CSU290 Lecture Notes Lecture 7 25 Sept Peter Dillinger. Functions on Lists/Conses

Size: px

Start display at page:

Download "CSU290 Lecture Notes Lecture 7 25 Sept Peter Dillinger. Functions on Lists/Conses"

Lydia Griffin
7 years ago
Views:

1 CSU290 Lecure Noes Lecure 7 25 Sep 2008 Funcions on Liss/Conses Peer Dillinger Do you recall wha a rue lis is? I is eiher or a sequence of conses in which he las cdr is. We can wrie a 211-syle daa definiion for rue liss like so: rue-lis: Cons all rue-lis Noe ha here is no buil-in predicae ALLP, bu we use "all" o refer o all objecs in he ACL2 universe. This daa definiion can guide us in wriing a recognizer for rue liss: ; TRUE-LISTP: all -> boolean ; Reurns boolean indicaing wheher he parameer is a rue lis. Here s a emplae we migh come up wih: ; Templae: (based on checking for rue-lis) ; (defun rue-lisp (x) ; (if ( x ) ; (if (and (consp x) ; (allp (car x))) (rue-lisp (cdr x))... ))) excep here is no buil-in ALLP. We could define i, (defun allp (x) (declare (ignore x)) ; ell ACL2 i am inenionally ignoring x ) bu i s jus as easy o assume everyhing belongs o he ype "all" and no even check: ; Templae: (based on checking for rue-lis) ; (defun rue-lisp (x) ; (if ( x ) ; (if (consp x) (rue-lisp (cdr x))... ))) Now, ; (See ess for examples.) (check (rue-lisp ) ) (check (rue-lisp (1)) ) (check (rue-lisp ("hi")) ) (check (rue-lisp (1. 2)) ) (check (rue-lisp 2 )) And now we can fill in he body of he funcion: (defun rue-lisp (x) (if ( x ) (if (consp x) (rue-lisp (cdr x)) )))

2 So if x is, i is a rue lis. If i is a cons, i is a rue lis if is cdr is a rue lis. If i is no and no a cons, which mus be he case if we reach he final "", i is no a rue lis. Finally, by he way, we canno make his definiion in ACL2 because rue-lisp is already defined. Bu his version is equivalen. (ACL2 jus proved so. We ll see how o do ha laer.) Le s wrie anoher funcion ha works on liss o ge a feel for how his should work in ACL2: ; MEM: all rue-lis -> boolean ; Reurns a boolean indicaing wheher he firs parameer appears as an ; elemen in he lis given by he second parameer. ; ("mem" is shor for "member") ; Templae (wihou considering oaliy): ; (defun mem (x l) ; (if ( l ) (mem... (cdr l))...)...) Examples: (check (mem 5 (4 5 6)) ) (check (mem 1 (4 5 6)) ) (check (mem 5 ) ) (check (mem (1) ((2) (1))) ) (check (mem (1) (2 1)) ) Now le us wrie he definiion wihou considering oaliy or uninended recursion: (defun mem (x l) (if ( l ) ; rivially, nohing belongs o he empy lis (if ( (car l) x) ; check if curren elemen is he one we re looking for (mem x (cdr l))))) ; see if same elemen is member of res of he lis This passes all hose ess, bu now le s consider oaliy. If we give i somehing ouside he inended domain, as in (mem 1 2) wha happens? Acually, i does erminae, bu here is exra recursion for aomic inpu ouside he conrac: (mem 1 2) { definiion of mem } (if ( 2 ) (if ( (car 2) 1) (mem 1 (cdr 2)))) { recall (car 2) } (mem 1 (cdr 2))

3 { (cdr 2) } (mem 1 ) { definiion of mem } (if ( )...) Anoher ineresing resul is ha (mem 2) Which is ineresing bu no inherenly problemaic, because 2 is no a rue lis and our conrac/descripion does no specify wha should be reurned in ha case. So MEM calls iself an exra ime before erminaing, and we wan o eliminae ha o keep he recursion as simple as possible. We shall make sure all aomic daa ouside he inended domain maps o a base case. To keep he number of IFs he same, we will le he funcion reurn any ime l is an aom. We could check (aom l), bu he sandard way o perform he same check on somehing ha we inend o be a lis is (endp l). (Recall ENDP is he same as ATOM.) Here is he new definiion: (defun mem (x l) (if (endp l) (if ( (car l) x) (mem x (cdr l))))) Now we have (check (mem 1 2) ) (check (mem 2) ) because any aom for l is reaed as he empy lis,. Wha abou (mem 2 (1 2. 3))? (1 2. 3) is, of course, no a rue lis. (mem 2 (1 2. 3)) (if (endp (1 2. 3)) (if ( (car (1 2. 3)) 2) (mem 2 (cdr (1 2. 3))))) (mem 2 (cdr (1 2. 3))) (mem 2 (2. 3)) (if (endp (2. 3)) (if ( (car (2. 3)) 2) (mem 2 (cdr (1 2. 3)))))

4 This case makes a recursive call even hough i is passed inpu ouside he inended domain. When we fill in he res of he design recipe for ACL2, we will see ha his is he behavior we wan. Basically, raher han checking ha he enire compound daa srucure (one buil wih cons pairs) mees he conrac before we do anyhing wih i, we proceed as if i does u we encouner par of i ha doesn mee he conrac. We will discuss his furher laer. Anoher way of viewing he way his definiion of MEM behaves is ha i reas any improper liss such as (1 2. 3) as if is las cdr were. Thus, (1 2. 3) is reaed as (1 2), (. ) as ( ), and 2 as. And now for somehing compleely differen... Boolean Logic Boolean logic is a sysem of reasoning wih wo values. These values could be 0 & 1 False & True & Non- In ACL2, generalized booleans are, of course, & Non-, bu righ now we are going o focus on he radiional mahemaical incepion of Boolean logic using he conceps of True and False. In boolean logic, we have some basic operaions on ruh values: Pronounciaion Mahemaical synax ACL2 synax Implies p -> q (implies p q) And p /\ q (and p q) Or p \/ q (or p q) If and only if p <-> q (iff p q) No p (no p) There are many ways for reasoning compleely abou boolean logic formulas. We will primarily focus on he easies mehod: ruh ables. We will also use hese o describe he meaning of hese operaions. Here s a ruh able describing he meaning of he binary operaions above: p q p/\q p\/q p->q p<->q T T T T T T T F F T F F F T F T T F F F F F T T Noice ha all possible combinaions of True (T) and False (F) for p and q are lised in he able. If we look a he p/\q, "p and q", column, we see ha if p is True and q is True, hen p /\ q is True. If p is True and q is False, hen p /\ q is False. If p is False and q is True, hen p /\ q is False. If p is False and q is False, hen p /\ q is False. And and Or behave as expeced. Implicaion is somewha unexpeced for some people. Le us le p be "Is i raining?" and q be "Is i cloudy?". p -> q would mean "Raining implies cloudy," or "If i is raining, hen i is cloudy." If i is raining and i is cloudy (firs row in he ruh able), hen he saemen is rue. If i is raining and i is no cloudy, he saemen is

5 false. (Here comes he ricky par.) If i is no raining, hen he saemen is rue no maer wheher i is cloudy (3rd and 4h rows). If he hypohesis of he implicaion, he firs argumen, is false, hen he saemen does no conradic anyhing, so even hough i does no really ell us much, i is rue. p <-> q is much like equaliy from a boolean sandpoin. Now a ruh able for boolean negaion: p p T F F T Tha was simple. Now le s consider consrucing a ruh able for p \/ q. If we wan o make i really easy, we can add a column for p: p q p p\/q T T F T T F F F F T T T F F T T The easy way o fill his in, afer filling in all possible combinaions of p and q values is o fill in he p column based on he p column and he ruh able for negaion. Then we can use he p and q columns and he ruh able for OR o fill in he values for p\/q. Noice anyhing abou hose values for p\/q? Look familiar? They re he same as p->q. So p->q is he same as p\/q. Is here an alernae form for p<->q, in erms of an AND or an OR and possible negaions? Noice ha when we negae one or boh inpus, he ruh able oupus are jus shifed around. In he case of OR, here are always hree rues and one false. In he case of AND, here are always hree falses and one rue. IFF has wo of each hough. If we allow more han one AND and/or OR, we can consruc IFF in erms of hose consrucs. In wha cases is IFF rue? Well, is rue if boh are rue or boh are no rue. In oher words, p<->q is he same as (p/\q) \/ ( p/\ q). Finally we have wo erms o learn: Tauology - a saemen ha is always rue. An example is p \/ p. Conradicion - a saemen ha is always false, such as p /\ p. Now you have learned enough boolean logic o play he boolean logic game in ACL2s and linked off he class web page.

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)

cooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins) Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer