A step up in expressiveness of decidab le fi x point logi c s Micha - - PowerPoint PPT Presentation

A step up in expressiveness

• f decidable fixpoint logics

Michael Benedikt1, Pierre Bourhis2, and Michael Vanden Boom1

1University of Oxford 2CNRS CRIStAL, Universit´

e Lille 1, INRIA Lille

LICS 2016 New York, USA

1 / 15

Fixpoint logics

Fixpoint logics can express dynamic, recursive properties. Example binary relation R, unary relation P “from w, it is possible to R-reach some P-element”

[Reach-P](w)

2 / 15

Fixpoint logics

Fixpoint logics can express dynamic, recursive properties. Example binary relation R, unary relation P “from w, it is possible to R-reach some P-element”

[lfpY,y . ∃z(Ryz ∧ (Pz ∨ Yz))](w)

2 / 15

LFP

LFP: extension of first-order logic with fixpoint formulas [lfpY,y.ψ(y, Y)](w) for ψ(y, Y) positive in Y (of arity m = ∣y∣). For all structures A, the formula ψ induces a monotone operation P(Am) ⟶ P(Am) V ⟼ ψA(V) ∶= {a ∈ Am ∶ A, a, V ⊧ ψ} ⇒ there is a unique least fixpoint [lfpY,y.ψ(y, Y)]A ∶= ⋃α ψα

A

ψ0

A ∶= ∅

ψα+1

A

∶= ψA(ψα

A)

ψλ

A ∶= ⋃ α<λ

ψα

A

3 / 15

LFP

LFP: extension of first-order logic with fixpoint formulas [lfpY,y.ψ(y, Y)](w) for ψ(y, Y) positive in Y (of arity m = ∣y∣). For all structures A, the formula ψ induces a monotone operation P(Am) ⟶ P(Am) V ⟼ ψA(V) ∶= {a ∈ Am ∶ A, a, V ⊧ ψ} ⇒ there is a unique least fixpoint [lfpY,y.ψ(y, Y)]A ∶= ⋃α ψα

A

ψ0

A ∶= ∅

ψα+1

A

∶= ψA(ψα

A)

ψλ

A ∶= ⋃ α<λ

ψα

A

Semantics of fixpoint operator: A, a ⊧ [lfpY,y.ψ(y, Y)](w) iff a ∈ ⋃α ψα

A

3 / 15

Examples

“from w, it is possible to R-reach some P-element”

[lfpY,y . ∃z(Ryz ∧ (Pz ∨ Yz))](w)

a1 a2 a3 ak ak+1

4 / 15

Examples

“from w, it is possible to R-reach some P-element”

[lfpY,y . ∃z(Ryz ∧ (Pz ∨ Yz))](w)

a1 a2 a3 ak ak+1

“from w, it is possible to R-reach x” , i.e. “(w, x) is in the transitive closure of R”

[lfpY,y . ∃z(Ryz ∧ (z = x ∨ Yz))](w)

(Free first-order variable x in the fixpoint predicate is called a parameter.)

4 / 15

Some decidable fragments of LFP (fixpoint extension of FO)

The family of “guarded” fixpoint logics has decidable satisfiability.

LFP

Lµ GFP GNFP UNFP Guarded fixpoint logic (GFP): Andr´ eka, van Benthem, N´ emeti ’95-’98; Gr¨ adel, Walukiewicz ’99 Unary negation fixpoint logic (UNFP): ten Cate, Segoufin ’11 Guarded negation fixpoint logic (GNFP): B´ ar´ any, ten Cate, Segoufin ’11

5 / 15

Guarded negation fixpoint logic (GNFP)

Let σ be a signature of relations and constants. Syntax of GNFP[σ] φ ∶∶= R t ∣ Y t ∣ φ ∧ φ ∣ φ ∨ φ ∣ ∃y(ψ(xy)) ∣ G(x) ∧ ¬ψ(x) ∣ [lfpY,y . G(y) ∧ φ(y, Y, Z)](t)

where Y only occurs positively in φ

where R and G are relations in σ or =, and t is a tuple over variables and constants.

6 / 15

Guarded negation fixpoint logic (GNFP)

Let σ be a signature of relations and constants. Syntax of GNFP[σ] φ ∶∶= R t ∣ Y t ∣ φ ∧ φ ∣ φ ∨ φ ∣ ∃y(ψ(xy)) ∣ G(x) ∧ ¬ψ(x) ∣ [lfpY,y . G(y) ∧ φ(y, Y, Z)](t)

where Y only occurs positively in φ

where R and G are relations in σ or =, and t is a tuple over variables and constants.

Restrictions on fixpoint operator: must define a guarded relation (tuples in the fixpoint must be guarded by an atom from σ or =) cannot use parameters

6 / 15

Satisfiability

These guarded fixpoint logics all have the tree-like model property (models with tree decompositions of bounded tree-width) ⇒ amenable to tree automata techniques

7 / 15

Satisfiability

These guarded fixpoint logics all have the tree-like model property (models with tree decompositions of bounded tree-width) ⇒ amenable to tree automata techniques Theorem (Gr¨

adel, Walukiewicz ’99; B´ ar´ any, Segoufin, ten Cate ’11; B´ ar´ any, Boja´ nczyk ’12)

Satisfiability and finite satisfiability are decidable for guarded fixpoint logics (2EXPTIME in general, EXPTIME for fixed-width formulas in GFP). Idea: Reduce to tree automaton emptiness test.

7 / 15

Examples

In GNFP:

[lfpY,y . ∃z(Ryz ∧ (Pz ∨ Yz))](w)

8 / 15

Examples

In GNFP:

[lfpY,y . y = y ∧ ∃z(Ryz ∧ (Pz ∨ Yz))](w)

8 / 15

Examples

In GNFP:

[lfpY,y . y = y ∧ ∃z(Ryz ∧ (Pz ∨ Yz))](w)

Not in GNFP:

[lfpY,y . y = y ∧ ∃z(Ryz ∧ (z = x ∨ Yz))](w)

8 / 15

Can we go further?

LFP

Lµ GFP GNFP UNFP

Recall the restrictions on the fixpoint operators in GNFP: must define a guarded relation cannot use parameters Which of these restrictions are essential for decidability?

9 / 15

Can we go further?

LFP

Lµ GFP GNFP UNFP

Recall the restrictions on the fixpoint operators in GNFP: must define a guarded relation cannot use parameters Which of these restrictions are essential for decidability? Answer: only first one!

9 / 15

GNFPUP

GNFPUP: extend GNFP with unguarded parameters in fixpoint

10 / 15

GNFPUP

GNFPUP: extend GNFP with unguarded parameters in fixpoint Syntax of GNFPUP[σ] φ ∶∶= Rt ∣ Yt ∣ φ ∧ φ ∣ φ ∨ φ ∣ ∃y(ψ(xy)) ∣ G(x) ∧ ¬ψ(x) ∣ [lfpY,y . G(y) ∧ φ(x, y, Y, Z)](t)

where Y only occurs positively in φ

where R and G are relations in σ or =, and t is a tuple over variables and constants.

10 / 15

GNFPUP

GNFPUP: extend GNFP with unguarded parameters in fixpoint Syntax of GNFPUP[σ] φ ∶∶= Rt ∣ Yt ∣ φ ∧ φ ∣ φ ∨ φ ∣ ∃y(ψ(xy)) ∣ G(x) ∧ ¬ψ(x) ∣ [lfpY,y . G(y) ∧ φ(x, y, Y, Z)](t)

where Y only occurs positively in φ

where R and G are relations in σ or =, and t is a tuple over variables and constants.

Example GNFPUP can express the transitive closure of a binary relation R using

[lfpY,y . ∃z(Ryz ∧ (z = x ∨ Yz))](w)

10 / 15

Expressivity of GNFPUP

LFP

Lµ GFP GNFP GNFPUP UNFP

GNFPUP also subsumes C2RPQs (conjunctive 2-way regular path queries) ∃xyz ( [R∗S](x, y) ∧ [S ∣ R](y, z) ∧ P(z) ) MQs and GQs [Rudolph, Kr¨

• tzsch ’13 ; Bourhis, Kr¨
• tzsch, Rudolph ’15]

11 / 15

Satisfiability for GNFPUP

GNFPUP still has tree-like models ⇒ still amenable to tree automata techniques Unlike other guarded logics, satisfiability testing for φ ∈ GNFPUP is non-elementary, with running time

• 22. . .2f(∣φ∣)

where the height of the tower depends only on the parameter depth: the number of nested parameter changes in the formula.

12 / 15

Satisfiability for GNFPUP

GNFPUP still has tree-like models ⇒ still amenable to tree automata techniques Unlike other guarded logics, satisfiability testing for φ ∈ GNFPUP is non-elementary, with running time

• 22. . .2f(∣φ∣)

where the height of the tower depends only on the parameter depth: the number of nested parameter changes in the formula. Theorem Satisfiability is decidable for φ ∈ GNFPUP in (n + 2)-EXPTIME, where n is the parameter depth of φ.

12 / 15

Skirting undecidability

It is known that satisfiability is undecidable for GFP (even without fixpoints) when certain relations are required to be transitive.

[Gr¨ adel ’99, Ganzinger et al. ’99]

13 / 15

Skirting undecidability

It is known that satisfiability is undecidable for GFP (even without fixpoints) when certain relations are required to be transitive.

[Gr¨ adel ’99, Ganzinger et al. ’99]

GNFPUP can express the transitive closure of a binary relation R using

[lfpY,y . ∃z(Ryz ∧ (z = x ∨ Yz))](w).

But it cannot enforce that R is transitive.

13 / 15

FO-definability

Theorem It is decidable whether [lfpY,y . G(y) ∧ ψ(x, y, Y)](w) ∈ GNFPUP can be expressed in FO (when ψ does not use any additional fixpoints). It is decidable whether a C2RPQ can be expressed in FO. Idea: Adapt automata for GNFPUP, and reduce to a boundedness question for cost automata (automata with counters).

14 / 15

Conclusion

We can allow unguarded parameters in guarded fixpoint logics.

Contributions We showed that:

▶ tree automata techniques can be used to analyze GNFPUP ▶ satisfiability is decidable for GNFPUP, and the key factor impacting the

complexity is the parameter depth

▶ some boundedness and FO-definability problems are decidable for GNFPUP

15 / 15

Conclusion

We can allow unguarded parameters in guarded fixpoint logics.

Contributions We showed that:

▶ tree automata techniques can be used to analyze GNFPUP ▶ satisfiability is decidable for GNFPUP, and the key factor impacting the

complexity is the parameter depth

▶ some boundedness and FO-definability problems are decidable for GNFPUP

Open question Is finite satisfiability decidable for GNFPUP?

15 / 15