Importation Closure that is Robust to Circular
Dependencies
Tara Athan
Athan Services (athant.com)
West Lafayette, IN, USA...
11 Jul 2013 2
Contents
● Motivation for Embracing Circular Importation
● Issues in the nteraction of Common Logic (CL)
Imp...
Motivation
● Distributed authoring of
knowledge bases
depends on merging of
smaller sets of
formulas or rules (texts)
● “T...
Current Standard CL Importation
Semantics: Has an Issue
● Original CL semantics
● {(ttl foo G),
(import foo)}
● I("(import...
CL Domain Restriction:
Proposed Replacement for
“cl:module”
● Text CL1
(txt
(inDiscourse N A)
(domain N
(outDiscourse A)
(...
CL Domain Restriction:
Syntactic Sugar (sort of)
● CL Text CL2:
(txt
(outDiscourse N A)
(domain N
(outDiscourse A)
(inDisc...
Interaction of CL Domain Restriction
and Importation
● Corpus CL3
{(ttl foo
(txt (outDiscourse A)
(forall (x) (A x)) ),
(t...
CL Importation – Proposed
Semantics by Importation Closure
● Original Text
● {(ttl foo (A)),
(import foo)}
● This approach...
Working With Infinite Importation
Closures
● Start with a finite
number of finite texts,
mentioning a finite
subset of the...
Language Family Specification
● Syntax - Minimal
● Some infinite lexical space
● Expressions: Propositions and some Weird ...
Syntax of L0
:
Propositions and Some Weird Things
● Statements are ...
● Propositional
Statements
– (A)
● Titling Statemen...
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 ...
Full Equivalence
● Corpora G1, G2 are superficially equivalent iff
some text in G1 is false iff some text in G2 is false
●...
Covers
● A corpus G1 is a cover of corpus G2 iff
G1 ≡F
G2'
where G2' is the importation closure of G2
● Significance: if G...
Algebraic Properties:
The Family L0
● Notation
– the text construction operator is called Q
– F, F0, F1, ... are text oper...
Algebraic Properties:
The Family L0
● Titling Separable
● Titling statements can be extracted from texts
{Q(G1, …, B, ...,...
Algebraic Properties:
Subfamily L0
● Text Operators are Compositionally Compact
● The closure under composition of a finit...
Algebraic Properties: L0
+
● Binary text construction forms a commutative, idempotent
monoid
– Q-associative (semigroup):
...
Algebraic Properties: L0
Ω
● Text Operators are distributive over polyadic text
construction
– F(Q(G1, G2, ...)) ≡F
Q( F(G...
Algebraic Properties: L0
+Ω
● Title-separable
● Compositionally-compact
●
Ω-
● Operators distribute over text construction...
Cover-Determination Algorithm
● Exract and simplify titling statements
● Simplify non-titling texts into “normal form”
Q(F...
Cover-Determination Algorithm
● Add to corpus if not fully-equivalent to any text
already in the corpus
● Repeat, applying...
Discussion
● It was discovered in the course of the analysis that
the original formulation of importation closure
could be...
Conclusions
● Importation that is robust to circular imports can
be defined theoretically and implemented
practically
● It...
References
● Information technology – Common Logic (CL): a
framework for a family of logic-based languages
● Neuhaus, F. a...
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Importation Closure that is Robust to Circular Dependencies

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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.

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Importation Closure that is Robust to Circular Dependencies

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 19. Algebraic Properties: L0 Ω ● Text Operators are distributive over polyadic text construction – F(Q(G1, G2, ...)) ≡F Q( F(G1), F(G2), ...)
  20. 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. 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. 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. 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. 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. 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 http://philebus.tamu.edu/pipermail/cl/attachmen ts/20130405/153ad554/attachment-0001.pdf

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