Download presentation

Presentation is loading. Please wait.

Published byGretchen Winzer Modified over 2 years ago

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, - - - -, ) –N 1 : 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 for positive integers N 1 : IP N (“type” declaration) N 1 : IP N (“type” declaration) N 1 = N \ {0} (relation definition) N 1 = N \ {0} (relation definition) for natural numbers for natural numbers N : IP Int N : IP Int N = Int \ { - - -, -4, -3, -2, -1} 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 _ e.g. _ op _ –Addition operator, +, (example) _ + _ : N x N N should be included in the signature part of schema _ + _ : N x N N should be included in the signature part of schema _ + _ = ג m,n : N succ n m in the predicate part of the schema Or m + n = succ n m in the predicate part of schema _ + _ : N x N N _ + _ : N x N N m + n = succ n m (where succ is successor ) m + n = succ n 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 (note : is a relation) _ = N (note : is a relation) _ =< _ = succ* (reflexive transitive closure of succ function) _ =< _ = succ* (reflexive transitive closure of succ function) succ* = succ 0 U succ 1 U succ 2 U ----- succ* = succ 0 U succ 1 U succ 2 U ----- succ 0 = id N = {(0,0), (1,1), - - - } succ 0 = id N = {(0,0), (1,1), - - - } succ 1 = { (0,1), (1,2), (2,3), - - - } succ 1 = { (0,1), (1,2), (2,3), - - - } succ 2 = { (0,2), (1,3), (2,4), - - - } succ 2 = { (0,2), (1,3), (2,4), - - - } succ 3 = { (0,3), (1,4), (2,5), - - - } succ 3 = { (0,3), (1,4), (2,5), - - - } etc. 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* ? 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 [ ] use double line,, on the top use double line,, on the top e.g. (union, difference, intersection ) e.g. (union, difference, intersection ) [ T ] [ T ] _ U _, _ \ _, _ _ : IP T x IP T IP T _ U _, _ \ _, _ _ : IP T x IP T IP T s1, s2 : IP T s1, s2 : IP T s1 U s2 = { x : T I x s1 \/ x s2 } 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 } 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 ] [ T ] _ SCARD _ : IP T x IP T N _ SCARD _ : IP T x IP T N \/ S1, S2 : IP T S1 SCARD S2 = # (S1\S2) \/ 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] [ S] U _, _ : IP ( IP S) IP S U _, _ : IP ( IP S) IP S \/ A : IP S ( IP S ) \/ A : IP S ( IP S ) U A = { x : S I a A x a } U A = { x : S I a A x a } 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, T3 ] _ ; _ : [(T1 T2) x (T2 T3)] (T1 T3) _ ; _ : [(T1 T2) x (T2 T3)] (T1 T3) R1 ; R2 = { t1: T1, t3 : T3 I t2: T2 R1 ; R2 = { t1: T1, t3 : T3 I t2: T2 (t1, t2) R1 /\ (t2, t3) R2 } (t1, t2) R1 /\ (t2, t3) R2 } R1 and R1 needs to be defined in Signature part?

9
Relations in Z Restrictions on domain and range of relations in Z [ T!, T2 ] [ T!, T2 ] _ _ : [ IP T1 x (T1 T2)] (T1 T2) _ _ : [ IP T1 x (T1 T2)] (T1 T2) _ _ : [(T1 T2 ) x IP T2] (T1 T2) _ _ : [(T1 T2 ) x IP T2] (T1 T2) \/ S : IP T1, R : T1 T2 \/ S : IP T1, R : T1 T2 S R = { t1 : T1, t2 :T2 I t1 S /\ (t1,t2) R (t1,t2) } S R = { t1 : T1, t2 :T2 I t1 S /\ (t1,t2) R (t1,t2) } \/ R : T1 T2, S : IP T2 \/ R : T1 T2, S : IP T2 R S = { t1: T1, t2 : T2 I (t1,t2) R /\ t2 S (t1,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 ] _ ( _ ) : ( T1 T2) x IP T1 IP T2 _ ( _ ) : ( T1 T2) x IP T1 IP T2 \/ R : T1 T2, S : IP T1 \/ R : T1 T2, S : IP T1 R ( S ) = { t1: T1, t2 : T2 I t1 S /\ (t1,t2) R t2} 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 R S = (dom S R) U S = [ (T1 \ dom S) R] U S [ T1, T2 ] [ T1, T2 ] _ _ : (T1 T2) x ( T1 T2) (T1 T2) _ _ : (T1 T2) x ( T1 T2) (T1 T2) \/ f, g : (T1 T2) \/ f, g : (T1 T2) f g = { {dom g} f } U g } 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 : N 1 T I dom f = 1, - - - -, #f }, where #f is the cardinality of sequence f. seq T = { f : N 1 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 } non-e-seq T = { f : seq T I #f >0 } – c) injective sequence (sequence with no repetition) inj_seq T = { f: N 1 T I dom f = 1, - - -, #f } inj_seq T = { f: N 1 T I dom f = 1, - - -, #f } = seq T (N 1 T) = seq T (N 1 T) –Example : file_Q file_Q inQ, OutQ : seq Files inQ, OutQ : seq Files # inQ = #OutQ # 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 ] [ T ] _ Con _ : seq T x seq T seq T _ Con _ : seq T x seq T seq T \/ s1, s2 : seq T \/ s1, s2 : seq T s1 Con s2 = s1 U { i : dom s2 ( i + #s1, s2 i ) } s1 Con s2 = s1 U { i : dom s2 ( i + #s1, s2 i ) } S2 i represents the ith elements of seq, s2.

Similar presentations

OK

MA/CSSE 474 Theory of Computation Decision Problems, Continued DFSMs.

MA/CSSE 474 Theory of Computation Decision Problems, Continued DFSMs.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on porter's five forces diagram Ppt on single phase and three phase dual converters 360 degree customer view ppt on mac Ppt on great indian astronauts Presentations ppt online viewer Ppt on value stream mapping in lean manufacturing Ppt on home security systems Ppt on safe drinking water in india Ppt on machine translation definition Ppt on use of body language in communication