The ISO Standard 24707 for Common Logic is under revision to fix certain issues in its semantics. One of these issues is in regard to the interpretation of circular importations, which is syntactically valid but semantically ambiguous. Elsewhere we have proposed a modification to the importation semantics that solves the theoretical problem. However, the importation closure of some finite collections (corpora) of texts leads to infinite corpora. For practical reasoning, we can work with finite covers- finite corpora that are equivalent in a particular sense to the original corpus. In this study we derive algebraic conditions for the applicability of an finite-cover determination algorithm to a general model-theoretic language implementing this approach to importation closure.

1. Importation Closure that is Robust to Circular
Dependencies
Tara Athan
Athan Services (athant.com)
West Lafayette, IN, USA
taraathan AT gmail.com
presented at RuleML 2013
7th
International Rule Challenge
July 11, 2013
Seattle, USA

2. 11 Jul 2013 2
Contents
● Motivation for Embracing Circular Importation
● Issues in the nteraction of Common Logic (CL)
Importation and Domain Restriction
● Sandbox Language Family Specification
● Syntax, Semantics and Algebraic Properties
● Algorithm
● Discussion
● Conclusions
● References

3. Motivation
● Distributed authoring of
knowledge bases
depends on merging of
smaller sets of
formulas or rules (texts)
● “Titling + Importation”
is a standard approach
● Assign a title to a text
● Refer to the title in an
importation statement
● Circular importation
references are
problematic
● Resolution by copying
(e.g. XIncude) can lead
to infinite loops
● Other semantics can
be ambiguous
● Forbidding circular
dependence
– Heavy burden on
syntactic validation

4. Current Standard CL Importation
Semantics: Has an Issue
● Original CL semantics
● {(ttl foo G),
(import foo)}
● I("(import foo)") =
I( G)
● Problem
● {(ttl foo (import foo)),
(import foo)}
● I("(import foo)") =
I("(import foo)")
● Fails to define a truth-
value for the second
text
● Possible Solutions
● Ignore multiple importations
● Forbid circular importation
● Redefine importation semantics

5. CL Domain Restriction:
Proposed Replacement for
“cl:module”
● Text CL1
(txt
(inDiscourse N A)
(domain N
(outDiscourse A)
(inDiscourse a)
(forall (x) (A x))
) )
● CL1 Logically
Equivalent to:
(txt
(inDiscourse N A)
(not (N A))
(N a)
(forall (x)
(if (N x) (A x))
)

6. CL Domain Restriction:
Syntactic Sugar (sort of)
● CL Text CL2:
(txt
(outDiscourse N A)
(domain N
(outDiscourse A)
(inDiscourse a)
(forall (x) (A x))
) )
● CL2 Logically
equivalent (≡) to:
(txt
(outDiscourse N A)
(N a)
(forall (x)
(if (N x) (A x))
)
● There is not a context-independent way to rewrite
the domain restriction statement.

7. Interaction of CL Domain Restriction
and Importation
● Corpus CL3
{(ttl foo
(txt (outDiscourse A)
(forall (x) (A x)) ),
(txt (inDiscourse N1 A)
(domain N1
(import foo))),
(txt (inDiscourse N2 A)
(domain N2
(import foo)))}
●
CL3 ≡ to:
(txt
(inDiscourse N1 N2)
(not (N1 A))
(forall (x)
(if (N1 x) (A x)
(not (N2 A))
(forall (x)
(if (N2 x) (A x)))
● Approach of ignoring duplicate importations does
not preserve intended semantics, is ambiguous

8. CL Importation – Proposed
Semantics by Importation Closure
● Original Text
● {(ttl foo (A)),
(import foo)}
● This approach solves
some problems:
– {(ttl foo (import foo)),
(import foo)}
● But not all problems:
– {(ttl foo (domain N
(import foo))),
(import foo)}
● Importation Closure
● {(ttl foo (A)),
(import foo),
(A)}
● I("(import foo)") = T
– {(ttl foo (import foo)),
(import foo)}
– {(ttl foo (domain N
(import foo))),
(import foo),
(domain N
(import foo)), ...

9. Working With Infinite Importation
Closures
● Start with a finite
number of finite texts,
mentioning a finite
subset of the
vocabulary/signature
● Want to determine
satisfaction from a the
truth values of a finite
set of expressions,
even if importation
closure is infinite
● Goal of Analysis
● Determine conditions
on a general model-
theoretic language
using importation
closure such that given
an interpretation,
satisfaction of a finite
set of finite texts can
be evaluated in finite
time.

10. Language Family Specification
● Syntax - Minimal
● Some infinite lexical space
● Expressions: Propositions and some Weird Things
● Semantics – Beyond Minimal
● Corpus Satisfaction by Importation Closure
● Interpretation of Expressions:
It's True if I Say It Is
● Algebraic Properties – Where the Action Is
● Rewriting of Expressions that Preserve "Full
Equivalence" (≡F
)

11. Syntax of L0
:
Propositions and Some Weird Things
● Statements are ...
● Propositional
Statements
– (A)
● Titling Statements
– (ttl foo G)
● Importation Statements
– (import foo)
● Texts are statements
and ...
● Polyadic Text
Construction
– (txt G1
G2
... Gn
)
● Unary Text Operators
– (F0 G)
● Corpora are sets of
texts
– {G1
, G2
, G3
, ...}

12. Semantics of L0
:
It's True if I Say It's True
● An “interpretation” I is
a specification of all
true texts
● A corpus is “satisfied”
by I if all texts in its
importation closure are
true in I
● A text is “satisfied” by I
if a corpus containing
only that text is
satisfied by I
● A corpus G is “self-
contained” if it has
“enough” titling
statements to
determine a
“canonical” importation
closure G'
● Corpora G1, G2 are
logically equivalent iff
G1 is satisfied exactly
when G2 is satisfied

13. Full Equivalence
● Corpora G1, G2 are superficially equivalent iff
some text in G1 is false iff some text in G2 is false
● Two corpora are fully equivalent (≡F
) iff they are
logically equivalent and superficially equivalent

14. Covers
● A corpus G1 is a cover of corpus G2 iff
G1 ≡F
G2'
where G2' is the importation closure of G2
● Significance: if G1 is a cover of G2, the truth values
in an interpretation I of the texts in G1 (not its
importation closure G1') determine the satisfaction
of G1 by I, and hence, the satisfaction of G2 by I
● Task: determine algebraic properties of language
that permit algorithmic determination of a finite
cover for any self-contained finite corpus

15. Algebraic Properties:
The Family L0
● Notation
– the text construction operator is called Q
– F, F0, F1, ... are text operators
– B is a titling text, G, G1, G2, ... are any texts
– F0 ≡F
F1 iff F0(G) ≡F
F1(G) for all G
● Composition of Text Operators is Closed
{F0(F1(G)))} ≡F
{(F0 o F1)(G)} = {(F2)(G)}
● Composition of Text Operators is Associative
{((F0 o F1) o F2)(G)} ≡F
{(F0 o (F1 o F2))(G)}

16. Algebraic Properties:
The Family L0
● Titling Separable
● Titling statements can be extracted from texts
{Q(G1, …, B, ..., Gn)} ≡F
{B, Q(G1, ..., Gn)}
{F(Q(G1, …, B, ..., Gn))} ≡F
{F(B), F(Q(G1, ..., Gn))}
● Substitution
● If {Gi} ≡F
{G*}
● Then
– {Q(G1, …, Gi, ..., Gn)} ≡F
{Q(G1, .. ,G*, ... Gn)}
– {F(Gi)} ≡F
{F(G*)}

17. Algebraic Properties:
Subfamily L0
● Text Operators are Compositionally Compact
● The closure under composition of a finite set of text
operators is finite.
– Given F1, ...FN, there exists F1, ... FM, (M>=N) such that
– Fi o Fj ≡F
Fk
– Whenever 1 <= i, j, k, <= M

18. Algebraic Properties: L0
+
● Binary text construction forms a commutative, idempotent
monoid
– Q-associative (semigroup):
Q(G0, Q(G1, G2)) ≡F
Q( Q(G0, G1), G2)
– Q-commutative: Q(G0, G1) ≡F
Q( G1, G2)
– Q-identity: (monoid) Q(G0, Q()) ≡F
G0
– Q-idempotent: Q( G0, G0)) ≡F
G0
● Polyadic Q is the composition of binary Q
– Q(G0, Q(G1, G2)) ≡F
Q( G0, G1, G2)

19. Algebraic Properties: L0
Ω
● Text Operators are distributive over polyadic text
construction
– F(Q(G1, G2, ...)) ≡F
Q( F(G1), F(G2), ...)

20. Algebraic Properties: L0
+Ω
● Title-separable
● Compositionally-compact
●
Ω-
● Operators distribute over text construction
● Commutative
● Idempotent
● Monoid
● Text construction is associative
● Empty text construction is identity

21. Cover-Determination Algorithm
● Exract and simplify titling statements
● Simplify non-titling texts into “normal form”
Q(F0(G0), ..., Fn(Gn), Gn+1, ..., Gm) or F(G0) or G0
where Gi is a propositional or importation statement
● Pick one importation statement, and find titling
statements for associated title
– No titling statement? Try again later (might be imported)
– Inconsistent titling statements? Is unsatisfiable
– Otherwise, continue ...
● Create new text by substitution for importation

22. Cover-Determination Algorithm
● Add to corpus if not fully-equivalent to any text
already in the corpus
● Repeat, applying once to each importation
statement, including those added to the corpus by
importation
● Given properties of L0
+Ω
, guaranteed to terminate
● Were any importation statement not resolved?
– Yes --> No finite cover exists, corpus is not self-contained
– No --> finite corpus obtained is cover

23. Discussion
● It was discovered in the course of the analysis that
the original formulation of importation closure
could be improved
● Originally all importation statements in a text were
resolved at once, because it seemed more efficient
– This leads to texts that are theoretically self-contained but
difficult to resolve in practice (if the titling statement for one
title is contained in a text to be imported at the same time)
– This also has some non-intuitive consequences regarding
satisfaction of segregation requirements.
● (txt (import M) (import N)(ttl P (import N))) is not equivalent to
● (txt (import M) (import P) )(ttl P (import N)))

24. Conclusions
● Importation that is robust to circular imports can
be defined theoretically and implemented
practically
● It is still best practice, from a performance point of
view, to avoid or minimize circular imports
● Further study will include application to CL
extension IKL
● For application to nonmonotonic logic within a
monotonic wrapper
● Expect need for truncation (Ibelieve he believes ...)

25. References
● Information technology – Common Logic (CL): a
framework for a family of logic-based languages
● Neuhaus, F. and P. Hayes, Common Logic and the
Horatio problem, Appl. Ontol. v. 7 pp. 211-231
● CL Draft Semantics
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