Summer 2011 Wednesday, 07/13

Formal Systems: The ELI System The ELI-system uses only three letters of the alphabet: E, L, I. It has a single axiom, EI, and the following rules: Rule 1: if you possess a string whose last letter is I, you can add on an L at the end. Rule 2: If you have Ex, then you may add Exx to your collection. Rule 3: If III occurs in one of the strings in your collection, you may make a new string with L in place of III. Rule 4: If LL occurs inside one of your strings, you can drop it.

Formal Systems A formal system is completely defined by: A set of types of formal tokens or pieces, e.g. the letters E, L, I. One or more allowable starting positions—that is, initial formal arrangements of tokens of these types, e.g. the axiom EI. A set of formal rules specifying how such formal arrangements may or must be changed into others, e.g. Rules 1-4. Notice that many different things could serve as the tokens of this formal system, e.g. the letters can be written in many different colors or with many different writing implements, or made out of plastic, etc.

Formal Systems Many games are formal systems. Take Chess: There are twelve types of pieces, six of each color. There is only one allowable starting position. There are rules specifying how the positions change, or how the pieces move, disappear (get captured), or change type (get promoted). Notice that many different things can serve as the chess pieces, e.g. even helicopters!

Self-Propelling Formal Systems A self-propelling formal system is a formal system that “moves” by itself, e.g. a set of chess pieces that hop around the board, abiding by the rules, all by themselves. More precisely, it’s a physical device or machine such that: 1. Some configurations of its parts or statescan be regarded as the tokens and positions of some formal system. 2. In its normal operation, it automatically manipulates these tokens in accord with the rules of that system.

Self-Propelling Formal Systems Remember the PQ− system: Axiom Schema: xP−Qx− is an axiom, whenever x is composed of hyphens only. Rule (essentially): If some formula is a theorem, you can manipulate it by adding a dash in the middle and a dash at the end (to get another theorem). What would we need to make this formal system self-propelling?

Self-Propelling Formal Systems 1.Things that play the roles of dashes, P and Q. 2.A controller. 3.An executer (something that executes the controller’s instructions). Let’s try it out.

Self-Propelling Formal Systems Here’s another way to do this: −P−Q−− −P−−Q−−− Instead of people playing the role of dashes, we have pieces of tape that can be patched to each other and be written on. The executive unit here is a writer/tape-gluer. (Of course, to really implement the formal system, we would have to have an infinite number of bits of tape...) −P−−−Q−−−−

Turing Machines The imaginary device we came up with is similar (indeed, essentially the same as) a Turing Machine, which can make any formal system self-propelling. A Turing machine includes: 1. An infinitely long tape that can contain symbols written on it. (nowdays: memory bits) 2. A head that moves around the tape, erasing and replacing symbols. 3. A mechanical controller that physically ensures that the head erases and replaces symbols according to some set of rules/instructions. (nowdays: prossesor/CPU)

Turing Machines At any one time, the Turing Machine is in one of a finite set of internal states (or configurations). The simple Turing-like Machine we’ve introduced has 3 states: Initial state: go-right, push dash on left, switch to state 2. State 1: stay-put, push dash on left, switch to state 2. State 2: stay-put, push dash on right, switch to 1.

Turing Machines An abstract mathematical model, not a real physical model. We can’t built it, since the tape would have to be infinitely long. But we can prove that any problem solvable by any digital computer can also be solved by a Turing Machine. This makes Turing Machines very interesting from the philosophical/logical point of view.

Turing Machines A very important use of Turing Machines is in understanding the intuitive notion of computation, or of a problem’s being solvable in a mechanical or automatic way. Turing conjectured that every algorithmic computation (roughly, one that’s conducted by following a set of clearly defined rules) could be performed by a Turing machine. So far, no counterexamples have been found.

Machine Tables The machine table for a Turing Machine is a complete set of instructions for the machine’s operations, or a specification of the machine’s software. You can think of it as a complex “job description” for a machine.

Machine Tables Imagine that we’re designing a piece of software to run a simple Coke machine. Coke costs 15¢ and the Machine only accepts nickels and dimes. The Coke machine has three states: ZERO, FIVE, TEN.

Input Present State Go into this State Produce this Output NickelZEROFIVE − NickelFIVETEN − NickelTENZEROCoke DimeZEROTEN − DimeFIVEZEROCoke DimeTENFIVECoke

Notice that we said nothing about how the internal states of the Coke machine were constructed. We understand what it is for the Coke machine to be in these states in terms of how the states interact with each other and with input to produce output. Any system of states which interact in the way we described counts as an implementation of the ZERO, FIVE, and TEN states. Machine Tables

Machine Functionalism All there is to having mental states, is implementing some extremely complicated program. In us, this program is implemented by a human brain, but it could also be implemented on other hardware, like a Martian brain or a digital computer.

Further Motivation for Machine Functionalism The brain’s activity could, in principle, be specified in terms of (extremely complicated and still unknown) causal relationships that could be captured in a machine table. The machine table could be implemented by various kinds of hardware, not just brains. We could, in principle, replace all the neurons in your brain by mechanical devices that do the same job. Once we’re done, you would still have mentality and would not even notice any difference!

Machine Functionalism: Qualification Programs have no inbuilt relation to real- world timing, which depends on how the program is implemented on a real device. We must expect biological computational strategies to be adopted to getting useful real- time results from available, slow, brain components. So machine functionalism, even if true, should not lead anyone to ignore facts about the biological brain.