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Problem Solving as Modeling

Introduction​

The modeling perspective on problem solving represents a fundamental shift from viewing problem solving as merely selecting actions to understanding it as constructing knowledge structures. This approach, which emerged from the evolution of Newell's ideas and advancements in knowledge-based systems, proposes that problem solving is fundamentally about building situation-specific models that organize and integrate knowledge to achieve goals.

Rather than being an input-output process (Traditional) or simply a means to select actions (GPS/Newell's early work), problem solving becomes a process of organizing knowledge through assumption-making that allows problem solvers to conclude their task is accomplished.

Core Insight

In the modeling view, problem solving is the creation of internal representations (models) that provide coherent understanding of a specific situation, guiding action and recognizing goal achievement.

Theoretical Background​

Evolution of Problem-Solving Theories​

The modeling perspective emerged as researchers recognized limitations of earlier approaches:

Key Theoretical Advances​

ApproachView of Problem SolvingLimitation
TraditionalStrengthening correct associationsCan't explain insight, reasoning
GestaltRestructuring perceptual organizationVague mechanisms
GPSHeuristic search through problem spaceWeak methods, no learning
Expert SystemsApplying domain-specific rulesStatic knowledge base
ModelingConstructing situation-specific understandingCurrent frontier

Core Concepts of the Modeling Perspective​

From Knowledge-Level to Model Construction​

At the knowledge level, Newell proposed that:

  • Agents perceive the world through knowledge
  • Goals correspond to desired knowledge states
  • Actions are means to obtain knowledge

The modeling perspective extends this by emphasizing that knowledge must be organized into coherent structures to support reasoning and goal achievement.

What Is a Model?​

A model in this context is:

Definition: A structured representation that captures the problem solver's current understanding of a specific problem instance, integrating:

  1. Domain knowledge: What is known about the problem domain
  2. Task knowledge: What the goal means and success criteria
  3. Case-specific information: Particular details of this instance
  4. Assumptions: Connections and inferences that make reasoning possible

Components of the Modeling Framework​

1. Domain Models​

Domain models contain knowledge about the systems and entities the problem concerns.

Characteristics​

Content:

  • Objects and their properties
  • Relationships between objects
  • Causal connections
  • Constraints and regularities
  • Typical behaviors

Structure:

  • Organized hierarchically or in networks
  • May include multiple abstraction levels
  • Contains general domain principles

Example: Medical Diagnosis Domain Model

Assumptions in Domain Models:

  • Statements about domain are connected (not isolated facts)
  • Entities have typical behaviors and exceptions
  • Causal relationships exist and can be reasoned about
  • Information can be at different abstraction levels
Example Domain Assumptions

In a medical domain model, assuming:

  • Symptoms are caused by underlying conditions (not random)
  • Anatomically connected systems can influence each other
  • Lab values outside normal ranges indicate abnormality
  • Treatment effects are predictable based on mechanism

2. Task Models​

Task models specify what the goal means and how to recognize achievement.

Characteristics​

Content:

  • Type of problem (diagnosis, design, planning, etc.)
  • What constitutes a solution
  • Success criteria and constraints
  • Acceptable solution properties

Structure:

  • Goal specifications
  • Success conditions
  • Evaluation criteria

Example: Diagnostic Task Model

Goal: Identify condition causing observed symptoms
Success Criteria:
  - Identified condition explains ALL symptoms
  - Condition is consistent with patient history
  - No contradicting evidence
Constraints:
  - Prefer single-disorder explanation (parsimony)
  - Consider likelihood (base rates)
  - Account for severity (triage)
Solution Form:
  - Primary diagnosis
  - Confidence level
  - Alternative hypotheses (if applicable)

Assumptions in Task Models:

  • Modeling a task makes more precise what accomplishing the goal means
  • Different task models embody different assumptions about solution nature
  • Task choice constrains possible solution approaches
Example Task Assumptions

Modeling diagnosis as "generate-and-test over components" assumes:

  • Fault can be localized to component(s)
  • Components have discrete states (working/broken)
  • Faults are independent (not emergent from interactions)

Different task model = different assumptions!

3. Problem-Solving Methods​

Methods tie domain and task models together, providing strategies for using domain knowledge to achieve task goals.

Characteristics​

Content:

  • Inference strategies
  • Search control knowledge
  • Heuristics for efficiency
  • Procedures for combining information

Role:

  • Bridge between domain and task
  • Specify how to use domain knowledge for task
  • Define competence of the system

Common Problem-Solving Methods:

MethodTask TypeStrategy
Heuristic ClassificationCategorization, DiagnosisMatch features to pre-defined categories
Case-Based ReasoningAnyRetrieve and adapt similar past cases
Constructive Problem SolvingDesign, PlanningBuild solution incrementally
Model-Based ReasoningDiagnosis, PredictionReason from first principles using causal model
Constraint SatisfactionConfiguration, SchedulingFind assignment satisfying all constraints

Example: Heuristic Classification Method​

Heuristic Classification Assumptions:

  • Solution exists in pre-enumerated set (differential)
  • Observable features can be abstracted
  • Abstract features map to solution categories
  • Categories can be refined to specific solutions
  • Method defines competence: Can classify if solution in differential

The Case Model​

Definition and Purpose​

A case model is the instantiation of the general framework (domain + task + method) for a specific problem instance.

Purpose:

  • Capture current understanding of the specific situation
  • Organize case-specific information with domain knowledge
  • Support reasoning toward goal
  • Recognize when goal is achieved

Construction Process​

Case Model Form​

The form of the case model is determined by the problem-solving method selected.

Examples:

MethodCase Model Form
Heuristic ClassificationDifferential list with supporting/contradicting evidence
Case-Based ReasoningRetrieved similar case + adaptations needed
Hierarchical PlanningGoal-subgoal tree with action sequences
Constraint SatisfactionVariables, domains, constraints, partial assignments
Model-Based DiagnosisSystem model + component states + failure hypotheses
Key Principle

The form of the case model structures what information is relevant, how it's organized, and how reasoning proceeds—fundamentally shaping the problem-solving process.

Competence Theory​

Definition​

A competence theory explains what the system can conclude and why its behavior is rational.

Components:

  1. Capabilities: What the system can determine/achieve
  2. Assumptions: What must be true for method to work
  3. Rationality Principle: Why actions are appropriate given goals
  4. Justification: Basis for "why" questions about behavior

Example: Heuristic Classification Competence​

System Using Heuristic Classification for Medical Diagnosis

Capabilities:

  • Can classify symptoms into disease categories
  • Can handle incomplete or uncertain information
  • Can narrow differential based on evidence

Assumptions:

  • Correct diagnosis is in the differential (pre-enumerated)
  • Symptoms correlate reliably with diseases (though imperfectly)
  • More evidence generally improves classification

Specialized Rationality Principle:

"Actions should reduce the size of the differential by gathering information that discriminates between alternatives."

Justification for Behavior:

  • Why order test X? → It discriminates between top 2 hypotheses
  • Why not consider disease Y? → It's not in differential (outside system's competence)
  • Why ask about symptom Z? → It's key feature for classification
Limitations of Competence

The competence theory makes explicit what the system can and cannot do. A heuristic classifier cannot diagnose novel diseases not in its differential—this is a known limitation, not a failure.

Competence vs. Performance​

AspectCompetencePerformance
FocusWhat can ideally be achievedWhat is actually achieved
ConstraintsMethod assumptionsMethod + resource constraints
Analysis LevelKnowledge levelSymbol/implementation level
ErrorsOutside competence = expected limitationWithin competence = system failure

Assumptions and Their Role​

The Central Role of Assumptions​

In the modeling view, assumptions are fundamental:

"Successful problem solving is a matter of making the right assumptions and exploring their consequences."

Types of Assumptions:

  1. Domain Assumptions: How the problem domain works

    • "Symptoms have underlying causes"
    • "Similar inputs produce similar outputs"

  2. Task Assumptions: What solving means

    • "Solution is in pre-enumerated set"
    • "Acceptable solution must satisfy all constraints"

  3. Method Assumptions: How to solve

    • "Heuristic matches are reliable enough"
    • "Decomposition into subgoals is possible"

  4. Case Assumptions: About this specific instance

    • "This symptom is reliable (not measurement error)"
    • "This patient is compliant with treatment"

Making Appropriate Assumptions​

Good assumptions:

  • Are explicit (not hidden)
  • Are justified by domain knowledge or task requirements
  • Are testable (can be checked if doubted)
  • Are revisable (can be changed if contradicted)

Poor assumptions:

  • Are implicit (unrecognized)
  • Are unjustified (arbitrary)
  • Are untestable (can't be verified)
  • Are rigid (maintained despite contradiction)
Common Assumption Errors
  • Assuming comprehensiveness: "I've considered all possibilities" (may have missed some)
  • Assuming independence: "These factors don't interact" (may actually interact)
  • Assuming uniformity: "This always works this way" (may have exceptions)
  • Assuming accuracy: "This measurement is precise" (may have error)

Actions as Knowledge Acquisition​

Reframing the Role of Actions​

Traditional view: Actions are the goal of problem solving

Modeling view: Actions are a resource for model construction

Actions in the Modeling Framework​

Purpose of Actions:

  1. Perception: Gather case-specific information
  2. Testing: Check model predictions
  3. Refinement: Improve model accuracy
  4. Validation: Confirm model adequacy

Example: Medical Diagnosis Actions

ActionTraditional ViewModeling View
Order lab testGet information for diagnosisInstantiate case model parameters
Physical examLook for diagnostic signsTest model predictions about physical findings
Ask about historyGather factsUnderstand patient's situation (context)
Trial treatmentAttempt to cureValidate diagnostic model (response confirms/disconfirms)
Key Insight

Actions are side effects in the process of maintaining internal organization (the model). The primary goal is the creation of a suitable case model; interaction with the world is a resource for achieving this.

Successful Problem Solving​

What Constitutes Success?​

In the modeling view, successful problem solving means:

"Making the right assumptions that allow one to conclude (in effect, only assume) that the task is accomplished."

Not:

  • Finding the objectively correct answer (may be unknowable)
  • Executing perfect actions (focus is understanding, not action)
  • Achieving certainty (satisfactory confidence suffices)

But Rather:

  • Constructing a model that coheres (internally consistent)
  • Making assumptions that are justified (based on evidence)
  • Reaching a state where one can reasonably conclude goal is met

Criteria for Model Adequacy​

A case model is adequate when:

  1. Completeness: All relevant aspects of problem are addressed
  2. Consistency: No internal contradictions
  3. Coherence: Elements fit together logically
  4. Justification: Assumptions are warranted
  5. Sufficiency: Supports goal recognition
  6. Efficiency: No unnecessary complexity

Example: Adequate Diagnostic Model

  • Explains all observed symptoms (completeness)
  • No conflicting evidence (consistency)
  • Mechanism is plausible (coherence)
  • Hypothesis is supported by tests (justification)
  • Diagnosis can be confidently stated (sufficiency)
  • Single disease explanation preferred over multiple (efficiency)

Comparison with Earlier Approaches​

DimensionSearch View (GPS)Modeling View
Primary metaphorNavigating mazeUnderstanding situation
GoalSelect actionsConstruct knowledge
Problem spaceStates and operatorsModels and assumptions
ProgressMoving toward goal stateIncreasing model adequacy
SuccessReaching goal stateConcluding goal is met
ActionsThe pointA means to model building

Modeling vs. Input-Output Processing​

KADS-I Approach (early expert systems):

  • Problem solving as transformation process
  • Input → Processing → Output
  • Static knowledge application

Modeling Approach:

  • Problem solving as model construction process
  • Iterative building of understanding
  • Dynamic knowledge organization

Advantages of Modeling:

  • Explains why certain actions taken
  • Accounts for uncertainty and revision
  • Handles incomplete information naturally
  • Provides deeper understanding

Practical Implications​

For System Design​

Knowledge-Based System Design:

  1. Identify domain → Build domain model
  2. Clarify task → Specify task model
  3. Select method → Choose problem-solving approach
  4. Define competence → State assumptions and capabilities
  5. Implement model construction → Enable case model building

Advantages:

  • Clear separation of knowledge types
  • Explicit competence boundaries
  • Easier maintenance and extension
  • Better explanation capabilities

For Human Problem Solving​

Implications for Learners:

  1. Understand domain structure (not just facts)
  2. Clarify what "solving" means for task type
  3. Learn appropriate methods for task-domain combinations
  4. Make assumptions explicit (don't let them hide)
  5. Build coherent understanding (not just collect facts)

Teaching Strategies:

  • Help students construct models, not memorize answers
  • Make assumptions explicit in worked examples
  • Show multiple representations (multiple models)
  • Encourage explanation and justification
  • Teach model evaluation and revision

For Expert Performance​

Expertise as Rich Models:

Experts differ from novices in:

  • Richer domain models (more connections, deeper understanding)
  • More refined task models (clearer success criteria)
  • Flexible methods (adapt approach to situation)
  • Better assumptions (more accurate, appropriate)
  • Faster model construction (efficient information gathering)

Example: Expert Physician

  • Recognizes patterns quickly (chunked domain knowledge)
  • Knows when each test is informative (task understanding)
  • Selects efficient diagnostic strategy (method expertise)
  • Makes warranted assumptions about typical vs. unusual cases
  • Constructs adequate model with minimal information

Future Directions​

Open Questions​

  1. How are models actually constructed in real-time?
  2. What triggers model revision vs. accommodation of new info?
  3. How do people balance competing assumptions?
  4. What individual differences exist in modeling ability?
  5. Can modeling be effectively taught?

Research Opportunities​

  • Computational modeling: Implement model construction processes
  • Protocol analysis: Study how experts build models
  • Educational interventions: Teach modeling explicitly
  • Individual differences: Identify effective modeling strategies
  • Development: How modeling ability develops with age/experience

Memory Aid: MADCAT Framework​

Remember problem solving as modeling with MADCAT:

  • Models: Domain, Task, Case
  • Assumptions: Central to reasoning
  • Dynamic construction: Not static application
  • Competence theory: Defines capabilities
  • Actions as information: Not goals themselves
  • Task specification: Defines success

Self-Assessment Questions​

  1. Conceptual Understanding: Explain how the modeling view differs from viewing problem solving as search. What are the key conceptual shifts?

  2. Model Components: For a problem domain of your choice, sketch: (a) a domain model, (b) a task model, and (c) a potential problem-solving method. How do these connect?

  3. Assumptions Analysis: Identify three assumptions you're making in approaching a current problem. Are they explicit or implicit? Justified or arbitrary? Testable?

  4. Competence Theory: For a simple problem-solving system (e.g., a thermostat, a GPS navigation system), specify its competence theory: capabilities, assumptions, and rationality principle.

  5. Actions as Information: Think of a recent problem you solved. How did your actions serve to build your understanding rather than being the goal themselves?

  6. Model Adequacy: What makes a case model "adequate"? How would you know when to stop gathering information and conclude you've solved a problem?

  7. Comparison: Compare the modeling view with Gestalt restructuring. Are they compatible? Do they explain different aspects of problem solving?

  8. Application: How might explicitly teaching students to construct models (rather than just apply procedures) improve learning in a subject you're familiar with?

External Resources​

Research Articles​

  • Breuker, J., & Van de Velde, W. (Eds.). (1994). CommonKADS Library for Expertise Modelling. Amsterdam: IOS Press. [Modeling framework]
  • Schreiber, G., et al. (2000). Knowledge Engineering and Management: The CommonKADS Methodology. MIT Press. [Comprehensive methodology]
  • Chandrasekaran, B. (1986). "Generic tasks in knowledge-based reasoning: High-level building blocks for expert system design." IEEE Expert, 1(3), 23-30. [Task-method framework]
  • Clancey, W. J. (1985). "Heuristic classification." Artificial Intelligence, 27(3), 289-350. [Heuristic classification method]

Educational Videos​

Wikipedia References​

Source PDFs: