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– Seminar in Software Engineering Cynthia Disenfeld

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Alloy It was developed at MIT by the Software Design Group under the guidance of Daniel Jackson. Alloy is a modelling language for software design. Find instances and counterexamples to define a correct model, find errors..

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Alloy in the Industry Daniel Jackson: – “Alloy is being used in a bunch of companies, but most of them are rather secretive about it. I do know that it's been used by Navteq (who make navigation systems), ITA Software (for understanding invariants in their flight reservation database), AT&T (for specifying and analyzing novel switch architectures), Northrop Grumman (some large military avionics models), Telcordia (who've delivered a prototype network configuration tool to the DoD based on Alloy).”

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Alloy in the Industry Alloy was used for specification and modelling of name servers network configuration protocols access control Telephony Scheduling document structuring key management Cryptography instant messaging railway switching filesystem synchonization semantic web

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Example 1. Shopping abstract sig Person {} sig Salesman extends Person {sellsTo: set Customer} sig Customer extends Person {buysFrom: lone Salesman}

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Set multipliers set : any number. one: exactly one. lone: zero or one. some: one or more.

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Example 1. Shopping abstract sig Person {} sig Salesman extends Person {sellsTo: set Customer} sig Customer extends Person {buysFrom: lone Salesman} pred show{} run show for 4

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Model:

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Correcting the specification fact {sellsTo = ~buysFrom}

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Check the specification is now correct assert factOk{all s:Salesman, c:Customer | c in s.sellsTo s in c.buysFrom } check factOk for 5

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Quantifiers all x: e | F some x: e | F no x: e | F lone x: e | F one x: e | F

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Operators + : union & : intersection - : difference in : subset = : equality

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Relational operators. : dot (Join) – {(N0), (A0)}. {(A0), (D0)} = {(N0), (D0)} -> : arrow (product) – s -> t is their cartesian product ^ : transitive closure * : reflexive-transitive closure ~ : transpose

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Logical Operators ! : negation && : conjunction (and) || : disjunction (OR) => : implication : if and only if

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Example: Stock sig Product {} sig Stock { amount: Product -> one Int }{ all p: Product | p.amount >= 0 } run show for 3 but 1 Stock run show for 3 Person, 3 Product, 1 Stock

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Example: Stock

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Sets univ: the set of all the elements in the current domain none: the empty set iden: the identity relation

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Sets

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Predicates We can find models that are instances of the predicates. As well, we can specify operations, by describing what changes in the state of the system (like in a Z specification we have preconditions on the previous state of the system and post conditions on the result).

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Predicates A customer buys from a certain salesman a certain product. This causes the stock of the product to be reduced by one.

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Predicates pred buy[s:Salesman, c:Customer, p:Product, st, st': Stock] { st'.amount = st.amount + {p->(p.(st.amount)-1)} } run buy for 2 Stock, 2 Person, 1 Product

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Predicates pred buy[s:Salesman, c:Customer, p:Product, st, st': Stock] { st'.amount = st.amount ++{p->(p.(st.amount)-1)} } run buy for 2 Stock, 2 Person, 1 Product

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Predicates

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pred buy[s:Salesman, c:Customer, p:Product, st, st': Stock] { c in s.sellsTo st'.amount = st.amount ++{p->(p.(st.amount)-1)} } run buy for 2 Stock, 2 Person, 1 Product

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Predicates

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Projection

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Functions fun newAmount[p:Product, st: Stock]: Product->Int{ p->(p.(st.amount)-1) } pred buy2[s:Salesman, c:Customer, p:Product, st, st': Stock] { c in s.sellsTo st'.amount = st.amount ++ newAmount[p, st] } run buy2 for 2 Stock, 2 Person, 1 Product

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Themes

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Transitive Closure Given the following model specification of a Tree: sig Tree { children: set Tree } To be able to say that a tree doesn’t appear later in it’s children: fact {all t:Tree | t not in t.^children}

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Alloy Analyzer To run it: java -jar /usr/local/cs236800/alloy/alloy4.jar

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Characteristics Based on first order logic (may be thought as subset of Z) finite scope check infinite model declarative automatic analysis structured data

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Models There are three basic levels of abstraction at which an Alloy model can be read OO paradigm set theory atoms and relations sig S extends E { F: one T } fact { all s:S | s.F in X }

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OO S is a class S extends its superclass E F is a field of S pointing to a T s is an instance of S. accesses a field s.F returns something of type T sig S extends E { F: one T } fact { all s:S | s.F in X }

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Set Theory S is a set (and so is E) S is a subset of E F is a relation which maps each S to exactly one T s is an element of S. composes relations s.F composes the unary relation s with the binary relations F, returning a unary relation of type T sig S extends E { F: one T } fact { all s:S | s.F in X }

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Atoms and relations S is an atom (and so is E) the containment relation maps E to S (among other things it does) F is a relation from S to T the containment relation maps S to s (among other things). composes relations s.F composes the unary relation s with the binary relations F, resulting in a unary relation t, such that the containment relation maps T to t sig S extends E { F: one T } fact { all s:S | s.F in X }

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Underlying method Relations may be expressed as boolean matrices, and operators between relations are operators on the matrices. Example: In the domain {o1, o2}, the relation {(o1, o1)(o2, o1)} may be represented by the matrix O1O2 O110 O210

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Underlying method Given the boolean representation of the relations, every operator on the relations can be represented as an operator on the matrices For example, the conjunction of R and S is conjuction of each cell i,j in R with i,j in S Also sets and scalars can be represented as relations: Sets are unidimensional relations (1 for each element that is in the set, 0 otherwise) Scalars are singleton sets.

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Underlying method The scope determines the size of the matrices. Given a model, an assertion and a scope, all the possible combinations within the scope, that satisfy the model are evaluated to see whether they contradict the assertion. This is done by representing the model and the assertion both in terms of boolean variables (turns out to be a big CNF formula) which is used as the input for a SAT solver.

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Modules It is possible to define and use other modules. For example: Linear Ordering open util/ordering[State] first returns the first State atom in the linear ordering. last returns the last State atom in the linear ordering. next maps each State atom (except the last atom) to the next State atom.

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Modules - Example open util/ordering[State] module TrafficLight abstract sig Color {} one sig Red, RedYellow, Green, Yellow extends Color {} sig State {color: Color} fact { first.color = Red } pred example {} run example for 5

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Modules - Example

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pred semaphoreChange[s,s': State] { s.color = Red => s'.color = RedYellow else s.color = RedYellow => s'.color = Green else s.color = Green => s'.color = Yellow else s'.color = Red } fact{ all s:State, s': s.next | semaphoreChange[s,s'] }

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Modules - Example

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Some Issues Hard to express recursion, have to find ways by means of transitive closure for instance. Kodkod solves several problems of Alloy: Interface to other tools Partial evaluation (e.g. Sudoku) Sharing common subformulas

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Summary Widely used Declarative language Iterative process Instances and counterexamples help find the correct model specification Possible reuse of modules

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Summary Translation to CNF formulas (by using finite scope) allows automatic verification It is possible to interpret models in different ways There still are limitations in the expression power of the language There are other limitations that Kodkod deals with them.

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Questions? Publications, Courses, Tutorials : Kodkod Software Abstractions – Logic, Language and Analysis. Daniel Jackson. The MIT Press 2006 References

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