Propositional Logic Proof Cost Analysis
Evaluating space and time complexity in formal proofs with rule efficiency metrics
Proof Comparison
We compare a 10-line proof against a 15-line proof using different inference rules:
10-Line Proof (Efficient)
1. Premise P → Q
2. Premise Q → R
3. HS (1,2) P → R
4. Premise P ∨ S
5. Absorption P → P
6. Premise ¬S
7. DS (4,6) P
8. MP (3,7) R
9. Addition R ∨ T
10. Conclusion R ∨ T
15-Line Proof (Inefficient)
1. Premise P → Q
2. Premise Q → R
3. LEM P ∨ ¬P
4. Case 1 Assume P
5. MP (1,4) Q
6. MP (2,5) R
7. Addition R ∨ T
8. Case 2 Assume ¬P
9. Premise P ∨ S
10. DS (9,8) S
11. Premise ¬S
12. Contradiction ⊥
13. Ex Falso R ∨ T
14. Case Close R ∨ T
15. Conclusion R ∨ T
Rule Efficiency Analysis
Space-Time Cost Evaluation
| Metric | Space Cost | Time Cost |
|---|---|---|
| Definition | Memory/storage requirements (proof length) | Computational steps (rule applications) |
| Measurement | • Lines in proof • Symbol complexity per line |
• Primitive inferences • Search operations |
| Comparison | 10-line proof wins | Depends on rule complexity |
Rule Characteristics
| Rule | Space Cost | Time Cost | Efficiency |
|---|---|---|---|
| Hypothetical Syllogism (HS) | Low | Low | Optimal for chaining |
| Addition | Moderate-High | Low | Can bloat proof |
| Law of Excluded Middle (LEM) | High | High | Case splitting doubles work |
| Absorption | Low | Low | Space-saving |
| Substitution | Variable | Variable | Pattern matching cost |
Cost Visualization
Space-Time Efficiency
Rule Application Frequency
Redundancy Check
flowchart TD
subgraph YourRules[Your Rules]
HS[Hypothetical Syllogism] -->|Derivable from MP| Core
Addition -->|Often avoidable| Core
LEM[Law of Excluded Middle] -->|Classical only| Core
Absorption -->|Compression rule| Space
end
Core[Modus Ponens + Axioms] --> Minimal
Space[Space-saving rules] --> Practical
Redundancy Analysis
| Rule | Redundant? | Recommendation |
|---|---|---|
| Hypothetical Syllogism | No | Keep for compact chains |
| Addition | Situationally | Use sparingly |
| LEM | No (classical) | Avoid when possible |
| Absorption | Situationally | Use for downstream steps |
| Substitution | No | Optimize implementation |
Conclusions & Recommendations
Proof Efficiency Findings
- The 10-line proof is more space-efficient than the 15-line version
- LEM significantly increases time cost due to case splitting
- HS and Absorption provide space-time tradeoffs that benefit most proofs
- Addition should be used judiciously as it can bloat proofs unnecessarily
Optimization Strategies
- Replace LEM with direct proofs when possible
- Use HS for implication chains to reduce line count
- Apply Absorption only when needed for downstream steps
- Minimize Addition - only use when necessary for conclusion
- Implement efficient substitution with pattern sharing
Verification Protocol
- Expand to primitive rules (MP + axioms) for step comparison
- Calculate inference density (conclusions per line)
- Profile worst-case branching factor
- Compare with alternative methods (natural deduction, tableaux)
- Benchmark against minimal proof lengths
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