Computation in Physical Systems

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    Computation in Physical Systems

    Piccinini, Gualtiero

    First published Wed Jul 21, 2010

    In our ordinary discourse, we distinguish between physical systems that perform

    computations, such as computers and calculators, and physical systems that don't, such as

    rocks. Among computing devices, we distinguish between more and less powerful ones. These

    distinctions affect our behavior: if a device is computationally more powerful than another, we

    pay more money for it. What grounds these distinctions? What is the principled difference, if

    there is one, between a rock and a calculator, or between a calculator and a computer?

    Answering these questions is more difficult than it may seem.

    In addition to our ordinary discourse, computation is central to many sciences. Computer

    scientists design, build, and program computers. But again, what counts as a computer? If a

    salesperson sold you an ordinary rock as a computer, you should probably get your money

    back. Again, what does the rock lack that a genuine computer has?

    How powerful a computer can you build? Can you build a machine that computes anything you

    wish? Although it is often said that modern computers can compute anything (i.e., any

    function of natural numbers, or equivalently, any function of strings of letters from a finite

    alphabet), this is not correct. Ordinary computers can compute only a tiny subset of all

    functions. Is it physically possible to do better? Which functions are physically computable?

    These questions are bound up with the foundations of physics.

    Computation is also central to psychology and neuroscience (and perhaps other areas of

    biology). According to the computational theory of cognition, cognition is a kind of

    computation: the behavior of cognitive systems is causally explained by the computations they

    perform. In order to test a computational theory of something, we need to know what counts

    as a computation in a physical system. Once again, the nature of computation lies at the

    foundation of empirical science.

    1. Abstract Computation and Concrete Computation 2. Accounts of Concrete Computation

    o 2.1 The Simple Mapping Accounto 2.2 Causal, Counterfactual, and Dispositional Accountso 2.3 The Semantic Accounto 2.4 The Syntactic Accounto 2.5 The Mechanistic Account

    3. Is Every Physical System Computational?o 3.1 Varieties of Pancomputationalismo 3.2 Unlimited Pancomputationalismo 3.3 Limited Pancomputationalism

    http://plato.stanford.edu/entries/computation-physicalsystems/#AbsComConComhttp://plato.stanford.edu/entries/computation-physicalsystems/#AccConComhttp://plato.stanford.edu/entries/computation-physicalsystems/#SimMapAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#CauCouDisAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#SemAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#SynAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#MecAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#EvePhySysComhttp://plato.stanford.edu/entries/computation-physicalsystems/#VarPanhttp://plato.stanford.edu/entries/computation-physicalsystems/#UnlPanhttp://plato.stanford.edu/entries/computation-physicalsystems/#LimPanhttp://plato.stanford.edu/entries/computation-physicalsystems/#LimPanhttp://plato.stanford.edu/entries/computation-physicalsystems/#UnlPanhttp://plato.stanford.edu/entries/computation-physicalsystems/#VarPanhttp://plato.stanford.edu/entries/computation-physicalsystems/#EvePhySysComhttp://plato.stanford.edu/entries/computation-physicalsystems/#MecAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#SynAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#SemAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#CauCouDisAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#SimMapAcchttp://plato.stanford.edu/entries/computation-physicalsystems/#AccConComhttp://plato.stanford.edu/entries/computation-physicalsystems/#AbsComConCom
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    o 3.4 The Universe as a Computing System 4. Physical Computability

    o 4.1 The Physical Church-Turing Thesis: Boldo 4.2 The Physical Church-Turing Thesis: Modesto 4.3 Hypercomputation

    Bibliography Other Internet Resources Related Entries

    1. Abstract Computation and Concrete Computation

    Computation may be studied mathematically by formally defining computational objects, such

    as algorithms and Turing machines, and proving theorems about their properties. The

    mathematical theory of computation is a well-established branch of mathematics. It deals with

    computation in the abstract, without worrying much about physical implementation.

    By contrast, most uses of computation in science and ordinary practice deal with concrete

    computation: computation in concrete physical systems such as computers and brains.

    Concrete computation is closely related to abstract computation: we speak of physical systems

    as running an algorithm or as implementing a Turing machine, for example. But the

    relationship between concrete computation and abstract computation is not part of the

    mathematical theory of computation per se and requires further investigation. Questions

    about concrete computation are the main subject of this entry. Nevertheless, it is important to

    bear in mind some basic mathematical results.

    The most important notion of computation is that ofdigitalcomputation, which Alan Turing,

    Kurt Gdel, Alonzo Church, Emil Post, and Stephen Kleene formalized in the 1930s. Their work

    investigated the foundations of mathematics. One crucial question was whether first order

    logic is decidable whether there is an algorithm that determines whether any given first

    order logical formula is a theorem.

    Turing (19367) and Church (1936) proved that the answer is negative: there is no such

    algorithm. To show this, they offered precise characterizations of the informal notion of

    algorithmically computable function. Turing did so in terms of so-called Turing machines

    devices that manipulate discrete symbols written on a tape in accordance with finitely many

    instructions. Other logicians did the same thing they formalized the notion of algorithmically

    computable function in terms of other notions, such as -definable functions and general

    recursive functions.

    To their surprise, all such notions turned out to be extensionally equivalent, that is, any

    function computable within any of these formalisms is computable within any of the others.

    They took this as evidence that their quest for a precise definition of algorithm oralgorithmically computable function had been successful. The resulting view that Turing

    http://plato.stanford.edu/entries/computation-physicalsystems/#UniComSyshttp://plato.stanford.edu/entries/computation-physicalsystems/#PhyComhttp://plato.stanford.edu/entries/computation-physicalsystems/#PhyChuTurTheBolhttp://plato.stanford.edu/entries/computation-physicalsystems/#PhyChuTurTheModhttp://plato.stanford.edu/entries/computation-physicalsystems/#Hyphttp://plato.stanford.edu/entries/computation-physicalsystems/#Bibhttp://plato.stanford.edu/entries/computation-physicalsystems/#Othhttp://plato.stanford.edu/entries/computation-physicalsystems/#Relhttp://plato.stanford.edu/entries/computation-physicalsystems/#Relhttp://plato.stanford.edu/entries/computation-physicalsystems/#Othhttp://plato.stanford.edu/entries/computation-physicalsystems/#Bibhttp://plato.stanford.edu/entries/computation-physicalsystems/#Hyphttp://plato.stanford.edu/entries/computation-physicalsystems/#PhyChuTurTheModhttp://plato.stanford.edu/entries/computation-physicalsystems/#PhyChuTurTheBolhttp://plato.stanford.edu/entries/computation-physicalsystems/#PhyComhttp://plato.stanford.edu/entries/computation-physicalsystems/#UniComSys
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    machines and other equivalent formalisms capture the informal notion of algorithm is now

    known as the Church-Turing thesis (more on this in Section 4). The study of computable

    functions, made possible by the work of Turing et al., is part of the mathematical theory of

    computation.

    The theoretical significance of Turing et al.'s notion of computation can hardly be overstated.

    As Gdel pointed out (in a lecture following one by Tarski):

    Tarski has stressed in his lecture (and I think justly) the great importance of the concept of

    general recursiveness (or Turing's computability). It seems to me that this importance is largely

    due to the fact that with this concept one has for the first time succeeded in giving an absolute

    definition of an interesting epistemological notion, i.e., one not depending on the formalism

    chosen. (Gdel 1946, 84)

    Turing also showed that there are universal Turing machines machines that can compute

    any function computable by any other Turing machine. Universal machines do this by

    executing instructions that encode the behavior of the machine they simulate. Assuming the

    Church-Turing thesis, universal Turing machines can compute any function computable by

    algorithm. This result is significant for computer science: you don't need to build different

    computers for different functions; one universal computer will suffice to compute any

    computable function. Modern digital computers approximate universal machines in Turing's

    sense: digital computers can compute any function computable by algorithm for as long as

    they have time and memory. (Strictly speaking, a universal machine has an unbounded

    memory, whereas digital computer memories can be extended but not indefinitely, so they are

    not unbounded.)

    The above result should not be confused with the common claim that computers can

    compute anything. This claim is false: another important result of computability theory is that

    most functions are notcomputable by Turing machines (and hence, by digital computers).

    Turing machines compute functions defined over denumerable domains, such as strings of

    letters from a finite alphabet. There are uncountably many such functions. But there are only

    countably many Turing machines; you can enumerate Turing machines by enumerating all lists

    of Turing machine instructions. Since an uncountable infinity is much larger than a countable

    one, it follows that Turing machines (and hence digital computers) can compute only a tiny

    portion of all functions (over denumerable domains, such as natural numbers or strings of

    letters).

    Turing machines and most modern computers are known as (classical) digital computers, that

    is, computers that manipulate strings of discrete, unambiguously distinguishable states. Digital

    computers are sometimes contrasted withanalogcomputers, that is, machines that

    manipulate continuous variables. Continuous variables are variables that can change their

    value continuously over time while taking any value within a certain interval. Analog

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    computers are used primarily to solve certain systems of differential equations (Pour-El 1974,

    Rubel 1993).

    Classical digital computers may also be contrasted with quantumcomputers. Quantum

    computers manipulate quantum states called qubits. Unlike the computational states of digitalcomputers, qubits are not unambiguously distinguishable from one another. This entry will

    focus primarily on classical digital computation. For more on quantum computation, see the

    entry on quantum computing.

    The same objects studied in the mathematical theory of computation Turing machines,

    algorithms, and so on are typically said to be implemented by concrete physical systems.

    This poses a problem: how can a concrete, physical system perform a computation when

    computation is defined by an abstract mathematical formalism? This may be called the

    problem of computational implementation.

    The problem of computational implementation may be formulated in a couple of different

    ways. Some people interpret the formalisms of computability theory as defining

    abstract objects. According to this interpretation, Turing machines, algorithms, and the like are

    abstract objects. But how can a concrete physical system implement an abstract object? Other

    people treat the formalisms of computability theory simply as abstract

    computational descriptions. But how can a concrete physical system satisfy an abstract

    computational description? Regardless of how the problem of computational implementation

    is formulated, solving it requires an account of concrete computation an account of what it

    takes for a physical system to perform a given computation.

    A closely related problem is that of distinguishing between physical systems such as digital

    computers, which appear to compute, and physical systems such as rocks, which appear not to

    compute. Unlike computers, ordinary rocks are not sold in computer stores and are usually not

    considered computers. Why? What do computers have that rocks lack, such that computers

    compute and rocks don't? (If indeed they don't?) In other words, what does it take for a

    computation to be implemented in a concrete physical system? Different answers to these

    questions give rise to different accounts of concrete computation.

    Questions on the nature of concrete computation should not be confused with questions

    about computational modeling. The dynamical evolution of many physical systems may be

    described by computational models. Computational models describe the dynamics of a system

    that are written into, and run by, a computer. The behavior of rocks as well as rivers,

    ecosystems, and planetary systems, among many others may well be modeled

    computationally. From this, it doesn't follow that the modeled systems are computing devices

    that they themselves perform computations. Prima facie, only relatively few and quite

    special systems compute. Explaining what makes them special or explaining away our

    feeling that they are special is the job of an account of concrete computation.

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    2. Accounts of Concrete Computation

    2.1 The Simple Mapping Account

    One of the earliest and most influential accounts of computation is due to Hilary Putnam. To a

    first approximation, the account says that anything that is accurately described by a

    computational description Cis a computing system implementing C.

    More precisely, Putnam sketches his earliest account in terms of Turing machines only,

    appealing to the machine tables that are a standard way of defining specific Turing

    machines. A machine table consists of one column for each of the (finitely many) internal

    states of the Turing machine and one row for each of the machine's symbol types. Each entry

    in the machine table specifies what the machine does given the pertinent symbol and internal

    state. Here is how Putnam explains what it takes for a physical system to be a Turing machine:

    A machine tabledescribes a machine if the machine has internal states corresponding to the

    columns of the table, and if it obeys the instruction in the table in the following sense: when

    it is scanning a square on which a symbol s1appears and it is in, say, state B, that it carries out

    the instruction in the appropriate row and column of the table (in this case, column B and

    row s1). Any machine that is described by a machine table of the sort just exemplified is a

    Turing machine. (Putnam 1960/1975a, 365; cf. also Putnam 1967/1975a, 4334)

    This account relies on several unexplained notions, such as square (of tape), symbol, scanning,

    and carrying out an instruction. Furthermore, the account is specified in terms of Turingmachine tables, but there are other kinds of computational description. A general account of

    concrete computation should cover other computational descriptions besides Turing machine

    tables. Perhaps for these reasons, Putnam soon followed by many others abandoned

    reference to squares, symbols, etc.; he substituted them with an appeal to a physical

    description of the system. The result of that substitution is what Godfrey-Smith (2009) dubs

    the simple mapping account of computation.

    According to the simple mapping account, a physical system S performs computation Cjust in

    case (i) there is a mapping from the states ascribed to S by a physical description to the statesdefined by computational description C, such that (ii) the state transitions between the

    physical states mirror the state transitions between the computational states. Clause (ii)

    requires that for any computational state transition of the form s1s2 (specified by the

    computational description C), if the system is in the physical state that maps onto s1, it then

    goes into the physical state that maps onto s2.

    One difficulty with the formulation above is that ordinary physical descriptions, such as

    systems of differential equations, generally ascribe uncountably many states to physical

    systems, whereas ordinary computational descriptions, such as Turing machine tables, ascribe

    at most countably many states. Thus, there are not enough computational states for the

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    physical states to map onto. One solution to this problem is to reverse the direction of the

    mapping, requiring a mapping of the computational states onto (a subset of) the physical

    states. Another, more common solution to this problem often left implicit is to select

    either a subset of the physical states or equivalence classes of the physical states and map

    those onto the computational states. When this is done, clause (i) is replaced by the following:(i) there is a mapping from a subset of(or equivalence classes of) the states ascribed to S by a

    physical description to the states defined by computational description C.

    The simple mapping account turns out to be very liberal: it attributes many computations to

    many systems. In the absence of restrictions on which mappings are acceptable, such

    mappings are relatively easy to come by. As a consequence, some have argued that every

    physical system implements every computation (Putnam 1988, Searle 1992). This thesis, which

    trivializes the claim that something is a computing system, will be discussed in Section 3.1.

    Meanwhile, the desire to avoid this trivialization result is one motivation behind otheraccounts of concrete computation.

    2.2 Causal, Counterfactual, and Dispositional Accounts

    One way to construct accounts of computation that are more restrictive than the simple

    mapping account is to impose a constraint on acceptable mappings. Specifically, clause (ii) may

    be modified so as to require that the conditional that specifies the relevant physical state

    transitions be logically stronger than a material conditional.

    As the simple mapping account has it, clause (ii) requires that for any computational state

    transition of the form s1s2(specified by a computational description), if the system is in the

    physical state that maps onto s1, it then goes into the physical state that maps onto s2. The

    second part of (ii) is a material conditional. It may be strengthened by turning it into a logically

    stronger conditional specifically, a conditional expressing a relation that supports

    counterfactuals.

    In a pure counterfactual account, clause (ii) is strengthened simply by requiring that the

    physical state transitions support certain counterfactuals (Maudlin 1989, Copeland 1996). In

    other words, the pure counterfactual account requires the mapping between computational

    and physical descriptions to be such that the counterfactual relations between the physical

    states are isomorphic to the counterfactual relations between the computational states.

    Different authors formulate the relevant counterfactuals in slightly different ways: (a) if the

    system had been in a physical state that maps onto an arbitrary computational state (specified

    by the relevant computational description), it would then have gone into a physical state that

    maps onto the relevant subsequent computational state (as specified by the computational

    description) (Maudlin 1989, 415), (b) if the system had been in a physical state that maps

    onto s1, it would have gone into a physical state that maps onto s2 (Copeland 1996, 341), (c) ifthe system were in a physical state that maps onto s1, it would go into a physical state that

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    maps onto s2 (Chalmers 1996, 312). Regardless of the exact formulation, none of these

    counterfactuals are satisfied by the material conditional of clause (ii) as it appears in the simple

    mapping account of computation. Thus, counterfactual accounts are stronger than the simple

    mapping account.

    An account of concrete computation in which the physical state transitions support

    counterfactuals may also be generated by appealing to causal or dispositional relations,

    assuming (as most people do) that causal or dispositional relations support counterfactuals.

    Appealing to causation or dispositions may also have advantages over pure counterfactual

    accounts in blocking unwanted computational implementations (Klein 2008, 145, makes the

    case for dispositional versus counterfactual accounts).

    In a causal account, clause (ii) is strengthened by requiring a causal relation between the

    physical states: for any computational state transition of the form s1s2 (specified by a

    computational description), if the system is in the physical state that maps onto s1, its physical

    state causes it to go into the physical state that maps onto s2 (Chrisley 1995, Chalmers 1995,

    1996, Scheutz 1999, 2001).

    To this causal constraint on acceptable mappings, David Chalmers (1995, 1996) adds a further

    restriction (in order to avoid pancomputationalism, which is discussed in Section 3): a genuine

    physical implementation of a computational system must divide into separate physical

    components, each of which maps onto the components specified by the computational

    formalism. As Godfrey-Smith (2009, 293) notes, this combination of a causal and

    a localizationalconstraint goes in the direction of mechanistic explanation (Machamer,

    Darden, and Craver 2000). An account of computation that is explicitly based on mechanistic

    explanation will be discussed in Section 2.5. For now, the causal account simpliciter requires

    only that the mappings between computational and physical descriptions be such that the

    causal relations between the physical states are isomorphic to the relations between state

    transitions specified by the computational description. Thus, according to the causal account,

    concrete computation is the causal structure of a physical process.

    In a dispositional account, clause (ii) is strengthened by requiring a dispositional relation

    between the physical states: for any computational state transition of the

    form s1s2 (specified by a computational description), if the system is in the physical state

    that maps onto s1, the system manifests a disposition whose manifestation is the transition

    from the physical state that maps onto s1 to the physical state that maps onto s2 (Klein 2008).

    In other words, the dispositional account requires the mapping between computational and

    physical descriptions to be such that the dispositional relations between the physical states are

    isomorphic to the relations between state transitions specified by the computational

    description. Thus, according to the dispositional account, concrete computation is the

    dispositional structure of a physical process.

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    The difference between the simple mapping account on the one hand and counterfactual,

    causal, and dispositional accounts on the other may be seen by examining a simple example.

    Consider a rock under the sun, early in the morning. During any time interval, the rock's

    temperature rises. The rock goes from temperature Tto temperature T+1, to T+2, to T+3. Nowconsider a NOT gate that feeds its output back to itself. At first, suppose the NOT gate receives

    0 as an input; it then returns a 1. After the 1 is fed back to the NOT gate, the gate returns a

    0 again, and so on. The NOT gate goes back and forth between outputting a 0 and

    outputting a 1. Now map physical statesTand T2 onto 0; then mapT+1 and T3 onto 1.

    According to the simple mapping account, the rock implements a NOT gate undergoing the

    computation represented by 0101.

    By contast, according to the counterfactual account, the rock's putative computational

    implementation is spurious, because the physical state transitions do not support

    counterfactuals. If the rock were put in state T, it may or may not transition into T+1

    depending on whether it is morning or evening and other extraneous factors. Since the rock's

    physical state transitions that map onto the NOT gate's computational state transitions do not

    support counterfactuals, the rock does not implement the NOT gate according to the

    counterfactual account.

    According to the causal and dispositional accounts too, this putative computational

    implementation is spurious, because the physical state transitions are not due to causal or

    dispositional properties of the rock and its states. Tdoes not cause T+1, nor does the rock have

    a disposition to go into T+1 when it is in T. Rather, the rock changes its state due to the action

    of the sun. Since the rock's physical state transitions that map onto the NOT gate's

    computational state transitions are not grounded in either the causal or dispositional

    properties of the rock and its states, the rock does not implement the NOT gate according to

    the causal and dispositional accounts.

    It is important to note that under the present family of accounts, there are mappings between

    any physical system and at least some computational descriptions. Thus, according to the

    present accounts, everything performs at least some computations (cf. Section 3.2). This still

    strikes some as overly inclusive. In computer science and cognitive science, there seems to be

    a distinction between systems that compute and systems that do not. To account for this

    distinction, one option is to retain the current account of computational implementation while

    restricting the class of descriptions that count as computational descriptions. Another option is

    to move beyond this account of implementation.

    2.3 The Semantic Account

    In our everyday life, we usually employ computations to process meaningful symbols, in orderto extract information from them. The semantic account of computation turns this practice

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    into a metaphysical doctrine: computation is the processing of representations or at least,

    the processing of appropriate representations in appropriate ways. Opinions as to which

    representational manipulations constitute computations vary a great deal (Fodor 1975,

    Cummins 1983, Pylyshyn 1984, Churchland and Sejnowski 1992, Shagrir 2006). What all

    versions of the semantic account have in common is that they take seriously the reference tosymbols in Putnam's original account of computation: there is no computation without

    representation (Fodor 1981, 180).

    The semantic account may be seen as imposing a further restriction on acceptable mappings.

    In addition to the causal restriction imposed by the causal account (mutatis mutandis for the

    counterfactual and dispositional accounts), the semantic account imposes a semantic

    restriction. Only physical states that qualify as representations may be mapped onto

    computational descriptions, thereby qualifying as computational states. If a state is not

    representational, it is not computational either.

    The semantic account is probably the most popular in the philosophy of mind, because it

    appears to fit its specific needs better than other accounts. Since minds and digital computers

    are generally assumed to manipulate (the right kind of) representations, they turn out to

    compute. Since most other systems are generally assumed notto manipulate (the relevant

    kind of) representations, they do not compute. Thus, the semantic account appears to

    accommodate some common intuitions about what does and does not count as a computing

    system. It keeps minds and computers in while leaving most everything else out, thereby

    vindicating the computational theory of cognition as a strong and nontrivial theory.

    The semantic account raises three important questions: how representations are to be

    individuated, what counts as a representation of the relevant kind, and what gives

    representations their semantic content.

    On the individuation of computational states, the main debate divides internalists from

    externalists. According to externalists, computational vehicles are symbols individuated by

    their wide cognitive contents paradigmatically, the things that the symbols stand for (Burge

    1986, Shapiro 1997, Shagrir 2001). By contrast, most internalists maintain that computational

    vehicles are symbols individuated by narrow cognitive contents (Segal 1991). Narrow contents

    are, roughly speaking, semantic contents defined in terms of intrinsic properties of the

    system. Cognitive contents, in turn, are contents ascribed to a system by a cognitive

    psychological theory. For instance, the cognitive contents of the visual system are visual

    contents, whereas the cognitive contents of the auditory system are auditory contents.

    To illustrate the dispute, consider two physically identical cognitive systemsA and B. Among

    the symbols processed byAis symbol S.A produces instances ofS wheneverA is in front of

    bodies of water, whenA is thinking of water, and whenAis forming plans to interact with

    water. In short, symbol S appears to stand for water. Every timeA processes S,

    system Bprocesses symbol S, which is physically identical to S. But system B lives in an

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    environment different fromA's environment. WheneverA is surrounded by water, B is

    surrounded by twater. Twater is a substance superficially indistinguishable from water but in

    fact physically different from it. Thus, symbol S appears to stand for twater (cf. Putnam

    1975b). So, we are assuming thatA and B live in relevantly different environments, such

    that S appears to stand for water while S

    appears to stand for twater. We are also assumingthatA is processing S in the same way that B is processing S. There is no intrinsic physical

    difference betweenA and B.

    According to externalists, whenA is processing S and B is processing S they are in

    computational states ofdifferenttypes. According to internalists,A and B are in computational

    states of the same type. In other words, externalists maintain that computational states are

    individuated in part by their reference, which is determined at least in part independently of

    the intrinsic physical properties of cognitive systems. By contrast, internalists maintain that

    computational states are individuated in a way that supervenes solely on the intrinsic physicalproperties of cognitive systems.

    So far, externalists and internalists agree on one thing: computational states are individuated

    by cognitive contents. This assumption can be resisted without abandoning the semantic

    account of computation. According to Egan (1999), computational vehicles are not

    individuated by cognitive contents of any kind, whether wide or narrow. Rather, they are

    individuated by their mathematicalcontents that is, mathematical functions and objects

    ascribed as semantic contents to the computational vehicles by a computational theory of the

    system. Since mathematical contents are the same across physical duplicates, Egan maintains

    that her mathematical contents are a kind of narrow content she is a kind of internalist.

    Let us now turn to what counts as a representation. This debate is less clearly delineated.

    According to some authors, only structures that have a language-like combinatorial syntax,

    which supports a compositional semantics, count as computational vehicles, and only

    manipulations that respect the semantic properties of such structures count as computations

    (Fodor 1975, Pylyshyn 1984). This suggestion flies in the face of computability theory, which

    imposes no such requirement on what counts as a computational vehicle. Other authors are

    more inclusive on what representational manipulations count as computations, but they have

    not been especially successful in drawing the line between computational and non-

    computational processes. Few people would include all manipulations of representations

    including, say, painting a picture and recording a speech as computations, but there is no

    consensus on where to draw the boundary between representational manipulations that

    count as computations and representational manipulations that do not.

    A third question is what gives representations their semantic content. There are three families

    of views. Instrumentalists believe that ascribing semantic content to things is just heuristically

    useful for prediction and explanation; semantic properties are not real properties of

    computational states (e.g., Dennett 1987, Egan forthcoming). Realists who are not naturalists

    believe semantic properties are real properties of computational states, but they are

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    irreducible to non-semantic properties. Finally, realists who are also naturalists believe

    semantic properties are both real and reducible to non-semantic properties, though they

    disagree on exactly how to reduce them (e.g., Fodor 2008, Harman 1987).

    The semantic account of computation is closely related to the common view that computationis information processing. This idea is less clear than it may seem, because there are several

    notions of information. The connection between information processing and computation is

    different depending on which notion of information is at stake. What follows is a brief

    disambiguation of the view that computation is information processing based on four

    important notions of information (cf. Piccinini and Scarantino forthcoming).

    1. Information in the sense of thermodynamics is closely related to thermodynamicentropy. Entropy is a property of every physical system. Thermodynamic entropy is,

    roughly, a measure of an observer's uncertainty about the microscopic state of a

    system after she considers the observable macroscopic properties of the system. The

    study of the thermodynamics of computation is a lively field with many implications in

    the foundations of physics (Leff and Rex 2003). In this thermodynamic sense of

    information, any difference between two distinguishable states of a system may be

    said to carry information. Computation may well be said to be information processing

    in this sense, but this has little to do with semantics properly so called. However, the

    connections between thermodynamics, computation, and information theory are one

    possible source of inspiration for the view that every physical system is a computing

    system (see Section 3.4).

    2. Information in the sense of communication theory is a measure of the averagelikelihood that a given message is transmitted between a source and a receiver

    (Shannon and Weaver 1949). This has little to do with semantics, too.

    3. Information in one semantic sense is approximately the same as natural meaning(Grice 1957). A signal carries information in this sense just in case it reliably correlates

    with a source (Dretske 1981). The view that computation is information processing in

    this sense is prima facie implausible, because many computations such as

    arithmetical calculations carried out on digital computers do not seem to carry any

    natural meaning. Nevertheless, this notion of semantic information is relevant here

    because it has been used by some theorists to ground an account of representation

    (Dretske 1981, Fodor 2008).

    4. Information in another semantic sense is just ordinary semantic content or non -natural meaning (Grice 1957). This is the kind of semantic content that most

    philosophers discuss. The view that computation is information processing in this

    sense is similar to a generic semantic account of computation.

    Although the semantic account of computation appears to fit the needs of philosophers of

    mind, it appears less suited to make sense of other sciences. Most pertinently, representation

    does not seem to be presupposed by the notion of computation employed in at least some

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    areas of cognitive science as well as computability theory and computer science the very

    sciences that gave rise to the notion of computation at the origin of the computational theory

    of cognition (Piccinini 2008a, Fresco 2010). If this is correct, the semantic account may not

    even be adequate to the needs of philosophers of mind at least those philosophers of mind

    who wish to make sense of the analogy between minds and the systems designed and studiedby computer scientists and computability theorists. Another criticism of the semantic account

    is that specifying the kind of representation and representational manipulation that is relevant

    to computation may require a non-semantic way of individuating computations (Piccinini

    2004). These concerns motivate efforts to account for computation in non-semantic terms.

    2.4 The Syntactic Account

    As we saw, the semantic account needs to specify which representations are relevant to

    computation. One view is that the relevant representations are language-like, that is, they

    have the kind of syntactic structure exhibited by sentences in a language. Computation, then,

    is the manipulation of language-like representations in a way that is sensitive to their syntactic

    structure and preserves their semantic properties (Fodor 1975).

    As discussed in the previous section, however, using the notion of representation in an

    account of computation involves some difficulties. If computation could be accounted for

    without appealing to representation, those difficulties would be avoided. One way to do so is

    to maintain that computation simply is the manipulation of language-like structures in

    accordance with their syntactic properties, leaving semantics by the wayside. The structures

    being manipulated are assumed to be language-like only in that they have syntactic properties

    they need not have any semantics. In this syntactic account of computation, the notion of

    representation is not used at all.

    The syntactic account may be seen as adding a restriction on acceptable mappings that

    replaces the semantic restriction proposed by the semantic account. Instead of a semantic

    restriction, the syntactic account imposes a syntactic restriction: only physical states that

    qualify as syntactic may be mapped onto computational descriptions, thereby qualifying as

    computational states. If a state lacks syntactic structure, it is not computational.

    What remains to be seen is what counts as a syntactic state. An important account of syntax in

    the physical world is due to Stephen Stich (1983, 150157). Although Stich does not use the

    term computation, his account of syntax is aimed at grounding a syntactic account of m ental

    states and processes. Stich's syntactic theory of mind is, in turn, his interpretation of the

    computational theories proposed by cognitive scientists in competition with Fodor's

    semantic interpretation. Since Stich's account of syntax is ultimately aimed at grounding

    computational theories of cognition, Stich's account of syntax also provides an (implicit)

    syntactic account of computation.

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    According to Stich, roughly speaking, a physical system contains syntactically structured

    objects when two conditions are satisfied. First, there is a mapping between the behaviorally

    relevant physical states of the system and a class of syntactic types, which are specified by a

    grammar that defines how complex syntactic types can be formed out of (finitely many)

    primitive syntactic types. Second, the behavior of the system is explained by a theory whosegeneralizations are formulated in terms of formal relations between the syntactic types that

    map onto the physical states of the system.

    The syntactic account of computation is not very popular. A common objection is that it seems

    difficult to give an account of primitive syntactic types that does not presuppose a prior

    semantic individuation of the types (Crane 1990, Jacquette 1991, Bontly 1998). In fact, it is

    common to make sense of syntax by construing it as a way to combine symbols, that is,

    semantically interpreted constituents. If syntax is construed in this way, it presupposes

    semantics. And if so, the syntactic account of computation collapses into the semanticaccount.

    Another objection is that language-like syntactic structure is not necessary for computation as

    it is understood in computer science and computability theory. Although computing systems

    surely can manipulate linguistic structures, they don't have to. They can also manipulate

    simple sequences of letters, without losing their identity as computers. (Computability

    theorists call any set of words from a finite alphabet a language, but that broad notion of

    language should not be confused with the narrower notion inspired by grammars in logic

    and linguistics that Stich employs in his syntactic account of computation.)

    2.5 The Mechanistic Account

    The mechanistic account (Piccinini 2007b, Piccinini and Scarantino forthcoming, Section 3)

    avoids appealing to both syntax and semantics. Instead, it accounts for concrete computation

    in terms of the mechanistic properties of a system. According to the mechanistic account,

    concrete computing systems are functional mechanisms of a special kind mechanisms that

    perform concrete computations.

    A functional mechanism is a system of organized components, each of which has functions to

    perform (cf. Craver 2007, Wimsatt 2002). When appropriate components and their functions

    are appropriately organized and functioning properly, their combined activities constitute the

    capacities of the mechanism. Conversely, when we look for an explanation of the capacities of

    a mechanism, we decompose the mechanism into its components and look for their functions

    and organization. The result is a mechanistic explanation of the mechanism's capacities.

    This notion of mechanism is familiar to biologists and engineers. For example, biologists

    explain physiological capacities (digestion, respiration, etc.) in terms of the functions

    performed by systems of organized components (the digestive system, the respiratory system,etc.).

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    According to the mechanistic account, a computation in the generic sense is the processing of

    vehicles according to rules that are sensitive to certain vehicle properties, and specifically, to

    differences between different portions of the vehicles. The processing is performed by a

    functional mechanism, that is, a mechanism whose components are functionally organized to

    perform the computation. Thus, if the mechanism malfunctions, a miscomputation occurs.

    Digital computation, analog computation, etc. turn out to be species of generic computation.

    They are differentiated by more specific properties of the vehicles being processed. If a

    computing system processes strings of discrete states, then it performs a digital computation.

    If a computing system processes continuous variables, then it performs an analog

    computation. If a computing system processes qubits, then it performs a quantum

    computation.

    When we define concrete computations and the vehicles that they manipulate, we need not

    consider all of their specific physical properties. We may consider only the properties that are

    relevant to the computation, according to the rules that define the computation. A physical

    system can be described more or less abstractly. According to the mechanistic account, an

    abstract description of a physical system is not a description of an abstract object but rather a

    description of a concrete system that omits certain details. Descriptions of concrete

    computations and their vehicles are sufficiently abstract as to be defined independently of the

    physical media that implement them in particular cases. Because of this, the mechanistic

    account calls concrete computations and their vehicles medium-independent.

    In other words, a vehicle is medium-independent just in case the rules (i.e., the input-output

    maps) that define a computation are sensitive only to differences between portions of the

    vehicles along specific dimensions of variation they are insensitive to any more concrete

    physical properties of the vehicles. Put yet another way, the rules are functions of state

    variables associated with a set of functionally relevant degrees of freedom, which can be

    implemented differently in different physical media. Thus, a given computation can be

    implemented in multiple physical media (e.g., mechanical, electro-mechanical, electronic,

    magnetic, etc.), provided that the media possess a sufficient number of dimensions of

    variation (or degrees of freedom) that can be appropriately accessed and manipulated and

    that the components of the mechanism are functionally organized in the appropriate way.

    Notice that the mechanistic account avoids pancomputationalism. First, physical systems that

    are not functional mechanisms are ruled out. Functional mechanisms are complex systems of

    components that are organized to perform functions. Any system whose components are not

    organized to perform functions is not a computing system because it is not a functional

    mechanism. Second, mechanisms that lack the function of manipulating medium-independent

    vehicles are ruled out. Finally, medium-independent vehicle manipulators whose

    manipulations fail to accord with appropriate rules are ruled out. The second and third

    constraints appeal to special functional properties manipulating medium-independent

    vehicles, doing so in accordance with rules defined over the vehicles that are possessed only

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    by relatively few physical systems. According to the mechanistic account, those few systems

    are the genuine computing systems.

    Another feature of the mechanistic account is that it accounts for the possibility of

    miscomputation a possibility difficult to make sense of under other accounts. To illustratethe point, consider an ordinary computer programmed to compute functionfon input i.

    Suppose that the computer malfunctions and produces an output different fromf(i). According

    to the causal (semantic) account, the computer just underwent a causal process (a

    manipulation of representations), which may be given a computational description and hence

    counts as computing some function g(i), where gf. By contrast, according to the mechanistic

    account, the computer simply failed to compute, or at least it failed to complete its

    computation correctly. Given the importance of avoiding miscomputations in the design and

    use of computers, the ability of the mechanistic account to make sense of miscomputation

    may be an advantage over rival accounts.

    A final feature of the mechanistic account is that it distinguishes and characterizes precisely

    many different kinds of computing systems based on the specific vehicles they manipulate and

    their specific mechanistic properties. The mechanistic account has been used to explicate

    digital computation (Piccinini 2007b), analog computation (Piccinini 2008b, Section 3.5),

    computation by neural networks (Piccinini 2008c), and other important distinctions such as

    hardwired vs. programmable and serial vs. parallel computation (Piccinini 2008b).

    3. Is Every Physical System Computational?

    Which physical systems perform computations? According to pancomputationalism, they all

    do. Even rocks, hurricanes, and planetary systems contrary to appearances are

    computing systems. Pancomputationalism is quite popular among some philosophers and

    physicists.

    3.1 Varieties of Pancomputationalism

    Varieties of pancomputationalism vary with respect to how manycomputations all, many, a

    few, or just one they attribute to each system.

    The strongest version of pancomputationalism is that every physical system

    performs everycomputation or at least, every sufficiently complex system implements a

    large number of non-equivalent computations (Putnam 1988, Searle 1992). This may be

    called unlimited pancomputationalism.

    The weakest version of pancomputationalism is that every physical system performs some (as

    opposed to every) computation. A slightly stronger version maintains that everything

    performs a fewcomputations, some of which encode the others in some relatively

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    unproblematic way (Scheutz 2001). These versions may be called limited

    pancomputationalism.

    Varieties of pancomputationalism also vary with respect to whyeverything performs

    computations the source of pancomputationalism.

    One alleged source of pancomputationalism is that which computation a system performs is a

    matter of relatively free interpretation. If whether a system performs a given computation

    depends solely or primarily on how the system is perceived, as opposed to objective fact, then

    it seems that everything computes because everything may be seen as computing (Searle

    1992). This may be called interpretivist pancomputationalism.

    Another alleged source of pancomputationalism is that everything has causal structure.

    According to the causal account, computation is the causal structure of physical processes

    (Chrisley 1995, Chalmers 1995, 1996, Scheutz 1999, 2001). Assuming that everything has

    causal structure, it follows that everything performs the computation constituted by its causal

    structure. This may be called causal pancomputationalism.

    Not everyone will agree that everything has causal structure. Some processes may be non-

    causal, or causation may be just a faon de parler that does not capture anything fundamental

    about the world (e.g., Norton 2003). But those who have qualms about causation can recover

    a view similar to causal pancomputationalism by reformulating the causal account of

    computation and consequent version of pancomputationalism in terms they like e.g., in

    terms of the dynamical properties of physical systems.

    A third alleged source of pancomputationalism is that every physical state carries information,

    in combination with an information-based semantics plus a liberal version of the semantic

    view of computation. According to the semantic view of computation, computation is the

    manipulation of representations. According to information-based semantics, a representation

    is anything that carries information. Assuming that every physical state carries information, it

    follows that every physical system performs the computations constituted by the manipulation

    of its information-carrying states (cf. Shagrir 2006). Both information-based semantics and the

    assumption that every physical state carries information (in the relevant sense) remain

    controversial.

    Yet another alleged source of pancomputationalism is that computation is the nature of the

    physical universe. According to some physicists, the physical world is computational at its most

    fundamental level. This view, which is a special version of limited pancomputationalism, will be

    discussed in Section 3.4.

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    3.2 Unlimited Pancomputationalism

    Arguments for unlimited pancomputationalism go back to Hinckfuss's pail, a putative

    counterexample to computational functionalism the view that the mind is the software of

    the brain. Hinckfuss's pail is named after its proponent, Ian Hinckfuss, but was first discussed inprint by William Lycan. A pail of water contains a huge number of microscopic processes:

    Now is all this activity not complex enough that, simply by chance, it might realize a human

    program for a brief period (given suitable correlations between certain micro-events and the

    requisite input-, output-, and state-symbols of the program)? (Lycan 1981, 39)

    Hinckfuss's implied answer to this question is that yes, a pail of water might implement a

    human program, and therefore any arbitrary computation, at least for a short time.

    Other authors developed more detailed arguments along the lines of Hinckfuss's pail. John

    Searle (1992) explicitly argues that whether a physical system implements a computation

    depends on how an observer interprets the system; therefore, for any sufficiently complex

    object and for any computation, the object can be described as implementing the

    computation. The first rigorous argument for unlimited pancomputationalism is due to Hilary

    Putnam (1988), who argues that every ordinary open system implements every abstract finite

    automaton (without inputs and outputs).

    Putnam assumes that electromagnetic and gravitational fields are continuous and that physical

    systems are in different maximal states at different times. He considers an arbitrarily chosen

    finite automaton whose table calls for the sequence of states ABABABA. He then considers an

    arbitrary physical system S over the arbitrarily chosen time interval from 12:00 to 12:07 and

    argues that S implements the sequenceABABABA. Since both the automaton and the physical

    system are arbitrary, the argument generalizes to any automaton and any physical state. Here

    is the core of Putnam's argument:

    Let the beginnings of the intervals during which S is to be in one of its stagesA or B be t1, t2,

    tn (in the example given, n = 7, and the times in question are t1 = 12:00, t2 = 12:01, t3 =

    12:02, t4 = 12:03, t5 = 12:04, t6 = 12:05, t7= 12:06). The end of the real-time interval duringwhich we wish Sto obey this table we calltn+1 (= t8 = 12:07, in our example). For each of the

    intervals tito ti+1, i= 1, 2, ,n, define a (nonmaximal) interval statesiwhich is the region in

    phase space consisting of all the maximal states with tit< t+1. (I.e., S is in si just in case S is

    in one of the maximal states in this region.) Note that the systemS is in s1 from t1 to t2,

    in s2 from t2 to t3, , insnfrom tn to tn+1. (Left endpoint included in all cases but not the right

    this is a convention to ensure the machine is in exactly one of thesiat a given time.)

    DefineA = s1s3s5s7; B = s2s4s6.

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    Then, as is easily checked, S is in stateA from t1 to t2, from t3 to t4, and from t5 to t6, and

    from t7 to t8, and in state B at all other times between t1 and t8. So Shas the table we

    specified, with the statesA,Bwe just defined as the realizations of the statesA,B described

    by the table. (Putnam 1988, 1223, emphasis original)

    In summary, Putnam picks an arbitrary physical system with continuous dynamics, slices up its

    dynamics into discrete time intervals, and then aggregates the slices so that they correspond

    to an arbitrary sequence of computational states. He concludes that every physical system

    implements every finite automaton.

    Putnam points out that his argument does not apply directly to computational theories of

    cognition, because cognitive systems receive specific physical inputs through their sensory

    organs and yield specific physical outputs through their motor organs. To determine which

    computations are implemented by a system with physical inputs and outputs, the inputs and

    outputs must be taken into account:

    Imagine that an objectSwhich takes strings of 1s as inputs and prints such strings as

    outputs behaves from 12:00 to 12:07 exactly as ifit had a certain [computational]

    description D. That is, Sreceives a certain string, say 111111, at 12:00 and prints a certain

    string, say 11, at 12:07, and there exists (mathematically speaking) a machine with

    description D which does this (by being in the appropriate state at each of the specified

    intervals, say 12:00 to 12:01, 12:01 to 12:02, , and printing or erasing what it is supposed to

    print or erase when it is in a given state and scanning a given symbol). In this case, S too can

    be interpretedas being in these same logical statesA,B,C, at the very same times and

    following the very same transition rules; that is to say, we can findphysicalstatesA,B,C

    which S possesses at the appropriate times and which stand in the appropriate causal relations

    to one another and to the inputs and the outputs. The method of proof is exactly the same

    Thus we obtain that the assumption that something is a realization of a given automaton

    description is equivalent to the statement that it behaves as if it had that

    description (Putnam 1988, 124, emphasis original).

    In summary, Putnam picks an arbitrary physical system with physically specified inputs and

    outputs and then matches it to an arbitrary finite automaton whose abstractly specified inputs

    and outputs map onto the physically specified inputs and outputs. He then slices up the

    physical system's internal dynamics as before, and then aggregates the slices so that they

    correspond to the sequence of computational states of the finite automaton. It follows that

    given any physical system and any finite automaton with isomorphic inputs and outputs, the

    physical system implements the computational system.

    Although this result is weaker than the result for systems without inputs and outputs, it is still

    striking because for any abstract input-output pair , there are infinitely many automata

    that yield output o given input i. Given Putnam's conclusion, any physical system with inputs

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    and outputs isomorphic to iand o implements all of the infinitely many automata with

    input iand output o.

    If unlimited pancomputationalism is correct, then the claim that a system S performs a certain

    computation becomes trivially true and vacuous or nearly so; it fails to distinguish S fromanything else (or perhaps from anything else with the same inputs and outputs). Thus,

    unlimited pancomputationalism threatens the computational theory of cognition. If cognition

    is computation simply because cognitive systems, like everything else, may be seen as

    performing computations, then it appears that the computational theory of cognition is both

    trivial and vacuous. By the same token, unlimited pancomputationalism threatens the

    foundations of computer science, where the objective computational power of different

    systems is paramount. The threat of trivialization is a major motivation behind responses to

    the arguments for unlimited pancomputationalism.

    The first thing to notice is that arguments for unlimited pancomputationalism rely either

    implicitly or explicitly on the simple mapping account of computation. They assume that an

    arbitrary mapping from a computational description Cto a physical description of a system is

    sufficient to conclude that the system implements C. In fact, avoiding unlimited

    pancomputationalism is a major motivation for rejecting the simple mapping account of

    computation. By imposing restrictions on which mappings are legitimate, other accounts of

    computation aim to avoid unlimited pancomputationalism.

    In one response to unlimited pancomputationalism, Jack Copeland (1996) argues that the

    mappings it relies on are illegitimate because they are constructed ex post facto after the

    computation is already given. In the case of kosher computational descriptions the kind

    normally used in scientific modeling the work of generating successive descriptions of a

    system's physical dynamics is done by a computer running an appropriate program (e.g., a

    weather forecasting program), not by the mapping relation. In the sort of descriptions

    employed in arguments for unlimited pancomputationalism, instead, the descriptive work is

    done by the mapping relation.

    An arbitrarily chosen computational description, such as those employed in arguments for

    unlimited pancomputationalism, does not generate successive descriptions of the state of an

    arbitrary system. If someone wants a genuine computational description of a physical system,

    she must first identify physical states and state transitions of the system, then represent them

    by a computational description (thereby fixing the mapping relation between the

    computational description and the system), and finally use a computer to generate subsequent

    representations of the state of the system, while the mapping relation stays fixed. By contrast,

    the arguments for unlimited pancomputationalism pick a computation first, then slice and

    aggregate the physical system to fit the computational description, and finally generate the

    mapping between the two. The work of describing the physical system is not done by the

    computational description but by whoever constructs the mapping. Copeland concludes that

    such ex post facto mappings are illegitimate.

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    In addition, both Chalmers (1995, 1996) and Copeland (1996) argue that the mappings invoked

    by unlimited pancomputationalism violate the counterfactual relations between the

    computational states. Consider again Putnam's slice-and-aggregate strategy for generating

    mappings. The mappings are constructed based on an arbitrary dynamical evolution of an

    arbitrary physical system. No attempt is made to establish what would happen to the physicalsystem had conditions been different. Chalmers and Copeland argue that this is illegitimate, as

    a genuine implementation must exhibit the same counterfactual relations that obtain between

    the computational states. This response leads to the counterfactual account of computation,

    according to which the counterfactual relations between the physical states must be

    isomorphic to the counterfactual relations between the computational states.

    Another possible response to unlimited pancomputationalism is that its mappings fail to

    construct an isomorphism between the causal structure of the physical system and the state

    transitions specified by the computational description. Consider Putnam's argument again. Themapping from the computational description to the physical description is chosen with no

    regard to the causal relations that obtain between the physical states of the system. Thus,

    after a computational description is mapped onto a physical description in that way, the

    computational description does not describe the causal structure of the physical system.

    According to several authors, non-causal mappings are illegitimate (Chrisley 1995, Chalmers

    1995, 1996, Scheutz 1999, 2001). Naturally, these authors defend the causal account of

    computation, according to which acceptable mappings must respect the causal structure of a

    system.

    Yet another response to unlimited pancomputationalism is implicitly given by Godfrey-Smith

    (2009). Although Godfrey-Smith is primarily concerned with functionalism as opposed to

    computation per se, his argument is still relevant here. Godfrey-Smith argues that for a

    mapping to constitute a genuine implementation, the microscopic physical states that are

    clustered together (to correspond to a given computational state) must be physically similarto

    one another there cannot be arbitrary groupings of arbitrarily different physical states, as in

    the arguments for unlimited pancomputationalism. Godfrey-Smith suggests that his similarity

    restriction on legitimate mappings may be complemented by the kind of causal and

    localizational restrictions proposed by Chalmers (1996).

    The remaining accounts of computation the semantic, syntactic, and mechanistic accounts

    are even more restrictive than the causal and counterfactual accounts; they impose further

    constraints on acceptable mappings. Therefore, like the causal and counterfactual accounts,

    they have resources for avoiding unlimited pancomputationalism.

    Such resources are not always straightforward to deploy. For example, consider the semantic

    account, according to which computation requires representation. If being a representation of

    something is an objective property possessed by relatively few things, then unlimited

    pancomputationalism is ruled out on the grounds that only the few items that constitute

    representations are genuine computational states. If, however, everything is representational

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    in the relevant way, then everything is computational (cf. Churchland and Sejnowski 1992,

    Shagrir 2006). If, in addition, whether something represents something else is just a matter of

    free interpretation, then the semantic account of computation gives rise to unlimited

    pancomputationalism all over again. Similar considerations apply to the syntactic and

    mechanistic accounts. For such accounts to truly avoid unlimited pancomputationalism, theymust not rely on free interpretation.

    3.3 Limited Pancomputationalism

    Limited pancomputationalism is much weaker than its unlimited cousin. It holds that every

    physical system performs one (or relatively few) computations. Which computations are

    performed by which system is deemed to be a matter of fact, depending on objective

    properties of the system. In fact, several authors who have mounted detailed responses to

    unlimited pancomputationalism explicitly endorse limited pancomputationalism (Chalmers

    1996, 331, Scheutz 1999, 191).

    Unlike unlimited pancomputationalism, limited pancomputationalism does not turn the claim

    that something is computational into a vacuous claim. Different systems generally have

    different objective properties; thus, according to limited pancomputationalism, different

    systems generally perform different computations. Nevertheless, it may seem that limited

    pancomputationalism still trivializes the claim that a system is computational. For according to

    limited pancomputationalism, digital computers perform computations in the same sense in

    which rocks, hurricanes, and planetary systems do. This may seem to do an injustice to

    computer science in computer science, only relatively few systems count as performing

    computations and it takes a lot of difficult technical work to design and build systems that

    perform computations reliably. Or consider the claim that cognition is computation. This

    computational theory of cognition was introduced to shed new and explanatory light on

    cognition. But if every physical process is a computation, the computational theory of

    cognition seems to lose much of its explanatory force (Piccinini 2007b).

    Another objection to limited pancomputationalism begins with the observation that any

    moderately complex system satisfies indefinitely many objective computational descriptions

    (Piccinini 2010). This may be seen by considering computational modeling. A computational

    model of a system may be pitched at different levels of granularity. For example, consider

    cellular automata models of the dynamics of a galaxy or a brain. The dynamics of a galaxy or a

    brain may be described using an indefinite number of cellular automata using different

    state transition rules, different time steps, or cells that represent spatial regions of different

    sizes. Furthermore, an indefinite number of formalisms different from cellular automata, such

    as Turing machines, can be used to compute the same functions computed by cellular

    automata. It appears that limited pancomputationalists are committed to the galaxy or the

    brain performing all these computations at once. But that does not appear to be the sense in

    which computers (or brains) perform computations.

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    In the face of these objections, limited pancomputationalists are likely to maintain that the

    explanatory force of computational explanations does not come from the claim that a system

    is computational simpliciter. Rather, explanatory force comes from the specific computations

    that a system is said to perform. Thus, a rock and a digital computer perform computations in

    the same sense. But they perform radically different computations, and it is the differencebetween their computations that explains the difference between them. As to the objection

    that there are still too many computations performed by each system, limited

    pancomputationalists have two main options: either to bite the bullet and accept that every

    system implements indefinitely many computations, or to find a way to single out, among the

    many computational descriptions satisfied by each system, the one that is ontologically

    privileged the one that captures the computation performed by the system. One way to do

    this is to postulate a fundamental physical level, whose most accurate computational

    description identifies the (most fundamental) computation performed by the system. This

    response is built into the view that the physical world is fundamentally computational (nextsection).

    As to those who remain unsatisfied with limited pancomputationalism, their desire to avoid

    limited pancomputationalism motivates the shift to more restrictive accounts of computation,

    analogously to how the desire to avoid unlimited pancomputationalism motivates the shift

    from the simple mapping account to more restrictive accounts of computation, such as the

    causal account. The semantic account may be able to restrict genuine computational

    descriptions to fewer systems than the causal account, provided that representations which

    are needed for computation according to the semantic account are hard to come by.Mutatis mutandis, the same is true of the syntactic and mechanistic accounts.

    3.4 The Universe as a Computing System

    Some authors argue that the physical universe is fundamentally computational. The universe

    itself is a computing system, and everything in it is a computing system too (or part thereof).

    Unlike the previous versions of pancomputationalism, which originate in philosophy, this ontic

    pancomputationalism originates in physics. It includes both an empirical claim and a

    metaphysical one. Although the two claims are logically independent, supporters of ontic

    pancomputationalism tend to make them both.

    The empirical claim is that all fundamental physical magnitudes and their state transitions are

    such as to be exactly described by an appropriate computational formalism without

    resorting to the approximations that are a staple of standard computational modeling. This

    claim takes different forms depending on which computational formalism is taken to describe

    the universe exactly. The two main options are cellular automata, which are a classical

    computational formalism, and quantum computing, which is non-classical.

    The earliest and best known version of ontic pancomputationalism is due to Konrad Zuse

    (1970, 1982) and Edward Fredkin, whose unpublished ideas on the subject influenced a

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    number of American physicists (e.g., Feynman 1982, Toffoli 1982, Wolfram 2002; see also

    Wheeler 1982, Fredkin 1990). According to some of these physicists, the universe is a giant

    cellular automaton. A cellular automaton is a lattice of cells; each cell can take one out of

    finitely many states and updates its state in discrete steps depending on the state of its

    neighboring cells. For the universe to be a cellular automaton, all fundamental physicalmagnitudes must be discrete, i.e., they must take at most finitely many values. In addition,

    time and space must be fundamentally discrete or must emerge from the discrete processing

    of the cellular automaton. At a fundamental level, continuity is not a real feature of the world

    there are no truly real-valued physical quantities. This flies in the face of most mainstream

    physics, but it is not an obviously false hypothesis. The hypothesis is that at a sufficiently small

    scale, which is currently beyond our observational and experimental reach, (apparent)

    continuity gives way to discreteness. Thus, all values of all fundamental variables, and all state

    transitions, can be fully and exactly captured by the states and state transitions of a cellular

    automaton.

    Although cellular automata have been shown to describe many aspects of fundamental

    physics, it is difficult to see how to simulate the quantum mechanical features of the universe

    using a classical formalism such as cellular automata (Feynman 1982). This concern motivated

    the development of quantum computing formalisms (Deutsch 1985, Nielsen and Chuang

    2000). Instead of relying on digits most commonly, binary digits or bits quantum

    computation relies on qudits most commonly, binary qudits or qubits. The main difference

    between a digit and a qudit is that whereas a digit can take only one out of finitely many

    states, such as 0 and 1 (in the case of a bit), a qudit can also take an uncountable number ofstates that are a superposition of the basis states in varying degrees, such as superpositions of

    0 and 1 (in the case of a qubit). Furthermore, unlike a collection of digits, a collection of qudits

    can exhibit quantum entanglement. According to the quantum version of ontic

    pancomputationalism, the universe is not a classical computer but a quantum computer, that

    is, not a computer that manipulates digits but a computer that manipulates qubits (Lloyd 2006)

    or, more generally, qudits.

    The quantum version of ontic pancomputationalism is less radical than the classical version.

    The classical version eliminates continuity from the universe, primarily on the grounds that

    eliminating continuity allows classical computers to describe the universe exactly rather than

    approximately. Thus, the classical version appears to be motivated not by empirical evidence

    but by epistemological concerns. Although there is no direct evidence for classical ontic

    pancomputationalism, in principle it is a testable hypothesis (Fredkin 1990). By contrast,

    quantum ontic pancomputationalism may be seen as a reformulation of quantum mechanics in

    the language of quantum computation and quantum information theory (qubits), without

    changes in the empirical content of the theory (e.g., Fuchs 2004, Bub 2005).

    But ontic pancomputationalists do not limit themselves to making empirical claims. They often

    make an additional metaphysical claim. They claim that computation (or information, in the

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    physical sense described in Section 2.3) is what makes up the physical universe. This point is

    sometimes made by saying that at the most fundamental physical level, there are brute

    differences between states nothing more need or can be said about the nature of the

    states. This view reverses the traditional conception of the relation between computation and

    the physical world.

    According to the traditional conception, which is presupposed by all accounts of computation

    discussed above, physical computation requires a physical substratum that implements it.

    Computation is an aspect of the organization and behavior of a physical system; there is no

    software without hardware. Thus, according to the traditional conception, if the universe is a

    cellular automaton, the ultimate constituents of the universe are the physical cells of the

    cellular automaton. It is legitimate to ask what kind of physical entity such cells are and how

    they interact with one another so as to satisfy their cellular automata rules.

    By contrast, according to the metaphysical claim of ontic pancomputationalism, a physical

    system is just a system of computational states. Computation is ontologically prior to physical

    processes, as it were. Hardware *is+ made of software (Kantor 1982, 526, 534). According

    to this non-traditional conception, if the universe is a cellular automaton, the cells of the

    automaton are not concrete, physical structures that causally interact with one another.

    Rather, they are software purely computational entities.

    Such a metaphysical claim requires an account of what computation, or software, or physical

    information, is. If computations are not configurations of physical entities, the most obvious

    alternative is that computations are abstract, mathematical entities, like numbers and sets. As

    Wheeler (1982, 570) puts it, the building element *of the universe+ is the elementary yes, no

    quantum phenomenon. It is an abstract entity. It is not localized in space and time. Under this

    account of computation, the ontological claim of ontic pancomputationalism is a version of

    Pythagoreanism. All is computation in the same sense in which more traditional versions of

    Pythagoreanism maintain that all is number or that all is sets (Quine 1976).

    Ontic pancomputationalism may be attacked on both the empirical and the ontological fronts.

    On the empirical front, there is little positive evidence to support ontic pancomputationalism.

    Supporters appear to be motivated by the desire for exact computational models of the world

    rather than empirical evidence that the models are correct. Even someone who shares this

    desire may well question why we should expect nature to fulfill it. On the metaphysical front,

    Pythagoreanism faces the objection that the abstract entities it puts at the fundamental

    physical level lack the causal and qualitative properties that we observe in the physical world

    or at least, it is difficult to understand how abstract entities could give rise to physical

    qualities and their causal powers (e.g., Martin 1997).

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    4. Physical Computability

    According to the Church-Turing thesis (CTT), any function that is intuitively computable is

    computable by some Turing machine (i.e., Turing-computable). Alternatively, CTT may be

    formulated as follows: any function that is naturally regardedas computable (Turing 19367,135) is Turing-computable. The phrases intuitively computable and naturally regarded as

    computable are somewhat ambiguous. When they are disambiguated, CTT takes different

    forms.

    In one sense, intuitively computable means computable by following an algorithm or

    effective procedure. An effective procedure is a finite list of clear instructions for generating

    new symbolic structures out of old symbolic structures. When CTT is interpreted in terms of

    effective procedures, it may be called Mathematical CTT, because the relevant evidence is

    more logical or mathematical than physical. Mathematical CTT says that any

    function computable by an effective procedure is Turing-computable.

    There is compelling evidence that Mathematical CTT is true (Kleene 1952, 62, 67; cf. also

    Sieg 2006):

    There are no known counterexamples. Diagonalization over Turing machines, contrary to what may be expected, does not

    yield a function that is not Turing-computable.

    Argument from confluence: all the formalisms proposed to capture the intuitive notionof computability by effective procedure formalisms such as general recursiveness

    (Gdel 1934), -definability (Church 1932, Kleene 1935), Turing-computability (Turing

    1936-7), and reckonability (Gdel 1936) turn out to capture the same class of

    functions.

    A Turing machine seems capable of reproducing any operation that a human being canperform while following an effective procedure (Turing 19367's main argument for

    CTT).

    In another sense, intuitively computable means computable by physical means. When CTT is

    so interpreted, it may be called Physical CTT(following Pitowsky 1990), because the relevant

    evidence is more physical than logical or mathematical.

    4.1 The Physical Church-Turing Thesis: Bold

    Physical CTT is often formulated in very strong forms. To a first approximation, Bold Physical

    CTTholds that any physical process anything doable by a physical system is computable

    by some Turing machine.

    Bold Physical CTT can be made more precise in a number of ways. Here is a representative

    sample, followed by references to where they are discussed:

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    A. Any physical process can be simulated by some Turing machine (e.g., Deutsch 1985,Wolfram 1985, Pitowsky 2002).

    B. Any function over denumerable domains (such as natural numbers) that is computableby an idealized computing machine that manipulates arbitrary real-valued quantities

    (as defined by Blum et al. 1998) is Turing-computable.C. Any system of equations describing a physical system gives rise to computable

    solutions (cf. Earman 1986, Pour-El 1999). A solution is said to be computable just in

    case, given computable real numbers as initial conditions, it returns computable real

    numbers as values. A real number is said to be computable just in case there is a

    Turing machine whose output effectively approximates it.

    D. For any physical system S and observable W, there is a Turing-computablefunctionf: NN such that for all timestN,f(t)=W(t) (Pitowsky 1990).

    Thesis (A) is ambiguous between two notions of simulation. In one sense, simulation is theprocess by which a digital computing system (such as a Turing machine) computes the same

    function as another digital computing system. This is the sense in which universal Turing

    machines can simulate any other Turing machine. If (A) is interpreted using this first notion of

    simulation, it entails that everything in the universe is a digital computing system. This is (a

    variant of) ontic pancomputationalism (Section 3.4).

    In another sense, simulation is the process by which the output of a digital computing system

    represents an approximate description of the dynamical evolution of another system. This is

    the sense in which computational models of the weather simulate the weather. If (A) is

    interpreted using this second notion of simulation, then (A) is true only if we do not care how

    close our computational approximations are. If we want close computational approximations

    as we usually do then (A) turns into the claim that any physical process can be

    computationally approximated to the degree of accuracy that is desired in any given case.

    Whether that is true varies from case to case depending on the dynamical properties of the

    system, how much is known about them, what idealizations and simplifications are adopted in

    the model, what numerical methods are used in the computation, and how many

    computational resources (such as time, processing speed, and memory) are available (Piccinini

    2007b).

    Thesis (B) is straightforwardly and radically false. Blum et al. (1989) set up a mathematical

    theory of computation over real-valued quantities, which they see as a fruitful extension of

    ordinary computability theory. Within such a theory, Blum et al. define idealized computing

    machines that perform addition, subtraction, multiplication, division, and equality testing as

    primitive operations on arbitrary real-valued quantities. They easily prove that such machines

    can compute all sets defined over denumerable domains by encoding their characteristic

    function as a real-valued constant (ibid., 405). Although they do not discuss this result as a

    refutation of Physical CTT, their work is often cited in discussions of physical computability and

    Physical CTT.

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    Theses (C) and (D) have interesting counterexamples that are consistent with some physical

    theories (cf. below and Pour-El 1999). These theoretical counterexamples may or may not

    occur in our concrete physical universe.

    Each of (A)(D) raises important questions pertaining to the foundations of computer science,physics, and mathematics. It is not clear, however, that any of these theses bears an

    interesting analogy to Mathematical CTT. Below are two reasons why.

    First, (A)(D) are falsified by processes that cannot be built and used as computing devices. The

    most obvious example is (B). Blum et al.'s result is equivalent to demonstrating that all

    functions over denumerable domains including the uncountably many functions that are

    not Turing-computable are computable by Blum et al.'s computing systems, which are

    allowed to mani