The Abstraction/Implementation Model of Problem Reformulation

ion Space The definition of an implementation link is as a representation map from an abstract theory to a concrete theory, wi th an abstraction function for mapping concrete objects to abstract objects. The diagram below shows an implementation link between bags and lists, where the abstract theory of bags and the concrete theories of lists are axiomatized using equational logic. Abstraction Map from lists to bags:ion Map from lists to bags: Abs(Nil) = NullBag Abs(MakeList(x)) = MakeBag(x) Abs(Append(Ll,L2) = BagUnion{Abs(Ll), Abs(L2)) In general there are many implementation links between two theories, going in both directions. It is assumed that a knowledge base has some of these implementation links predefined. Subsequent sections show how to automatically use predefined implementation links for behavioral abstraction when they are available, and also to generate new implementation links when given a novel problem constraint. The next page has the equational theory for lists. The equational theory for bags is obtained by renaming append to bagunion and making it commutative. The equational theory for sets is obtained by renaming bagunion to set union and making it idempotent (a set unioned with itself yields the same set). A different equational theory for sets is obtained by renaming bagunion to symmetric-set-difference and making it self-conjugate (a set differenced with itself yields the null set). This demonstrate how adding an equality to a theory can generate an abstract theory. These theories wil l be used in the example. Equa t iona l T h e o r y o f L is ts I I I A Simple Example This example wi l l be used to illustrate the methods described in subsequent sections. 1101001 Start String 01100 Operator Goal String The arrow puzzle 2 is a state space search problem where the states are bit strings and the operators are bit strings which are applied by bit-wise XORing with a state to yield a new state. The input sort is a StartString GoalString pair, and the output sort is a list(sequence) of operators. The input /output relation SOLVE is defined as: SOLV E({StartString, GoalString}, OpList) = Apply(Reduce(XOR,OpList), StartString) = GoalString Reduce takes a binary operator and a list of arguments, iteratively applying the operator to the accumulated result and the next element of the list. Apply takes an operator and bitwise XORs it with a bitstring. Reduce and Apply are defined using conditional equations as follows: if list = null then Reduce(binop,hst) = identity if list / null then Reduce(bmopjist) = car(list)binopReduce(binop,cdr(hst)) Apply (operator, Bit String) — operator XORBitStrtng The properties of XOR, denoted <8>, assumed to be defined in the knowledge base, are used to reformulate the problem: Commutative Associative Identity SelfConjugate The commutativity and associativity of XOR means that a list of operators can be arbitrarily re-ordered, which is used to lift the output sort from list of operators to bags of operators. An alternative representation for a bag is an exponential notation, which maps each operator into its number of occurrences in the bag. The self-conjugate property of XOR means that applying an operator twice is equivalent to not applying it at all. In other words, only the even/odd parity of the number of operator occurrences is relevant. Thus the exponential notation is collapsed to the characteristic function for a set by mapping the natural numbers to mod2. Similarly, StartString GoalString pairs can be collapsed into a single bit-string by XORing them together. This single bit-string represents the class of StartString GoalString pairs that have the same solutions. The following transformations illustrate these abstractions: