1/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

A logic for reasoning about counterfactualemotions

Emiliano Lorini and Francois SchwarzentruberIRIT, Toulouse, France

1/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

2/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Outline

1 IntroductionTopic of this talkState of the artIngredients

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives

2/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

3/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Example: “rock-paper-scissors” game

“rock-paper-scissors” game rule

� � � etc.

3/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

4/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Example: “rock-paper-scissors” game

“rock-paper-scissors” game rule

� � � etc.

Here Roberta regrets having played "paper".4/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

5/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Example: “rock-paper-scissors” game

“rock-paper-scissors” game rule

� � � etc.

Indeed... she could have won by playing rock.5/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

6/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Example: “rock-paper-scissors” game

Aim: to provide a logical framework for reasoning aboutcounterfactual emotions.

Applications: video games, robots assisting humans, etc.

6/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

7/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Outline

1 IntroductionTopic of this talkState of the artIngredients

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives

7/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

8/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

State of the art

No comprehensive formal model of counterfactual emotionslike:

regret, rejoicing;

guilt, shame.

8/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

9/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Outline

1 IntroductionTopic of this talkState of the artIngredients

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives

9/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

10/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Ingredients we need

Counterfactual emotions like regret are based on an agent’salteration of a factual situation and in the imagination of analternative situation that could have realized if somethingdifferent was done (Kahnemann & Miller, 1986)

We need:

Logic of agency for expressing counterfactual facts;

Imagination;

Time;

Preferences.

10/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

11/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Ingredients we will use

Counterfactual emotions like regret are based on an agent’salteration of a factual situation and in the imagination of analternative situation that could have realized if somethingdifferent was done (Kahnemann & Miller, 1986)

We will use:

Logic of agency for expressing counterfactual facts:Chellas’ STIT modal logic;

Imagination: modal epistemic logic;

Time (this work is preliminary... but it is a perspective);

Preferences: simple constructions with propositions.

11/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

12/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Topic of this talkState of the artIngredients

Table of Contents

1 Introduction

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives

12/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

13/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statementsWhy STIT?SyntaxSemanticsExpressing counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives13/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

14/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

CL, ATL VS STIT

In CL, ATL, we can express what agents can do;

In STIT, we also express what agents do.

→ STIT more suitable for counterfactual statements.

14/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

15/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statementsWhy STIT?SyntaxSemanticsExpressing counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives15/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

16/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Syntax

DefinitionLanguage LgroupSTIT:

ϕ ::= p | ϕ ∧ ϕ | ¬ϕ | [J]ϕ

where J ⊆ AGT .

As usual, 〈J〉ϕ def= ¬[J]¬ϕ.

[J]ϕ means “group J sees to it that ϕ” ≡ “group J choosesactions such that whatever group J does, ϕ is true”.

16/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

17/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Example

[J]ϕ means “whatever group J does, ϕ is true”.

[Roberta]paperRoberta

“Roberta chooses an action such that whatever Bob does,paperRoberta is true”

17/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

18/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Example: [∅] for expressing “necessarly”

[∅]ϕ means “whatever group AGT does, ϕ is true” ≡“necessarly, ϕ is true”.

[∅](paperRoberta ∨ scissorsRoberta ∨ rockRoberta)“necessarly we have (paperRoberta ∨ scissorsRoberta ∨ rockRoberta)”

18/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

19/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Example: expressing ability with 〈∅〉[J]

〈∅〉[Roberta]rockRoberta

“it is possible that Roberta chooses an action such that whatever Bobdoes, rockRoberta is true”

19/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

20/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Example: the dual operator 〈J〉

[J]ϕ means “whatever group J does, ϕ is true”〈J〉ϕ means “there exists actions for group J such that ϕ” ≡“group J allows that ϕ is true”.

〈Bob〉winRoberta

“the choose of Bob allows Roberta to win”

20/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

21/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statementsWhy STIT?SyntaxSemanticsExpressing counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives21/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

22/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Example of a groupSTIT-model

A groupSTIT-model is a strategic form game model.

22/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

23/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

groupSTIT-model

Definition

A groupSTIT-model is M = 〈W , R, V 〉 such that:

W 6= ∅;R : 2AGT → 2W×W such that:

for all G ⊆ AGT , RG is an equivalence relation over W ;for all G ⊆ AGT , RG ⊆ R∅;for all G ⊆ AGT , RG =

⋂a∈G R{a};

for all (xa)a∈AGT ∈ W AGT ,⋂

a∈AGT R{a}(xa) 6= ∅.

V : W → 2ATM .

A groupSTIT-model is a strategic form game model.

23/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

24/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Truth definitions

Definition

M, w |= [J]ϕ iff for all v ∈ RJ(w),M, v |= ϕ.

M, w |= [Roberta]paperRoberta.24/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

25/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Illustrating truth definitions

M, w |= [∅](paperRoberta ∨ scissorsRoberta ∨ rockRoberta).

25/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

26/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Illustrating truth definitions

M, w |= 〈∅〉[Roberta]rockRoberta.

26/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

27/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Illustrating truth definitions

〈Bob〉winRoberta means “there exists an action for Roberta suchthat winRoberta.”.

M, w |= 〈Bob〉winRoberta.27/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

28/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statementsWhy STIT?SyntaxSemanticsExpressing counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives28/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

29/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Example: “rock-paper-scissors” game

Here Roberta looses.

29/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

30/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Example: “rock-paper-scissors” game

But Roberta “could have prevented it” by playing "rock" instead.

30/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

31/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

We can see it on the STIT-model

M, w |= ¬winRoberta ∧ 〈Bob〉winRoberta.

31/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

32/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

We can see it on the STIT-model

M, w |= ¬winRoberta ∧ 〈Roberta〉winRoberta.

32/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

33/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Why STIT?SyntaxSemanticsExpressing counterfactual statements

Counterfactual statements in STIT

Definition

CHPJχdef= χ ∧ 〈J〉¬χ

CHPJ means “group J could have prevented χ to be true”:

χ is true;

Other agents allows ¬χ.

33/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

34/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicingAdding knowledgeAdding preferencesExpressing regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives

34/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

35/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Extending STIT with knowledge

DefinitionLanguage LKSTIT :

ϕ ::= p | ϕ ∧ ϕ | ¬ϕ | [J]ϕ | Kiϕ

where J ⊆ AGT and i ∈ AGT .

Kiϕ means “agent i knows ϕ.”

35/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

36/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Semantics of KSTIT

Definition

A KSTIT-model is M = (W , {RJ}J⊆AGT , {Ei}i∈AGT , V ) where:

(W , {RJ}J⊆AGT , V ) is a STIT-model;

For all i ∈ AGT , Ei is an equivalence relation.

Definition

M, w |= Kiϕ iff for all v ∈ Ei(w),M, v |= ϕ.

36/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

37/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicingAdding knowledgeAdding preferencesExpressing regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives

37/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

38/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Adding preferences

The special atom goodi identifies good states for agent i .

GOODiχdef= [∅](goodi → χ)

GOODiχ ≈ “χ is good for agent i”

DESiχdef= KiGOODiχ

DESiχ ≈ “χ is desirable for agent i”

38/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

39/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Some properties of agents’ preferences

Deductive closure:|=KSTIT (DESiχ1 ∧ DESi(χ1 → χ2)) → DESiχ2;

Positive introspection: |=KSTIT DESiχ ↔ KiDESiχ;

Negative introspection: |=KSTIT ¬DESiχ ↔ Ki¬DESiχ

39/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

40/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicingAdding knowledgeAdding preferencesExpressing regret and rejoicing

4 Decidability of the satisfiability problem

5 Perspectives

40/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

41/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Regret

Definition

REGRETiχdef= DESi¬χ ∧ KiCHPiχ

where CHPiχdef= χ ∧ 〈AGT \ {i}〉¬χ.

Agent i regrets that χ is true (REGRETiχ):

¬χ is desirable for i ;

i knows that it could have prevented χ to be true now.

41/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

42/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Regret: a little remark about action and effect

Explicit action I regret having played "paper".l l

Implicit action (effect) I regret my action such that ¬winRoberta.

42/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

43/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Rejoicing

REGRETiχdef= DESi¬χ ∧ KiCHPiχ

↓

rejoicing is the positive counterpart of regret (see, e.g.,Zeelenberg et al., 1996)

↓

Definition (Agent i rejoices over χ)

REJOICEiχdef= DESiχ ∧ KiCHPiχ

43/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

44/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Adding knowledgeAdding preferencesExpressing regret and rejoicing

Properties of regret and rejoicing

Positive and negative introspections of regret and rejoicing:|=KSTIT REGRETiχ ↔ KiREGRETiχ;|=KSTIT REJOICEiχ ↔ KiREJOICEiχ;|=KSTIT ¬REGRETiχ ↔ Ki¬REGRETiχ;|=KSTIT ¬REJOICEiχ ↔ Ki¬REJOICEiχ.

Regret implies desire frustration:|=KSTIT REGRETiχ → (DESi¬χ ∧ Kiχ);

Rejoice implies desire satisfaction:|=KSTIT REJOICEiχ → (DESiχ ∧ Kiχ).

44/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

45/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problemMotivationWhole STITSolution: a fragment of STIT

5 Perspectives

45/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

46/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Motivation for studying decidability

Is it possible for robots to reason about regret, rejoicing?

How difficult will it be for them?

46/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

47/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problemMotivationWhole STITSolution: a fragment of STIT

5 Perspectives

47/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

48/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Whole STIT is expressive

KSTIT is very expressive.

Example (Compassion)

I regret that you regret p.

and...

48/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

49/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Complexity of the whole STIT

Theorem

(Herzig & Schwarzentruber, 2008) The STIT-satisfiabilityproblem is:

NP-complete if card(AGT ) = 1;

NEXPTIME-complete if card(AGT ) = 2;

undecidable if card(AGT ) ≥ 3.

49/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

50/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Outline

1 Introduction

2 Chellas’ STIT for counterfactual statements

3 Counterfactual emotions: Regret and rejoicing

4 Decidability of the satisfiability problemMotivationWhole STITSolution: a fragment of STIT

5 Perspectives

50/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

51/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Solution: reducing expressivity

We deny:

I regret that you regret p.

We accept:

I regret (p ∨ q).

51/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

52/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Solution: a fragment dfSTIT

whole STIT

〈Sam〉([Bob]p ∧ [Sam]([Robert ]q ∨ 〈Bob, Robert〉p))

The fragment dfSTIT

[Bob](p ∨ q) ∧ 〈∅〉[Robert , Bob](p ∧ r)

52/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

53/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Definition of the fragment dfSTIT

DefinitionThe language LdfSTIT:

χ ::= ⊥ | p | χ ∧ χ | ¬χ (propositional formulas)

ψ ::= [J]χ | ψ ∧ ψ (STIT formulas conjunction)

ϕ ::= χ | ψ | ϕ ∧ ϕ | ¬ϕ | 〈∅〉ψ (STIT and “can” formulas)

where J ∈ 2AGT \ {∅}.

53/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

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IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

dfSTIT is decidable

Theorem

The dfSTIT-satisfiability problem is NP-complete whatevercard(AGT ).

LdfSTIT

LgroupSTIT

54/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

55/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

Extending STIT with knowledge

Definition

Language of dfKSTIT:

χ ::= ⊥ | p | χ ∧ χ | ¬χ (propositional formulas)

ψ ::= [J]χ | ψ ∧ ψ (STIT formulas conjunction)

ϕ ::= χ | ψ | ϕ ∧ ϕ | ¬ϕ | 〈∅〉ψ | Kiϕ (see-to-it, “can”,knowledge formulas )

where i ∈ AGT and J ∈ 2AGT \ {∅}.

55/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

56/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

MotivationWhole STITSolution: a fragment of STIT

dfKSTIT is decidable

Theorem

The satisfiability problem of dfKSTIT is:

NP-complete if card(AGT ) = 1;

PSPACE-complete if card(AGT ) ≥ 2.

56/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

57/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Perspectives

axiomatisation for the fragment?

add time?

improve modelisation of preferences?

find a more expressive fragment?

norms: guilt, shame?

57/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions

58/58

IntroductionChellas’ STIT for counterfactual statements

Counterfactual emotions: Regret and rejoicingDecidability of the satisfiability problem

Perspectives

Thank you!

58/58 Emiliano Lorini and Francois Schwarzentruber IRIT, Toulouse, FranceA logic for reasoning about counterfactual emotions