Resolving counterintuitive consequences in law using legal debugging

Abstract

There are cases in which the literal interpretation of statutes may lead to counterintuitive consequences. When such cases go to high courts, judges may handle these counterintuitive consequences by identifying problematic rule conditions. Given that the law consists of a large number of rule conditions, it is demanding and exhaustive to figure out which condition is problematic. For solving this problem, our work aims to assist judges in civil law systems to resolve counterintuitive consequences using logic program representation of statutes and Legal Debugging. The core principle of Legal Debugging is to cooperate with a user to find a culprit, a root cause of counterintuitive consequences. This article proposes an algorithm to resolve a culprit. Since the statutes are represented by logic rules but changes in law are initiated by cases, we adopt a prototypical case with judgement specified by a set of rules. Then, to resolve a culprit, we reconstruct a program so that it provides reasons as if we applied case-based reasoning to a new set of prototypical cases with judgement, which include a new set of facts relevant to a considering case.

Appendix: First-order definitions

Throughout this article, we present most definitions based on propositional representation for ease of exposition. In “Appendix”, we present some revision of definitions when considering a first-order representation.

First of all, since each claim must have a rule, we must add one constraint to a program that for every rule atom A occurring in a program, there must be a rule R in a program such that head(R) is substitutable to A.

We extend AA-CBR to support a first-order case base by introducing an abstraction to deal with a first-order case base as follows.

Definition 11

(Abstraction and subsumption)

• Let B be an atom, a set of atoms, a tuple of atoms, or a rule. An abstraction of B, denoted by abs(B), is obtained by constructing an injective mapping $\{c_1/v_1,\ldots ,$ $c_n/v_n\}$ for all constants in B and replacing every constant $c_i$ in B by a new distinct variable $v_i$ $(1 \le i \le n)$. For instance, $abs({}) = {}$.

• Let C and D be a set of atoms. We say C subsumes D () if there is a substitution $\theta$ such that $C\theta \subseteq D$ (Plotkin 1970). We denote such $C\theta$ by $[C]_D$. We say C is substitutable to D ($C \sqsupseteq D$) if there is a substitution $\theta$ such that $C\theta = D$. C non-substitutably subsumes D () if but $C \not \sqsupseteq D$.

Let CB be a finite set of first-order cases with judgement called a first-order case base. We assume that a first-order case base is consistent, that is no pair $(X,J_x),(Y,J_y) \in CB$ such that $abs(X) \sqsupseteq Y$ but $J_x \ne J_y$. We say a case with judgement $(N,{\bar{J}})$ is a trumping case with judgement to (C, J) w.r.t. CB when , and no other $(N',{\bar{J}}) \in CB$ such that and .

We also extend how AA-CBR predicts a judgement for a case N (called an analogous prediction) by constructing an analogous case base of CB w.r.t. N, denoted by .

An analogously predicted judgement of N is $J_0$ if $(\{\},J_0)$ is in the ground extension of an argumentation framework corresponding to $CB_N$. Otherwise, an analogously predicted judgement of N is $\bar{J_0}$ .

A theory construction for a first-order case base is revised as follows.

Definition 12

(Theory-cons in first-order) Given a ground atom $g_0$ with a rule predicate, a first-order critical case base CB with a default judgement $`{-}'$, and (CB, attacks) as its corresponding argumentation framework, is defined by

• (1) Let $pos_t(X,Y) = X\setminus [abs(Y)]_X$.

• (2) Let $head_t(X,Y)$ be a ground atom with a new reified rule predicate $p_{X,Y}$ and arguments from all arguments occurring in $pos_t(X,Y)$.

• (3) Let $var_t(A,B,C) = A_2$ when there is a substitution $\theta$ such that $abs(\langle A,B\rangle )\theta = (\langle A_2,[abs(B)]_C\rangle )$.

• (4) Let $neg_t(X,Y)= \{not~var_t(head_t(Z,X),pos_t(X,Y),Z)| (X,J) \leadsto (Y,{\bar{J}}) \in attack$ and $(Z,{\bar{J}}) \leadsto (X,J) \in attack\}$.

• (5) Then, .

Theorem 2

Let CBbe a first-order case base and $g_0$be a ground rule atom. is an analogous theory of CB w.r.t. $g_0$, namely an analogously predicted judgement of a case Nis $`{+}'$if and only if $T \cup N \vdash g$for some$abs(g_0) \sqsupseteq g$.

Proof

Let $CB_N$ be an analogous case base of CB w.r.t. N, $T_N$ be a program T inserting all ground fact atoms from C as fact rules, P be a ground instantiation of $T_N$, and $P_N = \{R \in P| pos(R) \subseteq N\}$. We get that there is g for some $abs(g_0) \sqsupseteq g$ such that $CB_N$ is an analogous case base of . Hence, is an analogous theory of CB w.r.t. $g_0$ $\square$

In Algorithm 2, one point that needs to be revised is to construct a new rule using $abs(p :- N)$ and $abs(Culp :- N\setminus [abs(C)]_N)$ instead of $p :- N$ and $Culp :- N\setminus C$ respectively. We also get a similar theorem to Theorem 1 with a proof in the same manner.

Theorem 3

Given a first-order program Trepresenting statutes, an initial counterintuitive consequence pand a case with judgement $cj_n = (N,J)$where Nis a set of relevant facts and Jis $`{+}'$if p shall be valid and $`{-}'$otherwise. Let . If , then the reconstructed theory $T'$according to Algorithm 2is ananalogous theory of $CB' = CB \cup \{cj_n\}$.

Proof

We get that since for $cj_n$, new rules are introduced corresponding to the definition of (Definition 12) for all $(C,{\bar{J}}) \in CB$ such that $cj_n$ is a trumping case with judgement to $(C,{\bar{J}})$ w.r.t. CB. Since $T' =$, $T'$ is an analogous theory of $CB'$. $\square$