In this section, we compareRefl to other non-monotonic formalisms that have been
applied to regulation. Examples include those based ondefault logic [26, 134], logic programming [105, 136], and defeasible logic [56, 60]. We will use default logic to illus- trate various issues, but the remarks apply to other systems as well. In Section 2.5.1,
we describe how rules in default logic can be translated to Refl, and consider the
interaction of the consistency checks with connectives. Section 2.5.2 describes how
references to list items can be expressed inReflvia the same mechanism that is used
to handle such references. We then compare the use of priorities in default logic to
the says predicate (Section 2.5.3), and identify some directions for future work.
2.5.1
Default Rules in
Refl
The following is an example of a rule in default logic: ϕ : φ1, ..., φn
(id).o
ψ
The rule is read as “If ϕ, and φi isconsistent for all 1≤i≤n, then ψ. Default rules
can be directly translated toRefl:
(id).o:ϕ∧Vn
i=1¬saysID¬φi 7→ψ
φi isconsistent is translated as¬saysID¬φi, i.e., the rulesdo not say¬φi. In Reiter’s
default logic [126], the consistency check is w.r.t. all the rules, and hence, in Refl,
we use the set of all identifiers ID. A limitation of default logic is that it does not
allow the mixing of consistency checks with the other connectives in the logic. For
example, it is not possible to state the equivalent of ϕ∨ ¬saysID¬φ. This limitation
is further magnified when we consider the addition of temporal operators. While
we can describe exceptions involving an eventuality (¬saysID¬3φ) in default logic,
exceptions that hold eventually (¬3saysID¬φ) cannot be described directly.
2.5.2
Lists
Lists are frequent in the CFR, and are used in a variety of ways. We discuss a simple
example here to illustrate how thesayspredicate can be used to preserve the structure
of lists. The following is an expanded version of the obligation from the CFR: (12) Except as specified in (4) every donation of blood or blood component must
be tested for evidence of infection due to the following disease agents: a. HIV Type 1
b. HIV Type 2 c. Hepatitis B d. ...
(12) conveys an obligation to perform tests for the detection of a list of disease, and the exceptions can apply to tests for any of the diseases. Such lists can be easily
represented in Refl:
(12).o:d(x)∧says{12a,12b,12c}(list(y))∧ ¬says{4}(¬3test(x, y))7→3test(x, y) (12a) HIV Type1(z) 7→list(z)
(12b) HIV Type2(z)7→list(z) (12c) HepB(z)7→list(z) ...
The predicate list(z) is read as “z is listed”. HIV Type1(z) is read as “z is the
disease agent HIV Type1”. We assume that there is a unique object that satisfies HIV Type1(z), i.e., it denotes a constant symbol. The formula says{12a,12b,12c}(list(y)) accesses the items that are listed by (12a)-(12c) and can be understood as the trans-
lation of the phrase the following disease agents. The predicate test(x, y) is read as
“x is tested for the disease agent y”. We note that a richer representation is needed
to accomodate reasoning about events, such as, tests and tests for a particular pur-
pose. In Chapter 3, we describe a case study using a prototype implementation that
extends Refl with an object-oriented representation of events.
While the list in (12a)-(12c) is simple, it illustrates the applicability of says for
constructions other than exceptions. In this example, the identifiers let us draw inferences from particular list items, thereby allowing us to preserve the structure of
the list in logic. To our knowlege, this use of identifiers is novel to Refl. Previous
non-monotonic formalisms based on default logic [26, 134] and defeasible logic [56, 60] make use of identifers to specify priorities over rules, and we will discuss this in the
following example. However, the priorites are commonly specified in the meta-logic, which limits its applicability to cases where rules are in conflict. To refer to list items or definitions in particular sections, we need to parametrize predicates, such as, list, with the identifier. This results in the need to maintain identifiers in the object
language for lists, and in the meta-language for priorities. The says predicate offers
a unified mechanism for these kinds of inter-sentential references.
2.5.3
Priorities
Let us consider an example of nested exceptions to illustrate the need for priorities in default logic:
(13) Except as specified in (14), every donation of blood or blood component must be tested for evidence of infection due to Hepatitis B.
(14) Except as specified in (15), you are not required to test donations of source plasma for evidence of infection due to Hepatitis B.
(15) If a source plasma donation is used for treatment, it must be tested for Hep- atitis B.
Such nested exceptions arise, for example, when a regulatory document is modified over time. For example, the regulatory base may initially consist of only (13) and (14). At some point, the regulator may realize that source plasma donations have unforseen uses, and exceptions, such as (15), may be added. The sentences are represented in
Refl as follows:
(13).o:d(x)∧ ¬says{14}¬3test(x)7→3test(x)
(14).p:d(x)∧sp(x)∧ ¬says{15}3test(x)7→ ¬3test(x) (15).o:d(z)∧sp(z)∧uf t(z)7→3test(x)
The predicate uf t(z) is read as “z is used for treatment”. And, the other predicates
donation sp(o) and used for treatment uf t(o). At the least fixed point, we obtain utterances from (13) and (15) – (13,3test(o)) and (15,3test(o)). If a donation is not used for treatment, we obtain utterances from either (13) or (14), as we discussed in Section 2.3.2.
These statements are represented in default logic without priorities as follows: d(x) : 3test(x) (13).o 3test(x) d(x)∧sp(x) : ¬3test(x) (14).p ¬3test(x) d(z)∧sp(z)∧uf t(z) (15).o 3test(x)
Since the consistency checks are w.r.t. all rules, in the case of a non-source plasma donation, the preconditions of both (13) and (14) will be ungrounded at the least fixed point. To avoid this, an order is specified over the rules [26], e.g., (15)>(14)>(13). With the order over rules, the statements are interpreted in an analogous manner to
the corresponding statements in Refl above. The use of identifiers, as a parameter
to the sayspredicate, lets us avoid the need for a separate specification of priorities.
We conclude this section by discussing some limitations of Refl. While the
says predicate generalizes default logic with priorities, rules have to refer to their
exceptions. However, there are examples in the CFR where the exceptions refer to the rule. Consider the following paraphrase of (14) and (15):
(16) You are not required to test donations of source plasma for evidence of infec- tion due to Hepatitis B.
(17) If a source plasma donation is used for treatment, exemption (16) no longer applies.
(16) and (17) would be expressed in Refl in a similar manner to (14) and (15), i.e.,
there would be a reference from (16) to (17) using says. However, the phrasing of
the sentences suggest that (17) refers to (16). Such references commonly arise in the
the permission provided by (16). Governatori and Rotolo [58, 59] analyse abrogation and annulment by using a temporal version of defeasible logic. The analysis involves the use of meta-rules which abrogate or annul other rules. However, the presence of
variables in Refl make matters more complicated. Only a subset of the utterances
produced are annulled. For example, (17) annuls the utterances produced by (16) for source plasma donations that are used for treatment. We speculate that a solution
in Reflwill involve the formalization of the word applies in (17) and its connection
tosays. We leave an investigation to future work.