Programming Methods. Part 4 4 November Alban Ponse. Programming Research Group Faculty of Science University of Amsterdam

Transcription

1 Programming Methods Part 4 4 November 2003 Alban Ponse Programming Research Group Faculty of Science University of Amsterdam

2 Today s program (November 4, 2003) Some details on recursion Conditionals and While-Loops Lab/working session in P.126, 13:30-16:30 (Exercises on blackboard; for the PGA-toolset see 2

3 Some details on recursion A procedure call instruction has the form c(pn). A procedure definition environment of size k is denoted by E p k and contains for each program name pn with n < k a defining identity pn = P n, where the programs P n may contain procedure call instructions c(pn) (thus recursion is allowed). Behavior extraction is given by c(pn); X = P n ; X. 3

4 Example of recursion Define E p 1 = {c(p0) = +a; c(p0); b}, so c(p0); X = +a; c(p0); b; X. Then we obtain the non-regular behavior (check!): c(p0);! = c(p0); b;! a b;!,. c(p0); b k ;! = c(p0); b k+1 ;! a b k+1 ;!,. 4

5 Conditionals and While-Loops PGLE, the sublanguage of PGLDg in which each test instruction is followed by either a termination instruction (!), or a goto-instruction (## n). The projection from PGLE to PGLDg is trivial: nothing changes. 5

6 Conditional Constructions and While-Loops Two classical, imperative programming constructions: Conditional construction. In if c then X else Y fi first the boolean condition c is evaluated. If this yields true, X is executed and otherwise Y. While-loop. In while c do X od first the boolean condition c is evaluated. If this yields true, X is executed and all thsi is repeated, otherwise termination follows. 6

7 Collatz conjecture (1937): The imperative program read(x); while x 1 do od if even(x) then x := x/2 fi else x := 3x + 1 terminates if x is a natural number greater than 0. E.g., 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1,. 7

8 PGLEc, four new kinds of instruction: 1. Positive conditional instruction: +a{ (a Σ) 2. Negative conditional instruction: a{ (a Σ) 3. Then/else seperator: }{ 4. End brace: } In +a{; b; }{; c; }, the boolean value of the test instruction +a determines whether b is executed (case true or c (case false). (Cf. if a then b else c fi.) 8

9 PGLEcw, PGLEc + four more kinds of instructions: 5. Positive while-loop header: +a{ (a Σ) 6. Negative while-loop header: a{ (a Σ) 7. Unconditional while-loop header { 8. End of while-loop: } In +a{ ; b; c; } the yield of the test instruction +a determines whether b; c is executed (case true) or termination follows (case false). (Cf. while a do b; c od.) And projection to PGLE...? 9

10 The Collatz program read(x); while x 1 do od if even(x) then x := x/2 else x := 3x + 1 fi interpretation in PGLEcw [read(x)]; +[x 1]{ ; +[even(x)]{; [x := x/2]; }{; [x := 3x + 1]; }; } 10

11 PGLEcw [read(x)]; +[x 1]{ ; +[even(x)]{; [x := x/2]; }{; [x := 3x + 1]; }; } projection to PGLE (& hence PGLDg) [read(x)]; 2; [x 1]; ## 8; [even(x)]; ## 3; [x := x/2]; ## 5; 3; [x := 3x + 1]; 5; ## 2; 8 11

12 Beware,, the program { ;+a; } perhaps seems behaviorally equivalent to { ; +a; ## 1; ## 4; 1; }; 4, but is not a PGLEcw-program: it does not satify the PGLE-requirement that a test instruction is followed by termination or a go-to. 12

13 So the two programming constructions if c then X else Y while c do X od fi can be expressed using 8 new kinds of instructions: +c{; X; }{; Y ; } respectively +c{ ; X; } or in case of the loop while true do X od by { ; X; } (even with 6, the 2 negative variants are not needed). (Even with less - all can be expressed in PGA - but that is not the issue here.) 13

14 Projection from PGLEcw to PGLE/PGLDg Also }{; 17; }; { ; a{; b; }; +c{; d{ ; +e{ is a correct PGLEcw program... (a sequence of instructions that satisfies all requirements), and defines a unique behavior. Which behavior? Projection semantics! PGLEcwa: some instructions are annotated with a natural number: 0 if they do not match, and otherwise with the position that they do match. (See also pp in JLAP 51(2), 2002.) 14

15 PGLEcwa, projection of PGLEcw using annotations: pglecw2pglecwa( a{; c; }{; b; }) = a{; c; }1{; b; }3 and pglecwa2pgle( a{; c; }1{; b; }3 = +a; ## 1; c; ## 3; 1; b; 3 Lab/working session: a program with 7 instr s: pglecw2pglecwa({ ; +a; ## 1; ## 4; 1; }; 4) = { 6; +a; ## 1; ## 4; 1; }1; 4 pglecwa2pgle({ 6; +a; ## 1; ## 4; 1; }1; 4) = 1; +a; ## 9; ## 12; 9; ## 1; 6;

16 Collatz conjecture (1937) revisited: Collatz program read(x); while x 1 do od if even(x) then x := x/2 else x := 3x + 1 fi interpretation in PGLEcw [read(x)]; +[x 1]{ ; +[even(x)]{; [x := x/2]; }{; [x := 3x + 1]; }; } 16

17 pglecw2pglecwa 1 [read(x)]; 2 +[x 1]{ 8; 3 +[even(x)]{; 4 [x := x/2]; 5 }3{; 6 [x := 3x + 1]; 7 }5; 8 }2 pglecwa2pgle (or pglecwa2pgldg) [read(x)]; 2; [x 1]; ## 8; [even(x)]; ## 3; [x := x/2]; ## 5; 3; [x := 3x + 1]; 5; ## 2; 8 17

18 Some Conclusions An imperative program has (at least) two characteristics: It can be projected to a sequence of instructions, a program object. Some execution mechanism is determined, by which a program defines a behavior. PGA is the underlying algebra of program objects, a real algebra with operations that allow manipulation with program objects (cf. PGA1-4). 18

19 We distinguish two possible worlds : PGA: sequences of instructions or program objects, e.g. a;! +a; #0; b;! (b; #4; c) ω operators: concatenation, ; repetition, ( ) ω BPPA: the world of behaviors, e.g. a S D a b S (b; #4; c) ω = P P = b c P operators: postcond. cmp., a action prefix, a 19

20 ... and we distinguish a number of programming languages. A collection PGLZ is a programming language if a projection pglz2pga is given (possibly in a stepwise fashion). The behavior of a PGLZ-program X is defined by pglz2pga(x), so X pglz = pglz2pga(x). 20

21 Embeddings. Relevant and difficult question: are the given program algebra projections correct? E.g.: X PGA in a certain canonical form, X = pglb2pga(pga2pglb(x))? Phrased differently: embeddings map programs into higher languages and can be used to support the correctness of projections. This might be of importance ( Collatz : 1 million U.S. $ (or more?), traffic regulation systems: life or death). 21

22 Main questions: Q. 1. What is an imperative program? (cf. Q. 4) A.1: an expression in a programming language (L, φ). Q.2. When can we identify 2 programs? A.2: if for their projections = ic holds or some other equivalence, e.g. structural equivalence; possibly ge. Q.3. When may we identify 2 programs? A.3: see answer A.2. Q.4. What is a programming language? A.4: a pair (L, φ) with L a collection of expressions and φ a projection to PGA. 22

23 Advices for your exam Never confuse instructions and behaviors: Thus S and D are no instructions, and a jump or an test instruction +a or a, or a termination instruction! cannot occur in a behavior. Behavior is about postconditional compositions en actions; instructions are composed with concatenation (and in PGA also with repetition). End of course 23