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Models and Analysis of Software Lecture 5 Introduction to Z [email protected] www.cs.put.poznan.pl/jnawrocki/mse/models/ Copyright, 2002 © Jerzy R. Nawrocki UML and formal models Use-case diagram Look-up Reader Change Add Remove J. Nawrocki, Models & ... Admin UML and formal models Class diagram PhoneDir 1 Init() Add(name,no) Lookup(name): Num Delete(name) J. Nawrocki, Models & ... Introduction Z resembles VDM • Model-based: basic types (integer, real, ..) and compound types (sets, sequences, ..) • Implicit specification (what?). • No explicit specification (how?). J. Nawrocki, Models & ... From the previous lecture.. Quantifiers That’s really different from Pascal! -- A prime number, n, is -- divisible only by 1 and n. IsPrime (n: N1) res: B post res k N1 (1 < k k < n) n mod k 0 J. Nawrocki, Models & ... From the previous lecture.. Pre-conditions Quotient (-6, 2) = 3 Quotient (a, b: Z) res: N pre b 0 post res = (abs a) div (abs b) J. Nawrocki, Models & ... From the previous lecture.. Sequences (I) -- CDs = sequence of Common Divisors CDs (a, b: N1) res: N1+ post res = [k | k N1 a mod k = 0 b mod k = 0] J. Nawrocki, Models & ... Plan of the lecture From the previous lecture.. Sets Characters and strings Type invariants Records Miscellaneous J. Nawrocki, Models & ... Sets Basic sets Basic sets or basic types? B - Boolean (true, false) N1 - positive integers (1, 2, 3, ..) N - natural numbers (including 0) Z - integers Q - rationals R - reals x BasicSet J. Nawrocki, Models & ... x BasicSet Sets Finite sets T-set a finite set of values of type T N-set a finite set of natural numbers R-set a finite set of reals R-set-set a finite set of finite sets of reals J. Nawrocki, Models & ... Sets Set values Only finite sets! {} empty set {0, 2, 4} explicit set value {2, ..., 5} = {2, 3, 4, 5} {2n | nN n<3} = {0, 2, 4} {E | B1, B2, ..., Bn Boolean_condition } {[a, b] | aN, bN b = aa a 3} J. Nawrocki, Models & ... Sets Finite set operators (I) Only finite sets! xS belongs to xS does not belong to card S cardinality of S S1 = S2 equals S1 S2 does not equal S1 S2 S1 is a subset of S2 S1 S2 S1 is a proper subset of S2 J. Nawrocki, Models & ... Sets Finite set operators (II) Only finite sets! S1 S2 union S1 S2 intersection S1\ S2 difference FS power set of S J. Nawrocki, Models & ... Sets A set of decimal digits of a number k Does not work! digit = {0, ..., 9} digits1(k: N) res: digit-set post res = {k mod 10} digits1(k div 10) J. Nawrocki, Models & ... Sets A set of decimal digits of a number k What if k=0? digits2(k: N) res: digit-set post (k=0 res { }) (k>0 res = {k mod 10} digits2(k div 10)) digits3(k: N) res: digit-set post (k=0 res = { 0 }) (k>0 res = digits2(k)) J. Nawrocki, Models & ... Plan of the lecture From the previous lecture.. Sets Characters and strings Type invariants Records Miscellaneous J. Nawrocki, Models & ... Characters and strings char - alfanumeric characters char* - possibly empty sequence of char char+ - nonempty sequence of char 'a' - a character literal "ABBA" - a string of chars (text) "S. Covey" = ['S', '.', ' ', 'C', 'o', 'v', 'e', 'y'] "S. Covey"(1)= 'S' J. Nawrocki, Models & ... Characters and strings Reversing a string -- Reversing a string of characters reverse(t: char*) res: char* post (t = [ ] res = [ ]) (t [ ] res = (tl t) [hd t] reverse("top") = "pot" J. Nawrocki, Models & ... Characters and strings Reversing a string -- Reversing a string of characters reverse(t: char*) res: char* post (t = [ ] res = [ ]) (t [ ] res = reverse(tl t) [hd t] reverse("top") = "pot" J. Nawrocki, Models & ... Important modification Characters and strings Integer to text conversion Can’t be simpler? d_seq= ['0', '1', '2', '3', '4', '5', '6', '7', '8', '9'] -- Integer to text conversion i2t(i: N) t: char+ post (i=0 t="0") (i>0 t=i2t1(i)) i2t1(i: N) t: char* post (i=0 t= [ ]) (i>0 t=i2t1(i div 10) [d_seq(i mod 10 + 1)]) J. Nawrocki, Models & ... Plan of the lecture From the previous lecture.. Sets Characters and strings Type invariants Records Miscellaneous J. Nawrocki, Models & ... Type invariants Declaration of invariants 0bb1 resembles 0b1 Id = T inv Pattern Boolean_condition Bit = N inv Bit 0 b b 1 Bit = {b | b N 0 b b 1} J. Nawrocki, Models & ... Type invariants Defining prime numbers More reusable and readable! Prime = N1 inv Prime i N1 (1<i i<a) a mod i 0 is_prime(a: N1) res: B post res = i N1 (1<i i<a) a mod i 0 Prime = N1 inv Prime is_prime(a) J. Nawrocki, Models & ... Type invariants Using prime numbers -- Checking if every even number between a and b -- can be represented as a sum of 2 prime numbers goldbach(a,b: N1) res: B pre a b post res = i N1 (a i i b i mod 2 = 0) x,y: Prime i= x+y Here the defined type is used. J. Nawrocki, Models & ... Plan of the lecture From the previous lecture.. Sets Characters and strings Type invariants Records Miscellaneous J. Nawrocki, Models & ... Records Record definition ‘FamilyN’ stands for ‘Family Name’ Rec:: Field1 : T1 Field2 : T2 ... Fieldn : Tn Worker:: FamilyN: char+ FirstN: char+ Hours: N J. Nawrocki, Models & ... Records Field selection Rec.Field WorkersFile = Worker* total_hours(w: WorkersFile) res: N post (w=[ ] res = 0) (w [ ] res = (hd w).Hours + total_hours(tl w) Selecting the field ‘Hours’. J. Nawrocki, Models & ... Plan of the lecture From the previous lecture.. Sets Characters and strings Type invariants Records Miscellaneous J. Nawrocki, Models & ... Unions T1 | T2 Enumerated types: Signal = RED | AMBER | GREEN J. Nawrocki, Models & ... Optional types nil - absence of a value Optional type: [ ] = | nil | nil [ ] or Optional type operator: Expression = nil if next(P) = nil .. J. Nawrocki, Models & ... Explicit functions func_name: T1 x T2 x .. x Tn T func_name(Id1, Id2, .., Idn) E pre B max: x x max (x, y, z) if (y x) (z x) then x elseif (x y) (z y) then y else z J. Nawrocki, Models & ... Polymorphic functions max [ @num ]: @num x @num x @num @num max (x, y, z) if (y x) (z x) then x elseif (x y) (z y) then y else z result = max [ result = max [ J. Nawrocki, Models & ... ] (1, 2, 3) ] (1.1, 2.2, 3.3) State state Id of field_list inv invariant_definition init initialisation end state maximum of max: init mk_maximum(m) m=0 end J. Nawrocki, Models & ... State state Id of Another example field_list inv invariant_definition init initialisation end state aircraft of speed: height: inv mk_aircraft(-,h) (h 0.0) init mk_aircraft(s,h) (s=0.0) (h= 0.0) end J. Nawrocki, Models & ... Implicit operations Op_name (Id1: T1, .., Idk:Tk) Idr: Tr ext Access_vars pre B post B’ Access_vars: rd or wr prefix MAX3() ext rd x, y, z: wr max: post (x max) (y max) (z max) (max {x, y, z}) J. Nawrocki, Models & ... Implicit operations Old state: variable MAX_NUM(n: ) ext wr max: post (n max) (max = max max = n) J. Nawrocki, Models & ... Error definitions PUT_YEAR(year: ) ext wr yr: pre year 1994 post yr = year errs yr2dXIX: 94 year year 99 yr= year+1900 yr2dXX: year < 94 yr = year+2000 J. Nawrocki, Models & ... Explicit operations o T OPER_NAME: T1 x .. x Tn OPER_NAME (Id1, Id2, .., Idn) Expression pre B o () MAX_NUM: MAX_NUM (n) if max < n then max:= n else skip J. Nawrocki, Models & ... Conditionals if B1 then ES1 elseif B2 then ES2 ... elseif Bn then ESn else ES J. Nawrocki, Models & ... cases Es: P1 ES1 ... Pn ESn others ES end Iteration statements for Id= E1 to E2 by Inc do St for Id in Sq do St for Id in reverse Sq do St for all Id E do St while B do St J. Nawrocki, Models & ... Summary Finite sets. Character string = sequence. Type invariants allow to define quite complicated types (e.g. prime numbers). Records allow do specify database-like computations. J. Nawrocki, Models & ... Homework • Specify a function digit 5 that returns a sequence of decimal digits of a number k (see functions digits3 and digits2). • Specify an example of a function that would be an implementation of a JOIN operation in a relational database. • Specify a polymorphic projection and selection operation. J. Nawrocki, Models & ... Further readings • A. Harry, Formal Methods Fact File, John Wiley & Sons, Chichester, 1996. J. Nawrocki, Models & ... Quality assessment 1. What is your general impression? (1 - 6) 2. Was it too slow or too fast? 3. What important did you learn during the lecture? 4. What to improve and how? J. Nawrocki, Models & ...
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