Dependence Logic

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Dependence Logic

• Jouko Väänänen
• University of Helsinki
• University of Amsterdam

LOGICCC - LINT
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LogCon
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LINT
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The dependence concept
Dependence of health on genes.
Dependence of future events on past decisions.
Dependence of moves of a player on previous moves.
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Arrows Theorem

• If the social welfare function respects
  unanimity and independence of irrelevant
  alternatives, it is a dictatorship.

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Question

• Can one add the dependence concept to
• first order logic (or other logics) in a coherent
  way?

What is the logic of dependence?
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Solution

• We consider the strongest form of dependence,
  namely functional determination z f(x1,...,xn),
  where x1,...,xn,z are individual variables.
• We denote it (x1,...,xn,z) and call it a
  dependence atom. Weaker forms of dependence are
  derived from this.
• In computer science x1xn ? z, where
  x1,...,xn,z are database fields. (Armstrong
  relation)

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Solution

• We consider the strongest form of dependence,
  namely functional determination z f(x1,...,xn),
  where x1,...,xn,z are individual variables.
• We denote it (x1,...,xn,z) and call it a
  dependence atom. Weaker forms of dependence are
  derived from this.
• In computer science x1xn ? z, where
  x1,...,xn,z are database fields. (Armstrong
  relation)

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Solution

• We consider the strongest form of dependence,
  namely functional determination z f(x1,...,xn),
  where x1,...,xn,z are individual variables.
• We denote it (x1,...,xn,z) and call it a
  dependence atom. Weaker forms of dependence are
  derived from this.
• In computer science x1xn ? z, where
  x1,...,xn,z are database fields. (Armstrong
  relation)

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Multitude

• Dependence does not manifest itself in a single
  play, event or observation.
• The underlying concept of dependence logic is a
  multitude a collection - of such plays, events
  or observations.
• These collections are called in this talk teams.
• They are the basic objects of our approach.

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Multitude

• Dependence does not manifest itself in a single
  play, event or observation.
• The underlying concept of dependence logic is a
  multitude a collection - of such plays, events
  or observations.
• These collections are called in this talk teams.
• They are the basic objects of our approach.

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Multitude

• Dependence does not manifest itself in a single
  play, event or observation.
• The underlying concept of dependence logic is a
  multitude a collection - of such plays, events
  or observations.
• These collections are called in this talk teams.
• They are the basic objects of our approach.

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Multitude

• Dependence does not manifest itself in a single
  play, event or observation.
• The underlying concept of dependence logic is a
  multitude a collection - of such plays, events
  or observations.
• These collections are called teams.
• They are the basic objects of our approach.

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Teams

• A set of records of stock exchange transactions
  of a particular dealer.
• A set of possible histories of mankind written as
  decisions and consequences.
• A set of chess games between Susan and Max, as
  lists of moves.

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Teams

• 1st intuition A team is a set of plays of a
  game.

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Teams

• 1st intuition A team is a set of plays of a
  game.
• 2nd intuition A team is a database.

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Towards a logic based on teams

• A set of plays satisfies x2gtx0 if move x2 is in
  each play greater than move x0.
• A set of plays satisfies (x1,...,xn,y) if move y
  is in each play determined by the moves
  x1,...,xn.

• A database satisfies x2gtx0 if field x2 is always
  greater than field x0.
• A database satisfies (x1,...,xn,y) if field y
  is functionally determined by the fields
  x1,...,xn.

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Towards a logic based on teams

• A set of plays satisfies x2gtx0 if move x2 is in
  each play greater than move x0.
• A set of plays satisfies (x1,...,xn,y) if move y
  is in each play determined by the moves
  x1,...,xn.

• A database satisfies x2gtx0 if field x2 is always
  greater than field x0.
• A database satisfies (x1,...,xn,y) if field y
  is functionally determined by the fields
  x1,...,xn.

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• Dependence atoms (x1,...,xn,z)
• First order logic
• Dependence logic

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Syntax of dependence logic
,?,?,?,?,?, ),(, xi

• tt
• Rt1...tn
• ??
• ? ?
• ??
• ?xi?
• ?xi?

xi , c, ft1tn
tt
(x1,...,xn,z)
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Assignment
Universe of the model
s
Variables
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Teams exact definition

• A team is just a set of assignments for a model.

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Teams exact definition

• A team is just a set of assignments for a model.
  (Propositional logic a set of valuations. Modal
  logic a set of possible worlds)
• Empty team ?.
• Database with no rows.
• No play was played.

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Teams exact definition

• A team is just a set of assignments for a model.
• Empty team ?.
• Database with no rows.
• No play was played.
• The team ? with the empty assignment.
• Database with no columns, and hence with at most
  one row.
• Zero moves of the game were played

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For the truth definition Negation Normal Form

• We push negations all the way
• to atomic formulas using de Morgan laws.
• Thus f will have the same meaning as f.

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Truth definition

• A team satisfies a formula if
• every assignment in the team does,
• and

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• A team satisfies Rt1tn if every team member
  does.

x0ltx1
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• A team satisfies Rt1tn if every team member
  does.

x1ltx0
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• A team satisfies Rt1tn if every team member
  does.

Note some X satisfy neither Rt1tn nor Rt1tn.
x1ltx0
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• A team satisfies tt if every team member does.

y
x
x1x2
xy
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• A team satisfies tt if every team member does.

x0x1
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• A team X satisfies (x1,...,xn,z) if in any two
  assignments in X, in which x1,...,xn have the
  same values, also z has the same value.

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• A team X satisfies (x1,...,xn,z) if in any two
  assignments in X, in which x1,...,xn have the
  same values, also z has the same value.

y
x
(x,y)
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• A team X satisfies (x1,...,xn,z) if in any two
  assignments in X, in which x1,...,xn have the
  same values, also z has the same value.

y
x
(x,y)
(x,y,z)
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An extreme case

• (x)
• x is constant in the team

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An extreme case

• (x)
• x is constant in the team

x
(x)
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• A team X satisfies f v ? if
• XY ? Z, where Y satisfies f and Z satisfies ?.

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• A team X satisfies f v ? if
• XY ? Z, where Y satisfies f and Z satisfies ?.

• Plays where rook or queen was sacrificed

Queen was sacrificed
Rook was sacrificed
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• (Rank,Salary) ?

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• (Rank,Salary) v (Rank,Salary)

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• (Rank,Salary) v (Rank,Salary)

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• A team X satisfies f ? ? if it satisfies f and ?.

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Quantifiers - modified assignment
xn
xi
xj
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• A team X satisfies ?xf if
• there is a team Y such that Y satisfies f and
  for every s in X we have s(a/x)?Y for some a.

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Team X can be supplemented with values for x so
that ? is satisfied.
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• A team X satisfies ?xf if
• there is a team Y such that Y satisfies f and
  for every s in X we have s(a/x)?Y for all a.

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Team X can be duplicated along x, by letting x
get all possible values, and then ? is satisfied.
X
Y
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Truth

• A sentence is true if satisfies it.

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Example even cardinality
Like Henkin (partially ordered) quantifiers.
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Conservative over FO
A team s satisfies a first order formula f iff
s satisfies f in the usual sense.
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Two important properties
Downward closure If a team satisfies a formula,
every subset does. (Hodges optimal!)
Empty set property The empty team satisfies
every formula.
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No Law of Excluded Middle
Suppose the universe has at least two elements.

• ?x (x) not true

?x (x) not true either
because it means ?x (x).
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A special axiom schema

• Comprehension Axioms
• ?x(????),
• if ? is FO.

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A special axiom schema

• Comprehension Axioms
• ?x(????),
• if ? is FO.

LEM Comprehension Axiom
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Armstrongs Axioms
Always (x,x) If (x,y,z), then (y,x,z). If
(x,x,y), then (x,y). If (x,z), then
(x,y,z). If (x,y) and (y,z), then (x,z).
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Incorrect rules
No absortion

• From ??? follows ?. Wrong!
• From(???)?(???) follows ??(???). Wrong!
• From(???)?(???) follows ??(???). Wrong!

Non-distributive
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Game theoretic semantics

• Dependence logic has two versions of the
  following games
• Semantic (evaluation) game
• Ehrenfeucht-Fraisse game

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Game theoretic semantics

• Dependence logic has two versions of the
  following games
• Semantic (evaluation) game
• Ehrenfeucht-Fraisse game
• Version 1 Players move assignments.
• Non-deterministic, imperfect information.

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Game theoretic semantics

• Dependence logic has two versions of the
  following games
• Semantic (evaluation) game
• Ehrenfeucht-Fraisse game
• Version 1 Players move assignments.
• Non-deterministic, imperfect information.
• Version 2 Players move teams.
• Deterministic, perfect information.

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Teams
Assignments
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Model theory of dependence logic

• Hodges 1997 For every formula f(x1,,xn) there
  is an existential second order sentence F(P) with
  P negative such that a team X satisfies f iff
  F(X) is true.

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Model theory of dependence logic

• Hodges 1997 For every formula f(x1,,xn) there
  is an existential second order sentence F(P) with
  P negative such that a team X satisfies f iff
  F(X) is true.

Theorem (Kontinen-V. 2008) The converse is also
true.
Answers a question of Hodges.
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Consequences

• A language for NP on finite models.
• Compactness.
• Löwenheim-Skolem.
• Separation (Interpolation).

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Classical negation

• The closure of dependence logic under classical
  negation has the exact strength of second order
  logic (Ville Nurmi, 2008).
• But we need negation to express Arrows Theorem?

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Intuitionistic negation
How about intuitionistic negation?

• Joint work with S. Abramsky.
• Definition X satisfies f?? iff every subteam of
  X which satisfies f also satisfies ?.
• Definition X satisfies ? iff X is the empty
  team.
• f is now equivalent to f??for atomic f.
• Intuitionistic negation (f??) is an alternative
  way to extend negation from atomic to non-atomic
  formulas.

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Intuitionistic negation
How about intuitionistic negation?

• Joint work with S. Abramsky.
• Definition X satisfies f?? iff every subteam of
  X which satisfies f also satisfies ?.
• Definition X satisfies ? iff X is the empty
  team.
• f is now equivalent to f??for atomic f.
• Intuitionistic negation (f??) is an alternative
  way to extend negation from atomic to non-atomic
  formulas.

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Intuitionistic negation
How about intuitionistic negation?

• Dependence atoms can now be defined in terms of
  constancy
• (x1,...,xn,z) ? ((x1) ? . ? (xn)) ?
  (z).
• Downward closure and the empty set property are
  preserved.
• Compactness fails.
• Goes beyond NP, unless NPco-NP.

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Intuitionistic negation
How about intuitionistic negation?

• Dependence atoms can now be defined in terms of
  constancy
• (x1,...,xn,z) ? ((x1) ? . ? (xn)) ? (z)
• Downward closure and the empty set property are
  preserved.
• Compactness fails.
• Goes beyond NP, unless NPco-NP

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Intuitionistic negation
How about intuitionistic negation?

• Dependence atoms can now be defined in terms of
  constancy
• (x1,...,xn,z) ? ((x1) ? . ? (xn)) ? (z)
• Downward closure and the empty set property are
  preserved.
• Compactness fails.
• Goes beyond NP, unless NPco-NP.

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Intuitionistic negation
How about intuitionistic negation?

• Dependence atoms can now be defined in terms of
  constancy
• (x1,...,xn,z) ? ((x1) ? . ? (xn)) ? (z)
• Downward closure and the empty set property are
  preserved.
• Compactness fails.
• Goes beyond NP, unless NPco-NP.

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We can prove Armstrongs Axioms
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Intuitionistic negation
Linear implication

• X satisfies f o ? iff for every team Y which
  satisfies f the team X ? Y satisfies ?.
• Downward closure is preserved.
• Compactness fails.
• Goes beyond NP unless NPco-NP.

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Galois connections

• Intuitionistic implication is the adjoint of
  conjunction
• Linear implication is the adjoint of disjunction.

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Proof

• Linear implication is the adjoint of disjunction.

X
Y
X ? Y
X ? Y
X ? Y
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Proof

• Linear implication is the adjoint of disjunction.

Z
X ? Y
X
Y
X ? Y
Z
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Lesson
The moral of the story

• One can add both intuitionistic and linear
  implication to dependence logic without losing
  the downward closure.
• Intuitionistic negation agrees with the original
  negation on the atomic level, and basic axioms of
  dependence become provable.
• Good (?) for proof theory, but bad (?) for model
  theory. Is there a reason for this?

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What is dependence logic good for?

• Language for NP.
• Tool for the study of more complex dependencies
  than just the Armstrong ones.
• A vehicle for uncovering the mathematics of
  dependence in a variety of contexts
• Data mining
• Social choice theory
• Logic for Rationality and Interaction (?)

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• J. Väänänen, Dependence Logic, Cambridge
  University Press, 2007.
• Logic for Interaction (LINT), ESF LogICCC

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Thank you!