1. Dimensions and Values for Legal Case Based Reasoning
Trevor Bench-Capon and Katie Atkinson
Department of Computer Science,
The University of Liverpool, UK

2. Case Based Reasoning in AI and Law
Rissland and Ashley: HYPO. Dimensions
relevant aspects of cases, ranging from extreme pro-plaintiff to extreme pro-defendant
Ashley and Aleven: CATO. Factors
points on dimensions. Boolean (present or absent), always favour the same side when present
Prakken and Sartor: Precedents as Rules
3 rules: if plaintiff factors then plaintiff, if defendant factors then defendant , priority
Horty 2012: Formalisation of precedents using factors
Horty and Rigoni 2017: Factors with magnitudes

3. Factors with Magnitude
Back to dimensions?
Extreme Pro D
Extreme Pro P
Switching Point
Determined by Precedents
Horty: 2 Factors: Pro-P up to switching point, Pro D after
Rigoni: Many Factors: Pro-P up to switching point, Pro D after
Similar to Prakken et al JLC 2015

4. Results v Reason Model (Larry Alexander and Grant Lamond)
If we represent cases in the manner of Prakken and Sartor, we have three rules:
If set of pro-p factors then plaintiff
If set of pro-d factors then defendant
A priority rule reflecting the outcome.
This is the results model.
Horty allows the winning side to use a subset of its factors
This is the reason model.

5. Adding Magnitude
If we have magnitudes there are two ways of broadening the rule to get the reason model:
As before, by using a subset of factors
By using weaker points on the dimension
In Horty the mapping to factors means the two models collapse to the results model
Rigoni’s multiple factors (ordered by strength) avoids this collapse
Which is a good thing!

6. Factors without magnitude
Rigoni recognises that not all factors have magnitude. He uses both booleans and dimension points in his representation of cases
Rigoni’s account works in many cases, but not where we have balance or trade off between factors
This is quite common, especially when factors have magnitude
Essentially Rigoni (and Horty) reduce to comparisons of subsets

7. Abstract Dialectical Frameworks
We store the knowledge in Abstract Dialectical Frameworks (ADFS). As we use them:
A tree of nodes representing statements (outcomes, issues, abstract factors and base level statements)
A set of truth conditions for each node. The truth conditions provide individually sufficient and jointly necessary conditions to ascribe a truth value to the parent in terms of its children

8. ADFs
ADFs were originally presented with statements as three-valued, but this has subsequently been generalised to allow truth values 0 …1, so we can represent magnitudes
We can interpret this extent to which the plaintiff is favoured, which can be mapped from an objective measure such as months or pounds (cf fuzzy logic)
The truth conditions collectively form a highly modularised logic program or KB
The statements can be interpreted using fuzzy logic style connectives (AND is max, OR is min)

9. 2 regular ADFs
Just as any set of relations with arbitrary arity can be rewritten as a set of binary relations, and an arbitrary SAT problem can be rewritten as 3-SAT, any ADF can be rewritten as a 2-regular ADF.
All non-leaf nodes have exactly 2 children A A BC DE B C D E B D E C

10. 5 types of Node Characterised by their children
Two factors: standard factor based reasoning
One dimension and one factor: Factor provides a context which may affect switching point
Two Dimensions: May be normal fuzzy logic, but often involves trade offs
One factor: Simply carries up to parent (other node is a dummy (acting as TRUE)
One dimension Either as above, or provides a threshold (other node is a constant)

11. Two Dimensions P favoured as we move North-East Dim 2 increasingly
Favours P
Dim 1 increasingly
Favours P

12. Plaintiff owns SE, Defendant owns NW
P2 Result P2
P2 Rule
P1 Rule P3 P1
P1 Result
P3 for plaintiff will confirm and extend P1 rule
P3 for defendant will confirm and extend P2 rule

13. Slope may change Slope may change
But is likely to be smooth: no kinks for special cases

14. Relation to Values Values are usually taken to express preferences
This sometimes true
But often a law is motivated by multiple values:
Automobile exception requires both urgency and privacy to be considered
Neither enjoy a fixed preference
May depend on context,
Or change as we move around the 2-D space

15. Consideration of Values
The ADF allows consideration of all relevant values
Preferences can be local rather than global
AND and OR say whether both or either of the values should be promoted
Dimensions allow trade off between values
Consideration of a value may motivate a rule or an antecedent in a rule

16. Summary
Representation as 2-regular ADF allows consideration to be restricted to the 2-D case
There are 5 types of nodes, characterised by its children
Some, but not all, dimensions can be reduced to factors, as in Horty and Rigoni
Sometimes geometric reasoning rather than propositional logic or set comparison is needed (especially for trade offs)
Values are required to enable consideration of all aspects a law is intended to promote, as well as to express preferences