Session No. 5 Software Support 167 PROCEDURAL EMBEDDING OF theoretically In terms of the notion of KNOWLEDGE IN PLANNER satisfiability. The problem of completeness is that, for a foundation Carl Hewitt of mathematics to be Intuitively satisfactory all the true formulas Artificial Intelligence Laboratory, should be proveable since a foundation M.I.T., Cambridge, Massachusetts, mathematics alms to be a theory of U.S.A. mathematical truth. Similar fundamental questions 0. Abstract must be faced by a foundation for Since the last IJCAI, the problem solving. However there are PLANNER problem solving formalism has some important differences since a continued to develop. Our eplstemolgy foundation of problem solving alms more for the foundations for problem solving to be a theory of actions and purposes has been extended. An overview of the than a theory of mathematical truth. A formalism Is given from an Information foundation for problem solving must processing viewpoint. A simple example specify a goal-orlented formalism In Is explained using snapshots of the which problems can be stated. state of the problem solving as the Furthermore there must be a formalism example Is worked, finally, current for specifying the allowable methods of applications for the formalism are solution of problems. As part of the Iisted. definition of the formalisms, the following elements must be defined: 1. The Structural Foundations of the data structure, the control Problem Solving structure, and the primitive We would like to develop a procedures. The problem of what are foundation for problem solving allowable data structures for facts analogous In some ways to the currently about the world Immediately arises. existing foundations for mathematics. Being a theory of actions, a foundation Thus we need to analyze the structure for problem solving must confront the of foundations for mathematics. A problem of change: How can account be foundation for mathematics must provide taken of the changing situation In the a definitional formalism In which world? In order for there to be mathematical objects can be defined and problem solving, there must be a their existence proved. For example problem solver. A foundation for set theory as a foundation provides problem solving must consider how much that objects must be built out of sets. knowledge and what kind of knowledge Then there must be a deductive problem solvers can have about formalism In which fundamental truths themselves. In contrast to the can be stated and the means provided to foundation of mathematics, the deduce additional truths from those semantics for a foundation for problem already established. Current solving should be defined In terms of mathematical foundations such as set properties of procedures. We would theory seem quite natural and adequate like to see mathematical Investigations for the vast body of classical on the adequacy of the foundations for mathematics. The objects and problem solving provided by PLANNER. reasoning of most mathematical domains In chapter C of the dissertation, we such as analysis and algebra can be have made the beginnings of one kind of easily founded on set theory. The such an Investigation. existence of certain astronomically To be more specific a large cardinals poses some problems for foundation for problem solving must set theoretic foundations. However, concern Itself with the following the problems posed seem to be of complex of topics. practical Importance only to certain category theorists. Foundations of PROCEDURAL EMBEDDING: How can mathematics have devoted a great deal "real world" knowledge be effectively of attention to the problems of embedded In procedures. What are good consistency and completeness. The ways to express problem solution problem of consistency Is Important methods and how can plans for the since If the foundations are solution of problems be formulated? inconsistent then any formula whatsoever may be deduced thus GENERALIZED COMPILATION: What trivializing the foundations. are good methods for transforming high Semantics for foundations of level goal-orlented language into mathematics are defined model specific efficient algorithms. 168 Session No. 5 Software Support DESCRIPTIONS are procedures VERIFICATION: Hovv can it be which recognize how well some candidate verified that a procedure does what Is fits the description. Intended. PATTERNS are PROCEDURAL ABSTRACTION: What descriptions which match configurations are Rood methods for abstracting of data. For example <either 4 general procedures from special cases. <atomic>> Is a procedure which will recognize something which is either 4 One approach to foundations for or is atomic. problem solving requires that there should be two distinct formalisms: DATA TYPES are patterns used In declarations of the allowable 1: A METHODS formalism which range and domain of procedures and specifies the allowable methods of Identifiers. More generally, data solutIon types have analogues In the form of procedures which create, destroy, 2: A PROBLEM SPECIFICATION recognize, and transform data. formalism In which to pose problems. GRAMMARS: The The problem solver Is expected to PROGRAMMAR language of Terry Wlnograd figure out how combine Its available represents the first step towards one methods In order to produce a solution kind of procedural analogue for natural which satisfies the problem language grammar. specification. One of the aims of the above formulation of problem solving is SCHEMATIC DRAWINGS have to clearly separate the methods of as their procedural analogue methods solution from the problems posed so for recognizing when particular figures that it Is Impossible to "cheat" and fit within the schemata. give the problem solver the methods for solving the problem along with the PROOFS correspond to statement of the problem. lie propose procedures for recognizing and to bridge the chasm between the methods expanding valid chains of deductions. formalism and the problem formalism. Indeed many proofs can fruitfully be Consider more carefully the two considered to define procedures which extremes In the specification of are proved to have certain properties. process ing: MODELS of PROGRAMS are A: Explicit processing (e.g. procedures for defining properties of methods) is the ability to specify and procedures and attempting to verify control actions down to the finest these properties. Models of programs detalls. can be defined by procedures which state the relations that must hold as B: implicit processing (e.g. control passes through the program. problems) is the ability to specify the end result desired and not have to say very much about how It should be PLANS are general, goal ach leved. oriented procedures for attempting to carry out some task. PLANNER attempts to provide a formalism In which a problem solver can bridge THEOREMS of the the continuum between explicit and OUANTIFICATIONAL CALCULUS have as their Implicit processing. We aim for a analogues procedures for carrying out maximum of flexibility so that whatever the deductions which are justified by knowledge is available can be the theorems. For example, consider a incorporated even If It Is fragmentary theorem of the form (IMPLIES x y). One and heurIstIc. procedural analogue of the theorem Is Our work on PLANNER has been an to consider whether x should be made a investigation in PROCEDURAL subgoal in order to try to prove ERISTEMOLOGY, the study of how something of the form y. Ira Goldstein knowledge can be embedded in has shown that the theorems of procedures. The THESIS OF PROCEDURAL elementary plane geometry have very EMBEDDING is that intellectual natural procedural analogues. structures should be analyzed through their PROCEDURAL ANALOGUES. We will DRAWINGS: The try to show what we mean through procedural analogue of a drawing is a examples: procedure for making the drawing. Session No. 5 Software Support 169 PROGRESSIVE REFINEMENT 170 Session No. 5 Software Support Rather sophisticated display processors and the FIND statement of PLANNER. The have been constructed for making primitive FIND will construct a list of drawings on cathode ray tubes, all the objects with certain properties. For example we can find RECOMMENDATIONS: five things which are on something PLANNER has primitives which allow which Is green by evaluating recommendations as to how disparate sections of goal oriented language <FIND 5 x should be linked together in order to <G0AL (ON x y)> accomplish some particular task. <GOAL (GREEN y)>> which reads "find 5 x's such that x Is GOAL TREES are represented by a ON y and y Is GREEN." snapshot of the instantaneous The patterns of looping and configuration of problem solving recursion represent common structural processes. methods used In programs. They specify how commands can be repeated One corollary of the thesis of Iteratively and recursively. One of procedural embedding Is that learning the main problems In getting computers entails the learning of the procedures to write programs Is how to use these in which the knowledge to be learned is structural patterns with the particular embedded. Another aspect of the thesis domain dependent commands that are of procedural embedding is that the available. It is difficult to decide process of going from general goal which/ if any, of the basic patterns is oriented language which Is capable of appropriate in any given problem. The accomplishing some task to a special problem of synthesizing programs out of purpose/ efficient/ algorithm for the canned loops Is formally Identical to task should Itself be mechanized. By the problem of finding proofs using expressing the properties of the mathematical Induction. We have special purpose algorithm in terms of approached the problem of constructing their procedural analogues/ we can use procedures out of goal oriented the analogues to establish that the language from two directions. The special purpose routine does in fact do first is to use canned loops (such as what it is intended to do. the FIND statement) where we assume a- We are concerned as to how a priorl the kind of control structure theorem prover can unify structural, that Is needed.