ [α]ψ). (*) The impliction in the other direction is lso derivble, s follows: ϕ (ψ (ϕ ψ)), becuse ϕ (ψ (ϕ ψ)) is propositionl tutology. By modl generliztion (necessittion) from this: [α](ϕ (ψ (ϕ ψ))). By two pplictions of the K xiom nd propositionl resoning from this: [α]ϕ ([α]ψ [α](ϕ ψ)). Since ϕ (ψ χ) is propositionlly equivlent to (ϕ ψ) χ, we get from this by propositionl resoning: Putting the two principles ( ) nd ( ) together we get: ([α]ϕ [α]ψ) [α](ϕ ψ). (**) [α](ϕ ψ) ([α]ϕ [α]ψ). (***)

28 6-28 CHAPTER 6. LOGIC AND ACTION Let us turn to the next xiom, the xiom for test. This xiom sys tht [?ϕ 1 ]ϕ 2 expresses n impliction: The xioms for sequence nd for choice: (test) [?ϕ 1 ]ϕ 2 (ϕ 1 ϕ 2 ) (sequence) [α 1 ; α 2 ]ϕ [α 1 ][α 2 ]ϕ (choice) [α 1 α 2 ]ϕ [α 1 ]ϕ [α 2 ]ϕ Exmple 6.35 As n exmple ppliction, we derive Here is the derivtion: [α; (β γ)]ϕ [α][β]ϕ [α][γ]ϕ. [α; (β γ)]ϕ (sequence) [α][β γ]ϕ (choice) (***) [α]([β]ϕ [γ]ϕ) [α][β]ϕ [α][γ]ϕ. These xioms together reduce PDL formuls without to formuls of multi-modl logic (propositionl logic extended with simple modlities [] nd ). Exmple 6.36 We show how this reduction works for the formul [(; b) (?ϕ; c)]ψ: [(; b) (?ϕ; c)]ψ (choice) [; b]ψ [?ϕ; c]ψ For the opertion there re two xioms: (sequence) (test) (mix) [α ]ϕ ϕ [α][α ]ϕ [][b]ψ [?ϕ][c]ψ [][b]ψ (ϕ [c]ψ). (induction) (ϕ [α ](ϕ [α]ϕ)) [α ]ϕ The mix xiom expresses the fct tht α is reflexive nd trnsitive reltion contining α, nd the xiom of induction cptures the fct tht α is the lest reflexive nd trnsitive reltion contining α. As ws mentioned before, ll xioms hve dul forms in terms of α, derivble by propositionl resoning. For exmple, the dul form of the test xiom reds The dul form of the induction xiom reds?ϕ 1 ϕ 2 (ϕ 1 ϕ 2 ). α ϕ ϕ α ( ϕ α ϕ).

29 6.8. AXIOMATISATION 6-29 Exercise 6.37 Give the dul form of the mix xiom. We will now show tht in the presence of the other xioms, the induction xiom is equivlent to the so-clled loop invrince rule: Here is the theorem: ϕ [α]ϕ ϕ [α ]ϕ Theorem 6.38 In PDL without the induction xiom, the induction xiom nd the loop invrince rule re interderivble. Proof. For deriving the loop invrince rule from the induction xiom, ssume the induction xiom. Suppose ϕ [α]ϕ. Then by modl generlistion: [α ](ϕ [α]ϕ). By propositionl resoning we get from this: ϕ (ϕ [α ](ϕ [α]ϕ)). From this by the induction xiom nd propositionl resoning: ϕ [α ]ϕ. Now ssume the loop invrince rule. We hve to estblish the induction xiom. By the mix xiom nd propositionl resoning: (ϕ [α ](ϕ [α]ϕ)) [α]ϕ. Agin from the mix xiom nd propositionl resoning: (ϕ [α ](ϕ [α]ϕ)) [α][α ](ϕ [α]ϕ). From the two bove, with propositionl resoning using (***): (ϕ [α ](ϕ [α]ϕ)) [α](ϕ [α ](ϕ [α]ϕ)). Applying the loop invrince rule to this yields: (ϕ [α ](ϕ [α]ϕ)) [α ](ϕ [α ](ϕ [α]ϕ)). From this we get the induction xiom by propositionl resoning: (ϕ [α ](ϕ [α]ϕ)) [α ]ϕ. This ends the proof.

30 6-30 CHAPTER 6. LOGIC AND ACTION Axioms for Converse re the following: Suitble xioms to enforce tht ˇ behves s the converse of ϕ [] ˇ ϕ ϕ [ˇ] ϕ Exercise 6.39 Show tht the xioms for converse re sound, by showing tht they hold in ny stte in ny LTS. 6.9 Expressive power: defining progrmming constructs The lnguge of PDL is powerful enough to express conditionl sttements, fixed loop sttements, nd condition-controlled loop sttements s PDL progrms. More precisely, the conditionl sttement if ϕ then α 1 else α 2 cn be viewed s n bbrevition of the following PDL progrm: (?ϕ; α 1 ) (? ϕ; α 2 ). The fixed loop sttement cn be viewed s n bbrevition of do n times α The condition-controlled loop sttement cn be viewed s n bbrevition of α; ; α } {{ } n times while ϕ do α (?ϕ; α) ;? ϕ. This loop construction expressed in terms of reflexive trnsitive closure works for finite repetitions only, for note tht the interprettion of while do α in ny model is the empty reltion. Successful execution of every progrm we re considering here involves termintion of the progrm. The condition controlled loop sttement cn be viewed s n bbrevition of repet α until ϕ α; (? ϕ; α) ;?ϕ.

31 6.10. OUTLOOK PROGRAMS AND COMPUTATION 6-31 Note how these definitions mke the difference cler between the while nd repet sttements. A repet sttement lwys executes n ction t lest once, nd next keeps on performing the ction until the stop condition holds. A while sttement checks continue condition nd keeps on performing n ction until tht condition does not hold nymore. If while ϕ do α gets executed, it my be tht the α ction does not even get executed once. This will hppen if ϕ is flse in the strt stte. In impertive progrmming, we lso hve the skip progrm (the progrm tht does nothing) nd the bort progrm (the progrm tht lwys fils): skip cn be defined s? (this is test tht lwys succeeds) nd bort s (this is test tht lwys fils). Tking stock, we see tht with the PDL ction opertions we cn define the whole repertoire of impertive progrmming constructs: inside of PDL there is full fledged impertive progrmming lnguge. Moreover, given PDL progrm α, the progrm modlities α ϕ nd [α]ϕ cn be used to describe so-clled postconditions of execution for progrm α. The first of these expresses tht α hs successful exection tht ends in n ϕ stte; the second one expresses tht every successful execution of α ends in ϕ stte. We will sy more bout the use of this in Section 6.10 below Outlook Progrms nd Computtion If one wishes to interpret PDL s logic of computtion, then nturl choice for interpreting the bsic ctions sttements is s register ssignment sttements. If we do this, then we effectively turn the ction sttement prt of PDL into very expressive progrmming lnguge. Let v rnge over set of registers or memory loctions V. A V -memory is set of storge loctions for integer numbers, ech lbelled by member of V. Let V = {v 1,..., v n }. Then V -memory cn be pictured like this: v 1 v 2 v 3 v 4 v 5 v 6 v 7 A V -stte s is function V Z. We cn think of V -stte s V -memory together with its contents. In picture: v 1 v 2 v 3 v 4 v 5 v 6 v If s is V -stte, s(v) gives the contents of register v in tht stte. So if s is the stte bove, then s(v 2 ) = 3.

32 6-32 CHAPTER 6. LOGIC AND ACTION Let i rnge over integer nmes, such s 0, 234 or nd let v rnge over V. Then the following defines rithmeticl expressions: ::= i v It is cler tht we cn find out the vlue [] s of ech rithmeticl expression in given V -stte s. Exercise 6.40 Provide the forml detils, by giving recursive definition of [] s. Next, ssume tht bsic propositions hve the form 1 2, nd tht bsic ction sttements hve the form v :=. This gives us progrmming lnguge for computing with integers s ction sttement lnguge nd formul lnguge tht llows us to express properties of progrms. Determinism To sy tht progrm α is deterministic is to sy tht if α executes successfully, then the end stte is uniquely determined by the initil stte. In terms of PDL formuls, the following hs to hold for every ϕ: α ϕ [α]ϕ. Clerly, the bsic progrmming ctions v := re deterministic. Termintion To sy tht progrm α termintes (or: hlts) in given initil stte is to sy tht there is successful execution of α from the current stte. To sy tht α lwys termintes is to sy tht α hs successful execution from ny initil stte. Here is PDL version: α. Clerly, the bsic progrmming ctions v := lwys terminte. Non-termintion of progrms comes in with loop constructs. Here is n exmple of progrm tht never termintes: while do v := v + 1. One step through the loop increments the vlue of register v by 1. Since the loop condition will remin true, this will go on forever. In fct, mny more properties beside determinism nd termintion cn be expressed, nd in very systemtic wy. We will give some exmples of the style of resoning involved. Consider the following problem concerning the out- Hore Correctness Resoning come of pebble drwing ction.

33 6.10. OUTLOOK PROGRAMS AND COMPUTATION 6-33 A vse contins 35 white pebbles nd 35 blck pebbles. Proceed s follows to drw pebbles from the vse, s long s this is possible. Every round, drw two pebbles from the vse. If they hve the sme colour, then put blck pebble into the vse (you my ssume tht there re enough dditionl blck pebbles outside of the vse). If they hve different colours, then put the white pebble bck. In every round one pebble is removed from the vse, so fter 69 rounds there is single pebble left. Wht is the colour of this pebble? It my seem tht the problem does not provide enough informtion for definite nswer, but in fct it does. The key to the solution is to discover n pproprite loop invrint: property tht is initilly true, nd tht does not chnge during the procedure. Exercise 6.41 Consider the property: the number of white pebbles is odd. Obviously, this is initilly true. Show tht the property is loop invrint of the pebble drwing procedure. Wht follows bout the colour of the lst pebble? It is possible to formlize this kind of resoning bout progrms. This formliztion is clled Hore logic. One of the seminl ppers in computer science is Hore s [Ho69]. where the following nottion is introduced for specifying wht computer progrm written in n impertive lnguge (like C or Jv) does: {P } C {Q}. Here C is progrm from formlly defined progrmming lnguge for impertive progrmming, nd P nd Q re conditions on the progrmming vribles used in C. Sttement {P } C {Q} is true if whenever C is executed in stte stisfying P nd if the execution of C termintes, then the stte in which execution of C termintes stisfies Q. The Hore-triple {P } C {Q} is clled prtil correctness specifiction; P is clled its precondition nd Q its postcondition. Hore logic, s the logic of resoning with such correctness specifictions is clled, is the precursor of ll the dynmic logics known tody. Hore correctness ssertions re expressible in PDL, s follows. If ϕ, ψ re PDL formuls nd α is PDL progrm, then {ϕ} α {ψ} trnsltes into ϕ [α]ψ. Clerly, {ϕ} α {ψ} holds in stte in model iff ϕ [α]ψ is true in tht stte in tht model. The Hore inference rules cn now be derived in PDL. As n exmple we derive the rule for gurded itertion: {ϕ ψ} α {ψ} {ψ} while ϕ do α { ϕ ψ}

34 6-34 CHAPTER 6. LOGIC AND ACTION First n explntion of the rule. The correctness of while sttements is estblished by finding loop invrint. Consider the following C function: int squre (int n) { int x = 0; int k = 0; while (k < n) { x = x + 2*k + 1; k = k + 1; } return x; } How cn we see tht this progrm correctly computes squres? By estblishing loop invrint: {x = k 2 } x = x + 2*k + 1; k = k + 1; {x = k 2 }. Wht this sys is: if the stte before execution of the progrm is such tht x = k 2 holds, then in the new stte, fter execution of the progrm, with the new vlues of the registers x nd k, the reltion x = k 2 still holds. From this we get, with the Hore rule for while: {x = k 2 } while (k < n) { x = x + 2*k + 1; k = k + 1; } {x = k 2 k = n} Combining this with the initilistion: { } int x = 0 ; int k = 0; {x = k 2 } while (k < n) { x = x + 2*k + 1; k = k + 1; } {x = k 2 k = n} This estblishes tht the while loop correctly computes the squre of n in x. So how do we derive the Hore rule for while in PDL? Let the premise {ϕ ψ} α {ψ} be given, i.e., ssume (6.1). (ϕ ψ) [α]ψ. (6.1) We wish to derive the conclusion i.e., we wish to derive (6.2). {ψ} while ϕ do α { ϕ ψ}, ψ [(?ϕ; α) ;? ϕ]( ϕ ψ). (6.2)