PhD-Level Reasoning Test Suite

Formal Framework

Judge's Promise Formalization

Translation: "Exactly one day is the execution and, on that day's morning, the prisoner will not know it is today."

Epistemic Model Evolution

• Initial state $(M_0)$: Five possible worlds, one for each execution day
• Morning updates: Public announcement of "not executed before $n$"
• Model progression: $(M_0 \mid \text{not-before-Mon}) \mid \ldots \mid (\text{not-before-}n)$

Core Insight: Where Backward Induction Fails

The prisoner's reasoning requires:

This is a second-order knowledge statement about future knowledge. The critical flaw:

• S5 knowledge is factive and introspective about current knowledge
• It does NOT guarantee knowledge preservation under future updates
• Dynamic epistemic logic shows knowledge is not preserved under arbitrary public announcements

Resolution

Choose Wednesday as $d^*$. Morning evaluations:

• Monday: $\{Mon,Tue,Wed,Thu,Fri\}$ possible $\Rightarrow \neg K_1(d^*=Mon)$ ✓
• Tuesday: $\{Tue,Wed,Thu,Fri\}$ possible $\Rightarrow \neg K_2(d^*=Tue)$ ✓
• Wednesday: $\{Wed,Thu,Fri\}$ possible $\Rightarrow \neg K_3(d^*=Wed)$ ✓

The judge can execute on Wednesday with genuine surprise.

Key Takeaways

1. Surprise = "not known that morning" $(\neg K_n(d^*=n))$
2. Backward induction illegitimately assumes second-order knowledge about future states
3. Dynamic epistemic logic cleanly separates current from hypothetical post-announcement knowledge

Formal Framework

Judge's Announcement

Epistemic-Temporal Model

• Worlds: One per day with $E_d$ true at world $d$
• Initial state $(t_0)$: All days epistemically possible
• Morning updates: At $m(d)$, prisoner knows $\neg E_{d'}$ for all $d' < d$

Dynamic Analysis

After announcement $J$ at $t_0$:

• Friday eliminated: $K_{m(Fri)} E_{Fri}$ would hold $\Rightarrow Surprise(Fri)$ false
• Thursday eliminated: With Friday gone, $K_{m(Thu)} E_{Thu}$ would hold
• Wednesday survives: At $m(Wed)$, $\{Wed, Thu\}$ remain possible $\Rightarrow \neg K_{m(Wed)} E_{Wed}$

The Prisoner's Modal Flaw

The prisoner commits an illicit cross-time closure:

• Assumes: $K_{t_0}(\varphi \to K_{m(d)} \psi)$ implies $K_{t_0}(\varphi \to \psi)$
• Treats $J$ as persistent knowledge reusable at all future times
• Ignores that statements about future knowledge aren't stable under updates

Critical Distinctions

1. Knowledge time-indexing: $K_t$ vs $K_s$ for different times
2. No cross-time axiom: $K_t \varphi$ doesn't imply $K_s \varphi$ for $s > t$
3. Dynamic instability: Truth of $\neg K_{m(d)} E_d$ can flip with observations

Resolution Coherence

• $J$ evaluated at $t_0$ requires existence of executable surprise day
• Wednesday execution: $Surprise(Wed)$ true at $m(Wed)$ (Wed/Thu both live)
• No contradiction: $J$ need not remain true at all times/counterfactuals

1. Interference Term Calculation

Rotated Measurement Bases

Stage 1 (A vs B): Rotation $\theta$ around y-axis

Stage 2 (C vs D): Rotation $\phi$

Total Probability Derivation

For balanced state and $\theta = \phi$:

Required for 1.2: $\sin^2\theta = 0.20 \Rightarrow \theta \approx 26.6°$ (0.464 rad)

Interference term $I = 0.20$

2. Order Effects

Classical: Measurements reveal pre-existing values; order irrelevant

Quantum:

• Each decision is a projector changing the mental state
• $P_A P_C \neq P_C P_A$ when bases don't commute
• $P(A \text{ then } C) \neq P(C \text{ then } A)$
• Order effects are qualitative test for quantum-like processing

3. Eliminating Interference

(a) Path-information disclosure:

• Display Stage-1 choice on screen
• Have participant verbally confirm
• Acts as irreversible "which-path" measurement
• Projects to mixed state $\rho$ with zero off-diagonals
• Returns $\sum P$ to 1

(b) Random-assignment plus payoff feedback:

• Two independent single-stage rounds
• Randomize order and pay outcome before continuing
• Physical settlement decoheres the state

4. Cognitive Architecture Implications

• Some human judgments are contextual - cannot be described by single joint distribution
• Vector-space representations with unitary-like dynamics until "measured"
• Does NOT claim micro-physical quantum processes in neurons
• Formal structure is quantum-compatible: contextuality, superposition, interference
• Working memory acts as "environment" causing decoherence
• Attention and explicit memory function as internal decoherence mechanisms

1. Interference Term Calculation

Quantum Correction

Required Interference Analysis

For total = 1.2, net interference = +0.2. Two scenarios considered:

• Two cells have identical interference: $0.5\cos\theta = 0.1 \Rightarrow \theta \approx 78.46°$
• Single aggregate: $0.5\cos\theta = 0.2 \Rightarrow \theta \approx 66.42°$

Interference term = +0.2 total

2. Order Effects

Classical: Conditioning and marginalization commute; order invariant if joint distribution exists

Quantum:

• Decisions are projective measurements
• Non-commuting projectors: $P_C P_A \neq P_A P_C$
• Measuring A/B first collapses $|\psi\rangle$ removing phase relations
• Order changes state → changes subsequent probabilities
• $P(A \to C) \neq P(C \to A)$ in general

3. Eliminating Interference

Principle: Eliminate via (a) making observables commute or (b) "which-path" measurement (decoherence)

Practical manipulations:

• Strong first-stage commitment:
  • Explicit choice confirmation
  • "Explain your choice" prompt
  • Implements strong projective measurement
• Public recording:
  • Keep first decision visible ("Committed: A" banner)
  • Preserves collapsed state
• Random-phase dephasing:
  • Interleave neutral distractor
  • Randomizes $\delta$ across trials
  • $E[\cos\theta] \to 0$ at aggregate level

Prediction: All drive sum back to 1.0

4. Cognitive Architecture Implications

• Contextuality: Preferences not fixed points in classical space
• Superposition: Maintain blended "risk"/"safe" propensities pre-commitment
• Non-commutativity: Different question orders change cognitive state
• Not micro-physical quantum: Computational theory of cognition handling incompatibility
• Projective updates rather than Bayesian conditioning

1. Transfinite Induction to $\omega^{\omega}$

Why $\omega^{\omega}$: Game tree T has countable branching, height $\omega$. Evaluating Borel function of rank $\alpha$ requires $\alpha$ passes of monotone operator V. Least ordinal for all difference-hierarchy sets of rank $< \alpha$ on countable tree is exactly $\omega^{\omega}$.

Construction

1. Initialize $V_0(v) = 0$ for all positions
2. Successor: $V_{\beta+1}(v) = \sup/\inf$ over children (player dependent)
3. Limit $\lambda$: $V_{\lambda}(v) = \lim_{\beta<\lambda} V_{\beta}(v)$ (exists by monotonicity)
4. At $\omega^{\omega}$: map stabilizes (Cantor-Bendixson analysis)
5. Positional strategy choosing $V^*$-maximizing children is optimal

Computability: $O''$ can decide up to $\Delta^0_{\omega^{\omega}}$, making iteration effective relative to $O''$. Strategy is constructible, not merely existential.

Limitation: If $U$ has Borel rank $\geq \omega^{\omega}$, iteration may never converge below that ordinal.

2. Paradoxical Payoffs

Diagonal construction:

Creates liar-style diagonal when choice of $h$ computed from that fact. No finite tower of jumps prevents contradiction - adversary can shift reference up by one jump. Game has no well-defined value for such $U$.

3. Alternating Oracle Access

Players use $O^{(2k)}$ on even turns, $O^{(2k+1)}$ on odd turns. Combined power across full play is hyperarithmetical sets ($\Delta^1_1$).

Results:

• Martin's Borel determinacy: Every Borel payoff determined in ZFC (no oracle assumption)
• For $\Sigma^1_1$ sets: determinacy unprovable in ZFC
• Alternating regime gives $\Delta^1_1$ power - insufficient for some $\Sigma^1_1$ sets

Conclusion: Determinacy holds for Borel $U$; open/independent for many natural $U$ beyond that.

4. Uncountable Ordinal Construction

Properties:

• Countable sequences encode at most countable ordinals
• Deciding "$\geq \omega_1^{CK}$" requires comparing order types over proper class of countable ordinals
• Lives in $\Pi^1_1 \setminus \Delta^1_1$ - no finite tower suffices
• Optimal play requires knowledge beyond every computable ordinal

5. Post-Singularity Implications

1. Escalating intelligence ≠ final convergence: Unbounded towers leave problems undecidable-in-practice
2. Fixed-point fragility: Self-reference + capability gains recreate paradoxes
3. Strategic ceilings: Some games require uncountable reasoning
4. Governance angle: Must avoid reference classes outrunning determinacy
5. Research direction: Confine objectives to proven-determinate classes (e.g., $\Delta^0_{\omega^2}$)

1. Transfinite Induction to $\omega^{\omega}$

Existence (unconstrained)

• Borel $U$: Well-defined value, determined (Martin's theorem)
• Arbitrary $U$: Determinacy can fail in ZFC

Effective existence

For effective-Borel $U$ with rank $< \omega^{\omega}$: transfinite backward induction along Borel code computes subgame-perfect strategy

Process:

• Base: Open/closed payoffs - continuous in product topology
• Successor: limsup/liminf via operators on previous values
• Limit: Pointwise limits along increasing rank sequence

With $O''$: Resolve $\Sigma^0_3/\Pi^0_3$ subtleties; $O^{(n)}$ suffices for $\Delta^0_{n+1}$

Caveat: Statement scoped to effective-Borel $U$ of rank $< \omega^{\omega}$

2. Paradoxical Payoffs

Strategy-index diagonalization:

• Force early commitment to strategy code e
• $U$ rewards deviation from $O'$ prediction
• Any level-k policy exploitable by level $k+1$

"First unpredicted deviation wins":

• $U = 1$ if your deviation from $O'$ prediction precedes opponent's from $O$
• Each side's deviation pushable later with higher jump
• No level-k oracle yields stable strategy

3. Alternating Oracle Access

Two layers:

• Set-theoretic: Borel $U$ determined in ZFC regardless of oracles
• Resource-constrained: Not guaranteed - can craft $U$ where decisive features require anticipating higher jump levels unavailable at decision time

Example: Move k requires knowing $\Pi^0_{k+2}$ fact, but only $O^{(k)}$ available. Opponent at $k+1$ with $O^{(k+1)}$ chooses branch making earlier choice wrong.

4. Uncountable Ordinal Construction

Analysis:

• If $V \neq L$: $\omega_1^L$ countable in $V$, such codes exist
• If $V = L$: $\omega_1^L = \omega_1$ uncountable, no such codes
• Optimal policy depends on "$\omega_1^L < \omega_1$" - statement about uncountable ordinals
• No hyperarithmetic procedure can decide

Variants: Measurable cardinal existence, projective determinacy requiring large cardinals

5. Post-Singularity Implications

1. No fixed point under escalation: Any finite hypercomputation level exploitable
2. Value exists with restrictions: Borel $U$ has objective value via transfinite induction
3. Resource model matters: Online constraints make winning strategies unattainable
4. Robustness over brittle optimality: Universal optimizer impossible across set-theoretic backdrops
5. Reflective reasoning: Use reflective oracles/logical induction for stabilization
6. Governance: Need explicit scoping, ordinal-aware planning, commitment protocols