Dependence logic

Dependence logic is a logical formalism, created by Jouko Väänänen,[1] which adds dependence atoms to the language of first-order logic. A dependence atom is an expression of the form  = ( t 1 … t n ) {\displaystyle =\\!\\!(t_{1}\ldots t_{n})} , where  t 1 … t n {\displaystyle t_{1}\ldots t_{n}} are terms, and corresponds to the statement that the value of  t n {\displaystyle t_{n}} is functionally dependent on the values of  t 1 , … , t n − 1 {\displaystyle t_{1},\ldots ,t_{n-1}} .

Dependence logic is a logic of imperfect information, like  branching quantifier logic or independence-friendly logic (IF logic): in other words, its game-theoretic semantics can be obtained from that of first-order logic by restricting the availability of information to the players, thus allowing for non-linearly ordered patterns of dependence and independence between variables. However, dependence logic differs from these logics in that it separates the notions of dependence and independence from the notion of quantification.

## Syntax

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The syntax of dependence logic is an extension of that of first-order logic. For a fixed signature σ = (Sfunc, Srel, ar), the set of all well-formed dependence logic formulas is defined according to the following rules:

### Terms

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Terms in dependence logic are defined precisely as in first-order logic.

### Atomic formulas

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There are three types of atomic formulas in dependence logic:

1. A relational atom is an expression of the form  R t 1 … t n {\displaystyle Rt_{1}\ldots t_{n}} for any n-ary relation  R {\displaystyle R} in our signature and for any n-tuple of terms  ( t 1 , … , t n ) {\displaystyle (t_{1},\ldots ,t_{n})} ;
2. An equality atom is an expression of the form  t 1 = t 2 {\displaystyle t_{1}=t_{2}} , for any two terms  t 1 {\displaystyle t_{1}} and  t 2 {\displaystyle t_{2}} ;
3. A dependence atom is an expression of the form  = ( t 1 … t n ) {\displaystyle =\\!\\!(t_{1}\ldots t_{n})} , for any  n ∈ N {\displaystyle n\in \mathbb {N} } and for any n-tuple of terms  ( t 1 , … , t n ) {\displaystyle (t_{1},\ldots ,t_{n})} .

Nothing else is an atomic formula of dependence logic.

Relational and equality atoms are also called first-order atoms.

### Complex formulas and sentences

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For a fixed signature σ, the set of all formulas  ϕ {\displaystyle \phi } of dependence logic and their respective sets of free variables  Free ( ϕ ) {\displaystyle {\mbox{Free}}(\phi )} are defined as follows:

1. Any atomic formula  ϕ {\displaystyle \phi } is a formula, and  Free ( ϕ ) {\displaystyle {\mbox{Free}}(\phi )} is the set of all variables occurring in it;
2. If  ϕ {\displaystyle \phi } is a formula, so is  ¬ ϕ {\displaystyle \lnot \phi } and  Free ( ¬ ϕ ) = Free ( ϕ ) {\displaystyle {\mbox{Free}}(\lnot \phi )={\mbox{Free}}(\phi )} ;
3. If  ϕ {\displaystyle \phi } and  ψ {\displaystyle \psi } are formulas, so is  ϕ ∨ ψ {\displaystyle \phi \vee \psi } and  Free ( ϕ ∨ ψ ) = Free ( ϕ ) ∪ Free ( ψ ) {\displaystyle {\mbox{Free}}(\phi \vee \psi )={\mbox{Free}}(\phi )\cup {\mbox{Free}}(\psi )} ;
4. If  ϕ {\displaystyle \phi } is a formula and  x {\displaystyle x} is a variable,  ∃ x ϕ {\displaystyle \exists x\phi } is also a formula and  Free ( ∃ v ϕ ) = Free ( ϕ ) ∖ { v } {\displaystyle {\mbox{Free}}(\exists v\phi )={\mbox{Free}}(\phi )\backslash \\{v\\}} .

Nothing is a dependence logic formula unless it can be obtained through a finite number of applications of these four rules.

A formula  ϕ {\displaystyle \phi } such that  Free ( ϕ ) = ∅ {\displaystyle {\mbox{Free}}(\phi )=\emptyset } is a sentence of dependence logic.

#### Conjunction and universal quantification

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In the above presentation of the syntax of dependence logic, conjunction and universal quantification are not treated as primitive operators; rather, they are defined in terms of negation and, respectively, disjunction and existential quantification, by means of De Morgan's Laws.

Therefore,  ϕ ∧ ψ {\displaystyle \phi \wedge \psi } is taken as a shorthand for  ¬ ( ¬ ϕ ∨ ¬ ψ ) {\displaystyle \lnot (\lnot \phi \vee \lnot \psi )} , and  ∀ x ϕ {\displaystyle \forall x\phi } is taken as a shorthand for  ¬ ( ∃ x ( ¬ ϕ ) ) {\displaystyle \lnot (\exists x(\lnot \phi ))} .

## Semantics

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The team semantics for dependence logic is a variant of Wilfrid Hodges' compositional semantics for IF logic.[2][3] There exist equivalent game-theoretic semantics for dependence logic, both in terms of imperfect information games and in terms of perfect information games.

### Teams

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Let  A = ( A , σ , I ) {\displaystyle {\mathcal {A}}=(A,\sigma ,I)} be a first-order structure and let  V = { v 1 , … , v n } {\displaystyle V=\\{v_{1},\ldots ,v_{n}\\}} be a finite set of variables. Then a team over A with domain V is a set of assignments over A with domain V, that is, a set of functions μ from V to A.

It may be helpful to visualize such a team as a database relation with attributes  v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} and with only one data type, corresponding to the domain A of the structure: for example, if the team X consists of four assignments  μ 1 , … , μ 4 {\displaystyle \mu _{1},\ldots ,\mu _{4}} with domain  { v 1 , v 2 , v 3 } {\displaystyle \\{v_{1},v_{2},v_{3}\\}} then one may represent it as the relation

v 1 v 2 v 3 μ 1 μ 1 ( v 1 ) μ 1 ( v 2 ) μ 1 ( v 3 ) μ 2 μ 2 ( v 1 ) μ 2 ( v 2 ) μ 2 ( v 3 ) μ 3 μ 3 ( v 1 ) μ 3 ( v 2 ) μ 3 ( v 3 ) μ 4 μ 4 ( v 1 ) μ 4 ( v 2 ) μ 4 ( v 3 ) {\displaystyle {\begin{array}{c|ccc}&v_{1}&v_{2}&v_{3}\\\\\hline \mu _{1}&\mu _{1}(v_{1})&\mu _{1}(v_{2})&\mu _{1}(v_{3})\\\\\mu _{2}&\mu _{2}(v_{1})&\mu _{2}(v_{2})&\mu _{2}(v_{3})\\\\\mu _{3}&\mu _{3}(v_{1})&\mu _{3}(v_{2})&\mu _{3}(v_{3})\\\\\mu _{4}&\mu _{4}(v_{1})&\mu _{4}(v_{2})&\mu _{4}(v_{3})\end{array}}}

### Positive and negative satisfaction

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Team semantics can be defined in terms of two relations  T {\displaystyle {\mathcal {T}}} and  C {\displaystyle {\mathcal {C}}} between structures, teams and formulas.

Given a structure  A {\displaystyle {\mathcal {A}}} , a team  X {\displaystyle X} over it and a dependence logic formula  ϕ {\displaystyle \phi } whose free variables are contained in the domain of  X {\displaystyle X} , if  ( A , X , ϕ ) ∈ T {\displaystyle ({\mathcal {A}},X,\phi )\in {\mathcal {T}}} we say that  X {\displaystyle X} is a trump for  ϕ {\displaystyle \phi } in  A {\displaystyle {\mathcal {A}}} , and we write that  A ⊨ X + ϕ {\displaystyle {\mathcal {A}}\models _{X}^{+}\phi } ; and analogously, if  ( A , X , ϕ ) ∈ C {\displaystyle ({\mathcal {A}},X,\phi )\in {\mathcal {C}}} we say that  X {\displaystyle X} is a cotrump for  ϕ {\displaystyle \phi } in  A {\displaystyle {\mathcal {A}}} , and we write that  A ⊨ X − ϕ {\displaystyle {\mathcal {A}}\models _{X}^{-}\phi } .

If  A ⊨ X + ϕ {\displaystyle {\mathcal {A}}\models _{X}^{+}\phi } one can also say that  ϕ {\displaystyle \phi } is positively satisfied by  X {\displaystyle X} in  A {\displaystyle {\mathcal {A}}} , and if instead  A ⊨ X − ϕ {\displaystyle {\mathcal {A}}\models _{X}^{-}\phi } one can say that  ϕ {\displaystyle \phi } is negatively satisfied by  X {\displaystyle X} in  A {\displaystyle {\mathcal {A}}} .

The necessity of considering positive and negative satisfaction separately is a consequence of the fact that in dependence logic, as in the logic of branching quantifiers or in IF logic, the law of the excluded middle does not hold; alternatively, one may assume that all formulas are in negation normal form, using De Morgan's relations in order to define universal quantification and conjunction from existential quantification and disjunction respectively, and consider positive satisfaction alone.

Given a sentence  ϕ {\displaystyle \phi } , we say that  ϕ {\displaystyle \phi } is true in  A {\displaystyle {\mathcal {A}}} if and only if  A ⊨ { ∅ } + ϕ {\displaystyle {\mathcal {A}}\models _{\\{\emptyset \\}}^{+}\phi } , and we say that  ϕ {\displaystyle \phi } is false in  A {\displaystyle {\mathcal {A}}} if and only if  A ⊨ { ∅ } − ϕ {\displaystyle {\mathcal {A}}\models _{\\{\emptyset \\}}^{-}\phi } .

### Semantic rules

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As for the case of Alfred Tarski's satisfiability relation for first-order formulas, the positive and negative satisfiability relations of the team semantics for dependence logic are defined by structural induction over the formulas of the language. Since the negation operator interchanges positive and negative satisfiability, the two inductions corresponding to  ⊨ + {\displaystyle \models ^{+}} and  ⊨ − {\displaystyle \models ^{-}} need to be performed simultaneously:

#### Positive satisfiability

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1. A ⊨ X + R t 1 … t n {\displaystyle {\mathcal {A}}\models _{X}^{+}Rt_{1}\ldots t_{n}} if and only if
1. R {\displaystyle R} is a n-ary symbol in the signature of  A {\displaystyle {\mathcal {A}}} ;
2. All variables occurring in the terms  t 1 , … , t n {\displaystyle t_{1},\ldots ,t_{n}} are in the domain of  X {\displaystyle X} ;
3. For every assignment  μ ∈ X {\displaystyle \mu \in X} , the evaluation of the tuple  ( t 1 , … , t n ) {\displaystyle (t_{1},\ldots ,t_{n})} according to  μ {\displaystyle \mu } is in the interpretation of  R {\displaystyle R} in  A {\displaystyle {\mathcal {A}}} ;
2. A ⊨ X + t 1 = t 2 {\displaystyle {\mathcal {A}}\models _{X}^{+}t_{1}=t_{2}} if and only if
1. All variables occurring in the terms  t 1 {\displaystyle t_{1}} and  t 2 {\displaystyle t_{2}} are in the domain of  X {\displaystyle X} ;
2. For every assignment  μ ∈ X {\displaystyle \mu \in X} , the evaluations of  t 1 {\displaystyle t_{1}} and  t 2 {\displaystyle t_{2}} according to  A {\displaystyle {\mathcal {A}}} are the same;
3. A ⊨ X + = ( t 1 … t n ) {\displaystyle {\mathcal {A}}\models _{X}^{+}=\\!\\!(t_{1}\ldots t_{n})} if and only if any two assignments  μ , μ ′ ∈ X {\displaystyle \mu ,\mu '\in X} whose evaluations of the tuple  ( t 1 , … , t n − 1 ) {\displaystyle (t_{1},\ldots ,t_{n-1})} coincide assign the same value to  t n {\displaystyle t_{n}} ;
4. A ⊨ X + ¬ ϕ {\displaystyle {\mathcal {A}}\models _{X}^{+}\lnot \phi } if and only if  A ⊨ X − ϕ {\displaystyle {\mathcal {A}}\models _{X}^{-}\phi } ;
5. A ⊨ X + ϕ ∨ ψ {\displaystyle {\mathcal {A}}\models _{X}^{+}\phi \vee \psi } if and only if there exist teams  Y {\displaystyle Y} and  Z {\displaystyle Z} such that
1. X = Y ∪ Z ′ {\displaystyle X=Y\cup Z'}
2. A ⊨ Y + ϕ {\displaystyle {\mathcal {A}}\models _{Y}^{+}\phi } ;
3. A ⊨ Z + ψ {\displaystyle {\mathcal {A}}\models _{Z}^{+}\psi } ;
6. A ⊨ X + ∃ x ϕ {\displaystyle {\mathcal {A}}\models _{X}^{+}\exists x\phi } if and only if there exists a function  F {\displaystyle F} from  X {\displaystyle X} to the domain of  A {\displaystyle {\mathcal {A}}} such that  A ⊨ X [ F / x ] + ϕ {\displaystyle {\mathcal {A}}\models _{X[F/x]}^{+}\phi } , where  X [ F / x ] = { μ [ F ( μ ) / x ] : μ ∈ X } {\displaystyle X[F/x]=\\{\mu [F(\mu )/x]:\mu \in X\\}} .

#### Negative satisfiability

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1. A ⊨ X − R t 1 … t n {\displaystyle {\mathcal {A}}\models _{X}^{-}Rt_{1}\ldots t_{n}} if and only if
1. R {\displaystyle R} is a n-ary symbol in the signature of  A {\displaystyle {\mathcal {A}}} ;
2. All variables occurring in the terms  t 1 , … , t n {\displaystyle t_{1},\ldots ,t_{n}} are in the domain of  X {\displaystyle X} ;
3. For every assignment  μ ∈ X {\displaystyle \mu \in X} , the evaluation of the tuple  ( t 1 , … , t n ) {\displaystyle (t_{1},\ldots ,t_{n})} according to  μ {\displaystyle \mu } is not in the interpretation of  R {\displaystyle R} in  A {\displaystyle {\mathcal {A}}} ;
2. A ⊨ X − t 1 = t 2 {\displaystyle {\mathcal {A}}\models _{X}^{-}t_{1}=t_{2}} if and only if
1. All variables occurring in the terms  t 1 {\displaystyle t_{1}} and  t 2 {\displaystyle t_{2}} are in the domain of  X {\displaystyle X} ;
2. For every assignment  μ ∈ X {\displaystyle \mu \in X} , the evaluations of  t 1 {\displaystyle t_{1}} and  t 2 {\displaystyle t_{2}} according to  A {\displaystyle {\mathcal {A}}} are different;
3. A ⊨ X − = ( t 1 … t n ) {\displaystyle {\mathcal {A}}\models _{X}^{-}=\\!\\!(t_{1}\ldots t_{n})} if and only if  X {\displaystyle X} is the empty team;
4. A ⊨ X − ¬ ϕ {\displaystyle {\mathcal {A}}\models _{X}^{-}\lnot \phi } if and only if  A ⊨ X + ϕ {\displaystyle {\mathcal {A}}\models _{X}^{+}\phi } ;
5. A ⊨ X − ϕ ∨ ψ {\displaystyle {\mathcal {A}}\models _{X}^{-}\phi \vee \psi } if and only if  A ⊨ X − ϕ {\displaystyle {\mathcal {A}}\models _{X}^{-}\phi } and  A ⊨ X − ψ {\displaystyle {\mathcal {A}}\models _{X}^{-}\psi } ;
6. A ⊨ X − ∃ x ϕ {\displaystyle {\mathcal {A}}\models _{X}^{-}\exists x\phi } if and only if  A ⊨ X [ A / x ] − ϕ {\displaystyle {\mathcal {A}}\models _{X[A/x]}^{-}\phi } , where  X [ A / x ] = { μ [ a / x ] : a ∈ A } {\displaystyle X[A/x]=\\{\mu [a/x]:a\in A\\}} and  A {\displaystyle A} is the domain of  A {\displaystyle {\mathcal {A}}} .

## Dependence logic and other logics

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### Dependence logic and first-order logic

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Dependence logic is a conservative extension of first-order logic:[4] in other words, for every first-order sentence  ϕ {\displaystyle \phi } and structure  A {\displaystyle {\mathcal {A}}} we have that  A ⊨ { ∅ } + ϕ {\displaystyle {\mathcal {A}}\models _{\\{\emptyset \\}}^{+}\phi } if and only if  ϕ {\displaystyle \phi } is true in  A {\displaystyle {\mathcal {A}}} according to the usual first-order semantics. Furthermore, for any first-order formula ϕ {\displaystyle \phi } ,  A ⊨ X + ϕ {\displaystyle {\mathcal {A}}\models _{X}^{+}\phi } if and only if all assignments  μ ∈ X {\displaystyle \mu \in X} satisfy  ϕ {\displaystyle \phi } in  A {\displaystyle {\mathcal {A}}} according to the usual first-order semantics.

However, dependence logic is strictly more expressive than first-order logic:[5] for example, the sentence

∃ z ∀ x 1 ∀ x 2 ∃ y 1 ∃ y 2 ( = ( x 1 , y 1 ) ∧ = ( x 2 , y 2 ) ∧ ( x 1 = x 2 ↔ y 1 = y 2 ) ∧ y 1 ≠ z ) {\displaystyle \exists z\forall x_{1}\forall x_{2}\exists y_{1}\exists y_{2}(=\\!\\!(x_{1},y_{1})\wedge =\\!\\!(x_{2},y_{2})\wedge (x_{1}=x_{2}\leftrightarrow y_{1}=y_{2})\wedge y_{1}\not =z)}

is true in a model  A {\displaystyle {\mathcal {A}}} if and only if the domain of this model is infinite, even though no first-order formula  ϕ {\displaystyle \phi } has this property.

### Dependence logic and second-order logic

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Every dependence logic sentence is equivalent to some sentence in the existential fragment of second-order logic,[6] that is, to some second-order sentence of the form

∃ R 1 … ∃ R n ψ ( R 1 , … , R n ) {\displaystyle \exists R_{1}\ldots \exists R_{n}\psi (R_{1},\ldots ,R_{n})}

where  ψ ( R 1 , … , R n ) {\displaystyle \psi (R_{1},\ldots ,R_{n})} does not contain second-order quantifiers. Conversely, every second-order sentence in the above form is equivalent to some dependence logic sentence.[7]

As for open formulas, dependence logic corresponds to the downwards monotone fragment of existential second-order logic, in the sense that a nonempty class of teams is definable by a dependence logic formula if and only if the corresponding class of relations is downwards monotone and definable by an existential second-order formula.[8]

### Dependence logic and branching quantifiers

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Branching quantifiers are expressible in terms of dependence atoms: for example, the expression

( Q H x 1 , x 2 , y 1 , y 2 ) ϕ ( x 1 , x 2 , y 1 , y 2 ) ≡ ( ∀ x 1 ∃ y 1 ∀ x 2 ∃ y 2 ) ϕ ( x 1 , x 2 , y 1 , y 2 ) {\displaystyle (Q_{H}x_{1},x_{2},y_{1},y_{2})\phi (x_{1},x_{2},y_{1},y_{2})\equiv {\begin{pmatrix}\forall x_{1}\exists y_{1}\\\\\forall x_{2}\exists y_{2}\end{pmatrix}}\phi (x_{1},x_{2},y_{1},y_{2})}

is equivalent to the dependence logic sentence  ∀ x 1 ∃ y 1 ∀ x 2 ∃ y 2 ( = ( x 1 , y 1 ) ∧ = ( x 2 , y 2 ) ∧ ϕ ) {\displaystyle \forall x_{1}\exists y_{1}\forall x_{2}\exists y_{2}(=\\!\\!(x_{1},y_{1})\wedge =\\!\\!(x_{2},y_{2})\wedge \phi )} , in the sense that the former expression is true in a model if and only if the latter expression is true.

Conversely, any dependence logic sentence is equivalent to some sentence in the logic of branching quantifiers, since all existential second-order sentences are expressible in branching quantifier logic.[9][10]

### Dependence logic and IF logic

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Any dependence logic sentence is logically equivalent to some IF logic sentence, and vice versa.[11]

However, the issue is subtler when it comes to open formulas. Translations between IF logic and dependence logic formulas, and vice versa, exist as long as the domain of the team is fixed: in other words, for all sets of variables  V = { v 1 … v n } {\displaystyle V=\\{v_{1}\ldots v_{n}\\}} and all IF logic formulas  ϕ {\displaystyle \phi } with free variables in  V {\displaystyle V} there exists a dependence logic formula  ϕ D {\displaystyle \phi ^{D}} such that

A ⊨ X + ϕ ⇔ A ⊨ X + ϕ D {\displaystyle {\mathcal {A}}\models _{X}^{+}\phi \Leftrightarrow {\mathcal {A}}\models _{X}^{+}\phi ^{D}}

for all structures  A {\displaystyle {\mathcal {A}}} and for all teams  X {\displaystyle X} with domain  V {\displaystyle V} , and conversely, for every dependence logic formula  ψ {\displaystyle \psi } with free variables in  V {\displaystyle V} there exists an IF logic formula  ψ I {\displaystyle \psi ^{I}} such that

A ⊨ X + ψ ⇔ A ⊨ X + ψ I {\displaystyle {\mathcal {A}}\models _{X}^{+}\psi \Leftrightarrow {\mathcal {A}}\models _{X}^{+}\psi ^{I}}

for all structures  A {\displaystyle {\mathcal {A}}} and for all teams  X {\displaystyle X} with domain  V {\displaystyle V} . These translations cannot be compositional.[12]

## Properties

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Dependence logic formulas are downwards closed: if  A ⊨ X ϕ {\displaystyle {\mathcal {A}}\models _{X}\phi } and  Y ⊆ X {\displaystyle Y\subseteq X} then  A ⊨ Y ψ {\displaystyle {\mathcal {A}}\models _{Y}\psi } . Furthermore, the empty team (but not the team containing the empty assignment) satisfies all formulas of dependence logic, both positively and negatively.

The law of the excluded middle fails in dependence logic: for example, the formula  ∃ y ( = ( y ) ∧ y = x ) {\displaystyle \exists y(=\\!\\!(y)\wedge y=x)} is neither positively nor negatively satisfied by the team  X = { ( x : 0 ) , ( x : 1 ) } {\displaystyle X=\\{(x:0),(x:1)\\}} . Furthermore, disjunction is not idempotent and does not distribute over conjunction.[13]

Both the compactness theorem and the Löwenheim–Skolem theorem are true for dependence logic. Craig's interpolation theorem also holds, but, due to the nature of negation in dependence logic, in a slightly modified formulation: if two dependence logic formulas  ϕ {\displaystyle \phi } and  ψ {\displaystyle \psi } are contradictory, that is, it is never the case that both  ϕ {\displaystyle \phi } and  ψ {\displaystyle \psi } hold in the same model, then there exists a first-order sentence  θ {\displaystyle \theta } in the common language of the two sentences such that  ϕ {\displaystyle \phi } implies  θ {\displaystyle \theta } and  θ {\displaystyle \theta } is contradictory with  ψ {\displaystyle \psi } .[14]

As for IF logic,[15] dependence logic can define its own truth operator:[16] more precisely, there exists a formula  τ ( x ) {\displaystyle \tau (x)} such that for every sentence  ϕ {\displaystyle \phi } of dependence logic and all models  M ω {\displaystyle {\mathcal {M}}_{\omega }} which satisfy Peano's axioms, if  ′ ϕ ′ {\displaystyle '\phi '} is the Gödel number of  ϕ {\displaystyle \phi } then

M ω ⊨ { ∅ } + ϕ {\displaystyle {\mathcal {M}}_{\omega }\models _{\\{\emptyset \\}}^{+}\\!\phi } if and only if  M ω ⊨ { ∅ } + τ ( ′ ϕ ′ ) . {\displaystyle {\mathcal {M}}_{\omega }\models _{\\{\emptyset \\}}^{+}\tau ('\phi ').}

This does not contradict Tarski's undefinability theorem, since the negation of dependence logic is not the usual contradictory one.

## Complexity

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As a consequence of Fagin's theorem, the properties of finite structures definable by a dependence logic sentence correspond exactly to NP properties. Furthermore, Durand and Kontinen showed that restricting the number of universal quantifiers or the arity of dependence atoms in sentences gives rise to hierarchy theorems with respect to expressive power.[17]

The inconsistency problem of dependence logic is semidecidable, and in fact equivalent to the inconsistency problem for first-order logic. However, the decision problem for dependence logic is non-arithmetical, and is in fact complete with respect to the  Π 2 {\displaystyle \Pi _{2}} class of the Lévy hierarchy.[18]

## Variants and extensions

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### Team logic

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Team logic[19] extends dependence logic with a contradictory negation ∼ ϕ {\displaystyle \sim \\!\\!\phi } . Its expressive power is equivalent to that of full second-order logic.[20]

### Modal dependence logic

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The dependence atom, or a suitable variant thereof, can be added to the language of modal logic, thus obtaining modal dependence logic.[21][22][23]

### Intuitionistic dependence logic

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As it is, dependence logic lacks an implication. The intuitionistic implication ϕ → ψ {\displaystyle \phi \rightarrow \psi } , whose name derives from the similarity between its definition and that of the implication of intuitionistic logic, can be defined as follows:[24]

A ⊨ X ϕ → ψ {\displaystyle {\mathcal {A}}\models _{X}\phi \rightarrow \psi } if and only if for all  Y ⊆ X {\displaystyle Y\subseteq X} such that  A ⊨ Y ϕ {\displaystyle {\mathcal {A}}\models _{Y}\phi } it holds that  A ⊨ Y ψ {\displaystyle {\mathcal {A}}\models _{Y}\psi } .

Intuitionistic dependence logic, that is, dependence logic supplemented with the intuitionistic implication, is equivalent to second-order logic.[25]

### Independence logic

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Instead of dependence atoms, independence logic adds to the language of first-order logic independence atoms  t 1 → ⊥ t 3 → t 2 → {\displaystyle {\vec {t_{1}}}\bot _{\vec {t_{3}}}{\vec {t_{2}}}} where  t 1 → {\displaystyle {\vec {t_{1}}}} ,  t 2 → {\displaystyle {\vec {t_{2}}}} and  t 3 → {\displaystyle {\vec {t_{3}}}} are tuples of terms. The semantics of these atoms is defined as follows:

A ⊨ X t 1 → ⊥ t 3 → t 2 → {\displaystyle {\mathcal {A}}\models _{X}{\vec {t_{1}}}\bot _{\vec {t_{3}}}{\vec {t_{2}}}} if and only if for all  s , s ′ ∈ X {\displaystyle s,s'\in X} with  t 3 → ⟨ s ⟩ = t 3 → ⟨ s ′ ⟩ {\displaystyle {\vec {t_{3}}}\langle s\rangle ={\vec {t_{3}}}\langle s'\rangle } there exists  s ″ ∈ X {\displaystyle s''\in X} such that  t 3 → ⟨ s ″ ⟩ = t 3 → ⟨ s ⟩ {\displaystyle {\vec {t_{3}}}\langle s''\rangle ={\vec {t_{3}}}\langle s\rangle } ,  t 1 → ⟨ s ″ ⟩ = t 1 → ⟨ s ⟩ {\displaystyle {\vec {t_{1}}}\langle s''\rangle ={\vec {t_{1}}}\langle s\rangle } and  t 2 → ⟨ s ″ ⟩ = t 2 → ⟨ s ′ ⟩ {\displaystyle {\vec {t_{2}}}\langle s''\rangle ={\vec {t_{2}}}\langle s'\rangle } .

Independence logic corresponds to existential second-order logic, in the sense that a non-empty class of teams is definable by an independence logic formula if and only if the corresponding class of relations is definable by an existential second-order formula.[26] Therefore, on the level of open formulas, independence logic is strictly stronger in expressive power than dependence logic. However, on the level of sentences these logics are equivalent.[27]

### Inclusion/exclusion logic

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Inclusion/exclusion logic extends first-order logic with inclusion atoms  t 1 → ⊆ t 2 → {\displaystyle {\vec {t_{1}}}\subseteq {\vec {t_{2}}}} and exclusion atoms  t 1 → ∣ t 2 → {\displaystyle {\vec {t_{1}}}\mid {\vec {t_{2}}}} where in both formulas  t 1 → {\displaystyle {\vec {t_{1}}}} and  t 2 → {\displaystyle {\vec {t_{2}}}} are term tuples of the same length. The semantics of these atoms is defined as follows:

* A ⊨ X t 1 → ⊆ t 2 → {\displaystyle {\mathcal {A}}\models _{X}{\vec {t_{1}}}\subseteq {\vec {t_{2}}}} if and only if for all  s ∈ X {\displaystyle s\in X} there exists  s ′ ∈ X {\displaystyle s'\in X} such that  t 1 → ⟨ s ⟩ = t 2 → ⟨ s ′ ⟩ {\displaystyle {\vec {t_{1}}}\langle s\rangle ={\vec {t_{2}}}\langle s'\rangle } ;
* A ⊨ X t 1 → ∣ t 2 → {\displaystyle {\mathcal {A}}\models _{X}{\vec {t_{1}}}\mid {\vec {t_{2}}}} if and only if for all  s , s ′ ∈ X {\displaystyle s,s'\in X} it holds that  t 1 → ⟨ s ⟩ ≠ t 2 → ⟨ s ′ ⟩ {\displaystyle {\vec {t_{1}}}\langle s\rangle \neq {\vec {t_{2}}}\langle s'\rangle } .

Inclusion/exclusion logic has the same expressive power as independence logic, already on the level of open formulas.[28] Inclusion logic and exclusion logic are obtained by adding only inclusion atoms or exclusion atoms to first-order logic, respectively. Inclusion logic sentences correspond in expressive power to greatest fixed-point logic sentences; hence inclusion logic captures (least) fixed-point logic on finite models, and PTIME over finite ordered models.[29] Exclusion logic in turn corresponds to dependence logic in expressive power.[30]

### Generalized quantifiers

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Another way of extending dependence logic is to add some generalized quantifiers to the language of dependence logic. Very recently there has been some study of dependence logic with monotone generalized quantifiers[31] and dependence logic with a certain majority quantifier, the latter leading to a new descriptive complexity characterization of the counting hierarchy.[32]

## See also

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* Game semantics
* Branching quantifier
* Independence-friendly logic

## Notes

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1. ^ Väänänen 2007
2. ^ Hodges 1997
3. ^ Väänänen 2007, §3.2
4. ^ Väänänen 2007, §3.2
5. ^ Väänänen 2007, §4
6. ^ Väänänen 2007, §6.1
7. ^ Väänänen 2007, §6.3
8. ^ Kontinen and Väänänen 2009
9. ^ Enderton 1970
10. ^ Walkoe 1970
11. ^ Väänänen 2007, §3.6
12. ^ Kontinen and Väänänen 2009 bis
13. ^ Väänänen 2007, §3
14. ^ Väänänen 2007, §6.2
15. ^ Hintikka 2002
16. ^ Väänänen 2007, §6.4
17. ^ Durand and Kontinen
18. ^ Väänänen 2007, §7
19. ^ Väänänen 2007, §8
20. ^ Kontinen and Nurmi 2009
21. ^ Sevenster 2009
22. ^ Väänänen 2008
23. ^ Lohmann and Vollmer 2010
24. ^ Abramsky and Väänänen 2009
25. ^ Yang 2010
26. ^ Galliani 2012
27. ^ Grädel and Väänänen
28. ^ Galliani 2012
29. ^ Galliani and Hella 2013
30. ^ Galliani 2012
31. ^ Engström
32. ^ Durand, Ebbing, Kontinen, Vollmer 2011

## References

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* Abramsky, Samson and Väänänen, Jouko (2009), 'From IF to BI'. Synthese 167(2): 207–230.
* Durand, Arnaud; Ebbing Johannes; Kontinen, Juha and Vollmer Heribert (2011), 'Dependence logic with a majority quantifier'. FSTTCS 2011: 252-263.
* Durand, Arnaud and Kontinen, Juha, 'Hierarchies in Dependence Logic'. ACM Transactions on Computational Logic, 2012.
* Enderton, Herbert B. (1970), 'Finite partially-ordered quantifiers'. Z. Math. Logik Grundlagen Math., 16: 393–397.
* Engström, Fredrik, 'Generalized quantifiers in dependence logic'. Journal of Logic, Language and Information, to appear.
* Galliani, Pietro (2012), 'Inclusion and Exclusion in Team Semantics - On some logics of imperfect information'. Annals of Pure and Applied Logic 163(1): 68-84.
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## External links

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* Galliani, Pietro. "Dependence Logic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy.
* Special issue of Studia Logica on "Dependence and Independence in Logic", containing a number of articles on Dependence Logic
* Presentations in Academy Colloquium Dependence Logic, Amsterdam, 2014