that problem-solving should be understood as the rule-governed transformation of symbol structures.
(3) The physical symbol system contains processes for manipulating symbols and symbol structures. We have already seen how a physical symbol system can contain processes for generating complex symbol structures from the basic building blocks provided by the system’s alphabet. But what really matters is what the system does with those complex symbol structures– just as what really matters in propositional logic are the rules that allow one complex formula to be derived from another.
The physical symbol system hypothesis is a hypothesis about intelligence and intelli- gent action. This means that it has to explain what thinking consists in– whether in human beings or in machines. And here we have the distinctive claim of the physical symbol system hypothesis. This is that thinking is simply the transformation of symbol structures according to rules. Any system that can transform symbol structures in a sophisticated enough way will qualify as intelligent. And when we fully understand what is going in agents that we uncontroversially take to be intelligent (such as human beings), what we will ultimately find is simply the rule-governed transformation of symbol structures.
This hypothesis about thinking is simple (and in many ways compelling). There are many different things that count as thinking. But not all of them really count as manifest- ations of intelligence. After all, even daydreaming is a type of thinking. Newell and Simon’s fundamental claim is that the essence of intelligent thinking is the ability to solve problems. Intelligence consists in the ability to work out, when confronted with a range of options, which of those options best matches certain requirements and constraints.
Intelligence only comes into the picture when there is what might abstractly be called a search-space. The notion of a search-space is very general. One example might be the position of one of the players halfway through a chess game– as in the situation being
B O X 6 . 1 Defining well-formed formulas (WFFs) in propositional logic
Propositional logic is the branch of logic that studies argument forms whose basic constituents are whole sentences or propositions. The basic building blocks of propositional logic (the alphabet) are infinitely many sentence symbols (P1, P2, P3. . .), together with a small set of logical connectives (the precise set varies, since the logical connectives are interdefinable and different authors take different sets of connectives as basic). One connective (“¬,” read as “not-”) is unary – that is, it applies to single formulas. Other connectives (such as “∧,” “∨,” and “),” read as “and,” “or,” and “if. . . then . . .” respectively) are binary – they connect pairs of formulas. The legitimate combinations of symbols in the alphabet might typically be defined as follows. (a) Any sentence symbol is a WFF.
(b) Ifφ is a WFF then ¬φ is a WFF.
(c) Ifφ and ψ are WFF, then φ ∧ψ is a WFF and so on for “∨” and “)”.
Note thatφ and ψ can stand here for any formula, not just for sentence symbols. So this definition gives us a recipe for creating WFFs of unlimited complexity. (The technical way of describing this is to say that (b) and (c) are recursive rules.)
analyzed by Newell and Simon inFigure 6.1. Each chess player has a large number of possible moves and a clearly defined aim– to checkmate her opponent. The possible moves define the search-space and the problem is deciding which of the possible moves will move her closest to her goal.
Another example (much studied by computer scientists and mathematicians) is a travel- ing salesman who starts in a particular city (say, Boston) and has to visit twenty other cities as quickly and efficiently as possible before eventually returning to Boston. Here we can think about the search-space in terms of all the possible routes that start and end in Boston and go through the twenty cities (perhaps visiting some more than once). The diagram at the top inFigure 6.2illustrates a simpler traveling salesman problem with only five cities.
Search-spaces are typically represented in terms of states. They are given by an initial state (the start state) and a set of permissible transformations of that start state. The search-space is composed of all the states that can be reached from the start state by applying the permissible transformations. The transformations can be carried out in any order. In the chess example, the start state is a particular configuration of the chess pieces and the permissible transformations are the legal moves in chess. In the traveling salesman example, the start state might be Boston, for example, and the permissible transformations are given by all the ways of getting directly from one city to another. Computer scientists standardly represent search-spaces in terms of trees. So, for example, the search-space for the traveling salesman problem is given by a tree whose Figure 6.1 Allen Newell and Herbert Simon studying a search-space.
first node is the starting city. InFigure 6.2the start city is a. There is a branch from the first node to a node representing each of the cities to which the start city is directly connected– i.e. cities b, c, d, and e. From each of those nodes there are further branches connecting each city to all the other cities to which it is directly connected. And so on. The diagram at the bottom ofFigure 6.2illustrates a part of the search-space for our five- city version of the traveling salesman problem.
What counts as solving a problem? Solving a problem is a matter of identifying a solution state. In the case of chess, the solution state is any configuration of the board on which the opponent’s king is in checkmate. In the traveling salesman case, the solution is
a b c d e 100 50 50 100 125 125 75 125 100 75 a b c d e 100 125 100 75 150 250 300 375 c d e c e e d a a abcdea abcdea 375 425 Search-space
An instance of the traveling salesman problem
Figure 6.2 A typical traveling salesman problem. The top diagram depicts the problem. A traveling salesman has to find the shortest route between five cities. The diagram below depicts part of the search-space. A complete representation of the search-space would show twenty-four different routes.
the shortest branch of the tree that ends with Boston and that has nodes on it corres- ponding to each of the twenty cities that the salesman needs to visit.
How should we think about the process of solving a problem? The most general characterization of problem-solving is as a process of searching through the search-space until a solution state is found. But everything here depends upon what counts as search. Brute force searches that follow each branch of the tree tend only to work for very simple problems. It does not take long for a problem-space to get so big that it cannot be exhaustively searched in any feasible amount of time.
The traveling salesman tree gets very complicated very quickly. If there are n cities, then it turns out that there are (n 1)! possible routes to take into account, where (N 1)! ¼ (N 1) (N 2) (N 3) . . . This is not too many for the five-city version of the problem depicted inFigure 6.2(it gives twenty-four different routes). But the problem gets out of control very quickly. In a twenty-city version there are approxi- mately 6 1016different ways for a traveling salesman to start in Boston and travel through the other nineteen cities visiting each exactly once. Checking one route per second, it would take more or less the entire history of the universe to search the problem-space exhaustively.
Here is a rather simpler example than the traveling salesman problem (which, by the way, computer scientists and mathematicians are still working on– no general solution is yet known). The foxes and the chickens problem is a version of a problem that Newell and Simon themselves used to illustrate their General Problem Solver (GPS) program.
The basic idea behind the GPS program is relatively straightforward. It uses the problem-solving technique known as means–end analysis. Means–end analysis is a three- stage process that is intended to converge on a solution state by reducing the difference between the current state and the goal state. Here is how it works.
1 Evaluate the difference between the current state and the goal state.
2 Identify a transformation that reduces the difference between current state and goal state.
3 Check that the transformation in (2) can be applied to the current state. 3a. If it can, then apply it and go back to step (1).
3b. If it can’t, then return to (2).
Means–end analysis is an example of what Newell and Simon call heuristic search. Heuristic search techniques are techniques for searching through a search-space that do not involve exhaustively tracing every branch in the tree until a solution is found. Heuristic search techniques trim the search-space down to make the search process more tractable.
Exercise 6.1 Explain how means–end analysis trims down the search-space.
Here is the problem of the foxes and the chickens– a type of problem that Newell and Simon showed could be solved by their GPS program. Imagine that there are three chickens and three foxes on one side of a river and they all need to get over to the other
side. The only way of crossing the river is in a boat that can only take two animals (or fewer). The boat can cross in either direction, but if at any moment the foxes outnumber the chickens then the outnumbered chickens will be eaten. The problem is to work out a way of getting all the chickens and foxes onto the other side of the river without any of the chickens being eaten.
We might think of each state as specifying which animals are on each bank and which in the boat (as well as the direction in which the boat is traveling). The start state obviously has all six on one bank (say the right bank) with nobody in the boat or on the other bank. The solution state is the state that has all six on the left bank, with nobody in the boat or on the other bank. The permissible transformations are defined by the rule that the boat cannot carry more than two animals.
The foxes and the chickens problem lends itself very clearly to the general approach to problem-solving that Newell and Simon propose. If we feed into the GPS program repre- sentations of the start state and the goal state(s), the program employs various strategies to transform the start state in a way that minimizes the difference from the goal state. The eventual solution is a series of representations, whose first member is a representation of the start state and whose final member is a representation of one of the goal states, and where each member is derived from its predecessor by a permissible transformation.
Each of these representations is itself a symbol structure. Newell and Simon’s point is that the GPS program reaches a solution by modifying the original symbol structure (representing the start state) until it arrives at a symbol structure that coincides with one of the goal states. The trick in writing the GPS program, of course, is building into it search strategies and sub-routines that will ensure that it reaches the goal state as efficiently as possible.
Exercise 6.2 Find a solution to the foxes and the chickens problem. Show how your solution can be represented as a process of what Newell and Simon call heuristic search.
We should observe, finally, that these rule-governed transformations are algorithmic in the sense discussed insection 1.2. According to our official definition, an algorithm is a finite set of unambiguous rules that can be applied systematically to an object or set of objects. The rules transform the objects in definite and circumscribed ways. To put it another way, algorithms are purely mechanical procedures. They can be followed blindly, without any exercise of judgment or intuition. Elementary school arithmetic provides plenty of examples of algorithms, such as the algorithms for multiplying pairs of numbers and for long division.
The algorithm requirement is not explicitly mentioned by Newell and Simon, but it is clearly required by their overall project. Part of what they are trying to do is to explain what intelligence consists in. The physical symbol system hypothesis is what might be called a reductive definition of intelligence. Definitions are not very useful if they tacitly appeal to what they are trying to explain. But some degree of intelligence is required to follow any rule that is not purely algorithmic, and so any definition of intelligence that appeals to transformations that are not purely algorithmic will end up being circular.