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The World’s Greatest Linked List

The World’s Greatest Linked List. Preliminary work by Jim McKim, Rensselaer-Hartford Somesh Mukherjee, Compunetrix. Contents. Specifying Data Structures Desirable Linked List Properties Mathematical ADT Spec - Stack Software ADT Spec - Stack Implementation Notes - Stack.

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The World’s Greatest Linked List

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  1. The World’s Greatest Linked List Preliminary work by Jim McKim, Rensselaer-Hartford Somesh Mukherjee, Compunetrix Winthrop University

  2. Contents Specifying Data Structures Desirable Linked List Properties Mathematical ADT Spec - Stack Software ADT Spec - Stack Implementation Notes - Stack Winthrop University

  3. Contents (Cont’d) Mathematical ADT Spec – Linked List Software ADT Spec – Linked List Implementation Notes – Linked List Conclusion Winthrop University

  4. Specifying Data Structures • Lots of early work on mathematical specs • Some work (Guttag) on mapping mathematical specs to software implementations • Stack is most famous example • Most mathematical specs of data structures do not map cleanly to code Winthrop University

  5. Specifying Data Structures (Cont’d) • E.g. Mathematical spec for “Linked” List is usually in terms of Head and Tail • But actual software linked lists are cursor driven • Goal: Develop a mathematical spec for a cursor driven list, showing a clear trail from the mathematics to a compilable software specification Winthrop University

  6. Specifying Data Structures (Big picture) • Do the same for all the other common data structures • Provide mathematical underpinnings roughly as strong as those for relational databases. • A second student, Shawn Pecze, is working on a Dispenser hierarchy Winthrop University

  7. Specifying Data Structures (Big picture) • NSF grant proposal in progress • Accessible to strong undergraduate and master’s level researchers Winthrop University

  8. Desirable Linked List Properties • Pleading note: Please take these more or less on faith due to time constraints, debate to come later • Want the desirable properties from the client’s perspective • Understandability • Correctness • Bidirectional (Symmetry) Winthrop University

  9. Desirable Linked List Properties (Understandability) • No more complex than necessary • Smallest possible number of special cases • Complete, rigorous spec, readable by the user Winthrop University

  10. Desirable Linked List Properties (Correctness) • Confidence that the implementation has been well tested against the spec • Compilable spec helps inspire that confidence • Confidence that an underlying mathematical theory is being implemented Winthrop University

  11. Desirable Linked List Properties (Bidirectional) • Can use bidirectional list where a unidirectional list would suffice, but not the other way around • Neither direction should be given preference (symmetry) • Think left and right instead of first and last Winthrop University

  12. Desirable Linked List Properties (Few special cases) • Cursor driven lists where cursor points to an item result in a number of special cases • Mainly ‘cause cursor can’t always point to an item • Empty list • After removal of one of the end items • Can we do better? Winthrop University

  13. Desirable Linked List Properties (Few special cases) • Cursor never points to an item • In typical case points between two items • First suggested by Wise in 1976 • JDK Linked List implementation actually uses this, but it is not part of the spec Winthrop University

  14. Desirable Linked List Properties (Few special cases) • If cursor points between two items, we have two current items, left_item and right_item • If all inserts and deletes are via the cursor then the cursor divides the list into two parts, left and right. • What kinds of structures are left and right? • Answer later... off to mathematics... Winthrop University

  15. new empty Mathematical ADT Spec (Stack) push(s, x) notempty top(push(s, x)) = x pop(push(s, x)) = s Winthrop University

  16. Connecting Mathematical ADT Spec to Software Spec • In Math, push and pop are functions; in software they’re usually commands that change state • So provide functional equivalents • In Math, equality means identity; in software, there is reference equality (=) and object equality (is_equal) • So provide both Winthrop University

  17. Connecting Mathematical ADT Spec to Software Spec • In Math, we are not concerned with types; in software typing helps build confidence in correctness • Use generic types • In Math, we have all of axiomatic set theory available to us; in software each class provides its own language • Use a small specification set to define everything else Winthrop University

  18. Software Spec for Stack in Eiffel class STACK[G] is_empty : BOOLEAN top : G require not is_empty body : STACK[G] -- functional equivalent of pop, newly created on each call require not is_empty Winthrop University

  19. Software Spec for Stack (Cont’d) top, body, and is_empty form the specification set for the stack. Everything else will follow from these Think of a nonempty stack as consisting of its top element and another stack Winthrop University

  20. Software Spec for Stack (Cont’d) make -- creation routine ensure is_empty is_equal(other : STACK[G]) : BOOLEAN require other /= Void ensure Result = (is_empty = other.is_empty and then (not is_empty implies top = other.top and body.is_equal(other.body))) Winthrop University

  21. Software Spec for Stack (Cont’d) plus_top ( t : G) : STACK[G] -- functional equivalent of push, newly created on each call ensure not Result.is_empty -- push(s,x) is not empty Result.top = t -- top(push(s,x) = x Result.body.is_equal(Current) -- pop(push(s,x)) = s Winthrop University

  22. Software Spec for Stack (Common features) push( t : G ) ensure is_equal(old plus_top(t)) pop require not is_empty ensure is_equal(old body) Winthrop University

  23. Software Spec for Stack (Common features) depth : INTEGER ensure is_empty implies Result = 0 not is_empty implies Result = 1 + body.depth Winthrop University

  24. Software Spec for Stack (Common features) replace(x : G) require not is_empty ensure not is_empty -- important that this one come first top = x body.is_equal(old body) Winthrop University

  25. Software Spec for Stack (Common features) pop_top : G -- Both a command and query require not is_empty ensure is_equal(old body) Result = old top Winthrop University

  26. Stack Implementation Notes • All the usual implementations of a stack will work. Just add implementations of body and plus_top • Possible to implement a version that matches spec • All require and ensure clauses shown are compilable and hence automatically checkable in Eiffel Winthrop University

  27. Mathematical ADT Spec (Linked List) • Recall the List can be thought of as consisting of the structures to the left and right of the cursor • What are these left and right structures? Winthrop University

  28. Mathematical ADT Spec (Linked List) • Yup, they’re stacks! • Each list, lst, may be viewed as an ordered pair of stacks • lst = (left, right) Winthrop University

  29. Mathematical ADT Spec (Linked List) new (empty, empty) put_left(lft, rgt, x) = (push(lft, x), rgt)) remove_left(put_left(lft,rgt,x)) = (lft,rgt) left_item(put_left(lft,rgt,x)) = x move_left(put_left(lft,rgt,x)) = (lft, push(rgt, x)) Winthrop University

  30. Linked List Software Spec class LINKED_LIST[G] -- Specification set left : STACK[G] -- Newly created on each call right : STACK[G] -- Newly created on each call Winthrop University

  31. Linked List Software Spec -- creation make ensure left.is_empty; right.is_empty left_item : G require not left.is_empty ensure Result = left.top Winthrop University

  32. Linked List Software Spec put_left(x) ensure left.is_equal(old left.plus_top(x)) right.is_equal(old right) Winthrop University

  33. Linked List Software Spec remove_left require not left.is_empty ensure left.is_equal(old left.body) right.is_equal(old right) Winthrop University

  34. Linked List Software Spec move_left require not left.is_empty ensure left.is_equal(old left.body) right.is_equal(old right.plus_top(x)) Winthrop University

  35. Linked List Software Spec length : INTEGER ensure Result = left.depth + right.depth off_left : BOOLEAN ensure Result = left.is_empty is_empty : BOOLEAN ensure Result = left.is_empty and right.is_empty Winthrop University

  36. Linked List Implementation Notes • All the usual implementations will work (e.g. Nodes, Shifting arrays) • Just add implementations of left and right • Can also directly implement the spec • Again all Require and Ensure clauses are compilable and automatically checkable Winthrop University

  37. Conclusion • We have a mathematical spec for a linked list • We have a clear path to a software specification of that list • Software spec is understandable to technical people who are not math wizards Winthrop University

  38. Conclusion • Special cases have been minimized (try an implementation to convince yourself of this) • Compilable and automatically checkable specs plus all the above add confidence that good programmers will implement correct code. Winthrop University

  39. Contact Info jcm@rh.edu www.rh.edu/~jcm/ 860-548-2458 Rensselaer-Hartford 275 Windsor St. Hartford, CT 06120 Winthrop University

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