236368 Emilia Katz, Shahar Dag

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Formal Specifications for Complex Systems (236368)

Tutorial #13

Algebraic Specification and Larch

236368 Emilia Katz, Shahar Dag

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Today:

• Larch Specification Language

• Initial and Final Algebras

• Larch Interface Language

• Examples

236368 Emilia Katz, Shahar Dag

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General Structure

trait_name : trait //data stucture behavior, properties

includes trait1 rename_list, trait2 rename_list, …

Introduces //operations declaration

operator_list

Asserts //axioms – operations definition

predicate_list

var_type generated by operator_list

var_type partitioned by observer_list

implies additional_claims

implies converts operation_list

implies converts operation_list exempting special_cases

236368 Emilia Katz, Shahar Dag

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Example (includes)trait1 : trait

introduces:__ ↔ __: T, T → bool

asserts∀x:T x↔x

trait2 : traitintroduces:

__ R __: T, T → boolasserts

∀x, y, z:T (x R y ∧ y R z) ⇒ x R z

trait3: traitincludes trait1 ( ≤ for ↔ ), trait2 ( ≤ for R )

What is the meaning of these traits?

trait1: reflexive relationtrait2: transitive relation

trait3: pre-order

What operations are defined for this trait?

≤ instead of ↔ , R

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Example - Set

Want to be able to:

• Create a new set• Add / remove elements from a set• Check whether an element is in the set• Get the size of the set• Get a union / intersection of two sets

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Example – Set (cont.)

settrait : traitintroduces

{} : → set_ ∈ _ : E , set → boolinsert : E , set → setdelete : E , set → setsize : set → int_ ∪ _ : set , set → set_ ⋂ _ : set , set → set

// to be continued…

can write “E x set” instead of “E, set” (another notation…)

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Example – Set (contd.2)

Define operations and connections between them:

• What does a newly created set look like?• What is the effect of adding / removing elements from a

set?• How is the size of a set defined?• What is a union / intersection of two sets?

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Example – Set (contd.3)

asserts∀ e , e1 : E , s , s1 : S¬( e ∈ {} );e ∈ insert(e1 , s) == e = e1 ⋁ e ∈ s;size( {} ) == 0;size( insert(e , s)) == if e ∈ s then size(s) else size(s) +

1;delete( e , {} ) == {};delete(e, insert(e1, s)) ==

if e=e1 then delete(e, s) else insert(e1, delete(e, s));s ∪ {} == s;s ∪ insert( e , s1 ) == insert( e , s ∪ s1 );s ⋂ {} == {};s ⋂ insert( e , s1 ) ==

if e ∈ s then insert( e , s ⋂ s1 ) else s ⋂ s1;

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Example – Set (contd.4)

generated by:set generated by {} , insert

partitioned by:set partitioned by ∈

Well-definedness of operations of the trait:

implies converts {} , ∈, insert, delete, size, ∪, ⋂ (all the operations are well-defined, no special cases)

236368 Emilia Katz, Shahar Dag

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Example – Set (contd.5)

delete(5 , insert(7 , insert(5 , {}))) == insert(7 , delete(5 , insert(5 , {}))) ==insert(7 , delete(5 , {})) ==insert(7 , {})

Is the following true?

set implies delete(5 , insert(7 , insert(5 , {}))) = insert(7 , {})

// axiom 2 about delete, the “else” part// axiom 2 about delete, the “then” part// axiom 1 about delete

=> The statement is true!

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Example – Set (contd.6)

Is the following true?

set implies insert(7 , insert(5 , {})) = insert(5 , insert(7 , {}))

No axioms to help us decide!

236368 Emilia Katz, Shahar Dag

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Initial and Final Algebras

Initial algebra:

insert(7 , insert(5 , {})) insert(5 , insert(7 , {})) since they cannot be proven equal from the axioms of set

Final algebra:

insert(7 , insert(5 , {})) = insert(5 , insert(7 , {})) since they cannot be distinguished by the observers

Larch keeps the decision open for the user of the trait (by the addition of partitioned by)

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Initial and Final Algebras

Question:What would the following statement mean:- set partitioned by size

Answer:We claim that two sets are equal if they are of the same size.

Is this good?No! it would mean that insert(5 , {}) = insert(7 , {}) which “breaks” the algebra as we can now prove false claims!-5 ∈ insert(5 , {}) -insert(5 , {}) = insert(7 , {})-=> 5 ∈ insert(7 , {}) -=> 5 ∈ {} !

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Larch Interface Language - LCL

• second layer of a Larch specification

• we will only show some of the main features of LCL

• termination requirement is implicit

• may use any sorts and operations defined in LSL traits

• the mapping of types to sorts (E for set…) is done when introducing the

used traits, by renaming the sorts to the correct types: uses trait (type

for sort, …)

• LCL manipulates objects (variables). They can be:

• mutable: its value can be changed (specified by var)

• immutable: its guaranteed to stay constant.

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LCL – The general formuses traits with [rename_list]procedure headerrequires Pmodifies Lensures Q

P – the precondition of the I/O assertion• Contains restrictions on the input • Prevents calls with illegal values• Must be fulfilled by the caller

L – the list of changeable objectsQ – the post condition

• Relating final values [primed (‘) version] to initial ones.• Must be established by the procedure

Note – implicit condition: the function must terminate!

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Exampleuses settrait with [set for set, integer for E]

procedure setinit(var s : set)modifies sensures s’ = {}

procedure setinsert(e : integer; var s : set)requires size( insert( e , s ) ) ≤ 100modifies sensures s’ = insert( e , s )

procedure setrem(e : integer; var s : set; var f : bool)modifies s , fensures s’ = delete( e , s ) ∧ f’ = ( e ∈ s)

function choose(s : set; var e : integer) : boolmodifies e , chooseensures if size( s ) > 0 then ( choose’ ∧ (e’ ∈ s)) else (¬choose’ ∧ (e’ = e))

Use Pascal-like syntax

corresponds to {} of settrait

corresponds to insert; add a restriction: size ≤100

Delete an element; report if it was in the set before

combination of delete and

return an arbitrary element

no corresp. operation

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setדוגמא ממבחן -

setבהינתן

והפעולות האריתמטיות (כמו שראינו)

וגם סימני היחס (<, <=, <, ...)

יש להגדיר:

maxהאיבר המקסימאלי בקבוצה -

secondהאיבר השני בגודלו -

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(המשך) setדוגמא ממבחן –

(פיתרון של סטודנט)maxניסיון ראשון לפיתרון

max: S → E max(s) = e . e∈S ∧ ¬∃a∈S . a>e

האם זה הוא פיתרון טוב?

לא(נתעלם מהרישום המקורב בו השתמשנו לדוגמא)

אנחנו רוצים הגדרה אינדוקטיבית בדומה לפעולות האחרות,כדי שנוכל להשתמש בה בהוכחות באינדוקציה ובאקסיומות

== max( insert( e , s ) )אחרות (ולא פיתרון מלוגיקה)if size(s)=0 then eelse if max(s) > e then max(s) else e

implies converts max exempting max( {} )

236368 Emilia Katz, Shahar Dag

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(המשך) setדוגמא ממבחן –

כבר צריך להיות קלsecondעכשיו לפתור את

second: S → E

second( s ) == max( delete( max( s ) , s ) ) (*)

implies converts second exemptingsecond( {} ),∀e∈E second( insert( e , {} ) )

האם השורה המסומנת ב * לא משנה את הקבוצה שלנו ?

לא, אנחנו רק מתארים כאן את הפעולות, שפת הממשק תדאג לקבוצה

מועד א2013שאלה ממבחן –

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מועד א2013שאלה ממבחן –

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