Classical Computation and the Computational Theory of Mind

Classical Computation and the Computational Theory of Mind

  • Definition of Classical Computation: Classical computation refers to the traditional processing of discrete symbolic representations according to explicit, step-by-step algorithmic rules. This theoretical framework underpins conventional electronic computers and forms a foundation for classical cognitive science.
  • Core Axiom of Classical Cognitive Science: The central hypothesis of cognitive science posits that the human mind is literally a computational system. Understanding cognitive processes requires identifying:
    • The exact computational definitions of mental processes.
    • What it means formally for an entity to be a computer.
    • The precise structures and representations used in mental calculations.
  • Central Questions of Mind-As-Computer Modeling:
    • What defines a computing system?
    • What does it mean to state that cognitive processes are computational?
    • What formal structure must an explanation take to account for mental operations?

Historical Mechanical Computers

  • The Antikythera Mechanism:

    • Origins: An ancient analog device discovered over a century ago in a Mediterranean shipwreck, constructed around 100 BC100\text{ BC}.
    • Historical Reconstruction: Decades of study by Michael Wright, a museum curator from London, revealed its structural composition. Despite popular cultural myths (such as fictional portrayals depicting it as a time machine), it served as an astronomical calculator.
    • Function: Designed to calculate and predict the positions of astronomical bodies known to the ancient Greeks, including the Sun, the Moon, and the five known planets (Mercury, Venus, Mars, Jupiter, Saturn).
    • Mechanical Design: Turned via a side crank, which drove pointer dials on a front face. The fastest pointer tracked the lunar cycle, making one revolution per month. Internally, a setup of differential gear wheels riding on other gear wheels implemented ancient Greek planetary theories.
    • Computational Classification: It functions as a special-purpose, non-programmable computer. Information about the physical state of the world (initial planetary positions) is set manually onto the gears; physical gear revolutions transform this data to output predictions of future planetary positions. It lacks general-purpose programmability.

  • Charles Babbage’s Difference Engine:

    • Development: Designed in the 1800s by Charles Babbage to compute mathematical tables and solve polynomial series of equations using addition and subtraction.
    • Mechanism: Operated via interconnecting gears, cams, rods, levers, and springs. Users set equation coefficients into aligned number wheels. Rotating the main drive handle engaged a helical arrangement of steel fingers that pulled register towers upward to execute automatic tens-carrying operations.
    • Execution: Babbage never completed a full physical build during his lifetime due to engineering limits, but functional models were constructed centuries later according to his exact mechanical drawings.
    • Classification: A fixed-hardware, single-purpose computer capable only of executing specific mathematical algorithms hardwired into its gear ratios.

  • Charles Babbage’s Analytical Engine:

    • Concept: Babbage’s design for the first general-purpose, fully programmable mechanical computer.
    • Programmability: Utilized punched paper cards to encode arbitrary operational programs and mathematical sequences, separating the physical processing mechanism from the program instructions.
    • Scale: Never constructed due to its massive theoretical size and complexity, but provided the conceptual blueprint for modern programmable hardware.

  • Ada Lovelace and the First Computer Program:

    • Historical Contribution: Ada Lovelace, a mathematician and long-time collaborator with Babbage, authored the first documented computer algorithm designed to run on the Analytical Engine (specifically for computing Bernoulli numbers).
    • Response to Scientific Skepticism: British Prime Minister John Russell expressed skepticism, arguing the machine was scientifically useless beyond accelerating rote arithmetic. Lovelace countered that symbolic calculation engine capabilities extend beyond numerical calculations.
    • Lovelace’s Symbolic Insight: She posited that if relationships between non-numerical entities (such as musical notes, pitch, and harmony) can be expressed via abstract mathematical rules, a computing engine could manipulate those symbols to execute non-numerical tasks (such as transposing music or composing complex harmonies).

Rules to Symbols: The Classical Computational Theory of Mind (CCTM)

  • Symbolic Representation and Mapping:

    • A computation requires physical state settings (e.g., gear angles, electronic voltages, neural firing rates) to correspond systematically to real-world objects, abstract concepts, or relationships.
    • Physical manipulation of symbols (operations on matter) tracks abstract relations in the external world.

  • The Classical Computational Theory of Mind (CCTM):

    • Literal Identity: CCTM states that the human mind is a physical computational system, not merely metaphorically analogous to one.
    • Discrete Symbols: Mental representations consist of discrete, countable symbols representing external entities. A discrete symbol (e.g., the digit 22) is categorical; it cannot exist as a continuous blend between states (e.g., a symbol cannot be half-22 and half-33). The physical form of a symbol is arbitrary relative to its referent.
    • Rule-Governed Processing (Algorithms): Cognition consists of executing step-by-step algorithms—a deterministic rulebook that transforms input symbols into output symbols.
    • Mindless Syntax: Algorithmic rules operate purely on the structural or formal properties (syntax) of symbols without requiring intrinsic semantic comprehension of what those symbols represent.

  • Algorithmic Step-by-Step Procedure Example (Dog Counting Task):

    • Task: Determine whether an image contains an even or odd number of dogs using symbolic operations.
    • Defined Symbols: Entities labeled dog, grass, spatial coordinates Pos(n)\text{Pos}(n), internal state registers guess (even\text{even} or odd\text{odd}), and location pointer nn.
    • Formal Algorithm:
    • Step 1 (Initialization): Set state guess =even= \text{even}, pointer n=1n = 1.
    • Rule A: If guess =even= \text{even} and state at Pos(n)=dog\text{Pos}(n) = \text{dog}, update guess →odd\rightarrow \text{odd}, set n→n+1n \rightarrow n + 1.
    • Rule B: If guess =odd= \text{odd} and state at Pos(n)=dog\text{Pos}(n) = \text{dog}, update guess →even\rightarrow \text{even}, set n→n+1n \rightarrow n + 1.
    • Rule C: If state at Pos(n)=grass\text{Pos}(n) = \text{grass}, retain guess, set n→n+1n \rightarrow n + 1.
    • Rule D: If n>MaxPosn > \text{MaxPos}, terminate and output guess.
    • Implication: This procedure yields correct results through pure step-by-step mechanical execution, requiring no high-level semantic decision-making or subjective understanding of canine biology or number theory.

Early Symbolic Artificial Intelligence

  • Logic Theorist (Herb Simon & Alan Newell, 1950s):

    • Description: Early artificial intelligence software designed to manipulate non-numerical logic symbols.
    • Mechanism: Encoded axioms of propositional logic and formal inference rules. For example, applying Modus Ponens: given facts AA and A  ⟹  BA \implies B, derive the symbolic state BB.
    • Achievements: Successfully generated formal proofs for the majority of foundational logic theorems in introductory logic texts.
    • Mind-Body Solution Claim: Simon and Newell asserted they created a physical thinking machine, demonstrating that physical substrate state changes (governed by physical law) executing symbolic transformations resolve the dualistic separation between physical matter and abstract cognition.

  • ELIZA (Joseph Weizenbaum, 1966):

    • Description: An early natural language processing program configured to simulate a Rogerian psychotherapist.
    • Mechanism: Relies entirely on simple pattern-matching and symbolic substitution rules:
    • Input pattern: "I am [sad-word]" →\rightarrow Output selection: "I am sorry to hear that you are [sad-word]" or "Do you think coming here will help you to not be [sad-word]?".
    • Input pattern: "I want [desire]" →\rightarrow Output selection: "Why do you want [desire]?" or "Suppose you got [desire] soon?".
    • Fallback rules (no matching keywords): "Could you elaborate on that?" or "I am glad you are telling me about that."
    • Psychological Effect: Users formed strong emotional attachments and attributed deep empathy and comprehension to ELIZA despite its purely mechanical pattern lookup table.
    • The Language Gedanken Experiment: A non-German speaker armed with a translated German rulebook could execute the ELIZA algorithm in German perfectly, generating correct conversational responses without understanding German vocabulary or meaning.

ACT-R Theory and Properties of Symbol Systems

  • ACT-R (Adaptive Control of Thought-Rational):

    • Declarative Memory: Network of symbolic structures representing facts, concepts, and relationships (e.g., 3+4=73 + 4 = 7, sequential rules like 3→4→53 \rightarrow 4 \rightarrow 5, personal facts like age).
    • Procedural Memory: Production rules structuring cognitive actions based on active declarative goals. Rules extract active inputs and generate cognitive outputs or step modifications.

  • Foundational Properties of Classical Symbol Systems:

    • Systematicity: Symbols are reusable across multiple structures. Re-arranging identical discrete symbols changes systemic semantic meaning in predictable ways (e.g., structural difference between "3 precedes 4" versus "4 precedes 3").
    • Productivity: A finite set of discrete symbols and syntactic rules can construct an infinite array of novel, arbitrarily complex combinations without maximum length limits (analogous to natural language recursive syntax).

Formalizing Computation: Turing Machines

  • Alan Turing: Developed the formal mathematical definition of computation to establish what can be calculated through systematic execution of rules.

  • Structural Components of a Turing Machine:

    • Memory Tape: An infinitely long tape divided into discrete cells/bins. Each bin stores a single discrete symbol from a finite alphabet.
    • Read/Write Head & State Indicator: Tracks the active location on the tape and maintains an internal variable representing the current state (task phase or active goal).
    • Lookup Table (Transition Function): A finite rulebook mapping the current state and symbol under the read head to:
    1. Writing a new symbol to the cell.
    2. Moving the read head one cell to the Left or Right.
    3. Transitioning to a new internal state.

  • The Church-Turing Thesis:

    • Definition: Asserts that any algorithm or computation that can be executed by a human clerk working systematically with paper and pencil can be computed by a Turing Machine.
    • Definition of Computability: Formally defines "computable" as any operation reducible to a Turing Machine lookup table.
    • Historical Terminology: Historically, "computer" designated a job title held by human workers who systematically carried out hand calculations via strict rulebooks.

  • Example Operations:

    • Even/Odd Binary Counting: Scans binary digits sequentially, toggling internal state registers between Even and Odd upon encountering 1s, terminating at tape boundaries.
    • Binary Addition: Evaluates two binary numbers separated by a + symbol on the tape, shifting states to maintain digit carry flags while rewriting output sum values.

  • Mapping to Marr's Levels of Analysis:

    • Level 2 (Algorithmic / Representational): Turing machine specifications operate here by explicitly detailing data formats (symbol sets) and exact transformational rules (lookup table).
    • Level 3 (Physical Implementation): The physical realization of the machine (e.g., electronic microchips, wooden pegs, Lego mechanisms).
    • Wood Implementation: Utilizes sliding wooden dowels set to three discrete physical depth positions (pushed out, half-way, pushed in) representing symbols, coupled to a pegboard encoding lookup rules.
    • Lego Implementation: Uses physical Lego blocks to mark tape states, read by mechanized sensor assemblies driven by external controllers.
    • Directionality across Marr's Levels: Cognitive analysis moves bidirectionally: top-down (Goal Level 1 →\rightarrow Algorithm Level 2 →\rightarrow Implementation Level 3) or bottom-up (Neurobiology Level 3 →\rightarrow Algorithm Level 2 $ Goal Level 1).\n\n- **Neural Realization of Logical and Turing Systems**:\n - **McCulloch & Pitts Neural Logic (1940s)**:\n - Demonstrated that biological neurons wired with specific excitatory and inhibitory thresholds implement standard Boolean logic gates (\text{AND},,\text{OR},,\text{NOT}$$).
    • AND Gate: A postsynaptic neuron fires if and only if two presynaptic excitatory inputs fire simultaneously.
    • NOT Gate: An inhibitory presynaptic connection suppresses firing regardless of excitatory input.
    • Implication: Networks of biological neurons can physically implement classical logic functions and Turing machine lookup tables.

Universal Turing Machines and Modern Computer Architectures

  • Custom vs. Universal Turing Machines:

    • Custom (Dedicated) Turing Machine: A machine designed with a hardcoded lookup table built for a single fixed algorithm (e.g., only binary addition).
    • Universal Turing Machine (UTM):
    • Accepts both input data and a encoded symbolic representation of an algorithm (a program) on its memory tape.
    • Reads the program off the tape and executes that program on the data.
    • Represents a general-purpose, fully software-programmable calculator.

  • Von Neumann Architecture:

    • Central Processing Unit (CPU): Fixed physical hardware designed to execute basic operations.
    • Memory Unit: A unified storage system holding both working data and executable programs.
    • Input/Output Systems: Channels through which data enters and exits the physical system.
    • Software Execution: Installing a modern application (e.g., downloading a foreign language learning application) writes software instructions into memory without changing physical hardware components.

Philosophical Implications and Cognitive Science Debates

  • Resolution of the Mind-Body Problem:

    • Cartesian Dualism: Posited a fundamental ontological split between physical matter (the brain) and non-physical mind-stuff.
    • CCTM Resolution: Physical operations (gear movements, electronic transitions, or neural action potentials) carrying out syntax-driven transformations on symbolic states instantiate thought using only physical matter and the laws of physics.

  • Cognitive Modularity vs. General-Purpose Processing:

    • General-Purpose Human Capacity: Humans can process novel verbal instructions to perform arbitrary rule-following tasks (resembling a Universal Turing Machine).
    • Modular Processing: Specific cognitive functions operate as specialized, unalterable "custom Turing machines" (e.g., hardwired facial recognition modules) that cannot be altered by conscious instruction.

  • Computational Explanation of Human Experience and Emotion:

    • Gary Marcus noted that while digital computers lack human emotions, emotions do not disprove CCTM.
    • Emotions possess internal semantic mappings (pointing to real-world conditions) and systematic, syntax-like transformational rules.
    • Emotional states can be modeled computationally as internal state variables that modulate behavioral processing rules.

  • Determinism, Ethics, and Probabilistic Execution:

    • Complex human domains like ethical decision-making can be framed computationally by formalizing moral frameworks into constraint rules.
    • Non-deterministic human behavior is modeled within classical computation through probabilistic state transitions and stochastic rule selection (as implemented in systems like ELIZA).