Modeling MAS
• model the agents’ mental attitudes
– beliefs: epistemic logic, doxastic logic
– goals, preferences: logic of desire, intention, preference
– hopes, fears, regrets, …
Logics for Agency and MultiAgent Systems
• model change
–
–
–
–
Introduction to Modal Logic
Jan Broersen
Andreas Herzig
Nicolas Troquard
events: dynamic logic PDL, dynamic epistemic logics DEL
abilities: coalition logic CL, alternating-time temporal logic ATL
actions, agency: STIT theory, ‘bringing it about’ logics
belief updates, belief revisions: KM theory, AGM theory
• model interaction and communication
–
–
–
–
ESSLLI’07, Dublin, Monday, August 13, 2007
group ability, group beliefs: ATL, common knowledge
commitments, norms: deontic logic
speech acts: speech act theory
protocols: dialogue games, …
Logics for Agency and
Multi-Agent Systems
Modeling MAS
• model the agents’ mental attitudes
– beliefs: epistemic logic, doxastic logic
– goals, preferences: logic of desire, intention, preference
– hopes, fears, regrets, …
– beliefs: epistemic logic, doxastic logic
– goals, preferences: logic of preferences
– hopes, fears, regrets, …
• model change
• model change
events: dynamic logic PDL, dynamic epistemic logics DEL
abilities: coalition logic CL, alternating-time temporal logic ATL
actions, agency: STIT theory, ‘bringing it about’ logics
belief updates, belief revisions: KM theory, AGM theory
–
–
–
–
• model interaction and communication
–
–
–
–
Jan Broersen, Andreas Herzig,
Nicolas Troquard
events: dynamic logic PDL, dynamic epistemic logics DEL
abilities: coalition logic CL, alternating-time temporal logic ATL
actions, agency: STIT theory, ‘bringing it about’ logics
belief updates, belief revisions: KM theory, AGM theory
• model interaction and communication
group ability, group beliefs: ATL, common knowledge
commitments, norms: deontic logic
speech acts: speech act theory
protocols: dialogue games, …
Logics for Agency and
Multi-Agent Systems
– speech acts: speech act theory
– commitments, obligations: deontic logic
– protocols: dialogue games, …
3
Logics for Agency and
Multi-Agent Systems
Overview of today’s introduction
1. What is logic, and what do we need it for in
multi-agent systems?
2. The basics of modal logic.
1.
2.
3.
4.
5.
6.
7.
3. Applications of modal logic to some reasoning
domains relevant for multi-agent systems
Jan Broersen, Andreas Herzig,
Nicolas Troquard
Jan Broersen, Andreas Herzig,
Nicolas Troquard
4
What is logic?
• Logic is the theory of valid principles of
inference, i.e. the theory that says what can be
derived from information we assume to be true.
• Thus, logic says nothing about how to establish
truth. Logic is only concerned with valid relations
between possible truths.
• Example: the logic principle ¬(φ ∧ ¬φ) says
nothing about whether p and ¬p are true or not,
but only that they cannot be true at the same
time.
Basic notions of logic
Normal multi-modal logics
Expressivity
Soundness and completeness
Combining logics: products
Decidability and complexity
Weak modal logics
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2
Modeling MAS
• model the agents’ mental attitudes
–
–
–
–
Jan Broersen, Andreas Herzig,
Nicolas Troquard
5
Logics for Agency and
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Jan Broersen, Andreas Herzig,
Nicolas Troquard
6
1
Why is Logic relevant for MAS?
(reason 1)
1.
•
•
•
Why is Logic relevant for MAS?
(reason 2)
Intelligent agents need to represent their environment
in order to be able to act intelligently ⇒ knowledge
representation
Assumption (McCarthy 1958, who also suggested the
term ‘Artificial Intelligence’): an efficient and intuitive
representation can be found using logic.
The idea: agents ‘only’ need to store basic facts and
assumptions, and can derive the rest through logic
inference.
For knowledge representation, we typically need
logics of ‘common sense’, concept description,
conditionality, decisions, etc.
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2.
•
•
•
7
What is ‘a’ logic, formally?
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Basic logic notions: semantics
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Logical consequence / deduction
• Propositional Logics (PL): a model is an
assignment of truth values to atomic
propositions
• Formulas are true on a model according to
truth-functionality of the connectives.
• A formula is satisfiable if there is a model
for it (a model that makes it true)
• A formula is valid if it is true for all possible
models
• If we want to use logic for knowledge representation we
need to know how to derive something!
• How can a subset of some language be a formalization
of reasoning patterns?
• Does the logical connective ‘→’ define logical
consequence? No!
• Deduction for PL:
ψ follows from φ, notation φ ` ψ, if addition of φ to the
axioms enables derivation of ψ
• Logical consequence for PL:
ψ follows from φ, notation φ ² ψ, if all models for φ are
models for ψ
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• Axiom schema’s, like φ ∨ ¬φ describe basic theorems of
the logic
• In a schema, we can perform uniform substitutions / the
φ are meta variables over the language
• Rules, like modus ponens, describe what theorems can
be derived from which other theorems
• A formula / schema whose negation is not derivable is
called ‘consistent’
Note: in the early days, (modal) logic was an axiomatic
enterprise. Model theory had to be developed to prove
logics are consistent!
– Semantically, using model theory
– Syntactically, using axiomatic systems
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Basic logic notions: axiomatics
• Formally a logic is a subset of some language
(Later: how can this be regarded a formalization of
inference patterns?)
• Propositional logic is a subset of the language
φ, ψ, ... := p | ¬φ | φ ∧ ψ
• Other logical connectives are introduced as
abbreviations
• The subset (i.e., logic) captures the logical invariants of
the reasoning domain
• The subset (i.e., logic) can be described in at least two
ways:
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In order to build agents we need to have a theory
about how we want them to function ⇒ agent / system
specification
Logic as a way of representing our ‘engineering’
knowledge.
If agents need to represent knowledge about other
agents, we can use these logics also for knowledge
representation.
For agent specification, we typically need logics of
action, time, knowledge, belief, norms, desires,
intentions, etc. ⇒ C&L, BDI and KARO
11
Jan Broersen, Andreas Herzig,
Nicolas Troquard
12
2
What is Modal Logic?
The language
What is Modal Logic?
Axiomatic answer
Given a set of proposition symbols P, and p
ranging over P, the well-formed formulas φ, ψ, ...
of the language LM are generated by the ‘BNF’:
At least the axioms:
– All (or enough) propositional axioms are in K
– ¤φ ∧ ¤(φ → ψ) → ¤ψ is in K
At least the rules:
φ, ψ, ... := p | ¬φ | φ ∧ ψ | φ ∨ ψ | ¤iφ | ♦iφ
– necessitation: if φ in K, derive that ¤φ in K.
– modus ponens: if φ in K, and φ → ψ in K, derive that ψ in K.
Intuitively: ¤iφ is ‘necessarily φ’
♦iφ is ‘possibly φ’
• Modal logics that subsume these, are called normal
• We denote that some formula φ is derivable in K by: `K φ
Multi-modal due to the indexes ‘i’
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What is Modal Logic?
Semantic answer
– binary relation between moments
– truth is relative to what counts as ‘the present’
• Action:
– binary relation between (system, agent) states
– truth is relative to what counts as ‘the current state of
affairs’
• Knowledge:
– binary relation between epistemically possible worlds
– truth is relative to what counts as ‘the current
epistemic state’
15
Logics for Agency and
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The model theory of modal logic:
Kripke Models
16
• Propositional logic connectives obey the standard
truth conditions
• The truth condition for M,w ² ¤φ is:
for all w’ such that (w,w’) ∈ R, it holds that M,w’ ² φ
• The truth condition for M,w ² ♦φ is:
there is a w’ such that (w,w’) ∈ R, and M,w’ ² φ
A model is a frame with in addition a
• valuation V: ATM → 2W of atomic propositions
For any formula φ of a modal language LM, truth of
φ in a world w ∈ W is denoted M,w ² φ
Jan Broersen, Andreas Herzig,
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Semantics
A Kripke frame consists of a
• (possibly infinite) set W of possible worlds
• a relation R (i.e. a possibly infinite set of tuples
(s,s’)) over W × W
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14
• Time:
– that fit well with an abstract representation
consisting of a binary relation over ‘worlds’.
– where ‘truth’ is defined relative to, a ‘current’
world ⇒ In ‘Modal Logic’ by Blackburn, de
Rijke, Venema (2001) this is called ‘the local
perspective’
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Typical application domains of
modal logics
• Modal logic: the theory of valid principles
of inference for domains,
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Intuitively:
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¤φ is ‘necessarily φ’
♦φ is ‘possibly φ’
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3
Some (non-)validities of basic
modal logic
Semantics
• A formula is true on a Kripke model if and only if it is true
in all states of that model
• A formula is valid on a Kripke frame if and only if it is
true on all models corresponding to that frame
• A formula is valid (notation: ² φ) with respect to a class
of frames if and only if it is valid on all frames in the
class.
• The set of validities of a class of frames is called ‘the
logic’ of the class of frames.
• K is the logic of the class of all Kripke frames.
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Logics over particular classes of
frames
All propositional validities (modal logic
subsumes propositional logic)
² ¤φ ∧ ¤ψ → ¤(φ ∧ ψ)
² ¤(φ ∧ ψ) → ¤φ ∧ ¤ψ
² ¤φ ∨ ¤ψ → ¤(φ ∨ ψ)
2 ¤(φ ∨ ψ) → ¤φ ∨ ¤ψ
² ¤(φ → ψ) → (¤φ → ¤ψ)
² ¤φ ∧ ¤(φ → ψ) → ¤ψ
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Well-known possible first-order
properties of frames
Now we are going to consider logics over subclasses of the general class of all Kripke models
⇒ logics stronger than K
We strengthen K by considering the logics over
models generated by frames with a certain (firstor second-order) property.
In stronger logics there are more validities,
formulas have more logical consequences.
Extreme case: the logic of the empty Kripke frame
is the strongest possible logic, where everything
is a validity, and everything is derivable.
• F = (S, R) is reflexive if ∀s ∈ S: (s, s) ∈ R.
• F = (S, R) is transitive if
∀s, t, u ∈ S: (s, t) ∈ R & (t, u) ∈ R ⇒ (s, u) ∈ R.
• F = (S, R) is symmetrical if
∀s, t ∈ S: (s, t) ∈ R ⇒ (t, s) ∈ R.
• F = (S, R) is euclidean if
∀s, t, u ∈ S: (s, t) ∈ R & (s, u) ∈ R ⇒ (t, u) ∈ R.
• F = (S, R) is serial if ∀s ∈ S, ∃ t ∈ S: (s, t) ∈ R.
• F = (S, R) is an equivalence relation if R is reflexive,
transitive and symmetrical
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Names for classes of models
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Extra validities
• T is the class of all reflexive Kripke models.
• S4 is the class of all reflexive-transitive
Kripke models.
• S5 is the class of all Kripke models with
accessibility relations that are equivalence
relations.
• KD45 is the class of all Kripke models with
serial, transitive and euclidean accessibility
relations.
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• Over the model class T we have e.g., the
extra validity: ϕ → ϕ
• Over S4 we have extra e.g.: ϕ → ϕ and
ϕ→ ϕ
• Over S5 we have extra e.g.: ϕ → ϕ and
ϕ → ϕ and ¬ ϕ → ¬ ϕ
• Over KD45 we have extra e.g.: ¬ ⊥ and
ϕ → ϕ and ¬ ϕ → ¬ ϕ
23
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4
Dependencies of frame properties
• R is symmetrical and transitive ⇒ R is euclidean.
• R is reflexive ⇒ R is serial.
• R is symmetrical, transitive and serial ⇔
R is reflexive and euclidean ⇔
R is an equivalence relation.
Two modal consequence relations, a local and a
global one:
• φ ²L ψ if ∀M,w: M,w ² φ implies M,w ² ψ
• φ ²G ψ if ∀M: M ² φ implies M ² ψ
and more are possible...
• we have: φ ²L ψ ⇒ φ ²G ψ
the global one is thus ‘stronger’
• the local one is most used or assumed
These give us relations between the logics over
the mentioned classes: inclusions of classes
give inclusions of logics.
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Differences between ²L and ²M
weakest:
stronger:
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27
Expressivity
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Jan Broersen, Andreas Herzig,
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Modal logic deduction
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Bisimulation invariance
A world w1 of model M1 is bisimilar to a world w2 in a model
M2 if:
1. V1(w1) = V2(w2)
2. ∀ w’1 ∈ W1: w1R1w’1 implies ∃ w’2 ∈ W2: w2R2w’2 and
w’1 and w’2 are bisimilar
3. ∀ w’2 ∈ W2: w2R2w’2 implies ∃ w’1 ∈ W1: w1R1w’1 and
w’2 and w’1 are bisimilar
Expressivity is about what semantic entities a
language is able to distinguish.
• On the level of worlds: bisimulation invariance:
modal formulas cannot distinguish between
bisimilar worlds.
• On the level of frames: correspondence theory:
what properties of frames can be characterized
by a modal logic?
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• Again, we might want to say: ψ follows from φ if
addition of φ to the axioms enables derivation of
ψ
• But, this would correspond to global
consequence, since, for instance p ` ¤p, by the
necessitation rule.
• For local consequence, which is most used, we
have to constrain application of the necessitation
rule to theorems only.
φ ²L ψ
φ ²G ψ
• For instance, we have: φ 2L ¤φ,
but we do have: φ ²G ¤φ
• And we have: φ ²L ψ implies ² φ → ψ (the
deduction property, Hilbert 1930)
but: φ ²G ψ does not imply ² φ → ψ
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Two semantic modal
consequence relations
Intuitively, worlds are bisimilar if from these worlds, looking
towards the accessible worlds, we see the same
possible continuations, modulo trivial copies.
29
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5
Correspondence theory
Correspondences
Example1: the validities ♦♦φ → ♦φ correspond to the
class of transitive frames (C = ∀ s,t,u: sRt ∧ tRu → sRu)
Example2: the validities φ → ♦φ correspond to the class of
reflexive frames (C = ∀ w: wRw)
Example3: the validities φ → ¤♦φ correspond to the class
of symmetric frames (C = ∀ s,t: sRt → tRs)
We already encountered: which modal validities
are generated by a given frame class? i.e.,
which validities φ hold on all frames with (first
order) property C?
But, what about the other way round? Which are
the frame classes that give rise to a given
validity schema? i.e., which classes of frames
(with first order property C) ‘satisfy’ the
validities φ?
If we have both ways, then the schema φ
corresponds to the FO property C on frames.
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Note that these formulas say nothing about similar
correspondences with classes of models: it is easy to
find a model that obeys φ → ♦φ, and that is not
reflexive.
31
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Some facts on correspondences
• Not all validity schema’s correspond to a first order
property of frames.
Examples:
(1) Löb’s axiom ¤(¤ φ → φ) → ¤φ
Corresponds with ‘R is transitive and well-founded’
(2) Mc Kinsey’s axiom: ♦φ → ♦ φ
• Not all first order properties of frames correspond
to a modal validity schema.
Example: irreflexivity C = ∀w: ¬(wRw)
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Soundness and Completeness
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Weak completeness
for modal logics
Soundness
•
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Soundness: if ` φ then ² φ
•
Weak completeness: if ² φ then ` φ
Equivalent formulations of weak completeness:
How do we prove soundness?
1.
2.
Prove that:
1. axioms are validities
2. rules preserve validity
3.
4.
if 0 φ then 2 φ
if ¬φ consistent then ∃ M,w : M,w 2 φ
(recall: consistency of φ is defined as 0 ¬φ)
if ¬φ consistent then ∃ M,w : M,w ² ¬φ
if φ consistent then ∃ M,w : M,w ² φ
Which is often simple.
So, we need to prove that for every formula consistency
implies satisfiability.
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6
Completeness of logical
consequence
Strong completeness
for modal logics
•
• Weak completeness is only with respect to
validities. What about logical consequence?
• if Φ ²L φ then Φ ` φ ?
• For the weakest notion of logical consequence
(²L), we have: ²L φ → ψ iff φ ²L ψ.
• It follows that for weak consequence and for
finite sets Φ we have that weak completeness
(² φ iff ` φ) implies: if Φ ²L φ then Φ ` φ
• For the stronger notions of consequence we
need stronger deductive systems!
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Equivalent formulations of strong completeness:
1. if Φ 0 φ then Φ 2 φ
2. if Φ ∪ {¬φ} consistent then ∃ M,w : M,w ² Φ and M,w 2 φ
if Φ ∪ {¬φ} consistent then ∃ M,w : M,w ² Φ ∪ {¬φ}
3. if Ψ consistent then ∃ M,w : M,w ² Ψ
So, we need to prove that for every arbitrary set of formulas
consistency implies satisfiability.
37
Proving strong completeness
39
Lindenbaum’s lemma
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1. Φ is S-consistent
2. no set Φ ∪ {φ}, with φ∉ Φ, is S-consistent
Property: all elements of S (i.e. all theorems)
belong to any maximal consistent set.
In particular, all theorems of K belong to all
maximal K-consistent sets.
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Lindenbaum’s lemma (proof)
• Every S-consistent set Φ can be extended to a
maximal S-consistent set Φω.
Proof Lindenbaum’s lemma
Enumerate the formulas of the modal language:
φ1, φ2, ….
Define
Φ0 = Φ
Φn+1 = Φn, if Φn ∪ {φ} is inconsistent,
Φn ∪ {φ}, otherwise
Φω = Υn≥0 Φn
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A set of formulas Φ is maximally S-consistent (is
an ‘MCS’) if:
• Build maximal consistent sets of formulas, using
the language structure and the axiomatization.
• The MCSs become worlds of a ‘canonical’
Kripke model.
• Build an appropriate relational structure such
that any consistent set of formulas is satisfied in
some world of the canonical model.
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Maximal consistent sets
How do we prove strong completeness?
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Strong completeness: if Φ ² φ then Φ ` φ for arbitrary
(thus possibly infinite) sets Φ
Φω is consistent, because it is build by adding
consistent formulas one at the time
(1)
Φω is maximally consistent.
Suppose not.
From definition MCS: there is a formula φm such
that φm ∉ Φω and Φω ∪ {φm} is consistent.
From definition Φm: Φm ∪ {φm} is inconsistent.
From definition Φω: Φm ⊂ Φω
From definition consistency: Φω is inconsistent
⇒ contradiction with (1)
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7
A canonical model
The truth lemma
Mc, sΦ ² φ iff φ ∈ Φ
MSCs are taken as worlds Sc, both atomic valuations πc
and the relation Rc are defined in terms of formula
membership over worlds Sc:
This ensures that every (infinite!) consistent set is
satisfiable.
Proof truth lemma
Easy direction: ⇐
By straightforward induction over formula
structure.
More difficult direction: ⇒
Mc = hSc, πc, Rci with:
Sc = {sΘ | Θ is an MCS}
sΘ ∈ Vc(p) iff p ∈ Θ
Rc = {(sΘ, sΨ) | ∀φ: φ ∈ Θ implies φ ∈ Ψ}
Note: formulas have become semantic entities (worlds)!
Maybe you recall this ‘trick’ from first-order logic
(Herbrand models) or algebra (term models).
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If Mc, sΦ ² φ then φ ∈ Φ
Assume: Mc,sΘ ² ¤ψ
Def: Θ/¤ = {φ | ¤φ ∈ Θ}
Thus, we get Θ/¤ from Θ by first selecting all formulas starting with a
box, and then removing these initial boxes.
Claim 1: Θ/¤ ∪ {¬ψ} is inconsistent
Proof Claim 1
Suppose not, i.e., Θ/¤ ∪ {¬ψ} is consistent
Then by Lindenbaum: there is a MCS Ψ such that Θ/¤ ∪ {¬ψ} ⊆ Ψ
Then, since Θ/¤ ⊆ Θ/¤ ∪ {¬ψ} ⊆ Ψ: (sΘ,sΨ) ∈ Rc.
From ¬ψ ∈ Ψ and the ‘easy direction’: Mc,sΨ ² ¬ψ,
Because (sΘ, sΨ) ∈ Rc: Mc,sΘ ² ¬¤ψ , which contradicts the assumption.
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Sahlqvist forms
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• General Sahlqvist forms have quite complicated
definitions.
• In the course, we only need simple Sahlqvist
forms.
• A positive formula is a modal formula where
every atomic proposition is within the scope of
an even number of negations
• Simple Sahlqvist forms: ♦i ….. ♦k i … lp → B
where B is a positive formula.
Example: add ♦♦φ → ♦φ as an axiom for a sound and
complete deductive system (K4) with respect to
transitive frames.
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From Claim1: some finite subset φ1, ..., φk, ¬ψ of Θ/¤ ∪ {¬ψ} is
inconsistent
So (prop. reasoning): ` φ1 → (φ2 → (...(φk → ψ)...))
So (necessitation rule): ` ¤(φ1 → (φ2 → (...(φk → ψ)...)))
(1)
So (property MCSs): ¤(φ1 → (φ2 → (...(φk → ψ)...))) ∈ Θ
Def: ξ = φ2 → (...(φk → ψ)...))
(2)
Since φ1, ..., φk ∈ Θ/¤ we have: ¤φ1, ..., ¤φk ∈ Θ
(3)
Since K is a theorem: ¤φ1 → (¤(φ1 → ξ) → ¤ξ) ∈ Θ
From (2) and (3), mp, and maximal consistency of Θ:
(4)
(¤(φ1 → ξ) → ¤ξ) ∈ Θ
From (1) and (4), mp, and maximal consistency of Θ: ¤ξ ∈ Θ
Go back to (1) and repeat the argument for φ2 and for φ3 etc. to get
¤ψ ∈ Θ in the end.
So, from the assumption Mc,sΘ ² ¤ψ we have showed ¤ψ ∈ Θ.
Simple Sahlqvist forms (definition)
Sahlqvist determined a general class of formulas C, such
that any schema φ whose instantiations are in C (we
say ‘it is in Sahlqvist form’),
1. corresponds to a first-order definable class of frames,
2. whose logic can be axiomatized by adding φ to the
axiomatization of the basic language.
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Proof continued
We only proof the case where φ is non-modal.
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Jan Broersen, Andreas Herzig,
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The (deductive) systems
T, S4, S5, KD45
•
•
•
•
T = K + axiom { ϕ → ϕ}
S4 = T + axiom { ϕ → ϕ}
S5 = S4 + axiom {¬ ϕ → ¬ ϕ}
KD45 = K + axioms {¬ ⊥ , ϕ →
¬ ϕ → ¬ ϕ}
Combining modal logics: Products
Given two frames (U0,R0) and (U1,R1), their
product frame is (U0 × U1, RH, RV), with
U0 × U1 the Cartesian product of the worlds from
the originating models, and
RH and RV accessibility relations defined by
(1) (w,x)RH(y,z) if and only if wR0y and x=z, and
(2) (w,x)RV(y,z) if and only if xR0z and w=y
ϕ,
So, the basic system K is extended with axioms
corresponding to the first-order properties of the
frames.
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Commutativity: HV φ ↔ VH φ
Confluence: ¬H¬V φ → V¬H¬φ
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Decidability
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Complexities
• the satisfiability problem for all
combinations of D,T,B,4,5 are decidable.
• The product S5 × S5 × S5 is undecidable.
• Recall:
NP ⊂ PSPACE ⊂ EXPTIME ⊂ NEXPTIME
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NP-complete: satisfiability for S5, KD45, K45
PSPACE-complete: satisfiability for K, KD, KT, K4,
KD4, S4, S5_n, LTL
EXPTIME-complete: PDL, Common knowledge,
global consequence in K
NEXPTIME-complete: satisfiability for S5 × S5
Jan Broersen, Andreas Herzig,
Nicolas Troquard
Neighborhood models
Weak modal logic
The well-known property K:
² ¤φ ∧ ¤(φ → ψ) → ¤ψ
S
This property is true in all (K)ripke models. If we want to
get rid of it, we have to leave Kripke semantics.
Some semantics weaker than Kripke semantics:
neighborhood semantics,
queer-world semantics,
selection function semantics, etc.
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53
v
s
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Neighborhood models
A generally applicable semantics for weak modal logics.
M = 〈W, V, N〉,
– where W is again a non-empty set of states and V is a
truth assignment function per state.
– N is a function W → ℘(℘(W) \ { ∅ }) \ { ∅ }: for every
w, N(w) is a non-empty collection of non-empty
subsets (neighborhoods) of W.
– M, w ² h i ϕ ⇔ ∃T ∈ N(w): t ∈ T iff M, t ² ϕ
– This semantics is quite weak...
– Extra conditions on neighborhoods make the
semantics generally applicable. Closure under
reachability of intersecting neighborhoods gives back
¤φ ∧ ¤ψ → ¤(φ ∧ ψ) , etc.
Logics for Agency and
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Jan Broersen, Andreas Herzig,
Nicolas Troquard
‘Applications’…
…of the modal theory to temporal,
action, epistemic and deontic
reasoning
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Temporal logic
• Xφ = next time in future φ
• Fφ = some time in future φ
• Gφ = globally in the future φ
Dynamic logic
( interpreted as a ♦φ )
( interpreted as a ♦φ )
( interpreted as a ¤φ )
• In dynamic logics we have a separate accessibility
relation for each action ⇒ multi-modal logic
• Simple version: accessibility is parameterized with
respect to a set of atomic actions: [a]φ means
‘performing action a guarantees that φ is made true’
• More complex: in dynamic logic, actions have a
structure:
α, β, ... ::= a | φ? | α; β | α + β | α*
The action connectives have their usual relational
interpretation.
much of the previous theory is not applicable anymore...
Possible axioms for ‘constructing’ a modal logic of time:
FFφ → Fφ (transitivity)
Gφ → Xφ
Gφ ↔ φ ∧ XGφ
φ ∧ G(φ → Xφ) → Gφ (not Sahlqvist)
Often: two temporal dimensions: duration and choice
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Bφ = in all worlds I deem possible according to what I
believe, it holds that φ (Bφ interpreted as a ¤φ )
•
•
•
•
(belief consistency)
(positive introspection)
(negative introspection)
Rumsfeld believes himself (abductively):
"I believe what I said yesterday. I don't know
what I said, but I know what I think, and, well, I
assume it's what I said."
KD45 most common for belief
S5 for knowledge
Groups: common knowledge
In what sense can agents know an action?
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Rumsfeld knows tautologies:
"We do know of certain knowledge that he
[Osama Bin Laden] is either in Afghanistan, or in
some other country, or dead. "
Possible properties for ‘constructing’ a modal logic of
belief:
¬(Bφ ∧ B¬φ)
Bφ → BBφ
¬Bφ → B¬Bφ
Jan Broersen, Andreas Herzig,
Nicolas Troquard
The ‘epistemic logic’ of Donald
Rumsfeld
Epistemic logic
•
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Logics for Agency and
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Deontic Logic
STIT Logic
• Obligation: Oφ = in all worlds that are deontically
optimal, it holds that φ (Oφ interpreted as a ¤φ )
• A key property of deontic logic is that
¬φ ∧ Oφ is consistent
• Permission: Pφ ≡def ¬O¬φ
• Prohibition: Fφ ≡def ¬Pφ
• Chisholm: φ → Oψ nor O(φ → ψ) express
conditional obligation ⇒ dyadic deontic logic
• STIT is an acronym for ‘Seeing To It That’
• Temporal logic of agency as opposed to atemporal logics for ‘Bringing It About That’.
• Central idea of all such logics: acting is ensuring
the actual world is among a set of possible
worlds ensured by performing the action.
• [A] φ means group A ensures that φ
• In STIT this is applied to histories: acting is
ensuring a certain set of histories.
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What we saw today
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Outlook
• Tuesday: What agents can do
• Modal logic has a well-developed mathematical
basis.
• Modal logic is applicable to a wide range of
reasoning domains.
• Modal logic has an appealing semantics, has
nice properties, and is widely used.
• The basic modal system is well-suited to base
more expressive logic theories on (agency,
concurrency, communication, ...)
Logics for Agency and
Multi-Agent Systems
Jan Broersen, Andreas Herzig,
Nicolas Troquard
– Logic of ability: Coalition Logic
• Wednesday: What agents do, what agents know
they (can) do
– Logic of agency: STIT
• Thursday: What agents want
– logic of intention: Cohen and Levesque
• Friday: What agents can plan
– ATL, Strategic STIT
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Multi-Agent Systems
Jan Broersen, Andreas Herzig,
Nicolas Troquard
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Thanks for listining
See you tomorrow
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