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**Introduction to Formal Methods**

Preconditions, Postconditions, and Invariants Revisited; Z language Example (Pressman)

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**What are formal methods?**

Formal methods are mathematically based. They are an attempt to deal with contradictions, ambiguities, vagueness, incomplete statements, and mixed levels of abstraction. They are most valuable for systems which have: --safety concerns (e.g., airplane systems, medical devices) --security concerns

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**--use set theory, logic to specify systems**

Formal techniques Formal techniques: --use set theory, logic to specify systems --increase probability of complete, consistent, unambiguous specifications --have high start-up costs; may require high overhead; some concepts (e.g., timing, reliability) difficult or impossible to capture in formal systems --do not replace more traditional approaches

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**Ten commandments of formal methods**

"Ten Commandments" of formal methods (Pressman, Software Engineering, A Practitioner's Approach): 1. Choose the appropriate notation 2. Formalize but don't overformalize 3. Estimate costs 4. Have a formal methods "guru" on call 5. Do not abandon traditional development methods 6. Document sufficiently 7. Don't compromise quality standards 8. Do not be dogmatic 9. Test, test, test, …. 10. Reuse

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**Preconditions, postconditions, invariants**

Earlier we looked at adding statements to ensure correct program behavior: precondition: logical condition that a caller of an operation guarantees before making the call postcondition: logical condition that an operation guarantees upon completion invariant: logical condition that is preserved by transformations These conditions are all expressed as logical statements --they can be quantified ( , ) --they can be used to support testing at different levels

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**A complete formal system **

What is Z? A complete formal system We will use an example formal specification language: Z system described through a set of "schemas”, which have data invariant(s) state(s) operations-- with precondition(s) / postcondition(s)

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Z example (1) Example (from Pressman, Software Engineering, A Practitioner’s Approach): “Block Handler” Used blocks Unused (free) blocks 11 12 Blocks released to queue when files deleted 2 5 8 11 7 Queued for entry into Unused

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**-------BlockHandler---------------------- used,free: BLOCKS **

Some Z notation Z specification: BlockHandler used,free: BLOCKS BlockQueue: seq P BLOCKS used free = used free = AllBlocks i: dom BlockQueue . BlockQueue i used i,j : dom BlockQueue . i j BlockQueue i BlockQueue j = set intersection union sequence contained in “then” and implies in empty set for all intersection

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**-------BlockHandler---------------------- used,free: BLOCKS **

Z example (2) Z specification: BlockHandler used,free: BLOCKS BlockQueue: seq P BLOCKS used free = used free = AllBlocks i: dom BlockQueue . BlockQueue i used i,j : dom BlockQueue . i j BlockQueue i BlockQueue j =

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**---------RemoveBlock-------------------------- BlockHandler **

#BlockQueue > 0, used’ = used \ head BlockQueue free’ = free head BlockQueue BlockQueue’ = tail BlockQueue AddBlock Ablocks? : BLOCKS Ablocks? used, used’ = used free’ = free BlockQueue’ = BlockQueue ^ (Ablocks?) Z example (3)

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**1. What if BlockQueue is replaced by BlockStack? **

Modifications 1. What if BlockQueue is replaced by BlockStack? 2. What are postconditions for the operations?

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