Typing Graph Manipulation Operations

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Typing Graph Manipulation Operations

• Jan Hidders
• University of Antwerp

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Outline

• Data Model
• Instance Graph / Schema Graph / Extension
  Relation
• Language
• Pattern / Addition / Deletion / Reduction
• Typing the Operations
• Patterns / Additons / Deletions / Reduction
• Further Research

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Instance Graph
Data Graph
Well-formed Data Graph

• Nodes are labeled or reachable from labeled nodes.

• No sharing of composite-value nodes

Instance Graph

• No duplicates of basic values

• No local duplicates of composite values

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Schema Graph

• Like Data Graph, but
• At most one class label
• No basic-value labels
• ISA edges
• No well-formedness
• E.g. sharing of comp. val.

• No constraints on attributes

• Recursive types allowed

• ISA between comp. values

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Extension Relation
Data Graph
Schema Graph
Def. An extension relation satisfies
class-name rule related if labeled with the same
class name.
attribute rule related if required by attr. type
in schema graph
isa rule related if required by attr. type in
schema graph
Def. A data graph D belongs to schema graph S if
the minimal extension relation - relates only
nodes with the same sort, - implies no extra
class names and - lets S cover D.
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Pattern

• Like Well-formed Data Graph, but
• Allow basic-value nodes w/o basic value label
• is edges

but no cycles of comp. val. nodes.
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Addition
An addition consists of a base pattern
and an extension pattern.

• Semantics For each embedding of the base pattern
  the data graph is extended such that the
  extension pattern also embeds with
• distinct extensions for each embedding
• distinct nodes for each additional node in ext.
  pattern.

• Well-formed addition
• only is edges from old to new nodes
• no is edges that merge old or new values
• no potentially recursive is edges

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Deletion
A deletion consists of a base pattern
with marked class names, nodes and edges.
Semantics For each embedding of the base pattern
the marked class names, nodes and edges are
removed.If nodes become unreachable they are
also removed.
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Reduction
Purpose Turn well-formed data graphs into
instance graphs. (no duplicates of basic values,
no local duplicates of composite values).

• Semantics Merge nodes if they
• represent the same basic value or
• represent the same composite value and are
  siblings (i.e., in the same attribute or named
  class),
• until no more nodes are merged.

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Typing Patterns
Problem Is there a well-formed data graph that
belongs to schema graph S in which the pattern J
embeds?
Note In the following object-class nodes are
assumed to be labeled.

• Without is edges
• Def. An extension relation from S to J supports J
  if
• - it relates only nodes with the same sort,
• - it is minimal on composed-value nodes and
• - it lets S cover J.
• Th. Existence of this is sufficient and necessary
  condition.
• Th. Problem in PTIME

• With is edges
• Extra requirements necessary for supp. ext. rel.
• Th. Problem coNP-complete

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Typing Additions (1/2)

• Problem Addition stays within schema graph?
• Without is edges

• Def. An addition is well-typed if for all the
  extension relations that support the base pattern
  it holds for their minimal extension to the
  extension pattern that it
• respects the sorts of the nodes,
• lets the schema graph cover the extension pattern
  and
• prevents the above three problems.

Th. Sufficient (but not necessary). Th. Deciding
well-typedness PSPACE complete.
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Typing Additions (2/2)
With is edges
Extensions caused by is edges cause similar
problems. These can be solved by adding extra
constraints for well-typedness. Th. Deciding
well-typedness still PSPACE complete.
Without composite-value nodes
Th. Deciding well-typedness in PTIME.
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Typing Deletions
Problem Deletion stays within schema graph?
Deletion
Schema Graph

• Def. A deletion is well-typed if for all
  combinations of deletions of class names for a
  single node
• the node is never forced back into a deleted
  class and
• the composite-value tree under this node remains
  covered.

Th. Sufficient (but not necessary). Th. Deciding
well-typedness PSPACE complete.
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Typing Reductions
Problem Reduction stays within schema graph?
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Further Research

• Find necessary well-typedness conditions.
• Investigate role of isa edges between
  composite-value class nodes.
• Extend for XML.