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

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

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

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.

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!

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

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

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)

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

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)

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

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

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

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

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

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