What it is not to Implement a Computation: A Critical Analysis of Chalmers’ Notion of Implemention

Abstract

In this paper, we introduce what can be called the “standard account of implementation” and briefly mention some objections raised against it. Then we carefully examine Chalmers' account of implementation and show that without a notion of “legitimate grouping of physical states” all sorts of physical systems would implement unintended computations. Specifically, we show how, despite Chalmers' attempts to overcome the difficulties inherent in defining physical state types, his definition of implementation still allows for unwanted implementations. Introduction: the standard account of implementation Over the years, several proposals of what it means to implement a computation have been advanced by philosophers (e.g., see Stabler 1987; Dietrich 1990; Chalmers 1994, 1996; Copeland 1996; Scheutz 1999, 2001). Since definitions of implemention are often cast in more general terms so as to not be forced to make commitments to particular notions of computation (or the notion of computation at all, for that matter), it is common to read about the “function(s) being implemented/realized by a physical system” instead of “the computation(s) implemented by a physical system”. Stabler (1987) presents, what could be called the “standard account of what it is to realize a function”: “We require first that the states of the system can be interpreted as representing the elements of the domain and range of the function, and we require that (in certain circumstances) if the system is in a state representing an element of the domain of the function, physical laws guarantee that it will go into a state representing the corresponding element of the range of the function.” (Stabler, 1987) Formally, this can be written as follows: Definition 1: Let S be a physical system and f a function. S computes f if, and only if, 1. there is an “interpretation” function I from a set PS of physical states of the system onto the union of the domain and range of f, such that 2. physical laws guarantee that (in certain specifiable circumstances) if the system is in a state i in PS, then the system will go into state o (in PS) such that I(o)=f(I(i)). Figure 1. The standard account of physical realization of a function: i and o are physical states of the physical system S, I is the interpretation function that maps these states onto the union of the range and the domain of f. While this definition is very general in that it includes various other accounts of implementation and physical realization as special cases, the standard account of implementation, as it stands, does not quite work unfortunately. It has been pointed out that it is “too liberal”, for it does not put any restriction on the interpretation function: without any restrictions and constraints, every system can be viewed as implementing every computation (e.g., see the well-known arguments advanced by Putnam 1988 and Searle 1990 to that effect). Finding constraints that prevent the standard account of implementation from being vacuous is crucial to computationalism, the view that mental processes can be seen to be computational processes, as otherwise – if everything can be viewed as computing every function – computationalism looses its explanatory force (e.g., Chalmers 1994; Scheutz 1999). One prima facie difficulty of the standard account is that “terms like ‘state’ and even ‘physical state’ tend to be used very loosely in this sort of context” (Stabler 1987, p. 3). Stabler demonstrates potential problems with the standard account by defining a special kind of physical state: assume the behavior F of a given physical system S can be described in a physical theory P (as long as certain background conditions C obtain that make this description applicable). Suppose further that an infinite sequence of times t1, t2, t3, ... is given. Infinitely many “physical states” can then be specified by stipulating that the system is in state pi if and only if it satisfies its description F at time ti. If the pi are then taken to be the computationally relevant states, the system will “compute” any function f over the natural number. Define the interpretation I (for an arbitrary function f over the natural numbers) to be I(pi)=i/2 if i is 0 or even, f((i-1)/2) otherwise Then S computes f by going through a sequence of states that are states by virtue of its description F being true of S at the respective time (under conditions C). In a sense, S does not really “compute f” but rather “enumerates” the pairs 〈i,f(i)〉 at any two successive