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Abstract Team Logic
Martin LΓΌck Leibniz UniversitΓ€t Hannover
- 25. Jahrestagung LOGINF, Jena, 22.10.19
Abstract Team Logic Martin Lck Leibniz Universitt Hannover 25. - - PowerPoint PPT Presentation
Abstract Team Logic Martin Lck Leibniz Universitt Hannover 25. - - PowerPoint PPT Presentation
Abstract Team Logic Martin Lck Leibniz Universitt Hannover 25. Jahrestagung LOGINF, Jena, 22.10.19 Team-based logic Classical semantics: Evaluate formulas in a single state of a system. Propositional assignment First-order structure with
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Team-based logic
Classical semantics: Evaluate formulas in a single state of a system. Propositional assignment First-order structure with assignment Point in Kripke structure Team semantics: Evaluate formulas in multiple states of a system. Set of propositional assignments Set of assignments in a first-order structure Set of points in a Kripke structure These sets are called teams.
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Team-based logic
Example team over N in first-order logic FO: π¦ π§ π¨ π‘1 2 1 2 π‘2 1 1 1 π‘3 1 Several interpretations: Databases: Set of rows in a table Epistemic: Set of states consistent with current knowledge Stochastic: Probability distribution
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Team-based logic
Historically, team semantics was introduced by Hodges (1997) as a compositional semantics for partially ordered quantifiers, e.g., βπ¦ βπ§/{π¦} π. βThere exists π§, independently of π¦, such that π.β makes no sense for single assignment
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Team-based logic
Definition (VÀÀnÀnen (2007))
Dependence logic is the extension of FO by the dependence atom =(π¦1, . . . , π¦n; π§). Read: βπ§ depends only on π¦1, . . . , π¦nβ Note: =(π§) means that π§ is constant in the team.
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Team-based logic
Definition (VÀÀnÀnen (2007))
Dependence logic is the extension of FO by the dependence atom =(π¦1, . . . , π¦n; π§). Read: βπ§ depends only on π¦1, . . . , π¦nβ Note: =(π§) means that π§ is constant in the team.
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Team-based logic
Formally: (π, π) =(β π¦; π§) :β βπ‘, π‘β² β π : if π‘(β π¦) = π‘β²(β π¦) then π‘(π§) = π‘β²(π§). π¦ π§ π¨ π‘1 2 4 2 π‘2 1 2 1 π‘3 1 2 Example: =(π¦; π§) is true in this team, but =(π§; π¨) is false.
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Team-based logic
Formally: (π, π) =(β π¦; π§) :β βπ‘, π‘β² β π : if π‘(β π¦) = π‘β²(β π¦) then π‘(π§) = π‘β²(π§). π¦ π§ π¨ π‘1 2 4 2 π‘2 1 2 1 π‘3 1 2 Example: =(π¦; π§) is true in this team, but =(π§; π¨) is false.
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Dependence logic
Dependence logic = FO + dependence atom. Formulas are in negation normal form. π ::= β | =(β π¦; π§) | π β§ π | π β¨ π | βπ¦π | βπ¦π Here: β first-order literal. Let π be a first-order structure and π β β(Var β π). (π, π) β :β βπ‘ β π : (π, π‘) β, β first-order literal, (π, π) π β§ π :β (π, π) π and (π, π) π, (π, π) π β¨ π :β βπ, π β π such that π = π βͺ π, (π, π) π, and (π, π) π, (to be continued)
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Dependence logic
Dependence logic = FO + dependence atom. Formulas are in negation normal form. π ::= β | =(β π¦; π§) | π β§ π | π β¨ π | βπ¦π | βπ¦π Here: β first-order literal. Let π be a first-order structure and π β β(Var β π). (π, π) β :β βπ‘ β π : (π, π‘) β, β first-order literal, (π, π) π β§ π :β (π, π) π and (π, π) π, (π, π) π β¨ π :β βπ, π β π such that π = π βͺ π, (π, π) π, and (π, π) π, (to be continued)
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Dependence logic
Dependence logic = FO + dependence atom. Formulas are in negation normal form. π ::= β | =(β π¦; π§) | π β§ π | π β¨ π | βπ¦π | βπ¦π Here: β first-order literal. Let π be a first-order structure and π β β(Var β π). (π, π) β :β βπ‘ β π : (π, π‘) β, β first-order literal, (π, π) π β§ π :β (π, π) π and (π, π) π, (π, π) π β¨ π :β βπ, π β π such that π = π βͺ π, (π, π) π, and (π, π) π, (to be continued)
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Dependence logic
Dependence logic = FO + dependence atom. Formulas are in negation normal form. π ::= β | =(β π¦; π§) | π β§ π | π β¨ π | βπ¦π | βπ¦π Here: β first-order literal. Let π be a first-order structure and π β β(Var β π). (π, π) β :β βπ‘ β π : (π, π‘) β, β first-order literal, (π, π) π β§ π :β (π, π) π and (π, π) π, (π, π) π β¨ π :β βπ, π β π such that π = π βͺ π, (π, π) π, and (π, π) π, (to be continued)
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Dependence logic
Dependence logic = FO + dependence atom. Formulas are in negation normal form. π ::= β | =(β π¦; π§) | π β§ π | π β¨ π | βπ¦π | βπ¦π Here: β first-order literal. Let π be a first-order structure and π β β(Var β π). (π, π) β :β βπ‘ β π : (π, π‘) β, β first-order literal, (π, π) π β§ π :β (π, π) π and (π, π) π, (π, π) π β¨ π :β βπ, π β π such that π = π βͺ π, (π, π) π, and (π, π) π, (to be continued)
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Semantics of β¨
Disjunction in team semantics: (π, π) π β¨ π :β βπ, π β π such that π = π βͺ π, (π, π) π, and (π, π) π, π¦ π§ π¨ π₯ π‘1 1 1 π‘2 2 2 1 4 π‘3 4 2 5 (N, π) (π¦ < 1) β¨ (π₯ > 3) (N, π) =(π¦) β¨ =(π¦)
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Semantics of β¨
Disjunction in team semantics: (π, π) π β¨ π :β βπ, π β π such that π = π βͺ π, (π, π) π, and (π, π) π, π¦ π§ π¨ π₯ π‘1 1 1 π‘2 2 2 1 4 π‘3 4 2 5 (N, π) (π¦ < 1) β¨ (π₯ > 3) (N, π) =(π¦) β¨ =(π¦) π π
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Semantics of β¨
Disjunction in team semantics: (π, π) π β¨ π :β βπ, π β π such that π = π βͺ π, (π, π) π, and (π, π) π, π¦ π§ π¨ π₯ π‘1 1 1 π‘2 2 2 1 4 π‘3 4 2 5 (N, π) (π¦ < 1) β¨ (π₯ > 3) (N, π) =(π¦) β¨ =(π¦)
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Semantics of β
We call π : π β β+(π) supplementing function. Example: π(π‘1) = {5} and π(π‘2) = {5, 6, 7}. π§ π‘1 π‘2 1 β π§ π¦ π‘1,1 5 π‘2,1 1 5 π‘2,2 1 6 π‘2,3 1 7 (π, π) βπ¦ π β βπ : π β β+(π) such that (π, π x
f ) π
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Semantics of β
We call π : π β β+(π) supplementing function. Example: π(π‘1) = {5} and π(π‘2) = {5, 6, 7}. π§ π‘1 π‘2 1 β π§ π¦ π‘1,1 5 π‘2,1 1 5 π‘2,2 1 6 π‘2,3 1 7 (π, π) βπ¦ π β βπ : π β β+(π) such that (π, π x
f ) π
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Semantics of β
We call π : π β β+(π) supplementing function. Example: π(π‘1) = {5} and π(π‘2) = {5, 6, 7}. π§ π‘1 π‘2 1 β π§ π¦ π‘1,1 5 π‘2,1 1 5 π‘2,2 1 6 π‘2,3 1 7 (π, π) βπ¦ π β βπ : π β β+(π) such that (π, π x
f ) π
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Semantics of β
π§ π‘1 π‘2 1 β π§ π¦ π‘1,0 π‘1,1 1 π‘1,2 2 . . . π‘2,0 1 π‘2,1 1 1 π‘2,2 1 2 . . . (π, π) βπ¦ π β (π, π x
π) π
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Boolean negation
Sometimes the Boolean negation (βΌ) is added: (π, π) βΌπ :β (π, π) π Β¬ is not the Boolean negation in team semantics! It is possible that, for example, π π¦ = 1 and π Β¬(π¦ = 1).
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Flatness
Classical formulas obey flatness: (π, π) π β βπ‘ β π : (π, {π‘}) π for π β FO. (Proof: By induction.) For example =(π¦) is not flat.
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Flatness
Classical formulas obey flatness: (π, π) π β βπ‘ β π : (π, {π‘}) π for π β FO. (Proof: By induction.) For example =(π¦) is not flat.
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Motivation
Based on team semantics, numerous logics have been defined: Propositional ... Modal ... First-order ...
+
... dependence (=(π¦; π§)) ... ... independence (π¦ β₯ π§) ... ... inclusion (π¦ β π§) ...
+
... logic. Many papers on expressiveness and complexity.
Logic Satisfiability Validity PL(=(Β· Β· Β· )) NP NEXP ML(=(Β· Β· Β· )) NEXP NEXP PL(β₯) NP NEXP-hard, in Ξ E
2
ML(β₯) NEXP Ξ E
2 -hard
PL(β) EXP co-NP ML(β) EXP co-NEXP-hard Slide 13
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Motivation
Based on team semantics, numerous logics have been defined: Propositional ... Modal ... First-order ...
+
... dependence (=(π¦; π§)) ... ... independence (π¦ β₯ π§) ... ... inclusion (π¦ β π§) ...
+
... logic. Many papers on expressiveness and complexity. This talk: Abstract framework to study team logic in. Fundamental principles for definition of team semantics? βTeamifyβ other connectives or even logics? Formulate results instead of re-proving from scratch everytime?
locality, various closure properties, expressiveness results ... Slide 13
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Motivation
Based on team semantics, numerous logics have been defined: Propositional ... Modal ... First-order ...
+
... dependence (=(π¦; π§)) ... ... independence (π¦ β₯ π§) ... ... inclusion (π¦ β π§) ...
+
... logic. Many papers on expressiveness and complexity. This talk: Abstract framework to study team logic in. Fundamental principles for definition of team semantics? βTeamifyβ other connectives or even logics? Formulate results instead of re-proving from scratch everytime?
locality, various closure properties, expressiveness results ... Slide 13
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Motivation
Based on team semantics, numerous logics have been defined: Propositional ... Modal ... First-order ...
+
... dependence (=(π¦; π§)) ... ... independence (π¦ β₯ π§) ... ... inclusion (π¦ β π§) ...
+
... logic. Many papers on expressiveness and complexity. This talk: Abstract framework to study team logic in. Fundamental principles for definition of team semantics? βTeamifyβ other connectives or even logics? Formulate results instead of re-proving from scratch everytime?
locality, various closure properties, expressiveness results ... Slide 13
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Motivation
Based on team semantics, numerous logics have been defined: Propositional ... Modal ... First-order ...
+
... dependence (=(π¦; π§)) ... ... independence (π¦ β₯ π§) ... ... inclusion (π¦ β π§) ...
+
... logic. Many papers on expressiveness and complexity. This talk: Abstract framework to study team logic in. Fundamental principles for definition of team semantics? βTeamifyβ other connectives or even logics? Formulate results instead of re-proving from scratch everytime?
locality, various closure properties, expressiveness results ... Slide 13
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Algebraic definition of semantics
Abstract definition in terms of universal algebra.
Definition
A signature π is a set of symbols, e.g., π = {Β¬, β§, β¨} βͺ Prop.
Definition
A π-algebra A = (π΅, (πβ³)β³βΟ) consists of a non-empty set π΅, the carrier, and for each β³ β π a map πβ³ : π΅arity(β³) β π΅. Think of elements π β π΅ as properties. Usually, we want π΅ = βπ for some βuniverseβ π, such that, e.g., πβ§ = β© and πβ¨ = βͺ.
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Algebraic definition of semantics
Abstract definition in terms of universal algebra.
Definition
A signature π is a set of symbols, e.g., π = {Β¬, β§, β¨} βͺ Prop.
Definition
A π-algebra A = (π΅, (πβ³)β³βΟ) consists of a non-empty set π΅, the carrier, and for each β³ β π a map πβ³ : π΅arity(β³) β π΅. Think of elements π β π΅ as properties. Usually, we want π΅ = βπ for some βuniverseβ π, such that, e.g., πβ§ = β© and πβ¨ = βͺ.
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Algebraic definition of semantics
Abstract definition in terms of universal algebra.
Definition
A signature π is a set of symbols, e.g., π = {Β¬, β§, β¨} βͺ Prop.
Definition
A π-algebra A = (π΅, (πβ³)β³βΟ) consists of a non-empty set π΅, the carrier, and for each β³ β π a map πβ³ : π΅arity(β³) β π΅. Think of elements π β π΅ as properties. Usually, we want π΅ = βπ for some βuniverseβ π, such that, e.g., πβ§ = β© and πβ¨ = βͺ.
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Algebraic definition of semantics
Definition
A π-team algebra is a π-algebra T = (π΅, (πβ³)β³βΟ) where π΅ = ββπ for some set π. Properties π are sets of teams, that is, sets of sets. Obtain βnaturalβ π: (ββπ)n β ββπ from π : (βπ)n β βπ?
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Algebraic definition of semantics
Definition
A π-team algebra is a π-algebra T = (π΅, (πβ³)β³βΟ) where π΅ = ββπ for some set π. Properties π are sets of teams, that is, sets of sets. Obtain βnaturalβ π: (ββπ)n β ββπ from π : (βπ)n β βπ?
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Teamification
Let π : (βπ)n β βπ and π: (ββπ)n β ββπ.
Definition
We call π a teamification of π if βπ(π1, . . . , πn) = π(βπ1, . . . , βπn) for all π1, . . . , πn β π. If this holds for all connectives, then β is an algebra homomorphism.
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Teamification
Theorem
A map π: (ββπ)n β ββπ is a teamification iff it preserves flatness. (A property π β ββπ is flat if π β π β βπ‘ β π : {π‘} β π.) (Flatness preserving: π1, . . . , πn flat β π(π1, . . . , πn) flat.)
Theorem
Team-semantical β§, β¨, βπ¦, βπ¦ are teamifications of classical counterparts. Same for β§, β¨, β¦, in modal team logic, and β§, β¨ in propositional team logic.
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Teamification
Theorem
A map π: (ββπ)n β ββπ is a teamification iff it preserves flatness. (A property π β ββπ is flat if π β π β βπ‘ β π : {π‘} β π.) (Flatness preserving: π1, . . . , πn flat β π(π1, . . . , πn) flat.)
Theorem
Team-semantical β§, β¨, βπ¦, βπ¦ are teamifications of classical counterparts. Same for β§, β¨, β¦, in modal team logic, and β§, β¨ in propositional team logic.
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Teamification
Theorem
A map π: (ββπ)n β ββπ is a teamification iff it preserves flatness. (A property π β ββπ is flat if π β π β βπ‘ β π : {π‘} β π.) (Flatness preserving: π1, . . . , πn flat β π(π1, . . . , πn) flat.)
Theorem
Team-semantical β§, β¨, βπ¦, βπ¦ are teamifications of classical counterparts. Same for β§, β¨, β¦, in modal team logic, and β§, β¨ in propositional team logic.
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Teamification
Example: Getting rid of negation normal form. How to define Β¬ on teams? π Β¬π :β π π for all non-empty π β π is a teamification of classical negation Β¬.
Proof.
βπΒ¬(π) = β(π β π) = {π β² β π | π β² β© π = β } = {π β² β π | βπ β π β² : π β© π = β } = {π β² β π | βπ β π β² : βπ β© βπ = {β }} = {π β² β π | βπ β π β² : π = β or π / β βπ} = πΒ¬(βπ)
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Teamification
Example: Getting rid of negation normal form. How to define Β¬ on teams? π Β¬π :β π π for all non-empty π β π is a teamification of classical negation Β¬.
Proof.
βπΒ¬(π) = β(π β π) = {π β² β π | π β² β© π = β } = {π β² β π | βπ β π β² : π β© π = β } = {π β² β π | βπ β π β² : βπ β© βπ = {β }} = {π β² β π | βπ β π β² : π = β or π / β βπ} = πΒ¬(βπ)
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Teamification
Problem: Teamifications are fully determined on flat arguments, but they are arbitrary on non-flat arguments! Formulas such as =(π¦) are non-flat.
Theorem
Let πi be a teamification of πi, π β {1, 2}. Then the following statements are equivalent: π1 = π2. π1(β π) = π2(β π) for all flat arguments β π.
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Teamification
Problem: Teamifications are fully determined on flat arguments, but they are arbitrary on non-flat arguments! Formulas such as =(π¦) are non-flat.
Theorem
Let πi be a teamification of πi, π β {1, 2}. Then the following statements are equivalent: π1 = π2. π1(β π) = π2(β π) for all flat arguments β π.
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Operators
Further sensible restrictions for connectives on arbitrary arguments? The connectives β§, β¨, βπ¦, βπ¦, β¦, are operators in the sense of JΓ³nsson and Tarski (1951). Every π-ary operator β³ is induced by some (π + 1)-ary βsuccessorβ relation πβ³: π β³(π1, . . . , πn) :β β(π, π1, . . . , πn) β πβ³ and πi πi for all π Unlike in classical logic, β§, βπ¦ and are operators in team semantics! Also, βΌΒ¬ is an operator. Natural class π β Operators β© Teamifications ?
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Operators
Further sensible restrictions for connectives on arbitrary arguments? The connectives β§, β¨, βπ¦, βπ¦, β¦, are operators in the sense of JΓ³nsson and Tarski (1951). Every π-ary operator β³ is induced by some (π + 1)-ary βsuccessorβ relation πβ³: π β³(π1, . . . , πn) :β β(π, π1, . . . , πn) β πβ³ and πi πi for all π Unlike in classical logic, β§, βπ¦ and are operators in team semantics! Also, βΌΒ¬ is an operator. Natural class π β Operators β© Teamifications ?
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Operators
Further sensible restrictions for connectives on arbitrary arguments? The connectives β§, β¨, βπ¦, βπ¦, β¦, are operators in the sense of JΓ³nsson and Tarski (1951). Every π-ary operator β³ is induced by some (π + 1)-ary βsuccessorβ relation πβ³: π β³(π1, . . . , πn) :β β(π, π1, . . . , πn) β πβ³ and πi πi for all π Unlike in classical logic, β§, βπ¦ and are operators in team semantics! Also, βΌΒ¬ is an operator. Natural class π β Operators β© Teamifications ?
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Operators
Further sensible restrictions for connectives on arbitrary arguments? The connectives β§, β¨, βπ¦, βπ¦, β¦, are operators in the sense of JΓ³nsson and Tarski (1951). Every π-ary operator β³ is induced by some (π + 1)-ary βsuccessorβ relation πβ³: π β³(π1, . . . , πn) :β β(π, π1, . . . , πn) β πβ³ and πi πi for all π Unlike in classical logic, β§, βπ¦ and are operators in team semantics! Also, βΌΒ¬ is an operator. Natural class π β Operators β© Teamifications ?
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Operators
Further sensible restrictions for connectives on arbitrary arguments? The connectives β§, β¨, βπ¦, βπ¦, β¦, are operators in the sense of JΓ³nsson and Tarski (1951). Every π-ary operator β³ is induced by some (π + 1)-ary βsuccessorβ relation πβ³: π β³(π1, . . . , πn) :β β(π, π1, . . . , πn) β πβ³ and πi πi for all π Unlike in classical logic, β§, βπ¦ and are operators in team semantics! Also, βΌΒ¬ is an operator. Natural class π β Operators β© Teamifications ?
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Transversals
Definition
A unary operator β³ is a transversal if, for all teams π, π, πβ³(π, π) β π = βοΈ
wβT
π(π₯) for some π such that πβ³({π₯}, π(π₯)) Successors of π are precisely the unions of successors of singletons in π. (Definition for π-ary transversals similar.)
Theorem
Transversals are flatness preserving.
Theorem
The connectives β§, β¨, βπ¦, βπ¦, β¦, are transversals, as well as all βclassicalβ atomic formulas in team semantics.
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Transversals
Definition
A unary operator β³ is a transversal if, for all teams π, π, πβ³(π, π) β π = βοΈ
wβT
π(π₯) for some π such that πβ³({π₯}, π(π₯)) Successors of π are precisely the unions of successors of singletons in π. (Definition for π-ary transversals similar.)
Theorem
Transversals are flatness preserving.
Theorem
The connectives β§, β¨, βπ¦, βπ¦, β¦, are transversals, as well as all βclassicalβ atomic formulas in team semantics.
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Transversals
Definition
A unary operator β³ is a transversal if, for all teams π, π, πβ³(π, π) β π = βοΈ
wβT
π(π₯) for some π such that πβ³({π₯}, π(π₯)) Successors of π are precisely the unions of successors of singletons in π. (Definition for π-ary transversals similar.)
Theorem
Transversals are flatness preserving.
Theorem
The connectives β§, β¨, βπ¦, βπ¦, β¦, are transversals, as well as all βclassicalβ atomic formulas in team semantics.
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Transversals
Definition
A unary operator β³ is a transversal if, for all teams π, π, πβ³(π, π) β π = βοΈ
wβT
π(π₯) for some π such that πβ³({π₯}, π(π₯)) Successors of π are precisely the unions of successors of singletons in π. (Definition for π-ary transversals similar.)
Theorem
Transversals are flatness preserving.
Theorem
The connectives β§, β¨, βπ¦, βπ¦, β¦, are transversals, as well as all βclassicalβ atomic formulas in team semantics.
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Transversals
Definition
A unary operator β³ is a transversal if, for all teams π, π, πβ³(π, π) β π = βοΈ
wβT
π(π₯) for some π such that πβ³({π₯}, π(π₯)) Successors of π are precisely the unions of successors of singletons in π. (Definition for π-ary transversals similar.)
Theorem
Transversals are flatness preserving.
Theorem
The connectives β§, β¨, βπ¦, βπ¦, β¦, are transversals, as well as all βclassicalβ atomic formulas in team semantics.
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Transversals
Conjunction β§: πβ§{π‘} = {οΈ {π‘}{π‘} }οΈ Disjunction β¨: πβ¨{π‘} = {οΈ {π‘}{π‘}, β {π‘}, {π‘}β }οΈ Existential quantifier βπ¦: πβx{π‘} = β+{π‘x
a | π β π}
Universal quantifier βπ¦: πβx{π‘} = {οΈ {π‘x
a | π β π}
}οΈ
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Transversals
Conjunction β§: πβ§{π‘} = {οΈ {π‘}{π‘} }οΈ Disjunction β¨: πβ¨{π‘} = {οΈ {π‘}{π‘}, β {π‘}, {π‘}β }οΈ Existential quantifier βπ¦: πβx{π‘} = β+{π‘x
a | π β π}
Universal quantifier βπ¦: πβx{π‘} = {οΈ {π‘x
a | π β π}
}οΈ
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Transversals
Conjunction β§: πβ§{π‘} = {οΈ {π‘}{π‘} }οΈ Disjunction β¨: πβ¨{π‘} = {οΈ {π‘}{π‘}, β {π‘}, {π‘}β }οΈ Existential quantifier βπ¦: πβx{π‘} = β+{π‘x
a | π β π}
Universal quantifier βπ¦: πβx{π‘} = {οΈ {π‘x
a | π β π}
}οΈ
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Transversals
Conjunction β§: πβ§{π‘} = {οΈ {π‘}{π‘} }οΈ Disjunction β¨: πβ¨{π‘} = {οΈ {π‘}{π‘}, β {π‘}, {π‘}β }οΈ Existential quantifier βπ¦: πβx{π‘} = β+{π‘x
a | π β π}
Universal quantifier βπ¦: πβx{π‘} = {οΈ {π‘x
a | π β π}
}οΈ
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Transversals
Application:
Theorem
Every FO(βΌ)-formula is equivalent to a Boolean combination of FO-formulas. Proof idea: Transversals* commute** with all Boolean operators!
Theorem
Satisfiability of the team-semantical extensions FO2(βΌ) of two-variable first-order logic FO2, GF(βΌ) of the guarded fragment GF of first-order logic, ML(βΌ) of modal logic ML is decidable. In fact logspace-complete for the class TIME (οΈ exppoly(n)(1) )οΈ .
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Transversals
Application:
Theorem
Every FO(βΌ)-formula is equivalent to a Boolean combination of FO-formulas. Proof idea: Transversals* commute** with all Boolean operators!
Theorem
Satisfiability of the team-semantical extensions FO2(βΌ) of two-variable first-order logic FO2, GF(βΌ) of the guarded fragment GF of first-order logic, ML(βΌ) of modal logic ML is decidable. In fact logspace-complete for the class TIME (οΈ exppoly(n)(1) )οΈ .
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Transversals
Application:
Theorem
Every FO(βΌ)-formula is equivalent to a Boolean combination of FO-formulas. Proof idea: Transversals* commute** with all Boolean operators!
Theorem
Satisfiability of the team-semantical extensions FO2(βΌ) of two-variable first-order logic FO2, GF(βΌ) of the guarded fragment GF of first-order logic, ML(βΌ) of modal logic ML is decidable. In fact logspace-complete for the class TIME (οΈ exppoly(n)(1) )οΈ .
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Transversals
Application:
Theorem
Every FO(βΌ)-formula is equivalent to a Boolean combination of FO-formulas. Proof idea: Transversals* commute** with all Boolean operators!
Theorem
Satisfiability of the team-semantical extensions FO2(βΌ) of two-variable first-order logic FO2, GF(βΌ) of the guarded fragment GF of first-order logic, ML(βΌ) of modal logic ML is decidable. In fact logspace-complete for the class TIME (οΈ exppoly(n)(1) )οΈ .
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Conclusion
Transversals are a natural class of flatness-preserving team-semantical connectives, and possess a number of nice properties. Future work: More classifications of team logics in the framework. Incorporate connectives that are not flatness preserving (e.g., temporal operators). Smallest unit: Atomic formulas. How to show, e.g., locality?
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