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Teaching the construction of correct programs using invariant based programming

Learn how to teach formal methods and correct programming using invariant based programming techniques. Experience report on teaching a first-year course at Åbo Akademi.

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Teaching the construction of correct programs using invariant based programming

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  1. Ralph-Johan Back Johannes Eriksson Linda Mannila Teaching the construction of correct programs usinginvariant based programming Åbo Akademi / Dept. of Information Technologies Turku, Finland SEEFM 07, Thessaloniki, Nov 2007

  2. SEEFM 07, Thessaloniki, Nov 2007 Formal methods in CS education • Formal methods are perceived as difficult and requiring mathematical sophistication • The CS curriculum is divided into “theory” and “practice” • Formal methods taught independently of programming courses • Students get impression that formal methods are not applicable in practice • Testing and debugging is therefore the main (only) programming method that they learn from CS studies

  3. SEEFM 07, Thessaloniki, Nov 2007 Overview of talk • A short introduction to invariant based programming • The Socos tool • Teaching formal methods at Åbo Akademi using invariant based programming • Experience report on a first year course on invariant based programming

  4. SEEFM 07, Thessaloniki, Nov 2007 Constructing correct programs “constructive approach” “a posteriori verification” “invariant based programming” Program code Contracts Invariants Verification conditions

  5. SEEFM 07, Thessaloniki, Nov 2007 A: Int[N] A: Int[N] Sorted(A,0,N) A=A0 Permutation(A,A0) Example: Sort an array! Start with a pre-/postcondition specification

  6. SEEFM 07, Thessaloniki, Nov 2007 Example: Sort an array! Structure according to invariants A: Int[N] A: Int[N] Sorted(A,0,N) A=A0 Permutation(A,A0)

  7. SEEFM 07, Thessaloniki, Nov 2007 sorted un-sorted Example: Sort an array! 0 k N Construct a loop A: Int[N] less than or equal to all A[k..N-1] ! A=A0 Sorted(A,0,N) Permutation(A,A0) k: Int 0≤k≤N LOOP Sorted(A,0,k) ∀i,j:Int • 0≤i<k ∧ k≤j<N ⇒ A[i]≤A[j]

  8. SEEFM 07, Thessaloniki, Nov 2007 0: Int 0≤0≤N Sorted(A,0,0) ∀i,j:Int • 0≤i<0 ∧ 0≤j<N ⇒ A[i]≤A[j] Example: Sort an array! What needs to be checked? A: Int[N] Add initial transition A=A0 ⇒ A: Int[N] ✔ A: Int[N] ✔ Permutation(A,A0) Permutation(A,A0) ✔ A=A0 Sorted(A,0,N) ✔ ✔ ✔ k: Int 0≤k≤N Sorted(A,0,k) ∀i,j:Int • 0≤i<k ∧ k≤j<N ⇒ A[i]≤A[j] k:=0

  9. SEEFM 07, Thessaloniki, Nov 2007 Trivial: Sorted(A,0,k) ∧ k=N ⇒ Sorted(A,0,N) Example: Sort an array! Add exit transition A: Int[N] Permutation(A,A0) A=A0 Sorted(A,0,N) k: Int [k=N] 0≤k≤N Sorted(A,0,k) ∀i,j:Int • 0≤i<k ∧ k≤j<N ⇒ A[i]≤A[j] k:=0

  10. SEEFM 07, Thessaloniki, Nov 2007 A: Int[N] Permutation(A,A0) k: Int 0≤k≤N Sorted(A,0,k) ∀i,j:Int • 0≤i<k ∧ k≤j<N ⇒ A[i]≤A[j] A’: Int[N] Permutation(A’,A0) k+1: Int 0≤k+1≤N Sorted(A’,0,k+1) ∀i,j:Int • 0≤i<k+1 ∧ k+1≤j<N ⇒ A’[i]≤A’[j] Example: Sort an array! Add loop transition k<N m=min(A,k,N) ∧ A’= A[ k←A[m], m←A[k] ] A: Int[N] ⇒ Permutation(A,A0) A=A0 Sorted(A,0,N) k: Int [k=N] 0≤k≤N Sorted(A,0,k) ∀i,j:Int • 0≤i<k ∧ k≤j<N ⇒ A[i]≤A[j] [k<N] k:=0 m:=min(A,k,N); A:=A[ k←A[m], m←A[k] ]; k:=k+1

  11. SEEFM 07, Thessaloniki, Nov 2007 0≤N-k Example: Sort an Array! Add a termination function A: Int[N] Variant decreases: N-(k+1) < N-k Bounded from below: 0≤k≤N ⇒0≤N-k Permutation(A,A0) A=A0 Sorted(A,0,N) k: Int [k=N] 0≤k≤N Sorted(A,0,k) ∀i,j:Int • 0≤i<k ∧ k≤j<N ⇒ A[i]≤A[j] [k<N] k:=0 A:=Swap(A,k,min(A,k,N)); k:=k+1

  12. SEEFM 07, Thessaloniki, Nov 2007 Socos: a prototype environment for: invariant based programming state-of-the-art automatic and interactive verification teaching formal methods

  13. SEEFM 07, Thessaloniki, Nov 2007 Invariant Diagrams in SOCOS

  14. SEEFM 07, Thessaloniki, Nov 2007 Verification in SOCOS • Three types of verification conditions: • Consistency(for transitions) • Completeness(for situations) • Termination(for loops) • Verification conditions are sent to external proof tools • Simplify (automatic proofs), • PVS (interactive proof checking) • Prover daemon works in the background • Monitors changed files • (Re)generates verification conditions and reruns proofs

  15. SEEFM 07, Thessaloniki, Nov 2007 Backends Higher assurance→ Testing Diagram is converted to a Python program, with run-time evaluation of invariants Static Checking Verification conditions are sent to Simplify, a fully automatic prover Full Verification PVS is used for full verification of the final components

  16. SEEFM 07, Thessaloniki, Nov 2007 Teaching invariant based programming at Abo Akademi • Spring 2005: Ph.D. course on invariant based programming -- testing out the idea • Spring 2007: a course on IBP for first year students • 2008 -- : IBP now part of standard CS curriculum • Planned next step: teaching IBP as a special math course in high school

  17. SEEFM 07, Thessaloniki, Nov 2007 New first year CS curriculum • Mathematics courses • algebra • probability theory • Computer Science courses • Introduction to CS • Python programming (to show that programming is fun) • Structured derivations (to teach mathematical and logical reasoning) • Invariant based programming (to teach how to construct programs that are correct) • Java programming • Systems design course Formal methods bundle

  18. SEEFM 07, Thessaloniki, Nov 2007 Invariant based programming course (spring 2007) • aimed at first or second year students • interactive, emphasizing student participation • 17 sessions a 90 min • 11 lectures • 6 practical excercises • Socos tool support only used in 4 last sessions • only automatic proofs (Simplify), no PVS proofs • 16 active participants • half with no background in formal methods

  19. SEEFM 07, Thessaloniki, Nov 2007 Collecting data about the course • pre- and postcourse questionairs • observations • hand-in assignments • final exam • 8 students selected for semi-structured interview

  20. SEEFM 07, Thessaloniki, Nov 2007 Main results - 1 • The students found the course • useful, • interesting, • somewhat fun and • of medium difficulty level. • On average, students found invariant based programming • rather easy to learn, • useful in practice and • made the general structure of the program more comprehensible

  21. SEEFM 07, Thessaloniki, Nov 2007 Main results - 2 • Difficulties were mainly in • constructing proofs and • finding the invariant for more complex programs • The programs written by the students show that • they had understood the idea behind IBP, and • were able to construct and prove simple invariant based programs.

  22. SEEFM 07, Thessaloniki, Nov 2007 Main results - 3 • The students appreciated the diagrammatic notation of IBP • most students are visual learners, • textual programming languages or pseudocode may not be the best way for expressing algorithms to these students • We had expected that identifying the invariants would be the most difficult task, but this was not the case • writing proofs by hand seemed to be most problematic, as they required much time and effort • formulating postconditions was also sometimes problematic • Students found that IBP provides good support for finding bugs during the program construction • instead of after the program is ready

  23. SEEFM 07, Thessaloniki, Nov 2007 Main results - 4 • Starting with informal reasoning in the course before introducing the formal framework was not appreciated • the students would have wanted the formal proof obligations to be introduced earlier • it seems that students who are not mathematically mature do not know how to reason ”informally” but first need to learn a formal approach with a fixed set of rules • Socos • supporting a formal method with a computer based tool in the course was very well received • the students preferred SOCOS over pen and paper, as the automation increased productivity. • unfamiliarity with the SOCOS syntax was the main cause of difficulty

  24. SEEFM 07, Thessaloniki, Nov 2007 Thank Youhttp://mde.abo.fi/SOCOS

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