Formal Specification of Cypher

Semantics Overview Ambiguous and Edge Cases Incomplete and Inconsistent Cases
Formal Speciﬁcation of Cypher
Nadime Francis
University of Edinburgh
Wednesday, May, 10th
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Property Graphs
Person, Postdoc
name : ‘Nadime’
institute : ‘UoE’
Person, Professor
name : ‘Leonid’
institute : ‘UoE’
knows
since : 2010
colleague
since : 2015
A property graph is a tuple G = (N, R, s, t, ι, λ, τ), where:
N ⊆ N: ﬁnite set of nodes
R ⊆ R: ﬁnite set of relationships
s : R → N: maps each relationship to its source
t : R → N: maps each relationship to its target
ι : (N ∪ R) × K → V: maps each x and k to x.k.
λ : N → 2L: associates a set of label to each node
τ : R → T : associates a type to each relationship
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Records and Tables
A record is a tuple with named ﬁelds : (a1 : v1, . . . , an : vn).
A table is a bag of uniform records.
Example:
(a : 1, b : 3), (a : ‘oCIM 2’, b : ‘London’),
(a : ‘oCIM’, b : ‘Walldorf’), (a : 1, b : 3)
a b
1 3
‘oCIM 2’ ‘London’
‘oCIM’ ‘Walldorf’
1 3
=
a b
‘oCIM’ ‘Walldorf’
1 3
‘oCIM 2’ ‘London’
1 3
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Operations and Expressions
An Example
MATCH (n : Person) − [: knows]−> (m : Person)
WHERE n.institute = m.institute
RETURN n.name, m.name, n.institute AS institute
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An Example
n m
{name : ‘Nadime’, institute : ‘UoE’} {name : ‘Leonid’, institute : ‘UoE’}
{name : ‘Paolo’, institute : ‘UoE’} {name : ‘Nadime’, institute : ‘UoE’}
{name : ‘Nadime’, institute : ‘UoE’} {name : ‘Stefan’, institute : ‘Neo’}
{name : ‘Alastair’, institute : ‘Neo’} {name : ‘Stefan’, institute : ‘Neo’}
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An Example
n m
{name : ‘Nadime’, institute : ‘UoE’} {name : ‘Leonid’, institute : ‘UoE’}
{name : ‘Paolo’, institute : ‘UoE’} {name : ‘Nadime’, institute : ‘UoE’}
{name : ‘Alastair’, institute : ‘Neo’} {name : ‘Stefan’, institute : ‘Neo’}
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An Example
n.name m.name institute
‘Nadime’ ‘Leonid’ UoE
‘Paolo’ ‘Nadime’ UoE
‘Alastair’ ‘Stefan’ Neo
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Q =
(α) MATCH (n : Person) − [: knows]−> (m : Person)
(β) WHERE n.institute = m.institute
(γ) RETURN n.name, m.name, n.institute AS institute
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Q =
Operations
[[op]]G : Tables → Tables
Semantics of a query by composition
Ex: [[Q]]G = [[α]]G ◦ [[β]]G ◦ [[γ]]G
Answers to Q on G: [[Q]]G ({})
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Q =
Operations
[[op]]G : Tables → Tables
Semantics of a query by composition
Ex: [[Q]]G = [[α]]G ◦ [[β]]G ◦ [[γ]]G
Answers to Q on G: [[Q]]G ({})
Expressions
[[exp]]G,u ∈ V where u is a record, giving binding to variables
Ex: [[β]]G (T) = u ∈ T | [[n.institute = m.institute]]G,u = true
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Pattern Matching
Rigid pattern satisfaction
Rigid path pattern: no variable length edge patterns.
Ex: (n : Person) − [: knows ∗ 2]−> () − [: likes]−> (m : Movie)
Unique way for a path p to satisfy a rigid pattern π wrt G, u.
Notation: (p, G, u) |= π
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Pattern Matching
Variable-length paths and free variables
rigid(π) = {π | π is rigid and π π }
Ex: () − [∗2]−> () − [∗4]−> () () − [∗1..3]−> () − [∗]−> ()
free(π, u): all names that occur in π and not in u
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Pattern Matching
Variable-length paths and free variables
rigid(π) = {π | π is rigid and π π }
Ex: () − [∗2]−> () − [∗4]−> () () − [∗1..3]−> () − [∗]−> ()
free(π, u): all names that occur in π and not in u
[[MATCH π]]G (T) =
π ∈rigid(π)
u∈T, p∈paths
(u, u )
u is uniform with free(π , u)
and (p, G, (u, u )) |= π
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Ambiguous and Edge Cases
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Nulls in Patterns
MATCH (n : Person {name : null})
RETURN (n)
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Nulls in Patterns
RETURN (n)
1 Every node n with a name property?
2 Every node n such that n.name IS NULL = true?
3 Nothing?
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Nulls in Patterns
RETURN (n)
1 Every node n with a name property?
2 Every node n such that n.name IS NULL = true?
3 Nothing!
Because Q is actually equivalent to:
MATCH (n : Person)
WHERE n.name = null
RETURN (n)
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Map Comparisons
When does {k1 : v1, . . . , kn : vn} = { 1 : w1, . . . , m : wm} return
true, false or null?
{name : null} = {}
{name : null} = {name : null}
{a : 1, b : 2} = {b : 2, a : 1}
{a : 1, a : 2} = {a : 2}
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Map Comparisons
When does {k1 : v1, . . . , kn : vn} = { 1 : w1, . . . , m : wm} return
true, false or null?
false
{name : null} = {}
true
{name : null} = {name : null}
true
{a : 1, b : 2} = {b : 2, a : 1}
true
{a : 1, a : 2} = {a : 2}
Neither purely syntactic, nor ∀k, m1.k = m2.k.
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Setting Properties using a Map
WITH {name : null} AS map
CREATE (n)
SET n = map
RETURN (n)
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Setting Properties using a Map
WITH {name : null} AS map
CREATE (n)
SET n = map
RETURN (n)
Returns n as {}.
The property map of n is not equal to the map it was set to.
In particular, n {.∗} = map returns false.
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MATCH with no Free Variables
MATCH ()
RETURN ∗
MATCH ()
RETURN 1
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MATCH with no Free Variables
Fail
MATCH ()
RETURN ∗
RETURN ∗ is not allowed with
no variable in scope.
Pass
MATCH ()
RETURN 1
Returns as many copies of 1
as nodes in the database.
After MATCH (), the active table is a bag containing multiple copies
of the empty record.
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Incomplete and Inconsistent Cases
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Repeating UNWINDs
UNWIND [1, 2, 3] AS r
UNWIND r AS s
RETURN s
UNWIND [[1, 2], 3] AS r
UNWIND r AS s
RETURN s
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Repeating UNWINDs
Fail
UNWIND r AS s
RETURN s
Type mismatch, expected List
but was Integer.
Pass
UNWIND [[1, 2], 3] AS r
UNWIND r AS s
RETURN s
Returns a column with 1, 2
and 3 as rows.
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Repeating UNWINDs
Fail
UNWIND r AS s
RETURN s
Type mismatch, expected List
but was Integer.
Pass
UNWIND [[1, 2], 3] AS r
UNWIND r AS s
RETURN s
Returns a column with 1, 2
and 3 as rows.
UNWIND [[1, 2], 3] AS r
UNWIND r AS s
UNWIND s AS t
UNWIND t AS u
RETURN u
Actually works, and returns a column with 1, 2 and 3 as rows.
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Violating Cyphermorphism
Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
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Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
Error: cannot use the same relationship variable ‘r’ for multiple
patterns.
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Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
patterns.
Q1 =
MATCH (x) − [r∗] − (y) − [r∗] − (z)
RETURN x, y, z
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Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
patterns.
Q1 =
MATCH (x) − [r∗] − (y) − [r∗] − (z)
RETURN x, y, z
Works and enforces the paths from x to y and from y to z to use
the same sequence of relationships, violating Cyphermorphism.
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Q0 =
MATCH (x) − [r] − (y) − [r] − (z)
RETURN x, y, z
patterns.
Q1 =
MATCH (x) − [r∗] − (y) − [r∗] − (z)
RETURN x, y, z
Works and enforces the paths from x to y and from y to z to use
the same sequence of relationships, violating Cyphermorphism.
It is not included in the query Q2 below:
Q1 =
MATCH (x) − [r∗] − (y) − [s∗] − (z)
RETURN x, y, z
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Two Small Issues: Nulls as Indices and Keys, and Naming
RETURN 1 AS ‘0‘, 0
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Fail
Multiple result columns with the same name are not supported.
Need to specify how expression naming is handled.
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Fail
[1, 2, 3][null] [1, 2, 3][null..4]
{name : ‘Nadime’} [null]
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Fail
Fail
[1, 2, 3][null] [1, 2, 3][null..4]
{name : ‘Nadime’} [null]
No error, never terminates.
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Thank you!
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Formal Specification of Cypher

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