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

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

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

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

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)))

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.

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

CL Importation – ProposedSemantics 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)), ...

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.

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)

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 G

1 G

2 ... G

n)

Unary Text Operators– (F0 G)

Corpora are sets of texts

– G1, G

2, G

3, ...

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

Full Equivalence

Corpora G1, G2 are superficially equivalent iffsome 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

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

Algebraic Properties: The Family L

0

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) ≡

FF1(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)

Algebraic Properties: The Family L

0

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*)

Algebraic Properties:Subfamily L

0

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

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)

Algebraic Properties: L0

Ω

Text Operators are distributive over polyadic text construction

– F(Q(G1, G2, ...)) ≡F Q( F(G1), F(G2), ...)

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

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 G0where 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

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

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)))

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 ...)

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 Semanticshttp://philebus.tamu.edu/pipermail/cl/attachments/20130405/153ad554/attachment-0001.pdf