Complexity of propositional logics in team semantics Movativation - - PowerPoint PPT Presentation

▶
complexity of propositional logics in team semantics
SMART_READER_LITE
LIVE PREVIEW

Complexity of propositional logics in team semantics Movativation - - PowerPoint PPT Presentation

Mar 08, 2023 132 likes •443 views

Complexity of propositional logics in team semantics Jonni Virtema Complexity of propositional logics in team semantics Movativation History Team Semantics Dependency atoms Jonni Virtema Complexity Results From 3SAT to University of

slide-1
SLIDE 1

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

1/ 14 Complexity of propositional logics in team semantics

Jonni Virtema

University of Helsinki, Finland jonni.virtema@gmail.com Joint work with Miika Hannula1, Martin L¨ uck2, Juha Kontinen3, and Heribert Vollmer2 Related to papers in GandALF 2016, MFCS 2015, Information and Computation 2016

1 The University of Auckland, 2 University of Hanover, 3 University of Helsinki,

8th of December, 2016 – Computational Logic Day 2016

slide-2
SLIDE 2

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

2/ 14 Core of Team Semantics

◮ In most studied logics formulae are evaluated in a single state of affairs.

E.g.,

◮ a first-order assignment in first-order logic, ◮ a propositional assignment in propositional logic, ◮ a possible world of a Kripke structure in modal logic.

◮ In team semantics sets of states of affairs are considered.

E.g.,

◮ a set of first-order assignments in first-order logic, ◮ a set of propositional assignments in propositional logic, ◮ a set of possible worlds of a Kripke structure in modal logic.

◮ These sets of things are called teams.

slide-3
SLIDE 3

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

2/ 14 Core of Team Semantics

◮ In most studied logics formulae are evaluated in a single state of affairs.

E.g.,

◮ a first-order assignment in first-order logic, ◮ a propositional assignment in propositional logic, ◮ a possible world of a Kripke structure in modal logic.

◮ In team semantics sets of states of affairs are considered.

E.g.,

◮ a set of first-order assignments in first-order logic, ◮ a set of propositional assignments in propositional logic, ◮ a set of possible worlds of a Kripke structure in modal logic.

◮ These sets of things are called teams.

slide-4
SLIDE 4

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

2/ 14 Core of Team Semantics

◮ In most studied logics formulae are evaluated in a single state of affairs.

E.g.,

◮ a first-order assignment in first-order logic, ◮ a propositional assignment in propositional logic, ◮ a possible world of a Kripke structure in modal logic.

◮ In team semantics sets of states of affairs are considered.

E.g.,

◮ a set of first-order assignments in first-order logic, ◮ a set of propositional assignments in propositional logic, ◮ a set of possible worlds of a Kripke structure in modal logic.

◮ These sets of things are called teams.

slide-5
SLIDE 5

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

3/ 14 Team Semantics: Motivation and History

Logical modelling of uncertainty, imperfect information, and different notions of dependence such as functional dependence and independence. Related to similar concepts in statistics, database theory etc. Historical development:

◮ Branching quantifiers by Henkin 1959. ◮ Independence-friendly logic by Hintikka and Sandu 1989. ◮ Compositional semantics for independence-friendly logic by Hodges 1997.

(Origin of team semantics.)

◮ Dependence logic by V¨

a¨ an¨ anen 2007.

◮ Modal dependence logic by V¨

a¨ an¨ anen 2008.

◮ Introduction of other dependency notions to team semantics such as

inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen.

◮ Generalized atoms by Kuusisto (derived from generalised quantifiers).

slide-6
SLIDE 6

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

3/ 14 Team Semantics: Motivation and History

Logical modelling of uncertainty, imperfect information, and different notions of dependence such as functional dependence and independence. Related to similar concepts in statistics, database theory etc. Historical development:

◮ Branching quantifiers by Henkin 1959. ◮ Independence-friendly logic by Hintikka and Sandu 1989. ◮ Compositional semantics for independence-friendly logic by Hodges 1997.

(Origin of team semantics.)

◮ Dependence logic by V¨

a¨ an¨ anen 2007.

◮ Modal dependence logic by V¨

a¨ an¨ anen 2008.

◮ Introduction of other dependency notions to team semantics such as

inclusion, exclusion, and independence. Galliani, Gr¨ adel, V¨ a¨ an¨ anen.

◮ Generalized atoms by Kuusisto (derived from generalised quantifiers).

slide-7
SLIDE 7

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

4/ 14 Propositional logic

Syntax of propositional logic: ϕ ::= p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) Semantics via propositional assignments: ”name” p q r s 1 1 s | = (q ∧ r) Team semantics / semantics via sets of assignments: ”name” p q r s 1 1 t 1 1 u 1 {s, t, u} | = q, {s, t} | = (p ∨ r)

slide-8
SLIDE 8

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

5/ 14 Team semantics

We want that for each formula ϕ of propositional logic and for each team X X | = ϕ iff ∀s ∈ X : s | = ϕ. We define that X | = p iff ∀s ∈ X : s(p) = 1 X | = ¬p iff ∀s ∈ X : s(p) = 0 X | = ϕ ∧ ψ iff X | = ϕ and X | = ψ X | = ϕ ∨ ψ iff Y | = ϕ and Z | = ψ, for some Y , Z ⊆ X such that Y ∪ Z = X.

slide-9
SLIDE 9

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

5/ 14 Team semantics

We want that for each formula ϕ of propositional logic and for each team X X | = ϕ iff ∀s ∈ X : s | = ϕ. We define that X | = p iff ∀s ∈ X : s(p) = 1 X | = ¬p iff ∀s ∈ X : s(p) = 0 X | = ϕ ∧ ψ iff X | = ϕ and X | = ψ X | = ϕ ∨ ψ iff Y | = ϕ and Z | = ψ, for some Y , Z ⊆ X such that Y ∪ Z = X.

slide-10
SLIDE 10

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

6/ 14 Extensions of propositional logic

We extend PL by adding atomic formulae that describe properties of teams. Dependence atoms: dep(p, q, r) ”the truth value of r is functionally determined by the truth values of p and q”. p q r s 1 1 t 1 1 u 1 {s, u} | = dep(p, r) , {s, t} | = dep(p, q) , {s, t, u} | = dep(q) , {s, t, u} | = dep(r) ∨ dep(r) .

slide-11
SLIDE 11

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

6/ 14 Extensions of propositional logic

We extend PL by adding atomic formulae that describe properties of teams. Inclusion atoms: (p1, p2) ⊆ (q1, q2) ”truth values that appear for p1, p2 also appear as truth values for q1, q2”. p q r s 1 1 t 1 1 u 1 {s, t} | = p ⊆ q, {s, t} | = q ⊆ r, {s, t, u} | = (p, q) ⊆ (r, q)

slide-12
SLIDE 12

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

6/ 14 Extensions of propositional logic

We extend PL by adding atomic formulae that describe properties of teams. Syntax of propositional dependence logic PD: ϕ ::= p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | dep(p1, . . . , pn, q) Syntax of propositional inclusion logic PLInc: ϕ ::= p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | (p1, . . . , pn) ⊆ (q1, . . . , qn) Syntax of propositional team logic PTL: ϕ ::= p | ¬p | (ϕ ∧ ϕ) | (ϕ ∨ ϕ) | ∼ϕ, with the semantics X | = ∼ϕ iff X | = ϕ.

slide-13
SLIDE 13

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

7/ 14 Important decision problems

Model checking: Input: A team X and a formula ϕ. Output: Does X | = ϕ hold? Satisfiability: Input: A formula ϕ. Output: Does there exists a non-empty team X s.t. X | = ϕ? Validity: Input: A formula ϕ. Output: Does X | = ϕ hold for every non-empty team X?

slide-14
SLIDE 14

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

8/ 14 Complexity results

Satisfiability Validity Model checking PL NP coNP NC1 PD NP NEXPTIME NP PLInc EXPTIME coNP P PTL AEXPTIME(poly) AEXPTIME(poly) PSPACE

slide-15
SLIDE 15

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

9/ 14 Source of hardness:

A well-known NP-complete problem: 3SAT: Input: A 3CNF-formula ϕ (e.g.,(p2 ∨ ¬p7) ∧ (¬p1 ∨ p3 ∨ p2) ∧ (p3 ∨ ¬p4 ∨ ¬p2) ∧ p2). Output: Does there exists an assignment s s.t. s | = ϕ?

slide-16
SLIDE 16

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

9/ 14 Source of hardness:

A well-known NP-complete problem: 3SAT: Input: A 3CNF-formula ϕ (e.g.,(p2 ∨ ¬p7) ∧ (¬p1 ∨ p3 ∨ p2) ∧ (p3 ∨ ¬p4 ∨ ¬p2) ∧ p2). Output: Does there exists an assignment s s.t. s | = ϕ? We may rewrite the above as follows: Input: A existentially prenex quantified QPL-sentence ϕ (e.g.,∃p1 . . . ∃p7

  • (p2 ∨ ¬p7) ∧ (¬p1 ∨ p3 ∨ p2) ∧ (p3 ∨ ¬p4 ∨ ¬p2) ∧ p2
  • ).

Output: Does ∅ | = ϕ hold?

slide-17
SLIDE 17

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

9/ 14 Source of hardness:

A well-known NP-complete problem: EQBF: Input: A sentence ϕ of the form ∃p1 . . . ∃pnψ, where ψ ∈ PL. Output: Does ∅ | = ϕ hold?

slide-18
SLIDE 18

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

9/ 14 Source of hardness:

A well-known NP-complete problem: EQBF: Input: A sentence ϕ of the form ∃p1 . . . ∃pnψ, where ψ ∈ PL. Output: Does ∅ | = ϕ hold? A well-known PSPACE-complete problem: QBF: Input: A sentence ϕ of the form ∃p1∀p2 . . . ∀pn−1∃pnψ, where ψ ∈ PL. Output: Does ∅ | = ϕ hold?

slide-19
SLIDE 19

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

10/ 14 From QBF to DQBF

A well-known PSPACE-complete problem: QBF: Input: A prenex quantified QPL-sentence ϕ (e.g., ∃p1∀p2∀p3∃p4ψ ). Output: Does ∅ | = ϕ hold? The formula ∃p1∀p2∀p3∃p4ψ may be equivalently written with the help of Skolem functions f1 ∈ {0, 1} and f2 : {0, 1}2 → {0, 1}: ∃f1∃f2∀p2∀p3 ψ

  • f1/p1, f2(p2, p3)/p4
slide-20
SLIDE 20

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

10/ 14 From QBF to DQBF

A well-known PSPACE-complete problem: QBF: Input: A prenex quantified QPL-sentence ϕ (e.g., ∃p1∀p2∀p3∃p4ψ ). Output: Does ∅ | = ϕ hold? The formula ∃p1∀p2∀p3∃p4ψ may be equivalently written with the help of Skolem functions f1 ∈ {0, 1} and f2 : {0, 1}2 → {0, 1}: ∃f1∃f2∀p2∀p3 ψ

  • f1/p1, f2(p2, p3)/p4
  • Formulae ϕ of the form ∃f1 . . . ∃fn∀p1 . . . ∀pkψ, where ψ ∈ PL and

arg(fi) ⊆ {p1, . . . , pn}, are called as DQBF-sentences. Moreover, if arg(fi) ⊆ arg(fi+1) for all i, we say that ϕ is simple.

slide-21
SLIDE 21

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

10/ 14 From QBF to DQBF

A well-known PSPACE-complete problem: QBF: Input: A prenex quantified QPL-sentence ϕ (e.g., ∃p1∀p2∀p3∃p4ψ ). Output: Does ∅ | = ϕ hold? The above PSPACE-complete problem can be reformulated as follows: SDQBF: Input: A simple DQBF-sentence ϕ. Output: Does ∅ | = ϕ hold?

slide-22
SLIDE 22

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

10/ 14 From QBF to DQBF

A well-known PSPACE-complete problem: QBF: Input: A prenex quantified QPL-sentence ϕ (e.g., ∃p1∀p2∀p3∃p4ψ ). Output: Does ∅ | = ϕ hold? The above PSPACE-complete problem can be reformulated as follows: SDQBF: Input: A simple DQBF-sentence ϕ. Output: Does ∅ | = ϕ hold? Not so well-known NEXPTIME-complete problem: DQBF: (Peterson, Reif, and Azhar 2001) Input: A DQBF-sentence ϕ. Output: Does ∅ | = ϕ hold?

slide-23
SLIDE 23

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

11/ 14 From DQBF to ADQBF

Example: DQBF

Essentially an instance of DQBF is as follows: ∃f1 . . . ∃fn∀p1 . . . ∀pkϕ(p1, . . . , pn, f1( c1), . . . , fn( cn)), where ϕ is a propositional formula and ci is some tuple of variables from p1, . . . , pk.

slide-24
SLIDE 24

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

11/ 14 From DQBF to ADQBF

Example: DQBF

Essentially an instance of DQBF is as follows: ∃f1 . . . ∃fn∀p1 . . . ∀pkϕ(p1, . . . , pn, f1( c1), . . . , fn( cn)), where ϕ is a propositional formula and ci is some tuple of variables from p1, . . . , pk.

Definition

A Σk-alternating qBf, Σk-ADQBF is a formula of the form (∃f 1

1 . . . ∃f 1 j1 )(∀f 2 1 . . . ∀f 2 j2 ) . . . (∃f k j1 . . . ∃f k jk )∀p1 . . . ∀pnϕ(p1, . . . , f i j (

ci

j ), . . . ),

where ϕ is a propositional formula and ci

j is some tuple of variables from

p1, . . . , pn.

slide-25
SLIDE 25

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

11/ 14 From DQBF to ADQBF

Definition

A Σk-alternating qBf, Σk-ADQBF is a formula of the form (∃f 1

1 . . . ∃f 1 j1 )(∀f 2 1 . . . ∀f 2 j2 ) . . . (∃f k j1 . . . ∃f k jk )∀p1 . . . ∀pnϕ(p1, . . . , f i j (

ci

j ), . . . ),

where ϕ is a propositional formula and ci

j is some tuple of variables from

p1, . . . , pn.

◮ Σk-ADQBF is ΣEXP k

  • complete odd k, and ΣEXP

k−1 -complete for even k. ◮ Πk-ADQBF is ΠEXP k

  • complete even k, and ΠEXP

k−1 -complete for odd k. ◮ ADQBF is AEXPTIME(poly)-complete.

slide-26
SLIDE 26

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

12/ 14 Connection between ADQBF and PTL

A Σk-ADQBF is a sentence (∃f 1

1 . . . ∃f 1 j1 )(∀f 2 1 . . . ∀f 2 j2 ) . . . (∃f k j1 . . . ∃f k jk )∀p1 . . . ∀pnϕ(p1, . . . , f i j (

ci

j ), . . . )

can be written as the following QPL[∼, dep(·)]-sentence

slide-27
SLIDE 27

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

12/ 14 Connection between ADQBF and PTL

A Σk-ADQBF is a sentence (∃f 1

1 . . . ∃f 1 j1 )(∀f 2 1 . . . ∀f 2 j2 ) . . . (∃f k j1 . . . ∃f k jk )∀p1 . . . ∀pnϕ(p1, . . . , f i j (

ci

j ), . . . )

can be written as the following QPL[∼, dep(·)]-sentence ∀p1 · · · ∀pn (∃q1

1 · · · ∃q1 j1) (Uq2 1 · · · Uq2 j2) (∃q3 1 · · · ∃q3 j3) . . . ( ∃qk 1 · · · ∃qk jk)

∼

  • ∼(p ∧ ¬p) ∧
  • 1≤i≤k

i is even

1≤l≤ji

dep

  • ci

l, qi l

  • ∨

1≤i≤k i is odd

1≤l≤ji

dep

  • ci

l, qi l

∧ θ

slide-28
SLIDE 28

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

12/ 14 Connection between ADQBF and PTL

∀p1 · · · ∀pn (∃q1

1 · · · ∃q1 j1) (Uq2 1 · · · Uq2 j2) (∃q3 1 · · · ∃q3 j3) . . . ( ∃qk 1 · · · ∃qk jk)

∼

  • ∼(p ∧ ¬p) ∧
  • 1≤i≤k

i is even

1≤l≤ji

dep

  • ci

l, qi l

  • ∨

1≤i≤k i is odd

1≤l≤ji

dep

  • ci

l, qi l

∧ θ

  • Dependence atoms can be eliminated from above by the use of ∼.

The quantifiers can be eliminated by a shift to satisfiability and by simulating existential quantifiers by ∨ and universal quantifiers by ∼ ∨ ∼.

slide-29
SLIDE 29

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

13/ 14

THANKS!

slide-30
SLIDE 30

Complexity of propositional logics in team semantics Jonni Virtema Movativation History Team Semantics Dependency atoms Complexity Results From 3SAT to ADQBF References

14/ 14 References

Jonni Virtema, Complexity of validity for propositional dependence logics, Information and Computation, Elsevier, 2016. Miika Hannula, Martin L¨ uck, Juha Kontinen, and Jonni Virtema, On quantified propositional logics and the exponential time hierarchy, proceedings of the 7th International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2016. Miika Hannula, Juha Kontinen, Jonni Virtema, and Heribert Vollmer, Complexity of Propositional Independence and Inclusion Logic, proceedings

  • f the 40th International Symposium on Mathematical Foundations of

Computer Science, MFCS 2015.