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Z -Toolkit

Z -Toolkit. Z specification language is based on formal system: Propositional and predicate calculus Set theory Relations and Functions Thus Z offers a set of facilities to include (or express) these concepts ---- we call the set of facilities the Z toolkit. Numbers and Operations in Z.

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Z -Toolkit

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  1. Z -Toolkit • Z specification language is based on formal system: • Propositional and predicate calculus • Set theory • Relations and • Functions • Thus Z offers a set of facilities to include (or express) these concepts ---- we call the set of facilities the Z toolkit.

  2. Numbers and Operations in Z • Z - language has 3 built-in number types • N : natural numbers (e.g. 0,1,2, - - - -, ) • N1 : positive integers (e.g. 1,2,3, - - - , ) • Int : integers (e.g. - - - , -2,-1,0,1,2, - - - , ) • Axiomatically expressed : (let IP represent power set) • for positive integers N1 : IP N (“type” declaration) N1 = N \ {0} (relation definition) • for natural numbers N : IP Int N = Int \ { - - -, -4, -3 , -2, -1} Rick ?

  3. Numbers and Operations in Z • Numerical operators • Defined as functions may use “lambda” notation • Binary operators defined with underscores on either side • e.g. _ op _ • Addition operator, + , (example) • _ + _ : N x N N should be included in thesignaturepart of schema • _ + _ = גm,n : N succn m in the predicate part of the schema • Or m + n = succn m in the predicate part of schema • _ + _ : N x N N m + n = succn m (where succ is successor ) total function

  4. Numbers and Operations in Z • Let’s look at the great than or equal, =<, operator over N. _ =< _ : N <-> N (note : <-> is a relation) _ =< _ = succ* (reflexive transitive closure of succ function) • succ* = succ0 U succ1 U succ2 U ----- • succ0 = id N = {(0,0), (1,1), - - - } • succ1 = { (0,1), (1,2), (2,3), - - - } • succ2 = { (0,2), (1,3), (2,4), - - - } • succ3 = { (0,3), (1,4), (2,5), - - - } • etc. • So, succ* contains all the pairs that satisfy the =< relation • The operator =< is thus defined in terms of a relation • Look at 2 =< 5 as an example; now look at above predicate. should _ =< _ be “equal to” or is an “element of”succ* ?

  5. Sets and Operators on Sets in Z • A Generic Definition is a definition that applies to sets of any type. • In schema representation: • use [ ] • use double line , , on the top e.g. (union, difference, intersection ) [ T ] _ U _ , _ \ _ , _ _ : IP T x IP T IP T s1, s2 : IP T s1 U s2 = { x : T I x s1 \/ x s2 } s1 \ s2 = { x : T I x s1 /\ x s2 } s1 s2 = { x : T I x s1 /\ x s2 }

  6. “Inventing” an Operator • Modified Example 9.1 in text: S1 and S2 be two sets. • Specify a SCARD operator that returns the cardinality of the set S1\S2. [ T ] _ SCARD _ : IP T x IP T N \/ S1, S2 : IP T S1 SCARD S2 = # (S1\S2)

  7. More Sets and Operators on Sets in Z • Subsets and proper subsets may be defined similarly as with unions and intersections, except subsets are defined as a “relation” between power sets, not a function. • Generalized union and generalized intersection is defined as follows: [ S] U _ , _ : IP ( IP S) IP S \/ A : IP S ( IP S ) U A = { x : S I a A x a } A = { x : S I a A x a } So, for S = {1,2,3} , IP S = { { }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }. And { {2,3} , {1,2,3} } = {2,3}

  8. Relations in Z • A Relation in Z between two sets, S1 and S2, may be expressed as S1 < > S2 in the signature part of the schema. So, a relation would be R1 : S1 < - > S2 • Consider the composition operator, ; , defined generically [ T1, T2, T3 ] _ ; _ : [(T1< >T2) x (T2< >T3)] (T1< >T3) R1 ; R2 = { t1: T1, t3 : T3 I t2: T2 (t1, t2) R1 /\ (t2, t3) R2 } R1 and R1 needs to be defined in Signature part?

  9. Relations in Z • Restrictionson domain and range of relations in Z [ T!, T2 ] _ _ : [ IP T1 x (T1 < >T2)] (T1 < > T2) _ _ : [(T1 < >T2 ) x IP T2] (T1 < > T2) \/ S : IP T1 , R : T1< >T2 S R = { t1 : T1, t2 :T2 I t1 S /\ (t1,t2) R (t1,t2) } \/ R : T1 < > T2 , S : IP T2 R S = { t1: T1 , t2 : T2 I (t1,t2) R /\ t2 S (t1,t2) }

  10. Relations in Z • The “image” operator , where the image of a Relation restricted to the set S as the domain. [ T1, T2 ] _ ( _ ) : ( T1 < > T2) x IP T1 IP T2 \/ R : T1 < > T2 , S : IP T1 R ( S ) = { t1: T1 , t2 : T2 I t1 S /\ (t1,t2) R t2}

  11. Functions in Z • Since functions are just special relations, all the previous operators for sets and relations can be used • Example with the “override” operator, • Recall that given two relations R and S each, over T1 x T2, • R S = (dom S R) U S = [ (T1 \ dom S) R] U S [ T1, T2 ] _ _ : (T1 T2) x ( T1 T2) (T1 T2) \/ f, g : (T1 T2) f g = { {dom g} f } U g }

  12. Sequences in Z • There are 3 types of sequences in Z • a) a finite sequence ( note: most practical systems are finite) • seq T = { f : N1 T I dom f = 1, - - - -, #f } , where #f is the cardinality of sequence f. • b) non-empty finite sequence • non-e-seq T = { f : seq T I #f >0 } • c) injective sequence (sequence with no repetition) • inj_seq T = { f: N1 T I dom f = 1, - - - , #f } • = seq T (N1 T) • Example : file_Q inQ, OutQ : seq Files # inQ = #OutQ

  13. Concatenaton of sequences in Z • Two sequences may be concatenatec or a sequence and a single element may be concatenated. • example”: [ T ] _ Con _ : seq T x seq T seq T \/ s1, s2 : seq T s1 Con s2 = s1 U { i : dom s2 ( i + #s1, s2i) } S2 i represents the ith elements of seq, s2.

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