SKILL: Human Problem Solving Theory

Core Insight

Intelligence = examining the right 50 nodes out of 10^100, not searching faster through millions. This skill provides L3 decision frameworks for when and how to apply selective search principles to agent systems.

DECISION POINTS

Agent Performance Diagnosis

IF agent examines >1000 nodes for problems with <10^6 states
├── THEN selectivity problem
├── Fix: Extract structural information for heuristics
└── NOT: Add more compute power

IF same problem solved easily with different representation
├── THEN problem space construction issue  
├── Fix: Test alternative state representations
└── NOT: Optimize search within current representation

IF agent succeeds on similar problems but fails here
├── Check: Does current problem space expose task structure?
├── If NO: Reconstruct problem space
└── If YES: Refine heuristics for this structure

IF resource usage grows unbounded
├── THEN strategy exceeds architectural constraints
├── Fix: Switch to constraint-compatible strategy
└── Example: Progressive deepening for limited memory

Search Strategy Selection

Working Memory: LIMITED + Backtracking: CHEAP
└── Use progressive deepening (depth-first with backtrack)

Working Memory: ABUNDANT + State comparison: NEEDED  
└── Use scan-and-search (breadth-first)

Task Structure Assessment:
├── Clear goal state + Measurable progress → Means-ends analysis
├── Hard constraints + Dependencies → Most-constrained-first
├── Recognizable patterns + Action opportunities → Opportunistic planning
└── Multiple valid paths + Resource limits → Constraint satisfaction

Heuristic Design Decision Tree

Task has well-defined goal state?
├── YES: Extract differences between current and goal
│   ├── Differences are measurable? → Means-ends analysis
│   └── Differences are qualitative? → Pattern-based selection
└── NO: Use structural constraints
    ├── Hard constraints exist? → Most-constrained-first
    ├── Known patterns exist? → Production rules (condition→action)
    └── Multiple operators available? → Extract structural ranking info

FAILURE MODES

Speed Fallacy Anti-Pattern

Detection Rule: If solution involves "faster hardware" or "more parallelism" for search problems Symptoms: Agent examines thousands of nodes, performance scales with compute Diagnosis: Missing selectivity through structural information extraction Fix: Analyze task structure for exploitable patterns, constraints, or goal-distance measures

Universal Strategy Trap

Detection Rule: If same search approach (usually means-ends) applied to all problems Symptoms: Works on clear-goal problems, fails on constraint satisfaction or opportunistic tasks Diagnosis: Strategy-task structure mismatch Fix: Match search pattern to task structure—means-ends for goal reduction, constraint propagation for CSPs, pattern recognition for opportunistic planning

Representation Blindness

Detection Rule: If optimization effort goes to search algorithms before testing representations Symptoms: Extensive tuning yields marginal gains, "obviously easy" problems remain hard Diagnosis: Wrong problem space for task structure Fix: Test 2-3 alternative state representations before optimizing search within any one

Architecture-Strategy Mismatch

Detection Rule: If strategy requires more working memory than system provides Symptoms: Thrashing, exponential memory growth, inability to backtrack effectively Diagnosis: Importing unlimited-memory strategies to constrained systems Fix: Use progressive deepening for memory-limited, scan-and-search only when memory permits full state tracking

Construction Neglect Pattern

Detection Rule: If problem space treated as "given" without explicit construction phase Symptoms: Agent can't initialize on new tasks, representation seems arbitrary Diagnosis: No systematic problem space construction from task environment Fix: Allocate design effort to representation selection proportional to search difficulty

WORKED EXAMPLES

Example 1: Cryptarithmetic Problem (SEND + MORE = MONEY)

Novice Approach:

• Problem space: All possible letter-to-digit assignments (10! = 3.6M states)
• Search: Try random assignments, check arithmetic
• Result: Examines thousands of invalid states

Expert Application:

• Problem space construction: States are partial assignments of letters to digits with constraints
• Structural information extraction:

- Column constraints (C1 + C2 + carryin = result + 10*carryout) - Most-constrained variable first (S and M have only 2 valid values each) - Constraint propagation (if S=9, then M=1, eliminating other M values)

• Decision point: Use constraint satisfaction, not means-ends (no single goal state)
• Search strategy: Most-constrained-first with forward checking
• Result: Solution in 50-100 nodes by eliminating impossible branches early

Key Insight: Same task, different problem space. Constraint-based representation exposes structure that assignment-enumeration hides.

Example 2: Tower of Hanoi (5 disks)

Representation Comparison:

Poor representation: States as disk positions [(disk1peg, disk2peg, ...)]

• 3^5 = 243 possible states, most invalid (large on small)
• No structural guidance for move selection
• Search examines invalid configurations

Good representation: States as peg configurations with implicit size ordering

• Only ~100 valid states (legal configurations)
• Move constraints embedded in representation
• Goal-distance heuristic: count disks not on target peg

Decision Logic Applied:

1. Task structure: Clear goal state (all disks on peg 3), subgoal decomposition possible
2. Strategy selection: Means-ends analysis appropriate (goal-directed with measurable progress)
3. Heuristic design: Difference = number of disks not on target peg
4. Architecture match: Recursive subgoaling fits human memory limits through problem decomposition

Search Performance: Expert finds solution in 31 moves (optimal) by examining ~50 nodes. Poor representation might examine 1000+ nodes and find suboptimal 100-move solution.

Example 3: Medical Diagnosis Task

Task Environment: Patient symptoms → disease identification → treatment

Problem Space Construction Decision:

• Option A: States = all possible diseases, operators = diagnostic tests
• Option B: States = symptom clusters, operators = differential diagnosis rules
• Option C: States = causal pathways, operators = evidence accumulation

Expert Choice: Hybrid approach

• Initial search: Pattern recognition (symptoms → candidate diseases)
• Refinement search: Most-constrained-first (tests that differentiate top candidates)
• Architecture consideration: Use production rules for fast pattern matching, deeper search only for ambiguous cases

Decision Framework Applied:

1. Recognizable patterns exist → Use opportunistic planning (symptom patterns trigger disease hypotheses)
2. Hard constraints exist → Most-constrained-first for disambiguation (tests that rule out maximum candidates)
3. Memory limits binding → Progressive deepening with early pattern termination

QUALITY GATES

• Problem space representation chosen explicitly (not assumed/inherited)
• Search strategy matches task structure (goal-directed vs. constraint-based vs. opportunistic)
• Heuristics extract information from problem structure (not generic distance metrics)
• Architecture constraints respected (memory limits, processing speed, backtracking costs)
• Selectivity demonstrated (examines <1% of theoretical search space for non-trivial problems)
• Failure modes have detection rules (can diagnose wrong representation vs. weak heuristics)
• Alternative representations tested when performance poor
• Structural information sources identified (constraints, patterns, goal-distances, dependencies)
• Strategy switches based on task structure assessment
• Problem space construction process explicit (not ad-hoc representation choice)

NOT-FOR Boundaries

Do NOT use this skill for:

• Pure optimization problems: Use mathematical optimization theory instead
• Machine learning model selection: Use statistical learning theory
• Parallel algorithm design: Use concurrent systems theory
• Database query optimization: Use relational algebra optimization
• Real-time control systems: Use control theory

Delegation Rules:

• For numerical optimization with continuous variables → Use convex-optimization or metaheuristics
• For pattern recognition with training data → Use machine-learning-fundamentals
• For concurrent search across multiple agents → Use distributed-systems-coordination
• For problems with probabilistic uncertainty → Use decision-theory-under-uncertainty
• For reactive systems with timing constraints → Use real-time-systems-design

Boundary Markers:

• Problem has discrete states and operators (not continuous optimization)
• Search space is large but finite (not infinite optimization landscapes)
• Solution quality depends on path/sequence (not just final state)
• Human-level intelligence provides useful benchmarks (not superhuman performance requirements)