Implementing CUE

NOTE: this is a working document attempting to describe CUE in a way relatable to existing graph unification systems. It is mostly redundant to the spec. Unless one is interested in understanding how to implement CUE or how it relates to the existing body of research, read the spec instead.

CUE is modeled after typed feature structure and graph unification systems such as LKB. There is a wealth of research related to such systems and graph unification in general. This document describes the core semantics of CUE in a notation that allows relating it to this existing body of research.

Background

CUE was inspired by a formalism known as typed attribute structures [Carpenter 1992] or typed feature structures [Copestake 2002], which are used in linguistics to encode grammars and lexicons. Being able to effectively encode large amounts of data in a rigorous manner, this formalism seemed like a great fit for large-scale configuration.

Although CUE configurations are specified as trees, not graphs, implementations can benefit from considering them as graphs when dealing with cycles, and effectively turning them into graphs when applying techniques like structure sharing. Dealing with cycles is well understood for typed attribute structures and as CUE configurations are formally closely related to them, we can benefit from this knowledge without reinventing the wheel.

Formal Definition

A CUE configuration can be defined in terms of constraints, which are analogous to typed attribute structures referred to above.

Definition of basic values

A basic value is any CUE value that is not a struct (or, by extension, a list). All basic values are partially ordered in a lattice, such that for any basic value a and b there is a unique greatest lower bound defined for the subsumption relation a ⊑ b.

Basic values
null
true
bool
3.14
string
"Hello"
>=0
<8
re("Hello .*!")

The basic values correspond to their respective types defined earlier.

Struct (and by extension lists), are represented by the abstract notion of a typed feature structure. Each node in a configuration, including the root node, is associated with a constraint.

Definition of a typed feature structures and substructures

A typed feature structure_ defined for a finite set of labels Label is directed acyclic graph with labeled arcs and values, represented by a tuple C = <Q, q0, υ, δ>, where

  1. Q is the finite set of nodes,
  2. q0 ∈ Q, is the root node,
  3. υ: Q → T is the total node typing function, for a finite set of possible terms T.
  4. δ: Label × Q → Q is the partial feature function,

subject to the following conditions:

  1. there is no node q or label l such that δ(q, l) = q0 (root)
  2. for every node q in Q there is a path π (i.e. a sequence of members of Label) such that δ(q0, π) = q (unique root, correctness)
  3. there is no node q or path π such that δ(q, π) = q (no cycles)

where δ is extended to be defined on paths as follows:

  1. δ(q, ϵ) = q, where ϵ is the empty path
  2. δ(q, l∙π) = δ(δ(l, q), π)

The substructures of a typed feature structure are the typed feature structures rooted at each node in the structure.

The set of all possible typed feature structures for a given label set is denoted as 𝒞Label.

The set of terms for label set Label is recursively defined as

  1. every basic value: P ⊆ T
  2. every constraint in 𝒞Label is a term: 𝒞Label ⊆ T a reference may refer to any substructure of C.
  3. for every n values t₁, ..., tₙ, and every n-ary function symbol f ∈ F_n, the value f(t₁,...,tₙ) ∈ T.

This definition has been taken and modified from [Carpenter, 1992] and [Copestake, 2002].

Without loss of generality, we will henceforth assume that the given set of labels is constant and denote 𝒞Label as 𝒞.

In CUE configurations, the abstract constraints implicated by υ are CUE expressions. Literal structs can be treated as part of the original typed feature structure and do not need evaluation. Any other expression is evaluated and unified with existing values of that node.

References in expressions refer to other nodes within the C and represent a copy of the substructure C' of C rooted at these nodes. Any references occuring in terms assigned to nodes of C' are be updated to point to the equivalent node in a copy of C'.

The functions defined by F correspond to the binary and unary operators and interpolation construct of CUE, as well as builtin functions.

CUE allows duplicate labels within a struct, while the definition of typed feature structures does not. A duplicate label l with respective values a and b is represented in a constraint as a single label with term &(a, b), the unification of a and b. Multiple labels may be recursively combined in any order.

Definition of subsumption and unification on typed attribute structure

For a given collection of constraints 𝒞, we define π ≡C π' to mean that typed feature structure C ∈ 𝒞 contains path equivalence between the paths π and π' (i.e. δ(q0, π) = δ(q0, π'), where q0 is the root node of C); and 𝒫C(π) = c to mean that the typed feature structure at the path π in C is c (i.e. 𝒫C(π) = c if and only if υ(δ(q0, π)) == c, where q0 is the root node of C). Subsumption is then defined as follows: C ∈ 𝒞 subsumes C' ∈ 𝒞, written C' ⊑ C, if and only if:

  • π ≡C π' implies π ≡C' π'
  • 𝒫C(π) = c implies𝒫C'(π) = c and c' ⊑ c

The unification of C and C', denoted C ⊓ C', is the greatest lower bound of C and C' in 𝒞 ordered by subsumption.

Like with the subsumption relation for basic values, the subsumption relation for constraints determines the mutual placement of constraints within the partial order of all values.

Evaluation function

The evaluation function is given by E: T -> 𝒞. The unification of two typed feature structures is evaluated as defined above. All other functions are evaluated according to the definitions found earlier in this spec. An error is indicated by _|_.

Definition of well-formedness

We say that a given typed feature structure C = <Q, q0, υ, δ> ∈ 𝒞 is a well-formed typed feature structure if and only if for all nodes q ∈ Q, the substructure C' rooted at q, is such that E(υ(q)) ∈ 𝒞 and C' = <Q', q, δ', υ'> ⊑ E(υ(q)).

The evaluation of a CUE configuration represented by C is defined as the process of making C well-formed.

References

Theory:

  • [1992] Bob Carpenter, "The logic of typed feature structures."; Cambridge University Press, ISBN:0-521-41932-8
  • [2002] Ann Copestake, "Implementing Typed Feature Structure Grammars."; CSLI Publications, ISBN 1-57586-261-1

Some graph unification algorithms:

  • [1985] Fernando C. N. Pereira, "A structure-sharing representation for unification-based grammar formalisms."; In Proc. of the 23rd Annual Meeting of the Association for Computational Linguistics. Chicago, IL
  • [1991] H. Tomabechi, "Quasi-destructive graph unifications.."; In Proceedings of the 29th Annual Meeting of the ACL. Berkeley, CA
  • [1992] Hideto Tomabechi, "Quasi-destructive graph unifications with structure- sharing."; In Proceedings of the 15th International Conference on Computational Linguistics (COLING-92), Nantes, France.
  • [2001] Marcel van Lohuizen, "Memory-efficient and thread-safe quasi-destructive graph unification."; In Proceedings of the 38th Meeting of the Association for Computational Linguistics. Hong Kong, China.

Implementation

The evaluation of a CUE configuration C is defined as the process of making C well-formed.

This section does not define any operational semantics. As the unification operation is communitive, transitive, and reflexive, implementations have a considerable amount of leeway in choosing an evaluation strategy. Although most algorithms for the unification of typed attribute structure that have been proposed are near O(n), there can be considerable performance benefits of choosing one of the many proposed evaluation strategies over the other. Implementations will need to be verified against the above formal definition.

Constraint functions

A constraint function is a unary function f which for any input a only returns values that are an instance of a. For instance, the constraint function f for string returns "foo" for f("foo") and _|_ for f(1). Constraint functions may take other constraint functions as arguments to produce a more restricting constraint function. For instance, the constraint function f for <=8 returns 5 for f(5), >=5 & <=8 for f(>=5), and _|_ for f("foo").

Constraint functions play a special role in unification. The unification function &(a, b) is defined as

  • a & b if a and b are two atoms
  • a & b if a and b are two nodes, respresenting struct
  • a(b) or b(a) if either a or b is a constraint function, respectively.

Implementations are free to pick which constraint function is applied if both a and b are constraint functions, as the properties of unification will ensure this produces identical results.

References

A distinguising feature of CUE's unification algorithm is the use of references. In conventional graph unification for typed feature structures, the structures that are unified into the existing graph are independent and pre-evaluated. In CUE, the typed feature structures indicated by references may still need to be evaluated. Some conventional evaluation strategies may not cope well with references that refer to each other. The simple solution is to deploy a breadth-first evaluation strategy, rather than the more traditional depth-first approach. Other approaches are possible, however, and implementations are free to choose which approach is deployed.