Strongly First-Order Dependencies in Team
Semantics
Pietro Galliani
TU Clausthal
Dependence Logic Academy Colloquium
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
A Small Problem
A common conversation
A Dependence Logic is First-Order Logic plus functional
dependence atoms.
B OK.
A And it is as expressive as the existential fragment of
Second-Order Logic.
B Why? Functional dependency is a first-order definable
property...
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
A Small Problem
A common conversation
A Dependence Logic is First-Order Logic plus functional
dependence atoms.
B OK.
A And it is as expressive as the existential fragment of
Second-Order Logic.
B Why? Functional dependency is a first-order definable
property...
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
A Small Problem
A common conversation
A Dependence Logic is First-Order Logic plus functional
dependence atoms.
B OK.
A And it is as expressive as the existential fragment of
Second-Order Logic.
B Why? Functional dependency is a first-order definable
property...
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
A Small Problem
A common conversation
A Dependence Logic is First-Order Logic plus functional
dependence atoms.
B OK.
A And it is as expressive as the existential fragment of
Second-Order Logic.
B Why? Functional dependency is a first-order definable
property...
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
The Answer
Actually. . .
Dependence Logic is First-Order Logic with team semantics
plus functional dependence atoms.
Team Semantics
Extends Tarski’s semantics to sets of assignments;
Second-order character (see rules for disjunctions and
existentials);
Equivalent to Tarski’s semantics wrt First Order sentences,
but allows for different extensions.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
The Answer
Actually. . .
Dependence Logic is First-Order Logic with team semantics
plus functional dependence atoms.
Team Semantics
Extends Tarski’s semantics to sets of assignments;
Second-order character (see rules for disjunctions and
existentials);
Equivalent to Tarski’s semantics wrt First Order sentences,
but allows for different extensions.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
What this talk is about
Team Semantics: extends Tarski, satisfaction wrt sets of
assignments: X = {s1 . . . sn }, M |=X φ iff . . .
Adding Dependency atoms: D = {D1 , . . . , Dn }, FO(D) can
be much stronger than FO (often, FO(D) ≡ Σ11 ), even if the
Di themselves are first-order (as properties of relations);
Main question: find nontrivial D s.t. FO(D) captures only
first order properties of relations (and, therefore,
FO(D) = FO wrt sentences).
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
What this talk is about
Team Semantics: extends Tarski, satisfaction wrt sets of
assignments: X = {s1 . . . sn }, M |=X φ iff . . .
Adding Dependency atoms: D = {D1 , . . . , Dn }, FO(D) can
be much stronger than FO (often, FO(D) ≡ Σ11 ), even if the
Di themselves are first-order (as properties of relations);
Main question: find nontrivial D s.t. FO(D) captures only
first order properties of relations (and, therefore,
FO(D) = FO wrt sentences).
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
What this talk is about
Team Semantics: extends Tarski, satisfaction wrt sets of
assignments: X = {s1 . . . sn }, M |=X φ iff . . .
Adding Dependency atoms: D = {D1 , . . . , Dn }, FO(D) can
be much stronger than FO (often, FO(D) ≡ Σ11 ), even if the
Di themselves are first-order (as properties of relations);
Main question: find nontrivial D s.t. FO(D) captures only
first order properties of relations (and, therefore,
FO(D) = FO wrt sentences).
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Why?
Answer One: Applications
People have been thinking about applications of team
semantics to database theory, belief representation, and even
physics (more about this later in this conference). It would be
useful to know which dependencies are “safe”.
Answer Two: the First Order / Second Order Barrier
Studying Team Semantics-based extensions of First Order
Logic gives us a new perspective about the boundary between
first and second order; and in this talk, we will attempt to
approach this boundary “from below”.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Why?
Answer One: Applications
People have been thinking about applications of team
semantics to database theory, belief representation, and even
physics (more about this later in this conference). It would be
useful to know which dependencies are “safe”.
Answer Two: the First Order / Second Order Barrier
Studying Team Semantics-based extensions of First Order
Logic gives us a new perspective about the boundary between
first and second order; and in this talk, we will attempt to
approach this boundary “from below”.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Outline
1
Team Semantics and Dependencies
2
Upwards Closed Dependencies
3
Boundedness
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Outline
1
Team Semantics and Dependencies
2
Upwards Closed Dependencies
3
Boundedness
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Tarski’s Semantics for First Order Logic
Assignments as States of Things
An assignment s is a function from variables to elements of a
model, representing a possible state of things. M |=s φ if and
only if φ is true of the state s.
Tarski Semantics (assuming Negation Normal Form)
M |=s R~t if and only if ~thsi ∈ R M ;
M |=s ¬R~t if and only if ~thsi 6∈ R M ;
M |=s t1 = t2 if and only if t1 hsi = t2 hsi;
M |=s t1 6= t2 if and only if t1 hsi = t2 hsi;
M |=s φ ∧ ψ if and only if M |=s φ and M |=s ψ;
M |=s φ ∨ ψ if and only if M |=s φ or M |=s ψ;
M |=s ∃v φ if and only if ∃m ∈ Dom(M) s.t. M |=s[m/v ] φ;
M |=s ∀v φ if and only if ∀m ∈ Dom(M), M |=s[m/v ] φ.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Tarski’s Semantics for First Order Logic
Assignments as States of Things
An assignment s is a function from variables to elements of a
model, representing a possible state of things. M |=s φ if and
only if φ is true of the state s.
Tarski Semantics (assuming Negation Normal Form)
M |=s R~t if and only if ~thsi ∈ R M ;
M |=s ¬R~t if and only if ~thsi 6∈ R M ;
M |=s t1 = t2 if and only if t1 hsi = t2 hsi;
M |=s t1 6= t2 if and only if t1 hsi = t2 hsi;
M |=s φ ∧ ψ if and only if M |=s φ and M |=s ψ;
M |=s φ ∨ ψ if and only if M |=s φ or M |=s ψ;
M |=s ∃v φ if and only if ∃m ∈ Dom(M) s.t. M |=s[m/v ] φ;
M |=s ∀v φ if and only if ∀m ∈ Dom(M), M |=s[m/v ] φ.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
From Assignments to Teams
Teams (= Belief Sets)
A team X is a set of assignments (possible states of things). It
may be thought of as the belief state of an agent.
Satisfaction in a Team
For all first order φ, M |=X φ if and only if M |=s φ for all s ∈ X .
Operations over Teams
Supplementation: X [H/v ] = {s[m/v ] : s ∈ X , m ∈ H(s)};
Duplication: X [M/v ] = {s[m/v ] : s ∈ X , m ∈ Dom(M)}.
From Teams to Relations
For X team, ~v variables, X (~v ) = {s(~v ) : s ∈ X }.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
From Assignments to Teams
Teams (= Belief Sets)
A team X is a set of assignments (possible states of things). It
may be thought of as the belief state of an agent.
Satisfaction in a Team
For all first order φ, M |=X φ if and only if M |=s φ for all s ∈ X .
Operations over Teams
Supplementation: X [H/v ] = {s[m/v ] : s ∈ X , m ∈ H(s)};
Duplication: X [M/v ] = {s[m/v ] : s ∈ X , m ∈ Dom(M)}.
From Teams to Relations
For X team, ~v variables, X (~v ) = {s(~v ) : s ∈ X }.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
From Assignments to Teams
Teams (= Belief Sets)
A team X is a set of assignments (possible states of things). It
may be thought of as the belief state of an agent.
Satisfaction in a Team
For all first order φ, M |=X φ if and only if M |=s φ for all s ∈ X .
Operations over Teams
Supplementation: X [H/v ] = {s[m/v ] : s ∈ X , m ∈ H(s)};
Duplication: X [M/v ] = {s[m/v ] : s ∈ X , m ∈ Dom(M)}.
From Teams to Relations
For X team, ~v variables, X (~v ) = {s(~v ) : s ∈ X }.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Team Semantics for First Order Logic
Team Semantics
M |=X φ if and only if M |=s φ for all s ∈ X .
If α literal, M |=X α iff for all s ∈ X , M |=s α;
M |=X φ ∧ ψ iff for all s ∈ X , M |=s φ and M |=s ψ;
M |=X φ ∨ ψ iff for all s ∈ X , M |=s φ or M |=s ψ;
M |=X ∃v φ iff for all s ∈ X , ∃m ∈ Dom(M) s.t. M |=s[m/v ] φ;
M |=X ∀v φ iff for all s ∈ X , ∀m ∈ Dom(M), M |=s[m/v ] φ.
Aside: Team Semantics and Game Theoretic Semantics
Teams correspond to sets of possible states in the subgames of
the semantic game, when players are allowed nondeterministic
strategies.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Team Semantics for First Order Logic
Team Semantics
M |=X φ if and only if M |=s φ for all s ∈ X .
If α literal, M |=X α iff for all s ∈ X , M |=s α;
M |=X φ ∧ ψ iff for all s ∈ X , M |=s φ and M |=s ψ;
M |=X φ ∨ ψ iff for all s ∈ X , M |=s φ or M |=s ψ;
M |=X ∃v φ iff for all s ∈ X , ∃m ∈ Dom(M) s.t. M |=s[m/v ] φ;
M |=X ∀v φ iff for all s ∈ X , ∀m ∈ Dom(M), M |=s[m/v ] φ.
Aside: Team Semantics and Game Theoretic Semantics
Teams correspond to sets of possible states in the subgames of
the semantic game, when players are allowed nondeterministic
strategies.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Team Semantics for First Order Logic
Team Semantics
M |=X φ if and only if M |=s φ for all s ∈ X .
If α literal, M |=X α iff for all s ∈ X , M |=s α;
M |=X φ ∧ ψ iff M |=X φ and M |=X ψ;
M |=X φ ∨ ψ iff for all s ∈ X , M |=s φ or M |=s ψ;
M |=X ∃v φ iff for all s ∈ X , ∃m ∈ Dom(M) s.t. M |=s[m/v ] φ;
M |=X ∀v φ iff for all s ∈ X , ∀m ∈ Dom(M), M |=s[m/v ] φ.
Aside: Team Semantics and Game Theoretic Semantics
Teams correspond to sets of possible states in the subgames of
the semantic game, when players are allowed nondeterministic
strategies.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Team Semantics for First Order Logic
Team Semantics
M |=X φ if and only if M |=s φ for all s ∈ X .
If α literal, M |=X α iff for all s ∈ X , M |=s α;
M |=X φ ∧ ψ iff M |=X φ and M |=X ψ;
M |=X φ ∨ ψ iff X = Y ∪ Z , M |=Y φ and M |=Z ψ;
M |=X ∃v φ iff for all s ∈ X , ∃m ∈ Dom(M) s.t. M |=s[m/v ] φ;
M |=X ∀v φ iff for all s ∈ X , ∀m ∈ Dom(M), M |=s[m/v ] φ.
Aside: Team Semantics and Game Theoretic Semantics
Teams correspond to sets of possible states in the subgames of
the semantic game, when players are allowed nondeterministic
strategies.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Team Semantics for First Order Logic
Team Semantics
M |=X φ if and only if M |=s φ for all s ∈ X .
If α literal, M |=X α iff for all s ∈ X , M |=s α;
M |=X φ ∧ ψ iff M |=X φ and M |=X ψ;
M |=X φ ∨ ψ iff X = Y ∪ Z , M |=Y φ and M |=Z ψ;
M |=X ∃v φ iff ∃H : X → P(Dom(M))\{∅} s.t. M |=X [H/v ] φ;
M |=X ∀v φ iff for all s ∈ X , ∀m ∈ Dom(M), M |=s[m/v ] φ.
Aside: Team Semantics and Game Theoretic Semantics
Teams correspond to sets of possible states in the subgames of
the semantic game, when players are allowed nondeterministic
strategies.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Team Semantics for First Order Logic
Team Semantics
M |=X φ if and only if M |=s φ for all s ∈ X .
If α literal, M |=X α iff for all s ∈ X , M |=s α;
M |=X φ ∧ ψ iff M |=X φ and M |=X ψ;
M |=X φ ∨ ψ iff X = Y ∪ Z , M |=Y φ and M |=Z ψ;
M |=X ∃v φ iff ∃H : X → P(Dom(M))\{∅} s.t. M |=X [H/v ] φ;
M |=X ∀v φ iff M |=X [M/v ] φ.
Aside: Team Semantics and Game Theoretic Semantics
Teams correspond to sets of possible states in the subgames of
the semantic game, when players are allowed nondeterministic
strategies.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Team Semantics for First Order Logic
Team Semantics
M |=X φ if and only if M |=s φ for all s ∈ X .
If α literal, M |=X α iff for all s ∈ X , M |=s α;
M |=X φ ∧ ψ iff M |=X φ and M |=X ψ;
M |=X φ ∨ ψ iff X = Y ∪ Z , M |=Y φ and M |=Z ψ;
M |=X ∃v φ iff ∃H : X → P(Dom(M))\{∅} s.t. M |=X [H/v ] φ;
M |=X ∀v φ iff M |=X [M/v ] φ.
Aside: Team Semantics and Game Theoretic Semantics
Teams correspond to sets of possible states in the subgames of
the semantic game, when players are allowed nondeterministic
strategies.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Satisfaction Conditions of First Order Formulas
Satisfaction Conditions in Team Semantics
kφ(~v )k = {(M, X (~v )) : M |=X φ(~v )}
Satisfaction Conditions of F.O. formulas are F.O. Definable
kφ(~v )k = {(M, R) : (M, R) |= ∀~v (R~v → φ(~v ))}.
However. . .
Not all first order properties of relations correspond to
satisfaction conditions of first order formulas in Team
Semantics. For example, M |=∅ φ for all φ, and thus
kφ(~v )k =
6 {(M, R) : R 6= ∅} = {(M, R) : (M, R) |= ∃~v R~v )}.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Satisfaction Conditions of First Order Formulas
Satisfaction Conditions in Team Semantics
kφ(~v )k = {(M, X (~v )) : M |=X φ(~v )}
Satisfaction Conditions of F.O. formulas are F.O. Definable
kφ(~v )k = {(M, R) : (M, R) |= ∀~v (R~v → φ(~v ))}.
However. . .
Not all first order properties of relations correspond to
satisfaction conditions of first order formulas in Team
Semantics. For example, M |=∅ φ for all φ, and thus
kφ(~v )k =
6 {(M, R) : R 6= ∅} = {(M, R) : (M, R) |= ∃~v R~v )}.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Satisfaction Conditions of First Order Formulas
Satisfaction Conditions in Team Semantics
kφ(~v )k = {(M, X (~v )) : M |=X φ(~v )}
Satisfaction Conditions of F.O. formulas are F.O. Definable
kφ(~v )k = {(M, R) : (M, R) |= ∀~v (R~v → φ(~v ))}.
However. . .
Not all first order properties of relations correspond to
satisfaction conditions of first order formulas in Team
Semantics. For example, M |=∅ φ for all φ, and thus
kφ(~v )k =
6 {(M, R) : R 6= ∅} = {(M, R) : (M, R) |= ∃~v R~v )}.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Extending First Order Logic by First Order Properties
What we just saw
There is no first order φ such that M |=X φ if and only if X 6= ∅.
A new atom
M |=X NE if and only if X 6= ∅.
Question
What kind of logic is FO(NE)? Is any sentence of FO(NE)
equivalent to some first order sentence? What about other
atoms, for example from Database Theory?
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Database-theoretic Dependencies
Dependence Logic FO(=(·, ·)): M |=X =(~y , ~z ) iff for all s, s′ ∈ X ,
s(~y ) = s′ (~y ) ⇒ s(~z ) = s′ (~z );
Exclusion Logic FO(|): M |=X ~y | ~z iff X (~y ) ∩ X (~z ) = ∅;
Inclusion Logic FO(⊆): M |=X ~y ⊆ ~z iff X (~y ) ⊆ X (~z );
Independence Logic FO(⊥): M |=X ~y ⊥ ~z iff
X (~y ~z ) = X (~y ) × X (~z ).
First Order?
These dependencies are first-order (as properties of relations),
but the resulting logics are not: for example, Independence
Logic = Σ11 and Inclusion Logic = GFP + .
The Question
Under which conditions is FO(D) equivalent to FO?
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Database-theoretic Dependencies
Dependence Logic FO(=(·, ·)): M |=X =(~y , ~z ) iff for all s, s′ ∈ X ,
s(~y ) = s′ (~y ) ⇒ s(~z ) = s′ (~z );
Exclusion Logic FO(|): M |=X ~y | ~z iff X (~y ) ∩ X (~z ) = ∅;
Inclusion Logic FO(⊆): M |=X ~y ⊆ ~z iff X (~y ) ⊆ X (~z );
Independence Logic FO(⊥): M |=X ~y ⊥ ~z iff
X (~y ~z ) = X (~y ) × X (~z ).
First Order?
These dependencies are first-order (as properties of relations),
but the resulting logics are not: for example, Independence
Logic = Σ11 and Inclusion Logic = GFP + .
The Question
Under which conditions is FO(D) equivalent to FO?
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Let’s be more precise. . .
Generalized Dependencies (Kuusisto 2013)
A dependency D of arity k is a class, closed under
isomorphisms, of models (Dom(M), R) for R k -ary.
M |=X D~y ⇔ (Dom(M), X (~y )) ∈ D.
First-Order Dependencies
D is first order if there is a F.O. sentence ψ ∗ (R) s.t.
(Dom(M), R) ∈ D ⇔ (Dom(M), R) |= ψ ∗ (R)
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
A Classification of Dependencies
Three Properties of Dependencies
Empty Team: M |=∅ D~y for all M and all ~y ;
Downwards Closure: If M |=X D~y and Y ⊆ X then M |=Y D~y ;
Union Closure: If M |=Xi D~y for all i ∈ I then M |=S Xi D~y .
i
Observation
All union closed dependencies and all nontrivial downwards
closed dependencies satisfy the Empty Team Property.
Another observation
These properties are preserved by Team Semantics: for
example, if D has downwards closure and φ ∈ FO(D) then φ
has downwards closure, that is,
M |=X φ, Y ⊆ X ⇒ M |=Y φ.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Classifying Dependencies
Dependency
=(y , z)
y |z
y ⊆z
y ⊥z
NE
Empty Team?
YES
YES
YES
YES
NO
Pietro Galliani
Downwards Cl.?
YES
YES
NO
NO
NO
Union Cl.?
NO
NO
YES
NO
NO
Strongly First-Order Dependencies in Team Semantics
Some Known Results
Three families of dependencies
1
DET = {D : D F.O. definable, empty team prop.};
2
D↓ = {D : D F.O. def, downwards closed, empty team pr.};
3
D∪ = {D : D F.O. def., union closed }.
Some Results (wrt formulas)
Kontinen, Väänänen 2009: Dependence Logic = FO(D↓ );
Galliani 2012: Independence Logic = FO(DET );
Galliani, Hella 2013: Inclusion Logic = FO(D ∪ ).
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Some Known Results
Three families of dependencies
1
DET = {D : D F.O. definable, empty team prop.};
2
D↓ = {D : D F.O. def, downwards closed, empty team pr.};
3
D∪ = {D : D F.O. def., union closed }.
Some More Results (wrt sentences)
Väänänen 2007: Dependence Logic = Σ11 ;
Grädel, Väänänen 2012: Independence Logic = Σ11 ;
Galliani, Hella 2013: Inclusion Logic = GFP + .
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Relations between Dependence Families (formulas)
FO(DET )
⊂
⊃
S
FO(D↓ )
FO(D )
⊃
⊂
FO
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Relations between Dependence Families (Sentences)
FO(DET ), FO(D↓ )
≡ Σ11
∪
S
FO(D )
≡ GFP+
∪
FO
Pietro Galliani
≡ FO
Strongly First-Order Dependencies in Team Semantics
Outline
1
Team Semantics and Dependencies
2
Upwards Closed Dependencies
3
Boundedness
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Strongly First Order Dependency Families
So far, we identified families of dependencies D such that
FO(D) is stronger than First Order Logic.
But can we find (non-trivial) families D such that
FO(D) = FO?
How much can we get away with adding to Team
Semantics before “jumping” beyond First Order Logic?
Strongly First Order Dependencies
Let D = {D1 , D2 , . . .} be a family of first order definable
dependencies. We say that D is strongly first order if every
sentence of FO(D) is equivalent to some first order sentence.
Open Problem
Find necessary and sufficient conditions for D to be strongly
first order.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Strongly First Order Dependency Families
So far, we identified families of dependencies D such that
FO(D) is stronger than First Order Logic.
But can we find (non-trivial) families D such that
FO(D) = FO?
How much can we get away with adding to Team
Semantics before “jumping” beyond First Order Logic?
Strongly First Order Dependencies
Let D = {D1 , D2 , . . .} be a family of first order definable
dependencies. We say that D is strongly first order if every
sentence of FO(D) is equivalent to some first order sentence.
Open Problem
Find necessary and sufficient conditions for D to be strongly
first order.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Constancy Logic
Constancy Logic
Constancy Logic FO(=(·)) is the fragment of Dependence
Logic containing only constancy atoms =(∅, ~z ) (also written just
=(~z )).
Galliani 2012
Let C be the set of all constancy dependencies. Then C is
strongly first order.
Can we find something more general?
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Constancy Logic
Constancy Logic
Constancy Logic FO(=(·)) is the fragment of Dependence
Logic containing only constancy atoms =(∅, ~z ) (also written just
=(~z )).
Galliani 2012
Let C be the set of all constancy dependencies. Then C is
strongly first order.
Can we find something more general?
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Upwards Closed Dependencies
Upwards Closed Dependencies
We say that a dependency D is upwards closed if
(Dom(M), R) ∈ D, R ⊆ S ⇒ (Dom(M), S) ∈ D
or, equivalently, if M |=X D~y , X ⊆ Y ⇒ M |=Y Dy .
Examples
Non-Emptiness: M |=X NE iff X 6= ∅;
n-bigness: M |=X |~y | ≥ n iff |X (~y )| ≥ n;
Totality: M |=X All(~y ) iff X (~y ) = Dom(M)|~y | ;
Intersection: M |=X ~y 6 | ~z iff X (~y ) ∩ X (~z ) 6= ∅;
κ-bigness: For κ infinite cardinal, M |=X |~y | ≥ κ iff |X (~y )| ≥ κ.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
On Upwards Closed Dependencies
Upwards Closure is Not Preserved
If D is a family of upwards closed dependencies then in general
FO(D) is not upwards closed: M |=X φ, X ⊆ Y 6⇒ M |=Y φ.
Theorem (Galliani 2013)
If D is a family of first order upwards closed dependencies and
constancy atoms then D is strongly first order: FO(D) = FO
over sentences.
Corollary
The set of the contradictory negations of inclusion, exclusion,
dependence and independence atoms is strongly first order.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
The Proof: Main Idea
I will sketch the proof of the case without constancy atoms.
Extending the proof to constancy atoms is easy.
An Observation
Why might we go beyond first order logic?
Disjunction: M |=X φ ∨ ψ iff ∃Y , Z s.t. X = Y ∪ Z , M |=Y φ,
M |=Z ψ;
Existential: M |=X ∃v φ iff ∃H s.t. M |=X [H/v ] φ.
A proof plan
Suppose that all dependencies are upwards closed. Find a way
to evaluate disjunction and existential quantification without
quantification over teams or functions, then the result follows.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Flattening
Flattening
Let φ ∈ FO(D). Then φf is the first order formula obtained by
replacing any dependence D~y with ⊤.
Note
If M |=X φ then M |=X φf and, for all s ∈ X , M |=s φf .
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
The Flattening Lemma
Flattening Lemma
If all D in φ are upwards closed, M |=X φ, X ⊆ Y and M |=s φf
for all s ∈ Y then M |=Y φ.
X
M |=X φ
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
The Flattening Lemma
Flattening Lemma
If all D in φ are upwards closed, M |=X φ, X ⊆ Y and M |=s φf
for all s ∈ Y then M |=Y φ.
Y
X
∀s ∈ Y , M |=s φf
M |=X φ
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
The Flattening Lemma
Flattening Lemma
If all D in φ are upwards closed, M |=X φ, X ⊆ Y and M |=s φf
for all s ∈ Y then M |=Y φ.
Y
X
M |=X φ
M |=Y φ
Pietro Galliani
∀s ∈ Y , M |=s φf
Strongly First-Order Dependencies in Team Semantics
Evaluating Disjunctions Greedily
M |=X φ ∨ ψ iff ∃Y , Z s.t. X = Y ∪ Z , M |=Y φ, M |=Z ψ
Let X1 = {s ∈ X , M |=s φf }, X2 = {s ∈ X , M |=s ψ f }. Then
M |=X φ ∨ ψ iff X = X1 ∪ X2 , M |=X1 φ and M |=X2 ψ.
X
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Evaluating Disjunctions Greedily
M |=X φ ∨ ψ iff ∃Y , Z s.t. X = Y ∪ Z , M |=Y φ, M |=Z ψ
Let X1 = {s ∈ X , M |=s φf }, X2 = {s ∈ X , M |=s ψ f }. Then
M |=X φ ∨ ψ iff X = X1 ∪ X2 , M |=X1 φ and M |=X2 ψ.
X
φ
Pietro Galliani
ψ
Strongly First-Order Dependencies in Team Semantics
Evaluating Disjunctions Greedily
M |=X φ ∨ ψ iff ∃Y , Z s.t. X = Y ∪ Z , M |=Y φ, M |=Z ψ
Let X1 = {s ∈ X , M |=s φf }, X2 = {s ∈ X , M |=s ψ f }. Then
M |=X φ ∨ ψ iff X = X1 ∪ X2 , M |=X1 φ and M |=X2 ψ.
X
φ
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Evaluating Disjunctions Greedily
M |=X φ ∨ ψ iff ∃Y , Z s.t. X = Y ∪ Z , M |=Y φ, M |=Z ψ
Let X1 = {s ∈ X , M |=s φf }, X2 = {s ∈ X , M |=s ψ f }. Then
M |=X φ ∨ ψ iff X = X1 ∪ X2 , M |=X1 φ and M |=X2 ψ.
X
φf
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Evaluating Disjunctions Greedily
M |=X φ ∨ ψ iff ∃Y , Z s.t. X = Y ∪ Z , M |=Y φ, M |=Z ψ
Let X1 = {s ∈ X , M |=s φf }, X2 = {s ∈ X , M |=s ψ f }. Then
M |=X φ ∨ ψ iff X = X1 ∪ X2 , M |=X1 φ and M |=X2 ψ.
X
φ
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Evaluating Disjunctions Greedily
M |=X φ ∨ ψ iff ∃Y , Z s.t. X = Y ∪ Z , M |=Y φ, M |=Z ψ
Let X1 = {s ∈ X , M |=s φf }, X2 = {s ∈ X , M |=s ψ f }. Then
M |=X φ ∨ ψ iff X = X1 ∪ X2 , M |=X1 φ and M |=X2 ψ.
X
ψ
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Evaluating Disjunctions Greedily
M |=X φ ∨ ψ iff ∃Y , Z s.t. X = Y ∪ Z , M |=Y φ, M |=Z ψ
Let X1 = {s ∈ X , M |=s φf }, X2 = {s ∈ X , M |=s ψ f }. Then
M |=X φ ∨ ψ iff X = X1 ∪ X2 , M |=X1 φ and M |=X2 ψ.
X
ψf
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Evaluating Disjunctions Greedily
M |=X φ ∨ ψ iff ∃Y , Z s.t. X = Y ∪ Z , M |=Y φ, M |=Z ψ
Let X1 = {s ∈ X , M |=s φf }, X2 = {s ∈ X , M |=s ψ f }. Then
M |=X φ ∨ ψ iff X = X1 ∪ X2 , M |=X1 φ and M |=X2 ψ.
X
ψ
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Evaluating Existentials Greedily
M |=X ∃v φ iff ∃H s.t. M |=X [H/v ] φ
For all s ∈ X , let K (s) = {m ∈ Dom(M) : M |=s[m/v ] φf }.
Then M |=X ∃v φ iff
For all s ∈ X , K (s) 6= ∅;
For Y = X [K /v ] = {s[m/v ] : s ∈ X , m ∈ K (s)}, M |=Y ψ.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Outline
1
Team Semantics and Dependencies
2
Upwards Closed Dependencies
3
Boundedness
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
X
M |=X φ
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
X
Y
M |=Y φ
|Y | ≤ k
Pietro Galliani
M |=X φ
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
Examples
Non-Emptiness: M |=X NE iff X 6= ∅:
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
Examples
Non-Emptiness: M |=X NE iff X 6= ∅: 1-bounded;
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
Examples
Non-Emptiness: M |=X NE iff X 6= ∅: 1-bounded;
n-bigness: M |=X |~y | ≥ n iff |X (~y )| ≥ n:
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
Examples
Non-Emptiness: M |=X NE iff X 6= ∅: 1-bounded;
n-bigness: M |=X |~y | ≥ n iff |X (~y )| ≥ n: n-bounded;
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
Examples
Non-Emptiness: M |=X NE iff X 6= ∅: 1-bounded;
n-bigness: M |=X |~y | ≥ n iff |X (~y )| ≥ n: n-bounded;
Totality: M |=X All(~y ) iff X (~y ) = Dom(M)|~y | :
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
Examples
Non-Emptiness: M |=X NE iff X 6= ∅: 1-bounded;
n-bigness: M |=X |~y | ≥ n iff |X (~y )| ≥ n: n-bounded;
Totality: M |=X All(~y ) iff X (~y ) = Dom(M)|~y | :
|M||~y | -bounded;
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
Examples
Non-Emptiness: M |=X NE iff X 6= ∅: 1-bounded;
n-bigness: M |=X |~y | ≥ n iff |X (~y )| ≥ n: n-bounded;
Totality: M |=X All(~y ) iff X (~y ) = Dom(M)|~y | :
|M||~y | -bounded;
Intersection: M |=X ~y 6 | ~z iff X (~y ) ∩ X (~z ) 6= ∅:
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness
Boundedness
A dependency D is k -bounded (in M) if whenever M |=X D~v ,
there exists a Y ⊆ X with |Y | ≤ k such that M |=Y D~v .
Examples
Non-Emptiness: M |=X NE iff X 6= ∅: 1-bounded;
n-bigness: M |=X |~y | ≥ n iff |X (~y )| ≥ n: n-bounded;
Totality: M |=X All(~y ) iff X (~y ) = Dom(M)|~y | :
|M||~y | -bounded;
Intersection: M |=X ~y 6 | ~z iff X (~y ) ∩ X (~z ) 6= ∅: 2-bounded.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
The Boundedness Theorem
Boundedness Theorem
Let D be a family of upwards closed dependencies, let
φ ∈ FO(=(·), D) contain upwards closed dependency instances
D1~y1 . . . Dk ~yk which are n1 -, . . . , nk - bounded. Then φ is
n = n1 + . . . + nk -bounded:
M |=X φ ⇒ ∃Y ⊆ X , |Y | ≤ n, s.t. M |=Y φ.
Proof.
Proof (sketch) First show that φ can be put in the form
∃z1 . . . zt (=(z1 ) ∧ . . . ∧ =(zt ) ∧ ψ), where ψ contains no
constancy atoms. Then proceed by induction on ψ.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness Theorem: Consequences
Corollary
If D is a family of constantly-bounded upwards closed
dependencies. Then All(y ) is not definable in FO(=(·), D):
there is no formula φ(y ) such that M |=X φ(y ) iff M |=X All(y ).
Corollary
All(y1 . . . yn ) is not definable in terms of All(y1 . . . yn−1 ).
Corollary
If D is k -bounded and upwards closed and φ(y ) ∈ FO(=(·), D)
characterizes n-bigness then φ contains at least ⌈ kn ⌉ instances
of D.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness Theorem: Consequences
Corollary
If D is a family of constantly-bounded upwards closed
dependencies. Then All(y ) is not definable in FO(=(·), D):
there is no formula φ(y ) such that M |=X φ(y ) iff M |=X All(y ).
Corollary
All(y1 . . . yn ) is not definable in terms of All(y1 . . . yn−1 ).
Corollary
If D is k -bounded and upwards closed and φ(y ) ∈ FO(=(·), D)
characterizes n-bigness then φ contains at least ⌈ kn ⌉ instances
of D.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Boundedness Theorem: Consequences
Corollary
If D is a family of constantly-bounded upwards closed
dependencies. Then All(y ) is not definable in FO(=(·), D):
there is no formula φ(y ) such that M |=X φ(y ) iff M |=X All(y ).
Corollary
All(y1 . . . yn ) is not definable in terms of All(y1 . . . yn−1 ).
Corollary
If D is k -bounded and upwards closed and φ(y ) ∈ FO(=(·), D)
characterizes n-bigness then φ contains at least ⌈ kn ⌉ instances
of D.
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Conclusions
What we have
Team Semantics is a rich semantical framework not only
for higher-order logics, but also for seeing first-order logic
from a different perspective!
We characterized a class of dependencies D such that
FO(D) = FO wrt sentences;
Non-definability results through boundedness.
What we still need
Necessary and sufficient conditions for FO(D) = FO?
Is there a single dependency D such that
FO(D) = FO(D↑ )?
Which properties of teams are definable in FO(All),
FO(NE), and other such logics?
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics
Conclusions
What we have
Team Semantics is a rich semantical framework not only
for higher-order logics, but also for seeing first-order logic
from a different perspective!
We characterized a class of dependencies D such that
FO(D) = FO wrt sentences;
Non-definability results through boundedness.
What we still need
Necessary and sufficient conditions for FO(D) = FO?
Is there a single dependency D such that
FO(D) = FO(D↑ )?
Which properties of teams are definable in FO(All),
FO(NE), and other such logics?
Pietro Galliani
Strongly First-Order Dependencies in Team Semantics