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Chapter 1 · Cheat Sheet

Cheat Sheet: The Dream of Thinking Machines

Cheat Sheet - The Dream of Thinking Machines

Milestones and what each made executable

YearWhoWhat became precise
c. 350 BCEAristotleValidity by form: a check that needs no knowledge of the subject
1642PascalMechanical addition with carry
1673LeibnizA machine meant to multiply; later (1703) binary arithmetic
1801JacquardA loom driven by punched cards: instructions as swappable data
1834 onBabbageAnalytical Engine: Store (1,000 × 50 digits), Mill, Operation and Variable cards
1843LovelaceMachines manipulate symbols, not only numbers; Note G's Bernoulli table
1854BooleLogic as algebra over 0 and 1
1936Turing; ChurchDefinition of computation; universal machine; undecidable problems
1937ShannonBoolean algebra describes relay circuits
1943McCulloch and PittsNeurons as threshold logic units
1945von NeumannStored program: instructions in the same memory as data
1948Shannon; WienerInformation in bits; feedback control (cybernetics)
1949HebbConnections strengthen when neurons fire together
1950TuringThe imitation game and a numerical forecast
1955–56McCarthy, Minsky, Rochester, Shannon"Artificial intelligence" named; Logic Theorist proves 38 of 52 theorems
1958RosenblattPerceptron: weights learned from examples
1966WeizenbaumELIZA: fluent pattern matching, no understanding
2025–26Jones and BergenGPT-4.5 with a persona prompt picked as human in 73% of five-minute games

Formulas

Boole:        x^2 = x        x AND y = xy     x OR y = x + y - xy     NOT x = 1 - x
Adder steps:  n + k + 2      (n digits, k trailing ones)
Threshold:    y = 1 if w1*x1 + w2*x2 >= theta, else 0
Shannon:      information = -log2(p) bits      p = 0.25 -> 2 bits
Turing test:  interrogator accuracy = 1 - (share of games the machine is picked as human)
Clean runs:   n >= ln(1 - c) / ln(1 - p)  ~  3/p at c = 0.95
Bound:        after n clean runs, rate < 1 - (1 - c)^(1/n)  ~  3/n at 95%

Numbers worth remembering

  • Turing's 1950 forecast: about 10⁹ binary digits of storage, five minutes, interrogator right no more than 70% of the time, within about fifty years.
  • Jones and Bergen (arXiv March 2025; PNAS May 2026), share of games picked as human: GPT-4.5 with the persona prompt 73%, without it 36%; LLaMa-3.1-405B 56% with, 38% without; ELIZA 23%; GPT-4o with minimal instructions 21%.
  • One threshold unit computes 14 of the 16 two-input functions; XOR and XNOR need two layers.
  • Dartmouth 1955: a "2 month, 10 man study"; $13,500 requested, $7,500 granted.
  • Clean runs for 95% confidence: 29 rule out 10%, 299 rule out 1%, 2,995 rule out 0.1%.
  • A perfect pass on τ-bench retail's 115 test tasks bounds the failure rate below 2.57%; on SWE-bench Verified's 500 tasks, below 0.60%.

Decision rule: enumerate, restrict or sample

If the component...ThenEvidence you can state
has few enough inputs to check (up to a few billion)enumerateverified on every input
can do something dangerous you can take awayrestrict: remove the permission, sandbox itthe failure is impossible by construction
accepts free text or runs generated codesamplefailure rate below x at confidence c after n clean runs

No tool decides "never does X" for arbitrary programs (Turing 1936, Rice 1953). Tools that claim to either restrict the programs they accept or sometimes answer "don't know".

Reading a behavioural claim

  • What were the conditions: duration, channel, prompt, who judged?
  • What baseline ran under the same conditions, and what happens when the prompt changes? (ELIZA scored 23%; GPT-4.5 fell from 73% to 36% without its persona prompt.)
  • Is the claim about sounding right or being right? Only task-level tests on your own cases measure the second.
Cheat Sheet — The Dream of Thinking Machines | KnowledgeOS