Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Reasoning about Agent Types
Fenrong Liu
Department of Philosophy, Tsinghua University
31 August, 2012, EASLLC Conference
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Ongoing joint project with Yanjing Wang
Some related publications:
Liu,F. and Wang, Y, Reasoning about agent types and the
hardest logic puzzle ever, to appear in Minds and Machines.
Liu, F. (2009). Diversity of agents and their interaction.
Journal of Logic, Language and Information, 18(1):23-53.
Liu, F. (2004).Dynamic Variations: Update and Revision for
Diverse Agents, Mol Thesis, ILLC publication, 2004.
Wang, Y. (2010). Epistemic Modelling and Protocol
Dynamics. PhD thesis, ILLC University of Amsterdam and
CWI.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Outline
1
Motivation
2
Type-based dynamic epistemic logics
3
Agent type in questions and answers
4
Handling arbitrary utterances
5
Formalizing HLPE
6
Conclusions and future work
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Motivation I: logical puzzles
In the book What’s the name of this book? Raymond Smullyan
introduces a series of puzzles featuring knights (truth tellers) and
knaves (liars).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Motivation I: logical puzzles
In the book What’s the name of this book? Raymond Smullyan
introduces a series of puzzles featuring knights (truth tellers) and
knaves (liars).
[Knights and Knaves] In a fictional island, all inhabitants
are either knights, who always tell the truth, or knaves, who
always lie. A visitor D from the outside world meets three
inhabitants A, B and C on the island. D asked them to tell
their types. Then A says: B is a knave. B says: C is a knave.
C says: A and B are knaves. Now, is it possible for the visitor
to find out the inhabitants’ types from their statements?
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Death or Freedom
A and B are standing at a fork in the road. Now comes
C . C knows that one of them is a Knight and the other
is a Knave, but C does not know who is who. C also
knows that one road leads to Death, and the other leads
to Freedom. C is allowed to ask a question to one of A
and B. How should he ask his question in such a way
that he will know the way to Freedom no matter what
the answer is?
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
The Hardest Logic Puzzle Ever (HLPE)
One variation of such puzzles is made famous by [Boolos, 1996],
where it is called the Hardest Logic Puzzle Ever (HLPE):
Three gods A, B, and C are called, in some order, True, False,
and Random. True always speaks truly, False always speaks
falsely, but whether Random speaks truly or falsely is a
completely random matter. Your task is to determine the
identities of A, B, and C by asking three yes/no questions;
each question must be put to exactly one god. The gods
understand English, but will answer all questions in their own
language, in which the words for yes and no are da and ja, in
some order. You do not know which word means which.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some features of these puzzles
Agents are of different types, truth teller, liars, or bluffer.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some features of these puzzles
Agents are of different types, truth teller, liars, or bluffer.
Agents communicate with each other according to their own
types. Who said what is important. In particular, in the
HlPE, the interpretation of an utterance depends on the type
of agents.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some features of these puzzles
Agents are of different types, truth teller, liars, or bluffer.
Agents communicate with each other according to their own
types. Who said what is important. In particular, in the
HlPE, the interpretation of an utterance depends on the type
of agents.
There are uncertainties about the types of agents, under
which we need to reason about knowledge correctly.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some features of these puzzles
Agents are of different types, truth teller, liars, or bluffer.
Agents communicate with each other according to their own
types. Who said what is important. In particular, in the
HlPE, the interpretation of an utterance depends on the type
of agents.
There are uncertainties about the types of agents, under
which we need to reason about knowledge correctly.
Announcing and asking quesions are the basic actions.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some features of these puzzles
Agents are of different types, truth teller, liars, or bluffer.
Agents communicate with each other according to their own
types. Who said what is important. In particular, in the
HlPE, the interpretation of an utterance depends on the type
of agents.
There are uncertainties about the types of agents, under
which we need to reason about knowledge correctly.
Announcing and asking quesions are the basic actions.
It calls for a formal framework to account for those puzzles.
We are aware of the exisiting work: Boolos (1996) ;Rabern
and Rabern (2008); Uzquiano (2010); Wheeler and Barahona
(2012); Wintein (2011).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some features of these puzzles
Agents are of different types, truth teller, liars, or bluffer.
Agents communicate with each other according to their own
types. Who said what is important. In particular, in the
HlPE, the interpretation of an utterance depends on the type
of agents.
There are uncertainties about the types of agents, under
which we need to reason about knowledge correctly.
Announcing and asking quesions are the basic actions.
It calls for a formal framework to account for those puzzles.
We are aware of the exisiting work: Boolos (1996) ;Rabern
and Rabern (2008); Uzquiano (2010); Wheeler and Barahona
(2012); Wintein (2011).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Motivation 2: type-based logics are needed
Dynamic epistemicl logic ([Plaza, 1989], [Gerbrandy,1998],
etc) studies the communication between agents. In particular,
the action of public announcement and its relationship with
knowledge.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Motivation 2: type-based logics are needed
Dynamic epistemicl logic ([Plaza, 1989], [Gerbrandy,1998],
etc) studies the communication between agents. In particular,
the action of public announcement and its relationship with
knowledge.
In PAL, it is assumed that all announcements are truthful,
thus always reliable. Little attention has been paid on agent
types.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Motivation 2: type-based logics are needed
Dynamic epistemicl logic ([Plaza, 1989], [Gerbrandy,1998],
etc) studies the communication between agents. In particular,
the action of public announcement and its relationship with
knowledge.
In PAL, it is assumed that all announcements are truthful,
thus always reliable. Little attention has been paid on agent
types.
In reality, agents are diverse. E.g. announcements made by
truth teller or liars should be taken differently.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Quick introduction to public announcement logic
Example (question-answer scenario)
Alice: Is this the way to the South-west University?
Bob: Yes, it is.
Before Bob’s answer, Alice does not know whether p is the case:
P
A
not P
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Quick introduction to public announcement logic
Example (question-answer scenario)
Alice: Is this the way to the South-west University?
Bob: Yes, it is.
Before Bob’s answer, Alice does not know whether p is the case:
P
A
not P
After the answer, Alice knows that p is the case:
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Quick introduction to public announcement logic
Example (question-answer scenario)
Alice: Is this the way to the South-west University?
Bob: Yes, it is.
Before Bob’s answer, Alice does not know whether p is the case:
P
A
not P
P
After the answer, Alice knows that p is the case:
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Observations
We can express the communication between Alice and Bob
with formula [!b p]Ka p, reads as ”after b announces p, a
knows that p”. In general, we write [!b φ]Ka ψ.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Observations
We can express the communication between Alice and Bob
with formula [!b p]Ka p, reads as ”after b announces p, a
knows that p”. In general, we write [!b φ]Ka ψ.
In PAL we only consider truthful public announcement, so
we omit ”b” , write it as [!φ]Ka ψ. Namely, we assume that
Bob always tells the truth.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Observations
We can express the communication between Alice and Bob
with formula [!b p]Ka p, reads as ”after b announces p, a
knows that p”. In general, we write [!b φ]Ka ψ.
In PAL we only consider truthful public announcement, so
we omit ”b” , write it as [!φ]Ka ψ. Namely, we assume that
Bob always tells the truth.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Public announcement logic (PAL)
Definition[Language of PAL] Let P be a set of proposition
letters and G a set of agents, with p ranging over P, a over G.
The language of public announcement logic is given by:
ϕ ::= > | p | ¬ϕ | ϕ ∧ ψ | Ka ϕ | [!ϕ]ϕ
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Public announcement logic (PAL)
Definition[Language of PAL] Let P be a set of proposition
letters and G a set of agents, with p ranging over P, a over G.
The language of public announcement logic is given by:
ϕ ::= > | p | ¬ϕ | ϕ ∧ ψ | Ka ϕ | [!ϕ]ϕ
A public announcement !ϕ of a true proposition ϕ turns the
current model (M, s) with actual world s into a model
(M!ϕ , s) whose worlds are just the set {w : M, w |= ϕ}.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Interpretation of the dynamic modality
The semantic clause for the dynamic modality is the following
M, s |= [!ϕ]ψ iff (if M, s |= ϕ, then M!ϕ , s |= ψ)
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Interpretation of the dynamic modality
The semantic clause for the dynamic modality is the following
M, s |= [!ϕ]ψ iff (if M, s |= ϕ, then M!ϕ , s |= ψ)
The ‘if clause’ indicates preconditions of making an announcement
φ, namely, φ is true, so it is a truthful announcement.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
System of PAL
1
2
3
4
5
6
7
8
9
10
11
12
All instantiations of propositional tautologies
Ka (ϕ → ψ) → (Ka ϕ → Ka ψ)
Ka ϕ → ϕ
Ka ϕ → Ka Ka ϕ
¬Ka ϕ → Ka ¬Ka ϕ
[!ϕ]p ↔ p
[!ϕ]¬ψ ↔ ¬[!ϕ]ψ
[!ϕ](ψ ∧ χ) ↔ [!ϕ]ψ ∧ [!ϕ]χ
[!ϕ]Ka ψ ↔ (ϕ → Ka [!ϕ]ψ)
From ` ϕ and ` ϕ → ψ infer ` ψ
From ` ϕ infer ` Ka ϕ
From φ ↔ ψ infer χ[ψ/φ] ↔ χ
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Completeness
Theorem
Public announcement logic is complete and decidable.
Note that the dynamic reduction axioms take every formula of our
dynamic language eventually to an equivalent formula inside the
static pure epistemic language.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Ideas related to agent types
Similar ideas have appeared in different contexts:
Agents’ strategies: Paul and Ramanujam (2011);
Diversity of agents: Liu(2004), Liu (2009);
Lying: van Ditmarsch et al. (2011); van Ditmarsch (2011);
Protocols: Hoshi (2009); Wang (2010).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Ideas related to agent types
Similar ideas have appeared in different contexts:
Agents’ strategies: Paul and Ramanujam (2011);
Diversity of agents: Liu(2004), Liu (2009);
Lying: van Ditmarsch et al. (2011); van Ditmarsch (2011);
Protocols: Hoshi (2009); Wang (2010).
We would like to develop the dynamic epistemic logic
systematically, taking agent types into account.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Our plan
A type language is introduced to describe complicated types.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Our plan
A type language is introduced to describe complicated types.
Agents may have uncertainties of other agents’ types.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Our plan
A type language is introduced to describe complicated types.
Agents may have uncertainties of other agents’ types.
The interpretation of an utterance depends on the type of
agents.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Our plan
A type language is introduced to describe complicated types.
Agents may have uncertainties of other agents’ types.
The interpretation of an utterance depends on the type of
agents.
Announcements and questions are basic communicative
actions.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Our plan
A type language is introduced to describe complicated types.
Agents may have uncertainties of other agents’ types.
The interpretation of an utterance depends on the type of
agents.
Announcements and questions are basic communicative
actions.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
A simple type-based logical language
Definition (Type language)
Given a fixed agent variable x and a fixed formula variable ϕ, the
set E of agent types η is recursively defined as:
η ::= ψ !x ϕ
ψ ::= > | ϕ | ¬ψ | ψ ∧ ψ | Kx ψ
where > stands for tautologies.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
A simple type-based logical language
Definition (Type language)
Given a fixed agent variable x and a fixed formula variable ϕ, the
set E of agent types η is recursively defined as:
η ::= ψ !x ϕ
ψ ::= > | ϕ | ¬ψ | ψ ∧ ψ | Kx ψ
where > stands for tautologies.
Note that x and ϕ are the only variables, thus Kx ϕ ∧ Ky ψ is not a
well-formed type.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
A simple type-based logical language
Definition (Type language)
Given a fixed agent variable x and a fixed formula variable ϕ, the
set E of agent types η is recursively defined as:
η ::= ψ !x ϕ
ψ ::= > | ϕ | ¬ψ | ψ ∧ ψ | Kx ψ
where > stands for tautologies.
Note that x and ϕ are the only variables, thus Kx ϕ ∧ Ky ψ is not a
well-formed type. Each agent type η can also be viewed as a
function assigning a precondition to each announcement made by
an agent of this type.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Defining agent types
We can use this type language to define agent types.
Type TT (truth teller): ϕ !x ϕ
Type LL (liar): ¬ϕ !x ϕ
Type LT (bluffer): > !x ϕ.
Type STT (subjective truth teller): Kx ϕ !x ϕ
Type SLL (subjective liar): Kx ¬ϕ !x ϕ.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Public announcement language with types
Definition (Public announcement language with types)
Given a finite set T ⊆ E of agent types, a finite set G of agent
names, a set P of basic proposition letters, the language PALTT is
defined as:
φ ::= > | p | η(a) | ¬φ | φ ∧ φ | Ka φ | [!a φ]φ
where p ∈ P, a ∈ G and η ∈ T.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Public announcement language with types
Definition (Public announcement language with types)
Given a finite set T ⊆ E of agent types, a finite set G of agent
names, a set P of basic proposition letters, the language PALTT is
defined as:
φ ::= > | p | η(a) | ¬φ | φ ∧ φ | Ka φ | [!a φ]φ
where p ∈ P, a ∈ G and η ∈ T. We call the announcement-free
fragment of PALTT the epistemic language with type formulas
(ELT ) and sometimes denote PALTT by ELT + [!a φ].
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Further explanations
As usual, we have the following abbreviations: ⊥ := ¬>,
φ ∨ ψ := ¬(¬φ ∧ ¬ψ), φ → ψ := ¬φ ∨ ψ, h!a ψiφ :=
¬[!a ψ]¬φ, K̂a φ := ¬Ka ¬φ. We also write KaW φ for
Ka φ ∨ Ka ¬φ, meaning that a knows whether φ.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Further explanations
As usual, we have the following abbreviations: ⊥ := ¬>,
φ ∨ ψ := ¬(¬φ ∧ ¬ψ), φ → ψ := ¬φ ∨ ψ, h!a ψiφ :=
¬[!a ψ]¬φ, K̂a φ := ¬Ka ¬φ. We also write KaW φ for
Ka φ ∨ Ka ¬φ, meaning that a knows whether φ.
η(a) expresses that agent a is of the type η and [!a ψ]φ says
that if a can announce ψ then after the announcement, φ
holds.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Further explanations
As usual, we have the following abbreviations: ⊥ := ¬>,
φ ∨ ψ := ¬(¬φ ∧ ¬ψ), φ → ψ := ¬φ ∨ ψ, h!a ψiφ :=
¬[!a ψ]¬φ, K̂a φ := ¬Ka ¬φ. We also write KaW φ for
Ka φ ∨ Ka ¬φ, meaning that a knows whether φ.
η(a) expresses that agent a is of the type η and [!a ψ]φ says
that if a can announce ψ then after the announcement, φ
holds.
Intuitively, an agent a of a type η can announce a concrete
proposition φ only when η(φ, a) holds.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
Definition (Semantics)
A model for the language of PALTT is a tuple
M = (S, {∼a | a ∈ G}, V , λ), where (S, {∼a | a ∈ G}, V ) is a
standard multi-agent S5 Kripke model:
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
Definition (Semantics)
A model for the language of PALTT is a tuple
M = (S, {∼a | a ∈ G}, V , λ), where (S, {∼a | a ∈ G}, V ) is a
standard multi-agent S5 Kripke model:
S is a non-empty set of possible worlds.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
Definition (Semantics)
A model for the language of PALTT is a tuple
M = (S, {∼a | a ∈ G}, V , λ), where (S, {∼a | a ∈ G}, V ) is a
standard multi-agent S5 Kripke model:
S is a non-empty set of possible worlds.
∼a ⊆ S × S is an equivalence relation over S.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
Definition (Semantics)
A model for the language of PALTT is a tuple
M = (S, {∼a | a ∈ G}, V , λ), where (S, {∼a | a ∈ G}, V ) is a
standard multi-agent S5 Kripke model:
S is a non-empty set of possible worlds.
∼a ⊆ S × S is an equivalence relation over S.
V : S → 2P is a valuation function assigning to each world a
set of basic propositions.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
Definition (Semantics)
A model for the language of PALTT is a tuple
M = (S, {∼a | a ∈ G}, V , λ), where (S, {∼a | a ∈ G}, V ) is a
standard multi-agent S5 Kripke model:
S is a non-empty set of possible worlds.
∼a ⊆ S × S is an equivalence relation over S.
V : S → 2P is a valuation function assigning to each world a
set of basic propositions.
The new component λ : S × G → T assigns to each agent on
each world a type in T.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Truth conditions
M, s Ka φ ⇔ ∀t : s ∼a t implies M, t φ
M, s η(a) ⇔ λ(s, a) = η
M, s [!a ψ]φ ⇔ M, s λ(s, a)(ψ, a) implies M|aψ , s φ
where M|aψ is defined as (S 0 , {∼0a | a ∈ G}, V 0 , λ0 ) where:
S 0 = {t | t ∈ S and M, t λ(t, a)(ψ, a)}
For each a ∈ G, t ∈ S 0 :∼0a =∼a |S 0 ×S 0 , V 0 (t) = V (t) and
λ0 (t) = λ(t).
Note that M|aψ is well-defined if S 0 is not empty, and
M, s λ(s, a)(ψ, a) in the clause of [!a ψ]φ guarantees that.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Knights and Knaves revisited
In a fictional island, all inhabitants are either knights, who
always tell the truth, or knaves, who always lie. A visitor D
from the outside world meets three inhabitants A, B and C on
the island. D asked them to tell their types. Then A says: B is
a knave. B says: C is a knave. C says: A and B are knaves.
Now, is it possible for the visitor to find out the inhabitants’
types from their statements?
Truth teller (TT) and Liar (LL) are the two possible types for
A, B and C .
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Knights and Knaves revisited
In a fictional island, all inhabitants are either knights, who
always tell the truth, or knaves, who always lie. A visitor D
from the outside world meets three inhabitants A, B and C on
the island. D asked them to tell their types. Then A says: B is
a knave. B says: C is a knave. C says: A and B are knaves.
Now, is it possible for the visitor to find out the inhabitants’
types from their statements?
Truth teller (TT) and Liar (LL) are the two possible types for
A, B and C .
A, B and C know their own type, but D knows nothing.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Knights and Knaves revisited
In a fictional island, all inhabitants are either knights, who
always tell the truth, or knaves, who always lie. A visitor D
from the outside world meets three inhabitants A, B and C on
the island. D asked them to tell their types. Then A says: B is
a knave. B says: C is a knave. C says: A and B are knaves.
Now, is it possible for the visitor to find out the inhabitants’
types from their statements?
Truth teller (TT) and Liar (LL) are the two possible types for
A, B and C .
A, B and C know their own type, but D knows nothing.
We write LLT for a world s where λ(s, A) = LL, λ(s, B) = LL
and λ(s, C ) = TT
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Modelling
LLL
D
LLT
D
LTT
D
D
TTT
LTL LTT
D
D
TTL
D
TLL
D
M1
D
D
TLT TLL
D
LTL
LTL
D
D
TLT
TLT
M2
M3
LTL
M4
In the initial M1 : A, B, and C know their own types (either
TT or LL) but D knows nothing.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Modelling
LLL
D
LLT
D
LTT
D
D
TTT
LTL LTT
D
D
TTL
D
TLL
D
M1
D
D
TLT TLL
D
LTL
LTL
D
D
TLT
TLT
M2
M3
LTL
M4
In the initial M1 : A, B, and C know their own types (either
TT or LL) but D knows nothing.
By the definition of the updated model, M2 = M1 |A
LL(B) keeps
the worlds s in M1 where M1 , s λ(s, A)(LL(B), A).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Update
That is: it keeps the worlds s satisfying one of the following
conditions:
λ(s, A) = TT and M1 , s LL(B),
λ(s, A) = LL and M1 , s ¬LL(B).
Since T = {LL, TT}, the above two conditions are equivalent to:
λ(s, A) = TT and λ(s, B) = LL (i.e., the worlds in the shape
of TL )
λ(s, A) = LL and λ(s, B) = TT (i.e., the worlds in the shape
of LT )
So M2 only contains TL and LT .
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Update
A similar reasoning works for M3 and M4 by the definition of
the updated model.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Update
A similar reasoning works for M3 and M4 by the definition of
the updated model.
Easy to see that LTL is the only world s in M1 s.t. all
announcements can be announced in the given order:
M1 , s h!A LL(B)ih!B LL(C )ih!C (LL(A) ∧ LL(B))i>
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Update
A similar reasoning works for M3 and M4 by the definition of
the updated model.
Easy to see that LTL is the only world s in M1 s.t. all
announcements can be announced in the given order:
M1 , s h!A LL(B)ih!B LL(C )ih!C (LL(A) ∧ LL(B))i>
M4 is a singleton model, it is clear that
M1 , LTL h!A LL(B)ih!B LL(C )ih!C (LL(A)∧LL(B))iKD (LL(A)∧TT(B)∧LL(C
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Update
A similar reasoning works for M3 and M4 by the definition of
the updated model.
Easy to see that LTL is the only world s in M1 s.t. all
announcements can be announced in the given order:
M1 , s h!A LL(B)ih!B LL(C )ih!C (LL(A) ∧ LL(B))i>
M4 is a singleton model, it is clear that
M1 , LTL h!A LL(B)ih!B LL(C )ih!C (LL(A)∧LL(B))iKD (LL(A)∧TT(B)∧LL(C
After the three announcements, agent D knows that A and C
are liars and B is a truth teller.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Axiomatization PALTT
Axiom Schemas
TAUT
MU
DISTK
T
4
5
!ATOM
!NEG
!CON
!K
Rules
(for arbitrary a, b ∈ G, p ∈ P ∪ PT )
V allVthe instancesVof tautologies
0
a∈G ( η∈T (η(a) ↔
η 0 6=η,η 0 ∈T ¬η (a)))
Ka (φ → ψ) → (Ka φ → Ka ψ)
Ka φ → φ
Ka φ → Ka Ka φ
¬Ka φ → Ka ¬Ka φ
[!a ψ]p ↔ (δψa → p)
[!a ψ]¬φ ↔ (δψa → ¬[!a ψ]φ)
[!a ψ](φ ∧ χ) ↔ ([!a ψ]φ ∧ [!a ψ]χ)
[!a ψ]Kb φ ↔ (δψa → Kb [!a ψ]φ)
GENK
RE
MP
Fenrong Liu Department of Philosophy, Tsinghua University
φ
Ka φ
φ↔ψ
χ[ψ/φ] ↔ χ
φ, φ → ψ
ψ
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some technical results
AT is sound and complete.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some technical results
AT is sound and complete.
ELT , PALT , and PALTT are equally expressive.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some technical results
AT is sound and complete.
ELT , PALT , and PALTT are equally expressive.
If T = {LL, TT} then [!a LL(a)]⊥ is valid. In such a case saying
that “I am a liar” is equal to saying something contradictory.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some technical results
AT is sound and complete.
ELT , PALT , and PALTT are equally expressive.
If T = {LL, TT} then [!a LL(a)]⊥ is valid. In such a case saying
that “I am a liar” is equal to saying something contradictory.
However, if T = {LL, TT, LT} then [!a LL(a)]⊥ is not valid any
more, instead [!a LL(a)]LT(a) becomes valid.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some technical results
AT is sound and complete.
ELT , PALT , and PALTT are equally expressive.
If T = {LL, TT} then [!a LL(a)]⊥ is valid. In such a case saying
that “I am a liar” is equal to saying something contradictory.
However, if T = {LL, TT, LT} then [!a LL(a)]⊥ is not valid any
more, instead [!a LL(a)]LT(a) becomes valid.
If there are a truth teller, a liar and a bluffer, then the truth
tellers and liars can not let others know their types without
the help of bluffers.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Some technical results
AT is sound and complete.
ELT , PALT , and PALTT are equally expressive.
If T = {LL, TT} then [!a LL(a)]⊥ is valid. In such a case saying
that “I am a liar” is equal to saying something contradictory.
However, if T = {LL, TT, LT} then [!a LL(a)]⊥ is not valid any
more, instead [!a LL(a)]LT(a) becomes valid.
If there are a truth teller, a liar and a bluffer, then the truth
tellers and liars can not let others know their types without
the help of bluffers.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Public question logic
Definition (Public question logic with types PQLTT )
Given T, P and G as before, the language PQLTT extends PALTT
with question operators and arbitrary answer operators:
φ ::= > | p | ¬φ | φ ∧ φ | Ka φ | η(a) | [!a φ]φ | [?a φ]φ | [!a ]φ
where η ∈ T and a ∈ G.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Public question logic
Definition (Public question logic with types PQLTT )
Given T, P and G as before, the language PQLTT extends PALTT
with question operators and arbitrary answer operators:
φ ::= > | p | ¬φ | φ ∧ φ | Ka φ | η(a) | [!a φ]φ | [?a φ]φ | [!a ]φ
where η ∈ T and a ∈ G.
Intuitively, [?a ψ]φ expresses that ‘After asking a whether ψ, φ
holds’, and [!a ]φ says that ‘No matter what answer a gives (to the
current question), afterwards φ holds’. Here we only focus on
yes/no questions.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
Definition (Semantics for PQLTT )
The semantics of PQLTT formulas on a model M = (S, ∼, V , λ) is
defined as the following w.r.t. a context
µ ∈ {#} ∪ {G × Form(PQLTT )} where Form(PQLTT ) is the set of
PQLTT formulas.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
Definition (Semantics for PQLTT )
The semantics of PQLTT formulas on a model M = (S, ∼, V , λ) is
defined as the following w.r.t. a context
µ ∈ {#} ∪ {G × Form(PQLTT )} where Form(PQLTT ) is the set of
PQLTT formulas.
Intuitively, µ is used to record the current question: it can be of
the form (a, φ) (a needs to answer whether φ) or simply # (there
is currently no question to be answered).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Truth condition
M, s
M, s
M, s
M, s
M, s
M, s
M, s
M, s
φ
µ >
µ p
µ ¬φ
µ φ ∧ ψ
µ K a φ
µ η(a)
µ [?a ψ]φ
M, s µ [!a ψ]φ
M, s µ [!a ]φ
⇔
⇔
⇔
⇔
⇔
⇔
⇔
⇔
M, s # φ
always
p ∈ V (s)
M, s 1µ φ
M, s µ φ and M, s µ ψ
∀t : s ∼a t implies M, t µ φ
λ(s, a) = η(a)
M, s (a,ψ) φ
µ = (a, χ), ψ = ±χ and
⇔
implies M|aψ , s # φ
M, s # λ(s, a)(ψ, a)
⇔ for all ψ : M, s µ [!a ψ]φ
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
where ψ = ±χ means ψ = χ or ψ = ¬χ. M|aψ is defined like
before as (S 0 , {∼0a | a ∈ G}, V 0 , λ0 ) with:
S 0 = {t | t ∈ S and M, t # λ(s, a)(ψ, a)}
For each a ∈ G, t ∈ S 0 : ∼0a =∼a |S 0 ×S 0 , V 0 (t) = V (t), and
λ0 (t) = λ(t).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
where ψ = ±χ means ψ = χ or ψ = ¬χ. M|aψ is defined like
before as (S 0 , {∼0a | a ∈ G}, V 0 , λ0 ) with:
S 0 = {t | t ∈ S and M, t # λ(s, a)(ψ, a)}
For each a ∈ G, t ∈ S 0 : ∼0a =∼a |S 0 ×S 0 , V 0 (t) = V (t), and
λ0 (t) = λ(t).
We say M|aψ is defined if {t | t ∈ S and M, t # λ(s, a)(ψ, a)} is
not empty.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Main ideas
Initially no question is asked (the use of # in the first clause).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Main ideas
Initially no question is asked (the use of # in the first clause).
When a question ?a ψ is asked, the question ψ and its answerer
a are recorded (see the use of (a, ψ) in the clause for [?a ψ]φ),
replacing the previously unanswered one, if there is any.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Main ideas
Initially no question is asked (the use of # in the first clause).
When a question ?a ψ is asked, the question ψ and its answerer
a are recorded (see the use of (a, ψ) in the clause for [?a ψ]φ),
replacing the previously unanswered one, if there is any.
A proposition can be announced by a (!a ψ) only if ψ is a
proper answer to the current question for a (the clause for
[!a ψ]φ). Thus no one can say anything before a question is
raised.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Main ideas
Initially no question is asked (the use of # in the first clause).
When a question ?a ψ is asked, the question ψ and its answerer
a are recorded (see the use of (a, ψ) in the clause for [?a ψ]φ),
replacing the previously unanswered one, if there is any.
A proposition can be announced by a (!a ψ) only if ψ is a
proper answer to the current question for a (the clause for
[!a ψ]φ). Thus no one can say anything before a question is
raised.
After an answer is given, the record is set to #.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Main ideas
Initially no question is asked (the use of # in the first clause).
When a question ?a ψ is asked, the question ψ and its answerer
a are recorded (see the use of (a, ψ) in the clause for [?a ψ]φ),
replacing the previously unanswered one, if there is any.
A proposition can be announced by a (!a ψ) only if ψ is a
proper answer to the current question for a (the clause for
[!a ψ]φ). Thus no one can say anything before a question is
raised.
After an answer is given, the record is set to #.
Any question can be addressed to any one, and the arbitrary
answer operator can be split into two answers, as
demonstrated by the following two valid formulas:
[?a φ]χ ↔ h?a φiχ
[?a φ][!a ]χ ↔ [?a φ]([!a φ]χ∧[!a ¬φ]χ)
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Death or Freedom with questions
Example (Death or Freedom with questions)
A and B are standing at a fork in the road. Now comes C . C
knows that one of them is a Knight and the other is a Knave, but
C does not know who is who. C also knows that one road leads to
Death, and the other leads to Freedom. C is allowed to ask a
question to one of A and B. How should he ask his question in
such a way that he will know the way to Freedom no matter what
the answer is?
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
A quick illustration
Let T = {LL, TT}. We can express the following questions with
this language:
?A ([?B FA ]h!B FA i>): ‘Will the other man tell me that your
path leads to Freedom?’
?A ([?A FA ]h!A FA i>): ‘Will you say ‘yes’ if you are asked
whether your path leads to Freedom?’
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
The model M can be pictured as :
FA , TT,LL
C
C
¬FA , TT,LL
¬FA , LL,TT
C
C
FA , LL,TT
We can verify that
M [?A ([?B FA ]h!B FA i>)][!A ]KCW FA ∧[?A ([?A FA ]h!A FA i>)][!A ]KCW FA .
This means, C can ask A one of the above two questions and find
out the way to Freedom.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Take the first conjunct and verify it at the world (FA , TT, LL):
M, (FA , TT, LL) [?A ([?B FA ]h!B FA i>)][!A ]KCW FA
⇐⇒ M, (FA , TT, LL) # [?A ([?B FA ]h!B FA i>)][!A ]KCW FA
⇐⇒ M, (FA , TT, LL) (A,[?B FA ]h!B FA i>) [!A ]KCW FA
⇐⇒ M, (FA , TT, LL) # [!A ([?B FA ]h!B FA i>)]KCW FA and
M, (FA , TT, LL) # [!A (¬[?B FA ]h!B FA i>)]KCW FA
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Consider the second conjunct of the final part (the first conjunct
can be verified similarly):
M, (FA , TT, LL) # [!A (¬[?B FA ]h!B FA i>)]KCW FA
⇐⇒ M, (FA , TT, LL) # TT(¬[?B FA ]h!B FA i>, A)
W
implies M|A
¬[?B FA ]h!B FA i> , (FA , TT, LL) # KC FA
⇐⇒ M, (FA , TT, LL) 1(B,FA ) h!B FA i>
W
implies M|A
¬[?B FA ]h!B FA i> , (FA , TT, LL) # KC FA
⇐⇒ M, (FA , TT, LL) 1# LL(FA , B)
W
implies M|A
¬[?B FA ]h!B FA i> , (FA , TT, LL) # KC FA
where M|A
¬[?B FA ]h!B FA i> keeps the worlds s in M such that
M, s λ(s, A)(¬[?B FA ]h!B FA i>, A).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Therefore the worlds satisfying one of the following conditions are
kept:
M, , TT, LL ¬[?B FA ]h!B FA i> or M, , LL, TT [?B FA ]h!B FA i>.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Therefore the worlds satisfying one of the following conditions are
kept:
M, , TT, LL ¬[?B FA ]h!B FA i> or M, , LL, TT [?B FA ]h!B FA i>.
Equivalently: M, , TT, LL 1B,FA h!B FA i> or
M, , LL, TT B,FA h!B FA i>.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Therefore the worlds satisfying one of the following conditions are
kept:
M, , TT, LL ¬[?B FA ]h!B FA i> or M, , LL, TT [?B FA ]h!B FA i>.
Equivalently: M, , TT, LL 1B,FA h!B FA i> or
M, , LL, TT B,FA h!B FA i>.
So M|A
¬[?B FA ]h!B FA i> only keeps the worlds (FA , TT, LL) and
(FA , LL, TT), thus
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Therefore the worlds satisfying one of the following conditions are
kept:
M, , TT, LL ¬[?B FA ]h!B FA i> or M, , LL, TT [?B FA ]h!B FA i>.
Equivalently: M, , TT, LL 1B,FA h!B FA i> or
M, , LL, TT B,FA h!B FA i>.
So M|A
¬[?B FA ]h!B FA i> only keeps the worlds (FA , TT, LL) and
(FA , LL, TT), thus
W
M|A
¬[?B FA ]h!B FA i> , (FA , TT, LL) # KC FA .
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
With arbitrary utterances
Definition (Public question language with types and utterances)
Let U be a finite set of utterances, the language PQLTT
U replaces
the announcements !a φ in PQLTT by utterances !a u:
φ ::= > | p | ¬φ | φ ∧ φ | Ka φ | η(a) | [!a u]φ | [?a φ]φ | [!a ]φ
where η ∈ T, u ∈ U and a ∈ G.
[!a u]φ expresses that, if a says u, then φ is true.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Model
A model M for PQLTT
U is a tuple: (S, {∼a | a ∈ G}, V , λ, I )
T
where I : S × Form(PQLTT
U ) × U → Form(PQLTU ) is a function
and I (s, φ, u) is the interpretation of an answer u on world s
given the question φ.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Model
A model M for PQLTT
U is a tuple: (S, {∼a | a ∈ G}, V , λ, I )
T
where I : S × Form(PQLTT
U ) × U → Form(PQLTU ) is a function
and I (s, φ, u) is the interpretation of an answer u on world s
given the question φ.
For example, if u = {yes, no}, we can define a function I
corresponding to the usual interpretation of yes and no as
answers to questions: I (s, φ, yes) = φ and I (s, φ, no) = ¬φ for
each s and each φ.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Model
A model M for PQLTT
U is a tuple: (S, {∼a | a ∈ G}, V , λ, I )
T
where I : S × Form(PQLTT
U ) × U → Form(PQLTU ) is a function
and I (s, φ, u) is the interpretation of an answer u on world s
given the question φ.
For example, if u = {yes, no}, we can define a function I
corresponding to the usual interpretation of yes and no as
answers to questions: I (s, φ, yes) = φ and I (s, φ, no) = ¬φ for
each s and each φ.
T
The semantics of PQLTT
U is mostly the same as that of PQLT ,
except for the formulas involving utterances, which depend on
the interpretation function.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Definition (Semantics for PQLTT
U)
The semantics of PQLTT
U formulas on the model
M = (S, {∼a | a ∈ G}, V , λ, I ) is defined exactly as the semantics
of PQLTT w.r.t. µ ∈ {#} ∪ G × Form(PQLTT
U ), except for the
following clauses:
M, s µ [!a u]φ
⇔
M, s µ [!a ]φ
⇔
µ = (a, χ) and
I (s, χ, u) = ±χ and
implies M|aχ,u , s # φ
M, s # λ(s, a)(I (s, χ, u), a)
for all u ∈ U : M, s µ [!a u]φ
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
where M|aχ,u is defined as (S 0 , {∼0a | a ∈ G}, V 0 , λ0 , I 0 ) where:
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
where M|aχ,u is defined as (S 0 , {∼0a | a ∈ G}, V 0 , λ0 , I 0 ) where:
S 0 = {t | t ∈ S and M, t # λ(t, a)(I (t, χ, u), a)}
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
where M|aχ,u is defined as (S 0 , {∼0a | a ∈ G}, V 0 , λ0 , I 0 ) where:
S 0 = {t | t ∈ S and M, t # λ(t, a)(I (t, χ, u), a)}
0
For each a ∈ G, t ∈ S 0 , u ∈ U, φ ∈ PQLTT
U : ∼a =∼a |S 0 ×S 0 ,
0
0
0
V (t) = V (t), λ (t) = λ(t), and I (t, φ, u) = I (t, φ, u).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
where M|aχ,u is defined as (S 0 , {∼0a | a ∈ G}, V 0 , λ0 , I 0 ) where:
S 0 = {t | t ∈ S and M, t # λ(t, a)(I (t, χ, u), a)}
0
For each a ∈ G, t ∈ S 0 , u ∈ U, φ ∈ PQLTT
U : ∼a =∼a |S 0 ×S 0 ,
0
0
0
V (t) = V (t), λ (t) = λ(t), and I (t, φ, u) = I (t, φ, u).
We say that M|aχ,u is defined if the set
{t | t ∈ S and M, t # λ(t, a)(I (t, χ, u), a)} is not empty.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Semantics
where M|aχ,u is defined as (S 0 , {∼0a | a ∈ G}, V 0 , λ0 , I 0 ) where:
S 0 = {t | t ∈ S and M, t # λ(t, a)(I (t, χ, u), a)}
0
For each a ∈ G, t ∈ S 0 , u ∈ U, φ ∈ PQLTT
U : ∼a =∼a |S 0 ×S 0 ,
0
0
0
V (t) = V (t), λ (t) = λ(t), and I (t, φ, u) = I (t, φ, u).
We say that M|aχ,u is defined if the set
{t | t ∈ S and M, t # λ(t, a)(I (t, χ, u), a)} is not empty.
It is easy to see that:
M, s µ h!a iφ
⇔ M, s µ ¬[!a ]¬φ
⇔ there exists a u ∈ U : M, s µ h!a uiφ
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Formalizing HLPE
Definition (Questioning strategy)
A questioning strategy π w.r.t. PQLTT
U is a tuple (Q, F , r , δ, L)
where
Q is a non-empty finite set of question states and r ∈ Q is
the initial state,
F is a non-empty finite set of final states such that F ∩ Q = ∅,
δ : Q × U → Q ∪ F is a transition function,
L : Q → G × Form(PQLTT
U ) essentially assigns to each
question state a question ?a φ expressible in PQLTT
U (formally
represented as a pair (a, φ)).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Example
Example
Given G = {A, B, C }, T = {TT, LL, LT} and U = {ja, da}, a simple
questioning strategy π: ‘asking them one by one if they are
bluffers’ is illustrated as follows:
r :?A LT(A)
ja
.
q1 :?B LT(B)
da 0
ja
da
.
0 q2 :?C LT(C )
ja
da
*4 f
where r :?A LT(A) means L(r ) = (A, LT(A)), similarly for other
nodes.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Example
Example
Given G = {A, B, C }, T = {TT, LL, LT} and U = {ja, da}, a simple
questioning strategy π: ‘asking them one by one if they are
bluffers’ is illustrated as follows:
r :?A LT(A)
ja
.
q1 :?B LT(B)
da 0
ja
da
.
0 q2 :?C LT(C )
ja
da
*4 f
where r :?A LT(A) means L(r ) = (A, LT(A)), similarly for other
nodes.
Let Seq(π) be all the potential question-answer sequences of π,
namely,
u
u
Seq(π) = {?a1 φ1 !a1 u1 . . .?an φn !an un | q0 →1 q1 · · · →n qn+1 ∈ P(π),
∀i : ai = LG (qi ), φi = LΦ (qi )}.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Formal definitions: puzzle and solution
T
A puzzle of PQLTT
U is a pair consisting of a PQLTU model and
a PQLTT
U formula as the goal: (M, φ). Intuitively, a puzzle
asks for a questioning strategy π such that φ is guaranteed
after executing π.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Formal definitions: puzzle and solution
T
A puzzle of PQLTT
U is a pair consisting of a PQLTU model and
a PQLTT
U formula as the goal: (M, φ). Intuitively, a puzzle
asks for a questioning strategy π such that φ is guaranteed
after executing π.
A questioning strategy π is a solution to a puzzle (M, φ) if
for all ?a1 φ1 !a1 u1 · · ·?an φn !an un ∈ Seq(π):
M [?a1 φ1 ](h!a1 i> ∧ [!a1 u1 ][?a2 φ2 ](h!a2 i> ∧ [!a2 u2 ][?a3 φ3 ](. . . [?an φn ]
(h!an i> ∧ [!an un ]φ)..)))
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Formal definitions: puzzle and solution
T
A puzzle of PQLTT
U is a pair consisting of a PQLTU model and
a PQLTT
U formula as the goal: (M, φ). Intuitively, a puzzle
asks for a questioning strategy π such that φ is guaranteed
after executing π.
A questioning strategy π is a solution to a puzzle (M, φ) if
for all ?a1 φ1 !a1 u1 · · ·?an φn !an un ∈ Seq(π):
M [?a1 φ1 ](h!a1 i> ∧ [!a1 u1 ][?a2 φ2 ](h!a2 i> ∧ [!a2 u2 ][?a3 φ3 ](. . . [?an φn ]
(h!an i> ∧ [!an un ]φ)..)))
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Boolos provides the following guidelines in (Boolos, 1996):
B1 Each god may get asked more than one question;
B2 Later questions may depend on previous ones and
their answers;
B3 Whether Random speaks truly or not depends on the
flip of a coin in his mind: if the coin comes down
heads, he speaks truly; if tails, falsely.
B4 Random will always answer ‘da’ or ‘ja’.
Rabern and Rabern (2008) first noticed that B3 may trivialize the
puzzle, and therefore proposed an alternative assumption B3’ that
we will follow in this work:
B3’ Whether Random answers ‘ja’ or ‘da’ depends on the
coin flip in his mind: if it comes down heads, he
answers ‘ja’; if tails, he answers ‘da’.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
To formalize the puzzle precisely, let us list the implicit (epistemic)
assumptions:
E0 A, B, and C are of the types in T = {TT, LL, LT} and
this is common knowledge (to all of the agents
including the questioner D).
E1 A, B, and C are of different types and this is
common knowledge.
E2 A, B, and C know each other’s types and this is
common knowledge.
E3 A, B, and C know the meaning of ‘da’ and ‘ja’ and
this is common knowledge.
E4 D does not know the types of A, B, C and this is
common knowledge.
E5 D does not know the exact meanings of ‘da’ and ‘ja’
but he knows that one means ‘yes’ and the other
means ‘no’, and this is common knowledge.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Moreover, we assume the following:
Q1 All questions are asked and answered publicly.
Q2 D does not mention himself in the questions.
LS We only consider solutions of length less than 4.
Q1 and Q2 may look unnecessary but they do play a role in the
analysis of HLPE within our framework: we only consider public
questions and answers in our technical preparations, and Q2 will
simplify our discussion.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
We fix U = {ja, da}, T = {TT, LL, LT} and G = {A, B, C , D}.
According to the assumptions E0-E5 we can build model M0 :
TT, LL, LT, JA
D
TT, LT, LL, JA
D
LT, LL, TT, JA
D
D
D
D
D
LL, TT, LT, JA
D
TT, LT, LL, DA
D
LT, LL, TT, DA
D
D
D
LT, TT, LL, JA
D
LL, LT, TT, JA
D
TT, LL, LT, DA
D
LT, TT, LL, DA
D
LL, LT, TT, DA
D
LL, TT, LT, DA
where JA at a world s denotes the interpretation that ja means yes,
and da means no at world s, i.e., I (s, φ, ja) = φ and
I (s, φ, da) = ¬φ for any PQLTT
U formula φ. Similarly, DA at world s
denotes that I (s, φ, ja) = ¬φ and I (s, φ, da) = φ for any φ.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Technical results
All the assumptions E0-E5 can be formalized and checked on
M0 . This shows that the model M0 complies with our
assumptions.
We can verify the existing solution proposed by (Rabern and
Rabern, 2008) by formally proving the following result (Embedded question lemma):
Let E ∗ be the function that takes a question q to
the question ‘If you were asked whether q would you
say “ja?”’. When either True or False are asked
E ∗ (q), a response of ‘ja’ indicates that the correct
answer to q is affirmative and a response of ‘da’
indicates that the correct answer to q is negative.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Solution
Rabern and Rabern (2008) proposed a three-step solution as
follows (we use Ea∗ (φ) as the short hand for formula [?a φ]h!a jai>):
r :?B EB∗ (LT(A))
ja
da
.
*
?C EC∗ (TT(C ))
ja
da
.
ja *
∗
0 ?C EC (LT(B)) da 4 @ f
ja
?A EA∗ (TT(A))
ja
da
0
.
da
?A EA∗ (LT(B))
In words, D first asks B whether A is a bluffer. Then depending on
the answer, either A or C must be a non-bluffer. Thus D can then
ask the non-bluffer about his own type and others’ types.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Solution
Rabern and Rabern (2008) proposed a three-step solution as
follows (we use Ea∗ (φ) as the short hand for formula [?a φ]h!a jai>):
r :?B EB∗ (LT(A))
ja
da
.
*
?C EC∗ (TT(C ))
ja
da
.
ja *
∗
0 ?C EC (LT(B)) da 4 @ f
ja
?A EA∗ (TT(A))
ja
da
0
.
da
?A EA∗ (LT(B))
In words, D first asks B whether A is a bluffer. Then depending on
the answer, either A or C must be a non-bluffer. Thus D can then
ask the non-bluffer about his own type and others’ types.
We can veryfy the above questioning strategy formally in our
framework.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Solution
Rabern and Rabern (2008) proposed a three-step solution as
follows (we use Ea∗ (φ) as the short hand for formula [?a φ]h!a jai>):
r :?B EB∗ (LT(A))
ja
da
.
*
?C EC∗ (TT(C ))
ja
da
.
ja *
∗
0 ?C EC (LT(B)) da 4 @ f
ja
?A EA∗ (TT(A))
ja
da
0
.
da
?A EA∗ (LT(B))
In words, D first asks B whether A is a bluffer. Then depending on
the answer, either A or C must be a non-bluffer. Thus D can then
ask the non-bluffer about his own type and others’ types.
We can veryfy the above questioning strategy formally in our
framework.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Possible new puzzles: alternative to E2
E2 A, B, and C know each other’s types and this is common
knowledge.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Possible new puzzles: alternative to E2
E2 A, B, and C know each other’s types and this is common
knowledge.
It is commonly known (to A, B, C , and D) that agents A, B
and C only know their own types.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Possible new puzzles: alternative to E2
E2 A, B, and C know each other’s types and this is common
knowledge.
It is commonly known (to A, B, C , and D) that agents A, B
and C only know their own types.
It is commonly known that A knows everyone’s type, but B
and C only know their own types.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Possible new puzzles: alternative to E2
E2 A, B, and C know each other’s types and this is common
knowledge.
It is commonly known (to A, B, C , and D) that agents A, B
and C only know their own types.
It is commonly known that A knows everyone’s type, but B
and C only know their own types.
It is commonly known that a bluffer knows everyone’s type,
but truth tellers and liars only know their own types.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Possible new puzzles: alternative to E2
E2 A, B, and C know each other’s types and this is common
knowledge.
It is commonly known (to A, B, C , and D) that agents A, B
and C only know their own types.
It is commonly known that A knows everyone’s type, but B
and C only know their own types.
It is commonly known that a bluffer knows everyone’s type,
but truth tellers and liars only know their own types.
A knows everyone’s type, but B and C only know their own
types and doubt whether A indeed knows their types. D is not
sure whether any of the three know all the types of each other.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
A simple example
Example (HLPE with ignorance)
A (subjective) liar, a (subjective) truth teller and a bluffer are
living on an island. They know their own types but do not know
others’ types. Moreover, it is commonly known that they are of
different types. They understand English but can only answer
questions in their own language, in which the words for yes and no
are da and ja, in some order. Now the question is: can you
determine their types by asking questions such that they are always
able to answer ja or da.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Model M1
STT, SLL, LT, JA
A,D
STT, LT, SLL, JA
C ,D
LT, STT, SLL, JA
B,D
B,D
B,D
LT, SLL, STT, JA
A,D
C ,D
C ,D SLL, LT, STT, JA A,D
SLL, STT, LT, JA
D
D
D
STT, SLL, LT, DA
A,D
STT, LT, SLL, DA
C ,D
LT, STT, SLL, DA
B,D
B,D
B,D
LT, SLL, STT, DA
C ,D
A,D
C ,D SLL, LT, STT, DA A,D
SLL, STT, LT, DA
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Technical result
Let θ(a) be the formula KD SLL(a) ∨ KD STT(a) ∨ KD LT(a)
and θ = θ(A) ∧ θ(B) ∧ θ(C ). The puzzle is then formalized as
(M1 , θ).
We have proved that there is no solution to (M1 , θ).
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Summary
Motivated by logic puzzles and the current status of PAL, we
introduced the following five logical languages:
ELT Epistemic language (with type formulas),
PALT = ELT + [!φ] Public announcement language (with type
formulas),
PALTT = ELT + [!a φ] Public announcement language with
types,
PQLTT = ELT + [!a φ] + [?a φ] + [!a ] Public question language
with types,
T
PQLTT
U = EL + [!a u] + [?a φ] + [!a ] Public question language
with types and arbitrary utterances.
In PALTT , PQLTT , and PQLTT
U , who says what is important due to
the types of speakers.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
We studied several variations of the Knight and Knave puzzles
within the logical frameworks that we developed.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
We studied several variations of the Knight and Knave puzzles
within the logical frameworks that we developed.
We formalized HLPE and verified a classic solution. It was
also shown that puzzles involving only objective truth tellers
and liars are usually simpler than those with subjective types
and epistemic uncertainties.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
We studied several variations of the Knight and Knave puzzles
within the logical frameworks that we developed.
We formalized HLPE and verified a classic solution. It was
also shown that puzzles involving only objective truth tellers
and liars are usually simpler than those with subjective types
and epistemic uncertainties.
We proposed a harder puzzle in which the gods in the original
HLPE are replaced by humans who do not know each other’s
types. We showed that there is no solution to it.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
We studied several variations of the Knight and Knave puzzles
within the logical frameworks that we developed.
We formalized HLPE and verified a classic solution. It was
also shown that puzzles involving only objective truth tellers
and liars are usually simpler than those with subjective types
and epistemic uncertainties.
We proposed a harder puzzle in which the gods in the original
HLPE are replaced by humans who do not know each other’s
types. We showed that there is no solution to it.
For more details of the proofs, see our paper:
Fenrong Liu and Yanjing Wang, Reasoning about Agent Types and
the hardest logic puzzle ever, to appear in Minds and Machines.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Future Work
From knowledge to belief.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Future Work
From knowledge to belief.
Richer agent types.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
The End
Thanks!
Contact me: [email protected]
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
Boolos, G. (1996). The hardest logic puzzle ever. The Harvard
Review of Philosophy, 6:62–65.
Hoshi, T. (2009). Epistemic Dynamics and Protocol Information.
PhD thesis, Stanford.
Liu, F. (2009). Diversity of agents and their interaction. Journal of
Logic, Language and Information, 18(1):23–53.
Paul, S. and Ramanujam, R. (2011). Neighbourhood structure in
large games. In Proceedings of TARK XIII, TARK XIII, pages
121–130, New York, NY, USA. ACM.
Rabern, B. and Rabern, L. (2008). A simple solution to the
hardest logic puzzle ever. Analysis, 68:105–112.
Uzquiano, G. (2010). How to solve the hardest logic puzzle ever in
two questions. Analysis, 70:39–44.
van Ditmarsch, H. (2011). The Ditmarsch tale of
wonders—dynamics of lying. Manuscript.
van Ditmarsch, H., van Eijck, J., Sietsma, F., and Wang, Y.
(2011). On the logic of lying. In van Eijck, J. and Verbrugge,
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types
Motivation Type-based dynamic epistemic logics Agent type in questions and answers Handling arbitrary utterances Formaliz
R., editors, Games, Actions and Social Software, pages 41–72.
Springer.
Wang, Y. (2010). Epistemic Modelling and Protocol Dynamics.
PhD thesis, ILLC University of Amsterdam and CWI.
Wheeler, G. R. and Barahona, P. (2012). Why the hardest logic
puzzle ever cannot be solved in less than three questions.
Journal of Philosophical Logic, 41(2):493–503.
Wintein, S. (2011). On the behavior of true and false. Minds and
Machines, 22(1):1–24.
Fenrong Liu Department of Philosophy, Tsinghua University
Reasoning about Agent Types