April 23, 2015

The Cyc Canonicalizer

It is useful to consider two representational levels of knowledge in Cyc. Before knowledge is exploited by Cyc to solve problems it must first be conceived and articulated and then stored internally as data structures accessible by the inference engine. Knowledge is articulated (e.g., by human knowledge engineers, by a front end NL module that is processing text, by a machine learning program, …) at the epistemological level (EL) as formulae in a notation very similar to first-order logic (FOL). Knowledge is stored internally at the heuristic level (HL), primarily as formulae in conjunctive-normal form (CNF).

FOL is appropriate for the EL because it is widely known and used among knowledge engineers, and it is both expressive and precise. CNF is appropriate for the HL because it both maximizes the modularity of the knowledge and supports fast inference algorithms (e.g., clause resolution). Separating the articulation representation from the internal representation benefits both the knowledge engineers, who need only learn a stable representation language, and the system designers, who are free to modify the internal data structures as needed without changing the EL and thus without disrupting the efforts of the knowledge engineers. The translation of formulae from the El to the HL is performed automatically by the CycL canonicalizer.

Canonicalization Objectives:

(a) compression: mapping equivalent EL formulae into a common HL construct

(b) correctness: only equivalent EL formulae are mapped into a common HL construct

(c) speed

The raison d’etre of the canonicalizer is to soundly translate equivalent EL formulae into a single HL construct. This avoids redundantly adding one formula that is simply a rephrasing of another formula. For example, the following are all equivalent EL formulae that map into a single HL construct:

(#$implies     (#$and (#$isa ?x #$Dog) (#$isa ?y #$Dog) (#$different ?x ?y))    (#$or (#$sociallyDominates ?x ?y) (#$sociallyDominates ?y ?x)))  (#$implies     (#$and (#$isa ?x #$Dog) (#$isa ?y #$Dog) (#$different ?y ?x))    (#$or (#$sociallyDominates ?x ?y) (#$sociallyDominates ?y ?x)))   (#$implies     (#$and (#$isa ?y #$Dog) (#$isa ?x #$Dog) (#$different ?x ?y))    (#$or (#$sociallyDominates ?x ?y) (#$sociallyDominates ?y ?x)))  (#$implies     (#$and (#$isa ?y #$Dog) (#$isa ?x #$Dog) (#$different ?y ?x))    (#$or (#$sociallyDominates ?y ?x) (#$sociallyDominates ?x ?y)))  (#$or    (#$sociallyDominates ?x ?y)     (#$sociallyDominates ?y ?x)    (#$not (#$isa ?x #$Dog))    (#$not (#$isa ?y #$Dog))    (#$not (#$different ?x ?y)))

The second differs from the first only in the order of the arguments to the symmetric predicate #$different. The third differs from the first only in the order of the antecedent literals. Achieving a single HL construct also involves canonicalizing the order of arguments to a commutative relation, in this case #$and. The fourth differs from the first only by variable naming. The fifth differs from the first by a few applications of standard logical transformations (e.g., DeMorgan’s Law). Attempting to add all five formulae to the Cyc KB results in only one HL construct.

An important secondary objective of the canonicalizer is speed; it is not uncommon for users to load transcripts containing thousands of EL formulae, each of which must be canonicalized. Therefore, canonicalization must be as fast as possible. However, the goal of efficient translation is directly at odds with the goal of detecting and removing duplication (which can involve arbitrary theorem proving). To balance this trade-off, the canonicalizer uses a limited set of rules for simplifying and transforming formulae that has been tuned with hundreds of thousands of examples; not all equivalent formulae simplify to the same expression. These rules reflect a pragmatic compromise between efficiently translating EL formulae into HL constructs and detecting and eliminating duplication, and they are truth-preserving. Thus the canonicalizer does not (cannot) guarantee to map all equivalent EL formulae into a single HL construct (after all, FOL is undecidable); however, it can guarantee soundness (only truly equivalent formulae are mapped into a common HL construct), and it can guarantee a relatively fast response.

Implementation note: canonicalizing the order of arguments to commutative relations obviates rules that implement commutativity, such as:

(#$implies (#$different ?x ?y) (#$different ?y ?x))

This requires that every component of the inference engine (n.b. the KB indexing mechanisms) assume responsibility to correctly handle commutativity.

Canonicalizer components:

(a) critic: test for syntactic and semantic well-formedness

(b) translation pre-processor: handle exceptions

(c) translator: map assertions into CNF, queries into DNF

(d) translation post-processor: handle rule macros

The critic ensures that a given EL formula is both syntactically and semantically well-formed, as defined by the CycL specification. The syntactic check simply involves parsing the formulae with the BNF grammar. The semantic check involves enforcing both arity constraints and argument constraints (e.g., #$arg1Isa). Formulae that are not syntactically well-formed canonicalize to nil; formulae that are not semantically well-formed canonicalize to #$True, #$False, or nil.

Relations in CycL are declared to be of either fixed or variable arity. The arity of fixed arity relations is declared and constant (but may be microtheory specific). Variable arity relations (e.g., #$and, #$different) accept zero or more arguments. The critic ensures the semantic constraint that each relation referenced in a formula has a declared arity (variable or a non-negative integer) and that the number of arguments satisfies the arity.

Relations in CycL are typed; one or more typing constraints can be defined for each argument. Furthermore, the type of the result of functions in CycL can be defined. All relation typing declarations are microtheory-specific. The critic invokes the arg-type module to insure that all argument-typing constraints applicable to a formula are satisfied.

Implementation note: because the critic enforces microtheory-specific constraints, the behavior of the canonicalizer is microtheory-specific. That is, a formula is canonicalized with respect to a designated microtheory, and canonicalizing the same formula in two different microtheories may produce different results.

After ensuring that an EL formula is well-formed, the canonicalizer transforms the formula into an HL construct. This transformation basically converts a FOL formula into clausal normal form (e.g., CNF). However, special support for particular syntactic constructs require special processing. The translation pre-processor handles those special transformations that are easiest to accomplish prior to conversion to clausal normal form; the post-processor handles those special transformations easiest to accomplish after conversion to clausal normal form.

Two special constructs handled by the pre-processor are exceptions: #$ExceptWhen and #$ExceptFor literals are transformed into #$abnormal assertions. For example

(#$ExceptWhen <formula-1> <formula-2>)

denotes that <formula-2> is default true, but is not warranted when <formula-1> holds; this formula is transformed by the pre-processor into

(#$implies <formula-1> (#$abnormal <variables> <kb-assertion>))

where <variables> are the EL variables appearing in <formula-2>, and <kb-assertion> is the HL construct, already asserted in the Cyc knowledge base, derived from canonicalizing <formula-2>.

The translator maps formulae into clausal normal form: assertions are mapped into CNF; queries are mapped into DNF. This uses a standard algorithm (see “Logical Foundations of Artificial Intelligence” by Genesereth and Nilsson); the approach is outlined below as six steps:

(1) remove implications:

(#$implies <formula-1> <formula-2>) -> (#$or (#$not <formula-1>) <formula-2>)

(2) distribute negations

(#$not (#$and <formula-1> ...)) -> (#$or (#$not <formula-1>) ...) (#$not (#$or <formula-1> ...)) -> (#$and (#$not <formula-1>) ...) (#$not (#$forAll <var> <formula>)) -> (#$thereExists <var> (#$not <formula>))  (#$not (#$thereExists <var> <formula>)) -> (#$forAll <var> (#$not <formula>))

(3) standardize variables

(#$forAll <var> <formula>) -> (#$forAll <unique var name> <formula>) (#$thereExists <var> <formula>) -> (#$thereExists <unique var name> <formula>)

(4) remove existential quantifiers

Substitute a unique skolem function in for every existentially quantified formula.

(5) remove universal quantifiers

Since every variable is now unique and universally quantified, all universal quantifiers can be dropped.

(6) distribute disjunctions and conjunctions to put into CNF or DNF

These six steps produce one or more clauses for each well-formed EL formula being canonicalized. The translation post-processor performs a series of additional transformations that are convenient to apply to clauses.

The first step of the post-processor handles a class of specially-supported constructs referred to as “rule macros”. Each rule macro is a compressed construct (typically a single literal) that denotes or expands into a non-atomic (e.g., disjunctive) formula. Typically, there are special inference modules that support each rule macro. For example, the formula

(#$implies (#$isa ?x #$Poodle) (#$isa ?x #$Dog))

is transformed by the translator into the CNF:

(#$or (#$not (#$isa ?x #$Poodle)) (#$isa ?x #$Dog))

which is then transformed by the post-processor into the literal:

(#$genls #$Poodle #$Dog).

Here, #$genls is a rule macro; (#$genls ?x ?y) denotes, or expands into,

(#$implies (#$isa ?x #$Poodle) (#$isa ?x #$Dog))

The inference engine has a special HL module that reasons efficiently with #$genls literals (see [:genls HL module]).

The next step of the of the translator post-processor is to canonicalize the components of the clauses. This involves three steps:

(a) canonicalizing the order of commutative arguments, including a decomposing heuristic technique for canonicalizing the order of non-atomic terms (e.g., literals) and using a simple lexical technique for ordering atomic terms

(b) reifying function terms

(c) replacing EL variables with HL variables (aka KB variables or inference variables)

The first step is performed in order to avoid redundantly adding to the KB two formulae which simply rephase each other. The second step is performed in order to support efficient indexing of assertions in the KB. The third step is performed in order to support an effient unification algorithm.

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