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Converse elimination in the algebra of binary relations

Dimitri Surinx

Hasselt University

Joint work with Jan Van den Bussche

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Calculus of relations

Natural set of (primitive) operations on binary relations (graphs) [Peirce, Schr¨

• der, Tarski] over some domain V

id = {(x, x) | x ∈ V } r ∪ s = {(x, y) | (x, y) ∈ r ∨ (x, y) ∈ s} r ◦ s = {(x, y) | ∃z : (x, z) ∈ r ∧ (z, y) ∈ s} rc = {(x, y) ∈ V 2 | (x, y) ∈ r} r−1 = {(x, y) | (y, x) ∈ r} r+ = the transitive closure of r

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Calculus of relations

Derived operators di = {(x, y) ∈ V 2 | x = y} = idc all = V 2 = id ∪ di r ∩ s = {(x, y) | (x, y) ∈ r ∧ (x, y) ∈ s} = (rc ∪ sc)c r − s = {(x, y) | (x, y) ∈ r ∧ (x, y) ∈ s} = r ∩ sc π1(r) = {(x, x) | ∃y : (x, y) ∈ r} = (r ◦ all) ∩ id π2(r) = {(x, x) | ∃y : (y, x) ∈ r} = (all ◦ r) ∩ id

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Fragments

A fragment is a set of primitive or derived operations The most basic fragment N is {id, ∪, ◦} N(F) denotes N extended with the operations in F We say that F has an operator if the operator is in F or if it can be constructed from other operators in F E.g. r ∩ s = r − (r − s) E.g. π1(r) = (r ◦ r−1) ∩ id

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Queries

Fix a binary relational vocabulary Γ (Label set) Expressions over a fragment F are built from relation names in Γ using the

• perations in F

⇒ Map instances of Γ (graphs) to binary relations Well established logic: N(−1, c) ≡ FO3 (Tarski & Givant)

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Boolean queries

Boolean queries (graph properties): test nonemptiness of expression results Examples: Transitivity: all − (all ◦ ((R ◦ R) − R) ◦ all) = ∅ S-T connectivity: S ◦ R+ ◦ T = ∅ R2 ◦ R−1 ◦ R2 = ∅ matches the pattern

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Converse elimination

Which boolean queries expressed using −1 can be equivalently expressed without using −1? E.g. R2 ◦ R−1 ◦ R2 = ∅ is equivalent to π1(R2 ◦ π2(π1(R2) ◦ R)) = ∅ Formally: a fragment F admits converse elimination if every boolean query in N(F) can be expressed in N(F − {−1})

Theorem ([Fletcher et al.])

If F has projection, but neither intersection nor transitive closure, then F admits converse elimination Note: the theorem clearly applies to the example above

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Motivation

Comparing different computational models/logics as is done in complexity theory and finite model theory E.g. π allows looking back (π2) but not navigation back. Is looking back always sufficient to eliminate converse?

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Remaining questions

Theorem ([Fletcher et al.])

If F has projection, but neither intersection nor transitive closure, then F admits converse elimination

1 What if intersection is present? 2 What if transitive closure is present? 3 Expression complexity of converse elimination? Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Remaining questions

Theorem ([Fletcher et al.])

If F has projection, but neither intersection nor transitive closure, then F admits converse elimination

1 What if intersection is present?

⇒ Converse elimination fails: (R2 ◦ R−1 ◦ R2) ∩ R = ∅ is not expressible in the most powerful fragment without converse (N(c, +))

2 What if transitive closure is present?

⇒ We will prove that it also fails

3 Expression complexity of converse elimination?

⇒ We will prove an exponential blowup in degree

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Converse elimination fails in the presence of transitive closure

Theorem

The query that checks for the existence of the graph pattern is not expressible in the largest language without converse. Formally, R2 ◦ (R ◦ R−1)+ ◦ R2 = ∅ is not expressible in N(c, +)

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Bisimulation game for N(c)

Restricted version of the 3-pebble game for FO3 Intuition: relax invariant condition of partial isomorphism so that converse relations need not be preserved by Duplicator Let G1 = (G1, a1, b1) and G2 = (G2, a2, b2) be pointed graphs. A k-round bisimulation game on G1 and G2 proceeds as follows: The spoiler picks an i ∈ {1, 2} and a node xi in Gi The duplicator responds with a node xj (j = i) in Gj Continue two k − 1-round subgames on:

(G1, a1, x1) and (G2, a2, x2) (G1, x1, b1) and (G2, x2, b2)

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

k-bisimilarity and indistinguishability of pointed graphs

The duplicator wins the game if for any newly started subgame on (G1, y1, z1) and (G2, y2, z2) we have (y1, z1) ∈ R(G1) iff (y2, z2) ∈ R(G2) for every R ∈ Γ y1 = z1 iff y2 = z2 If the duplicator has a winning strategy for the k-round bisimulation game

• n G1 and G2, we say that G1 and G2 are k-bisimilar

Theorem ([Fletcher et al.])

(G1, a1, b1) and (G2, a2, b2) are k-bisimilar iff for any expression e ∈ N(c) with degree at most k: (a1, b1) ∈ e(G1) ⇔ (a2, b2) ∈ e(G2) * degree(e) is the maximum number of nested compositions

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Family of graphs to prove inexpressibility

Let Q : R2 ◦ (R ◦ R−1)+ ◦ R2 and define the family Gm

1 and Gm 2 :

Gm

1

y1 ym+1 w1 wm+1 t1 tm+1 u1 um+1 Gm

2

w′

1

w′

m+1

v1 vm+1

⇒ Q(Gm

1 ) = ∅ and Q(Gm 2 ) = ∅

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Bisimilarity result on Gm

1 and Gm 2

Theorem

For any a1, b1 ∈ Gm

1 there exists a2, b2 ∈ Gm 2 such that (G1, a1, b1) and

(Gm

2 , a2, b2) are m/2 − 1-bisimilar

Main technical contribution. Proof is long and technical

Corollary

A boolean query in N(c) with degree at most m/2 − 1 cannot be true on Gm

1 and false on Gm 2 simultaneously

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Proof that converse elimination fails in presence of TC

Corollary

A boolean query in N(c) with degree at most m/2 − 1 cannot be true on G m

1 and false on G m 2 simultaneously

Recall: Q = R2 ◦ (R ◦ R−1)+ ◦ R2 Suppose e = ∅ with e ∈ N(c, +) expresses Q = ∅. For any m define km = |Gm

1 | = |Gm 2 |.

For any n there exists en without + equivalent to e on graphs G with domain size at most n. (ekm = ∅ is equivalent to Q = ∅ on Gm

1 and

Gm

2 )

⇒ Can ensure that degree(en) is logarithmic in n

⇒ exists l such that degree(ekl) ≤ l/2 − 1

⇒ By the Corollary ekl is nonempty on both Gl

1 and Gl 2

⇒ Contradicts that ekl = ∅ is equivalent to Q = ∅ on Gl

1 and Gl 2

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Exponential blowup in degree for converse elimination

Consider the family Qn : R2 ◦ (R ◦ R−1)n ◦ R2 = ∅ Degree of Qn is O(log(n)) Converse elimination for this family cannot be done in degree o(n)

Corollary

Converse elimination in the language with projection, but with neither intersection nor transitive closure is exponential in the degree

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

Directions for further research

1 We know that converse elimination is exponential in the degree. Can

we establish a similar result in the length?

2 Another interesting derived operator: Residuations [Pratt. Origins of

calculus of binary relations] C/B is the maximal X such that X ◦ B ⊆ C B\C is the maximal X such that B ◦ X ⊆ C

How does N(/, \) compare to N(c)? Satisfiability and equivalence problem in N(/, \)?

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)

References

This presentation is part of a larger project on the calculus of relations:

G.H.L. Fletcher, M. Gyssens, D. Leinders, D. Surinx, J. Van den Bussche, D. Van Gucht,

• S. Vansummeren, and Y. Wu.

Relative expressive power of navigational querying on graphs. Information Sciences, 298:390–406, 2015. G.H.L. Fletcher, M. Gyssens, D. Leinders, J. Van den Bussche, D. Van Gucht, and

• S. Vansummeren.

Similarity and bisimilarity notions appropriate for characterizing indistinguishability in fragments of the calculus of relations. Journal of Logic and Computation, 25(3):549–580, 2015. G.H.L. Fletcher, M. Gyssens, D. Leinders, J. Van den Bussche, D. Van Gucht,

• S. Vansummeren, and Y. Wu.

The impact of transitive closure on the expressiveness of navigational query languages on unlabeled graphs. Annals of Mathematics and Artificial Intelligence, 73(1–2):167–203, 2015.

• D. Surinx, G.H.L. Fletcher, M. Gyssens, D. Leinders, J. Van den Bussche, D. Van Gucht,
• S. Vansummeren, and Y. Wu.

Relative expressive power of navigational query on graphs using transitive closure. Logic Journal of the IGPL, 23(5):759–788, 2015.

Converse elimination in the algebra of binary relations Dimitri Surinx (Hasselt University)