An Algebra for Program Designs

1. An Algebra for Program Designs Tony Hoare Moscow July 2011

2. With ideas from • Ian Wehrman • John Wickerson • Stephan van Staden • Peter O’Hearn • Bernhard Moeller • Georg Struth • Rasmus Petersen • …and others

3. Summary denotationalmodels algebraic laws deduction rules operational rules

4. Part 1Algebra and Hoare logic • Some familiar algebraic laws • their application to program designs • derivation of Hoare logic from them

5. Subject matter: designs • variables (p, q, r) stand for programs, designs, specifications,… • they all describe what happens inside/around a computer that is executing a program. • The program itself is the most precise. • The specification is the most abstract. • Designs come in between.

6. Binary relation: p ⊑ q • Everything described by p is also described by q , e.g., • spec p implies spec q • prog p satisfies spec q • prog p more determinate than prog q • stepwise development is • spec ⊒ design ⊒ program • stepwise analysis is the reverse • program ⊑ design ⊑ spec

7. p ⊑ q • below • lesser • stronger • lower bound • more precise • …deterministic • included in  • antecedent => • above • greater • weaker • upper bound • more abstract • ...non-deterministic • containing (sets) • consequent (pred)

8. ⊑ is a partial order • ⊑ transitive • p ⊑ r if p ⊑ q and q ⊑ r • needed forstepwise development/analysis • ⊑ antisymmetric and reflexive • p = r iff p ⊑ r and r ⊑ p • needed for abstraction

9. Binary operator: p ; q • sequential composition of p and q • anexecution of p;q consists of • all events x from an execution of p • together with all events y from an execution of q • strong sequence: x must precede y • weak sequence: y must not precede x • the algebraic laws will apply to both

10. Hoare triple: {p} q {r} • defined as p;q ⊑ r • starting in the final state of an execution of p, q ends in the final state of some execution of r • p and r may be arbitrary designs. • example: {..x+1 ≤ n} x:= x + 1 {..x ≤ n} • where ..b (finally b) describes all executions that end in a state satisfying a single-state predicate b .

11. Partial correctness • disregards unending executions • ..b is re-interpreted as including them all • ‘if the execution terminates, it will end in a state satisfying b‘. • definition of triple stays the same • partial correctness logic is the same as total correctness logic.

12. monotonicity • Law: ( ; is monotonic) : • p;q ⊑ p’;q’ if p ⊑ p’ and q ⊑ q’ • justifies modular design/evolution • p’ and q’ may be developed independently • Theorem (rule of consequence): • p’ ⊑ p & {p} q {r} & r ⊑ r’ implies {p’} q {r’} • Law is also provable from the theorem

13. associativity • Law (; is associative) : • (p;q);q’ = p;(q;q’) • Theorem (sequential composition): • {p} q {s} & {s} q’ {r} implies{p} q;q’ {r} • half the law provable from theorem

14. Unit(skip):  • a program that does nothing • Law ( is the unit of ;): • p;= p = ;p • Theorem (nullity) • {p}  {p} • a quarter of the law is provable from theorem

15. concurrent composition: p | q • execution of (p|q) consists of • all events x of an execution of p, • and all events y of an execution of q • same laws apply to both: • interleaving: x precedes or follows y • true concurrency: x neither precedes nor follows y. • Laws: | is associative, commutative and monotonic

16. Separation Logic • Law (locality): • (s|p) ; q ⊑ s |(p;q) (left locality) • p ; (q|s) ⊑ (p;q) | s (right locality) • a weak version of associativity • a weak version of distribution • Theorem (frame rule) : • {p} q {r} implies {p|s} q {r|s} • in Hoare logic, & replaces | , with side-condition that q does not make s false • Left locality provable from the theorem!

17. Concurrency law • Law (; exchanges with *) • (p|q) ; (p’|q’) ⊑ (p;p’) | (q;q’) • a weak kind of mutual distribution • Theorem (| compositional) • {p} q {r} & {p’} q’ {r’} implies {p|p’} q|q’ {r|r’} • the law is provable from the theorem

18. Exchange law p p’ | | q q’ p|q;p’|q’

19. p|q;p|q⊑p;p’|p;q ; p p;p’|p;q p’ | q q’ ; p|q;p’|q’

20. Regular language model • p, q, r,… are sets of strings (languages). • p ⊑ q is inclusion of languages • p;q is (lifted) concatenation of strings • p|q is (lifted) interleaving of strings

21. Left locality • Theorem: (s|p) ; q ⊑ s |(p;q) in lhs: s interleaves with just p , and all of q comes at the end. in rhs: s interleaves with all of p;q so lhs is a special case of rhs • right locality is similar

22. Exchange • Theorem: (p|q) ; (p’|q’) ⊑ (p;p’) | (q;q’) • in lhs: all of p and q comes before all of p’ and q’ . • in rhs: p may interleave with q’ and p’ with q • the lhs is a special case of the rhs.

23. Conclusion • regular expressions satisfy all our laws for ⊑ , ; , and | • and other operators introduced later

24. Part 2. more operators and laws • Complete lattices • Iteration, recursion, fixed points • Subroutines, abstraction

25. Subject matter • variables (p, q, r) stand for programs, designs, specifications,… • they are all descriptions of what happens inside and around a computer that is executing a program. • the differences between programs and specs are often defined from their syntax.

26. Specification syntax includes • disjunction (or) to express abstraction, or to keep options open • ‘it may be painted green or blue’ • conjunction (and) to combine requirements • it must be cheaper than x and faster than y • negation (not) for safety and security • it must not explode • implication to define contracts • if the user observes the protocol, so will the system

27. Program syntax excludes • disjunction • non-deterministic programs difficult to test • conjunction • inefficient to find a computation satisfying both • negation • Incomputable • implication • there is no point in executing it

28. programs include • sequential composition (;) • concurrent composition (|) • iteration • recursion • interfaces • transactions • assignments, inputs, outputs, jumps,… • So let’s include these in our specification/designs

29. Bottom  • A specification that has no implementation like the false predicate • A program that has no execution e.g., because of some syntactic error • Define  as the least solution of _ ⊑ q • r ⊑ q implies ⊑ r • Law ( is the zero of ;) : •  ; p =  = p ;  • Theorem : • {p}  {q}

30. Top ⊤ • a program with a run-time error • for which the programmer is responsible • e.g., subscript error, division by zero, divergence,… • defined as the least solution of q ⊑ _ • Law: it is a zero of ; • ⊤; p = ⊤ = p ;⊤ if p ≠  • Theorem: none

31. Non-determinism (or): p ⊔ q • describes all executions that either satisfy p or satisfy q . • The choice is not (yet) determined. • It may be determined later • in development of the design • or in writing the program • or by the compiler • or even at run time

32. lub (join): ⊔ • Define p⊔q as least solution of p ⊑ _ & q ⊑ _ • Theorem • p ⊑ r & q ⊑ r iffp⊔q⊑ r • Theorem • ⊔ is associative, commutative, monotonic, idempotent and increasing • it has unit ⊥ and zero ⊤

33. glb (meet): ⊓ • Define p⊓qas greatest solution of _ ⊑ p & _ ⊑ q

34. Distribution • Law ( ; distributive through ⊔ ) • p ; (q⊔q’) = p;q ⊔ p;q’ • (q⊔q’) ; p = p;q ⊔ p’;q • Theorem (non-determinism) • {p} q {r} & {p} q’ {r} implies {p} q⊔q’ {r} • i.e., to prove something of q⊔q’ prove the same thing of both q and q’ • quarter of law provable from theorem

35. Conditional: p if b else p’ • Define p ⊰b⊱ p’ as b.. ⊓ p ⊔ not(b).. ⊓ p’ • where b.. describes all executions that begin in a state satisfying b . • Theorem. p ⊰b⊱ p’ is associative, idempotent, distributive, and • p ⊰b⊱ q = q ⊰not(b)⊱ p (symm) • (p ⊰b⊱ p’ ) ⊰c⊱ (q⊰b⊱ q’) = (p ⊰c⊱ q) ⊰b⊱ (p’ ⊰c⊱ q’) (exchange)

36. Transaction • Defined as (p ⊓..b) ⊔ (q ⊓..c) • where ..b describes all executions that end satisfying single-state predicate b . • Implementation: • execute p first • test the condition b afterwards • terminate if b is true • backtrack on failure of b • and try an alternative q with condition c.

37. Transaction (realistic) • Let r describe the successful executions of a transaction t . • r is tested when execution of t is complete. • any successful execution of t is committed • a single failed execution of t is undone, • and q is done instead. • Define: (t if r else q) = t if t ⊑ r = (t ⊓ r) ⊔ q otherwise

38. Complete Lattice • Let S be an arbitrary set of designs • Define ⊔S as its least upper bound • ∀s∊ S . s ⊑ ⊔S • ∀s∊ S . s ⊑ r ⇒ r ⊑ ⊔S (all r ∊ PR) • everything is an upper bound of { } , so ⊔{ } =  • a case where ⊔S ∉ S

39. similarly • ⊓S is greatest lower bound of S • ⊓ { } = ⊤

40. Iteration (Kleene *) • q* is least solution of • (ɛ ⊔ (q; _) ) ⊑ _ • q* =def⊔{s| ɛ ⊔ q; s ⊑ s} • ɛ ⊔ q; q* ⊑ q* • ɛ ⊔ q; q’ ⊑ q’ implies q* ⊑ q’ • q*=⊔ {qⁿ | n ∊ Nat} • Theorem (invariance): • {p}q*{p} if {p}q{p}

41. Infinite replication • !p is the greatest solution of p|_ ⊑ _ • as in the pi calculus • all executions of !p are infinite • or possibly empty

42. Recursion • Let F(_) be a monotonic function between programs. • Theorem (Knaster-Tarski): all functions defined by monotonic operators are monotonic. • μF is weakest solution of F(_) ⊑ _ • νF is strongest solution of _ ⊑ F(_) • Theorem (Knaster-Tarski): These solutions exist.

43. Interfaces • Let q be the body of a subroutine • Let s be its specification • Let (q => s) assert that q meets s • Programmer error (⊤) if incorrect • Caller of subroutine may assume s • Implementer may execute q

44. Subroutine: q => s • Define (q=>s) as least solution of q ⊑ _ & _ ⊑ s • Theorem: (q=>s) = q if q ⊑ s = ⊤ otherwise

45. Basic statements/assertions • skip  • bottom  • top ⊤ • assignment: x := e(x) • assertion: assert b • assumption: assume b • finally ..b • initially b..

46. more • assign thru pointer: [a] := e • output: c!e • input: c?x • points to: a|-> e • a |-> _ =def exists v . a|-> v • throw • catch

47. Laws(examples) • assume b =def b..⊓ • assert b =defb..⊓ ⊔ not(b).. • x := e(x) ; x := f(x) = x := f(e(x)) • in languages without interleaving

48. more • p|-> _ ; [p] := e ⊑ p|-> e • in separation logic • c!e | c?x = x := e • in CSP but not in CCS or Pi • throw x ; (catch x; p) = p

49. Part 3Unifying Semantic Theories • Six familiar semantic definition styles. • Their derivation from the algebra • and vice versa.

50. algebraic laws deduction rules operational rules