Superiority

Superiority relations resolve conflicts between competing rules.

The Problem

When two rules conclude opposite things, we have a conflict:

(given bird)
(given penguin)
(normally r1 bird flies)
(normally r2 penguin (not flies))

Both r1 and r2 fire. Without superiority, we get ambiguity — neither flies nor (not flies) is provable.

Declaring Superiority

(prefer r2 r1)    ; r2 beats r1

Now when both rules fire, r2 wins and (not flies) is provable.

Superiority Chains

Shorthand for multiple superiority relations:

(prefer r3 r2 r1)   ; r3 > r2 > r1

Expands to:

(prefer r3 r2)
(prefer r2 r1)

Transitivity

Superiority is not automatically transitive. If you need r3 > r1, declare it explicitly:

(prefer r3 r2)
(prefer r2 r1)
(prefer r3 r1)    ; Must be explicit

Conflict Resolution Algorithm

When evaluating a defeasible conclusion:

1. Find all rules that could prove the literal
2. Find all rules that could prove the complement (attackers)
3. For each attacker with satisfied body:
   • If no defender is superior to it → blocked
   • If some defender is superior → attack fails
4. If all attacks fail → conclusion is provable

Example: Three-Way Conflict

(given a)
(given b)
(given c)

(normally r1 a result)
(normally r2 b (not result))
(normally r3 c result)

(prefer r1 r2)    ; r1 beats r2
(prefer r3 r2)    ; r3 beats r2

Analysis:

• r2 attacks result
• Both r1 and r3 are superior to r2
• The attack is defeated
• +d result

Symmetric Conflicts

If two rules are equally superior over each other (or neither is superior), ambiguity results:

(given trigger)
(normally r1 trigger a)
(normally r2 trigger (not a))
; No superiority

Result: Neither a nor (not a) is provable.

Defeating Defeaters

Defeaters can be overridden by superiority:

(given bird)
(given healthy)

(normally r1 bird flies)
(except d1 bird flies)          ; Defeater blocks flies
(normally r2 healthy flies)

(prefer r2 d1)                  ; Healthy birds overcome the defeater

If both bird and healthy are true, r2 beats d1 and flies is provable.

Strict Rules and Superiority

Strict rules always win over defeasible rules, regardless of superiority:

(given p)
(always r1 p q)              ; Strict
(normally r2 p (not q))      ; Defeasible
(prefer r2 r1)               ; This has no effect!

Result: +D q (strict rule wins)

Superiority only affects conflicts between:

• Defeasible rules
• Defeasible rules and defeaters
• Defeaters

Best Practices

Use Specificity

More specific rules should be superior:

(normally r1 bird flies)
(normally r2 penguin (not flies))
(prefer r2 r1)    ; Penguin is more specific than bird

Document Reasoning

Use comments to explain why one rule beats another:

; Medical override: confirmed diagnosis beats symptoms
(prefer r-diagnosis r-symptoms)

Avoid Cycles

Don't create circular superiority:

; BAD — creates a cycle
(prefer r1 r2)
(prefer r2 r1)

This leads to undefined behavior.