Computational Aspects of Logics in Team Semantics [-.5cm]

Computational Aspects of Logics in Team Semantics Juha Kontinen University of Helsinki

Photo of the Kumpula Science Campus of the University of Helsinki by Samuli Junttila.

Background Dependence logic was introduced by Jouko V¨ a¨ an¨anen in 2007. It extends first-order logic (FO) by dependence atoms. In dependence logic formulas are evaluated on sets of assignments (teams) instead of single assignments as in FO. Dependence statements such as x depends on y do not make sense for single assignments. Team Semantics was invented by Hodges in 1997 as a result of developing a compositional semantics for Independence Friendly Logic of Hintikka and Sandu. After the introduction of dependence logic, team semantics has developed into an active semantical framework in which various dependency notions (e.g., dependence, independence, inclusion) have be formalised and studied with interesting connections to many areas such as Database Theory, Statistics, and Social Choice Theory. Juha Kontinen (University of Helsinki) Computational Aspects of Logics in Team Semantics[-.5cm]

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Outline of the talk

1

First-order team semantics I I I

2

Modal team semantics I I I I

3

Modal dependence, independence and inclusion logic The classical negation and modal team logic Team bisimulations Complexity of model checking, satisfiability and validity problems

Propositional team semantics I

4

Dependence, independence, inclusion, and exclusions logics Basic properties and characterizations of expressive power Other complexity results

Complexity of model checking, satisfiability and validity problems

Conclusion

Juha Kontinen (University of Helsinki) Computational Aspects of Logics in Team Semantics[-.5cm]

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Concrete notions of dependence and independence Team semantics is a framework in which various notions of dependence and independence can be studied. Dependence and independence occur in contexts such as: dependence of a move of a player in a game on some previous moves; dependence of an attribute of a database on other attributes; dependence/independence of a choice of an agent on choices of other agents; linear dependence/independence of a vector v of vectors v1 , ..., vn ; Independence of random variables X and Y ; dependence of an outcome an experiment e0 on the outcomes of e1 , ..., en .


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Dependence Logic In FO, formulas are formed using the connectives ∨, ∧, ¬, and quantifiers ∃ and ∀.

Definition Dependence logic, FO(=(. . .)), extends the syntax of FO by dependence atoms =(x1 , . . . , xn ). We consider also independence, inclusion, and exclusion atoms (and the corresponding logics) that replace dependence atoms respectively by y⊥x z, x ⊆ y and x|y.


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Assignments and teams

The semantics of dependence logic is defined using the notion of a team. Teams: Let A be a set and V = {x1 , . . . , xk } a finite set of variables. A team X with domain V is a set of assignments s : {x1 , . . . , xk } → A. A is called the co-domain of X (the universe of a model).


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Interpretation of the dependence atoms

Let A be a structure and X a team with co-domain Dom(A) and domain V s.t. {x1 , ..., xn } ⊆ V . A |=X =(x1 , ..., xn ), if and only if, for all s, s0 ∈ X: ^ s(xi ) = s0 (xi ) =⇒ s(xn ) = s0 (xn ). 0)). This is related to the fact that MDL is exponentially more succint than ML(>), i.e., some formulas of MDL can be expressed in ML(>) only by exponentially longer formulas.


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Complexity of model checking of MDL Theorem ([EL12]) The problem MC(MDL) is NP-complete. Membership: K, T |= ϕ can be decided by a non-deterministic top-down algorithm in polynomial time. The non-deterministic steps occur only when evaluating subformulas of the form ψ ∨ θ and 3ψ. Hardness: We reduce again from 3-SAT. Let Θ be an instance of 3-SAT: ^ Θ := (li1 ∨ li2 ∨ li3 ) 1≤i≤m

with variables p1 , ..., pn . Define a Kripke structure K = (W, R, π) over atomic propositions r1 , ..., rn , p1 , ..., pn as follows:


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W := R := π(si ) ∩ {rj , pj } :=

{s1 , . . . , sm }, ∅,   {rj , pj } {rj }  ∅

if pj occurs in Ci positively, if pj occurs in Ci negatively, if pj does not occur in Ci .

Let ψ be the formula ψ :=

_

ri ∧ =(pi ),

1≤i≤n

and T = W . Then it holds that Θ is a member of 3-SAT iff K, T |= ψ. A concise classification of the complexity of model checking for fragments of MDL can be found in [EL12].


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Complexity of satisfiability for MDL Theorem ([Sev09]) The satisfiability problem for MDL is NEXPTIME-complete. Sketch of the membership proof: Given ψ ∈ MDL, by downward closure it suffices to check whether ψ is satisfied by some K and a singleton team {a}. Every ψ ∈ MDL is equivalent to a formula of the form ∃f1 ...∃fm ϕ

(2)

arising by replacing subformulas =(p1 , ..., pn ) by ∃f (f (p1 , ..., pn−1 ) ↔ pn ).


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Proof cont.

Formula (2) is equivalent to >α1 ,..,αm ϕ[f1 /α1 , ..., fm /αm ], where αi vary over propositional encodings of fi0 s having length exponential in |ψ|. The idea for the algorithm is now to non-determinitically guess α1 , ..., αm , and a K of size at most 2|ψ| and {a} and to check whether K, a |= ϕ[f1 /α1 , ..., fm /αm ]. This shows containment in NEXPTIME.


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Sketch of the hardness proof For ML the the PSPACE hardness of satisfiability is shown by reduction from QBF. For MDL we reduce analogously from a variant of QBF called DQBF which is known to be NEXPTIME-complete [PRA01].

Definition (DQBF) Let φ be of the form φ := ∀p1 · · · ∀pn ∃q1 · · · ∃qm θ, where θ is a propositional formula, and C = (~c1 , . . . , ~cm ) a constraint, where ~ci is a tuple of variables from {p1 , . . . , pn }. We say that φ is true under constraint C if there exist f1 , . . . , fm with fi : {0, 1}|ci | → {0, 1} s.t. for each assignment s : {p1 , . . . , pn } → {0, 1}, s f1 (s(~c1 ))/q1 , . . . , fm (s(~cm ))/qm |= θ. DQBF now denotes the collection of (φ, C) s.t. φ is true under constraint C.


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More on the complexity of satisfiability in the modal framework

The complexity of satiafiability remain the same if MDL is extended by > or ∨ is removed from it. A concise classification of the complexity of satisfiability for fragments of MDL can be found in [LV10]. The problem SAT(MIL) is also NEXPTIME complete. This has been shown via a translation of MIL formulas to a prefix-class of FO (with quantified unary predicates) whose satisfiability problem is known to be NEXPTIME complete [KMSV14].


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Complexity Results for modal team semantics

Logic

SAT

VAL

MC

ML MDL MIL MINC ML(∼)

PSPACE NEXPTIME NEXPTIME EXPTIME [HKMV15] ?

PSPACE NEXPTIME [Han16] ΠE 2 -hard [Han16] coNEXPTIME-hard ?

PTIME NP NP PTIME PSPACE

The results on MC(MINC) and VAL(MINC) are due to [HKMV16], and PSPACE-completeness of MC(ML(∼)) is due to [M¨ ul14].


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Complexity results for propositional team semantics

We consider extensions of propositional logic PL with team semantics ϕ ::= p | ¬p | (ϕ ∨ ϕ) | (ϕ ∧ ϕ) by =(~ p, q) | p~ ⊆ ~q | ~r ⊥p~ ~q |∼ ϕ. For a set X of propositional assignments, satisfaction, X |= ψ, is defined exactly as in the modal team semantics setting.


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Propositional dependence logic PL(=(...)) The previous result on the complexity of MC(MDL) and downward closure property imply that MC(PL(=(...))) and SAT(PL(=(...))) are both NP-complete.

Theorem ([Vir14]) The validity problem of PL(=(...)) is NEXPTIME-complete. Note that, by downward closure, ϕ ∈ PL(=(...)) with variables p1 , ..., pn is valid iff X f |= ψ, where X f := {s | s : {p1 , ..., pn } → {0, 1}}. Therefore, the validity of ψ can be checked by checking if X f |= ψ. The size of X f is exponential in |ψ| and model checking can be done in NP hence validity is in NEXPTIME.


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Propositional dependence logic PL(=(...))

Sketch of hardness: We reduced again from DQBF: Let (φ, C) be such that φ := ∀p1 · · · ∀pn ∃q1 · · · ∃qm θ, and C = (~c1 , . . . , ~cm ), where ~ci = (si1 , ..., siki ) ⊆ {p1 , . . . , pn }. Now (φ, C) ∈ DQBF iff the formula _ ^ =(si1 , ..., siki , qi ) ( =(si1 , ..., siki , qi ) ∧ θ) ∨ i

i

is valid. It has recently been shown that the validity problem of MDL is also NEXPTIME-complete [Han16].


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Propositional team logic PL(∼)

Most connectives studied in propositional team semantics can be defined in PL(∼):

X |= ϕ 6 ψ

⇔ X |= ϕ or X |= ψ,

X |= ϕ ⊗ ψ

⇔

∀Y, Z: if Y ∪ Z = X, then Y |= ϕ or Z |= ψ,

X |= ϕ → ψ

⇔

∀ Y ⊆ X : if Y |= ϕ, then Y |= ψ,

X |= max(p1 , . . . , pn )

⇔

{(s(p1 ), . . . , s(pn )) | s ∈ X} = {0, 1}n .


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Exponential time hierarchy and AEXPTIME(poly)

The exponential-time hierarchy corresponds to the class of problems that can be recognized by an exponential-time alternating Turing machine with constantly many alternations. AEXPTIME(poly) = “alternating exponential time with polynomially many alternations”.


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Characterization of AEXPTIME(poly)

Theorem A set A ∈ AEXPTIME(poly) iff there exists a polynomial f and a deterministic polynomial time oracle machine M s.t. x ∈ A ⇐⇒ Q1 A1 . . . Qf (n) Af (n) Q1 ~y1 . . . Qf (n) ~yf (n) s.t. M accepts (x, ~y1 , . . . , ~yf (n) ) with oracle (A1 , . . . , Af (n) ), where Qi alternate between ∃ and ∀, Ai ⊆ {0, 1}f (n) , and yi ∈ {0, 1}f (n) .


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Simulation of oracle quantification in PL(∼)

The whole computation is encoded in a team. We use ⊗ to simulate universal quantification of oracles and words. We use ∨ to simulate existential quantification of oracles and words.


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Complexity of PL(∼) Theorem SAT(PL(∼)) is AEXPTIME(poly)-complete.

Proof. Hardness is proved by utilizing the above characterization of AEXPTIME(poly). For membership, guess a possibly exponential size team X and do APTIME model checking. PL(∼) and its fragments can be associated with a class of second-order QBF formulas providing complete problems for the levels of the exponential time hierarchy and AEXPTIME(poly) [L¨ uc16b].


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Complexity results for propositional team semantics

Logic

SAT

VAL

MC

PL PL(=(...)) PL(⊥) PL(⊆) PL(∼)

NP NP NP EXPTIME AEXPTIME(poly)

coNP NEXPTIME in ΠE 2 coNP AEXPTIME(poly)

NC1 NP NP PTIME PSPACE

For the results on PL(⊥) and PL(∼), and VAL(PL(⊆)), see [HKVV15]. The results on SAT(PL(⊆)) and MC(PL(⊆)) can be found in [HKMV15] and [HKMV16], respectively.


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Other timely topics in team semantics

Axiomatic characterizations: goal to find axioms and inference rules that govern the notions of dependence and independence: I

I I

Complete axiomatization for the FO-consequences of FO(=(. . .)) and FO(⊥)-sentences [KV13, Han15a], Several systems for propositional/modal logics, e.g., [YV16, L¨ uc16a], Nontrivial already for atoms (cf. implication problems in database theory) [HK16],

Multiteam semantics, approximate dependencies, probabilistic dependencies and logics [DHK+ 16, V¨ a¨ a17], Counting in team semantics [GH16].


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Thank you for your attention!


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References I Arnaud Durand, Miika Hannula, Juha Kontinen, Arne Meier, and Jonni Virtema, Approximation and dependence via multiteam semantics, Foundations of Information and Knowledge Systems - 9th International Symposium, FoIKS 2016, Linz, Austria, March 7-11, 2016. Proceedings (Marc Gyssens and Guillermo Ricardo Simari, eds.), Lecture Notes in Computer Science, vol. 9616, Springer, 2016, pp. 271–291. Arnaud Durand and Juha Kontinen, Hierarchies in dependence logic, ACM Transactions on Computational Logic 13 (2012), no. 4, 1–21. Arnaud Durand, Juha Kontinen, Nicolas de Rugy-Altherre, and Jouko V¨ a¨ an¨ anen, Tractability frontier of data complexity in team semantics, Proceedings Sixth International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2015, Genoa, Italy, 21-22nd September 2015. (Javier Esparza and Enrico Tronci, eds.), EPTCS, vol. 193, 2015, pp. 73–85.


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References II Johannes Ebbing, Juha Kontinen, Julian-Steffen M¨ uller, and Heribert Vollmer, A fragment of dependence logic capturing polynomial time, Logical Methods in Computer Science 10 (2014), no. 3. Johannes Ebbing and Peter Lohmann, Complexity of model checking for modal dependence logic, SOFSEM 2012: Theory and Practice of Computer Science (Berlin, Heidelberg), Lecture Notes in Computer Science, vol. 7147, Springer, 2012, pp. 226–237. Pietro Galliani, Inclusion and exclusion dependencies in team semantics on some logics of imperfect information, Annals of Pure and Applied Logic 163 (2012), no. 1, 68–84. Pietro Galliani and Lauri Hella, Inclusion logic and fixed point logic, Computer Science Logic 2013 (CSL 2013) (Dagstuhl, Germany) (Simona Ronchi Della Rocca, ed.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 23, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2013, pp. 281–295. Juha Kontinen (University of Helsinki) Computational Aspects of Logics in Team Semantics[-.5cm]

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References III Erich Gr¨ adel and Stefan Hegselmann, Counting in team semantics, 25th EACSL Annual Conference on Computer Science Logic, CSL 2016, August 29 - September 1, 2016, Marseille, France (Jean-Marc Talbot and Laurent Regnier, eds.), LIPIcs, vol. 62, Schloss Dagstuhl Leibniz-Zentrum fuer Informatik, 2016, pp. 35:1–35:18. Pietro Galliani, Miika Hannula, and Juha Kontinen, Hierarchies in independence logic, Computer Science Logic 2013 (CSL 2013) (Dagstuhl, Germany) (Simona Ronchi Della Rocca, ed.), Leibniz International Proceedings in Informatics (LIPIcs), vol. 23, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2013, pp. 263–280. Erich Gr¨ adel, Model-checking games for logics of imperfect information, Theor. Comput. Sci. 493 (2013), 2–14. Erich Gr¨ adel, Games for inclusion logic and fixed-point logic, Dependence Logic: Theory and Applications (Samson Abramsky, Juha Kontinen, Jouko V¨ a¨ an¨ anen, and Heribert Vollmer, eds.), Springer International Publishing, Cham, 2016, pp. 73–98. Juha Kontinen (University of Helsinki) Computational Aspects of Logics in Team Semantics[-.5cm]

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References IV Erich Gr¨ adel and Jouko A. V¨ a¨ an¨ anen, Dependence and independence, Studia Logica 101 (2013), no. 2, 399–410. Miika Hannula, Axiomatizing first-order consequences in independence logic, Ann. Pure Appl. Logic 166 (2015), no. 1, 61–91. , Hierarchies in inclusion logic with lax semantics, Logic and Its Applications - 6th Indian Conference, ICLA 2015, Mumbai, India, January 8-10, 2015. Proceedings (Mohua Banerjee and Shankara Narayanan Krishna, eds.), Lecture Notes in Computer Science, vol. 8923, Springer, 2015, pp. 100–118. , The entailment problem in modal and propositional dependence logics, CoRR abs/1608.04301 (2016). Miika Hannula and Juha Kontinen, A finite axiomatization of conditional independence and inclusion dependencies, Inf. Comput. 249 (2016), 121–137. Juha Kontinen (University of Helsinki) Computational Aspects of Logics in Team Semantics[-.5cm]

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References V Lauri Hella, Antti Kuusisto, Arne Meier, and Heribert Vollmer, Modal inclusion logic: Being lax is simpler than being strict, Mathematical Foundations of Computer Science 2015 - 40th International Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Proceedings, Part I (Giuseppe F. Italiano, Giovanni Pighizzini, and Donald Sannella, eds.), Lecture Notes in Computer Science, vol. 9234, Springer, 2015, pp. 281–292. Lauri Hella, Antti Kuusisto, Arne Meier, and Jonni Virtema, Model checking and validity in propositional and modal inclusion logics, CoRR abs/1609.06951 (2016). M. Hannula, J. Kontinen, J. Virtema, and H. Vollmer, Complexity of Propositional Logics in Team Semantics, ArXiv e-prints (2015).


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References VI Lauri Hella, Kerkko Luosto, Katsuhiko Sano, and Jonni Virtema, The expressive power of modal dependence logic, Advances in Modal Logic 10, invited and contributed papers from the tenth conference on ”Advances in Modal Logic,” held in Groningen, The Netherlands, August 5-8, 2014 (London), College Publications, 2014, pp. 294–312. Lauri Hella and Johanna Stumpf, The expressive power of modal logic with inclusion atoms, Proceedings Sixth International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2015, Genoa, Italy, 21-22nd September 2015. (Javier Esparza and Enrico Tronci, eds.), EPTCS, vol. 193, 2015, pp. 129–143. Juha Kontinen, Antti Kuusisto, Peter Lohmann, and Jonni Virtema, Complexity of two-variable dependence logic and if-logic, Inf. Comput. 239 (2014), 237–253.


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References VII Juha Kontinen, Antti Kuusisto, and Jonni Virtema, Decidability of predicate logics with team semantics, 41st International Symposium on Mathematical Foundations of Computer Science, MFCS 2016, August 22-26, 2016 - Krak´ ow, Poland (Piotr Faliszewski, Anca Muscholl, and Rolf Niedermeier, eds.), LIPIcs, vol. 58, Schloss Dagstuhl Leibniz-Zentrum fuer Informatik, 2016, pp. 60:1–60:14. Juha Kontinen, Julian-Steffen M¨ uller, Henning Schnoor, and Heribert Vollmer, Modal independence logic, Advances in Modal Logic 10, invited and contributed papers from the tenth conference on ”Advances in Modal Logic,” held in Groningen, The Netherlands, August 5-8, 2014 (London), College Publications, 2014, pp. 353–372. Juha Kontinen, Julian-Steffen M¨ uller, Henning Schnoor, and Heribert Vollmer, A Van Benthem Theorem for Modal Team Semantics, 24th EACSL Annual Conference on Computer Science Logic (CSL 2015) (Dagstuhl, Germany), Leibniz International Proceedings in Informatics (LIPIcs), vol. 41, Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik, 2015, pp. 277–291. Juha Kontinen (University of Helsinki) Computational Aspects of Logics in Team Semantics[-.5cm]

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References VIII Juha Kontinen and Ville Nurmi, Team logic and second-order logic, Fundamenta Informaticae 106 (2011), no. 2-4, 259–272. Jarmo Kontinen, Coherence and computational complexity of quantifier-free dependence logic formulas, Studia Logica 101 (2013), no. 2, 267–291. Juha Kontinen and Jouko V¨ a¨ an¨ anen, On definability in dependence logic, Journal of Logic, Language and Information 18 (2009), no. 3, 317–332. Juha Kontinen and Jouko A. V¨ a¨ an¨ anen, Axiomatizing first-order consequences in dependence logic, Ann. Pure Appl. Logic 164 (2013), no. 11, 1101–1117. Martin L¨ uck, Axiomatizations for propositional and modal team logic, 25th EACSL Annual Conference on Computer Science Logic, CSL 2016, August 29 - September 1, 2016, Marseille, France (Jean-Marc Talbot and Laurent Regnier, eds.), LIPIcs, vol. 62, Schloss Dagstuhl Leibniz-Zentrum fuer Informatik, 2016, pp. 33:1–33:18. Juha Kontinen (University of Helsinki) Computational Aspects of Logics in Team Semantics[-.5cm]

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References IX , Complete problems of propositional logic for the exponential hierarchy, CoRR abs/1602.03050 (2016). Peter Lohmann and Heribert Vollmer, Complexity results for modal dependence logic, Proceedings 19th Conference on Computer Science Logic, Lecture Notes in Computer Science, vol. 6247, Springer Berlin / Heidelberg, 2010, pp. 411–425. Julian-Steffen M¨ uller, Satisfiability and model checking in team based logics, Ph.D. thesis, Leibniz Universit¨ at Hannover, 2014. Ville Nurmi, Dependence logic: Investigations into higher-order semantics defined on teams, Ph.D. thesis, University of Helsinki, 2009. G. Peterson, J. Reif, and S. Azhar, Lower bounds for multiplayer noncooperative games of incomplete information, Computers & Mathematics with Applications 41 (2001), no. 7-8, 957 – 992.


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References X Raine R¨ onnholm, The expressive power of k-ary exclusion logic, Logic, Language, Information, and Computation - 23rd International Workshop, WoLLIC 2016, Puebla, Mexico, August 16-19th, 2016. Proceedings (Jouko A. V¨ a¨ an¨ anen, ˚ Asa Hirvonen, and Ruy J. G. B. de Queiroz, eds.), Lecture Notes in Computer Science, vol. 9803, Springer, 2016, pp. 375–391. Merlijn Sevenster, Model-theoretic and computational properties of modal dependence logic, Journal of Logic and Computation 19 (2009), no. 6, 1157–1173. Jouko V¨ a¨ an¨ anen, Dependence logic, London Mathematical Society Student Texts, vol. 70, Cambridge University Press, Cambridge, 2007. MR 2351449 (2009c:03026) Jouko V¨ a¨ an¨ anen, Modal dependence logic, New Perspectives on Games and Interaction (Robert van Rooij Krzysztof Apt, ed.), Texts in Logic and Games, vol. 5, Amsterdam University Press, 2008, pp. 237–254. Juha Kontinen (University of Helsinki) Computational Aspects of Logics in Team Semantics[-.5cm]

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References XI

Jouko V¨ a¨ an¨ anen, The logic of approximate dependence, Rohit Parikh on Logic, Language and Society (Can Ba¸skent, Lawrence S. Moss, and Ramaswamy Ramanujam, eds.), Springer International Publishing, Cham, 2017, pp. 227–234. Jonni Virtema, Complexity of validity for propositional dependence logics, Proceedings Fifth International Symposium on Games, Automata, Logics and Formal Verification, GandALF 2014, Verona, Italy, September 10-12, 2014., EPTCS, no. 161, 2014, pp. 18–31. Fan Yang and Jouko V¨ a¨ an¨ anen, Propositional logics of dependence, Ann. Pure Appl. Logic 167 (2016), no. 7, 557–589.


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Computational Aspects of Logics in Team Semantics [-.5cm]

Computational Aspects of Logics in Team Semantics [-.5cm]

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