Modal Logics for Multi-Agent Systems - CiteSeerX

Modal Logics for Multi-Agent Systems

Lecture Overview 1. Classical Logic

Jürgen Dix and Wojtek Jamroga

2. Modal Logic Department of Computer Science Clausthal University of Technology, Germany

3. Action and Time 4. Combining Modalities

European Agent Systems Summer School (Durham 2007)

Jürgen Dix and Wojtek Jamroga ·


5. Modalities in Action

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1. Classical Logic

Chapter 1. Classical Logic

Pointers to Other Courses

John Lloyd: Quantified Modal Logic for Building Agents (Thursday morning) Alessio Lomuscio & Wojciech Penczek: Verification of Multiagent Systems via Model Checking (Thursday afternoon)



1.1 1.2 1.3 1.4 1.5

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Classical Logic Why Logic? Sentential Logic (SL) Sudoku Calculi for SL Wumpus in SL



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1. Classical Logic

1. Classical Logic

In this chapter:

1.1 Why Logic?

We make the case for using formal logic as a representation formalism for reasoning about the world. We present classical propositional logic to introduce the standard language and formalism. The semantics of propositional logic is the basis of all other logics. It can be used to model many things: we show its versatility by modeling Sudoku-puzzles. We also describe two calculi for this logic and consider briefly correctness and completeness issues. We illustrate how to use propositional logic with the well-known Wumpus world.


1. Classical Logic


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1. Classical Logic

1. Why Logic?

Why logic at all?

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1. Why Logic?

Why logic at all?

framework for thinking about systems, makes one realise many assumptions, . . . and then we can: investigate them, accept or reject them, weaken some of them and still use a part of the formal and conceptual machinery,


1. Why Logic?


: check specification against implementation, executable specification,

verification

planning as model checking

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1. Classical Logic

1. Classical Logic

1. Why Logic?

Symbolic AI: Symbolic representation, e.g. sentential or first order logic. Agent as a theorem prover. Traditional: Theory about agents. Implementation as stepwise process (Software Engineering) over many abstractions. Symbolic AI: View the theory itself as executable specification. Internal state: Knowledge Base (KB), often simply called D (database).


1. Classical Logic


2. Sentential Logic (SL)

1.2 Sentential Logic (SL)

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1. Classical Logic


European Agent Systems Summer School (Durham 2007) 10/253


Definition 1.1 (Sentential Logic LSL , Language L ⊆ LSL )

Definition 1.2 (Semantics, Valuation, Model)

The language LSL of propositional (or sentential) logic consists of and >: the constants falsum and verum, p, q, r, x1 , x2 , . . . xn , . . .: a countable set AT of SL-constants, ¬, ∧, ∨, →: the sentential connectives (¬ is unary, all others are binary operators), (, ): the parentheses to help readability. In most cases we consider only a finite set of SL-constants. They define a language L ⊆ LSL . The set of L -formulae FmlL is defined inductively.

A valuation v for a language L ⊆ LSL is a mapping from the set of SL-constants defined by L into the set {true, false} with v() = false, v(>) = true. Each valuation v can be uniquely extended to a function v¯ : FmlL → {true, false} so that: true, if v(p) ¯ = false, v(¬p) ¯ = false, if v(p) ¯ = true. true, if v(ϕ) ¯ = true and v(γ) ¯ = true, v(ϕ ¯ ∧ γ) = false, else true, if v(ϕ) ¯ = true or v(γ) ¯ = true, v(ϕ ¯ ∨ γ) = false, else



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1. Classical Logic


Definition (continued) v(ϕ ¯ → γ) = true, if v(ϕ) ¯ = false or (v(ϕ) ¯ = true and v(γ) ¯ = true), false, else Thus each valuation v uniquely defines a v. ¯ We call v¯ L -structure. A structure determines for each formula if it is true or false. If a formula φ is true in structure v¯ we also say Av is a model of φ. From now on we will speak of models, structures and valuations synonymously. Semantics The process of mapping a set of L -formulae into {true, false} is called semantics.


1. Classical Logic



Definition 1.4 (Consequence Set Cn(T ))

A formula ϕ follows from T if for all models v of T (i.e. v |= T ) also v |= ϕ holds. We write: T |= ϕ. We call CnL (T ) =def {ϕ ∈ FmlL : T |= ϕ}, or simply Cn(T ), the semantic consequence operator.



1. Classical Logic


Definition 1.3 (Validity of a Formula, Tautology) 1

2

3

A formula ϕ ∈ FmlL holds under the valuation v if v(ϕ) ¯ = true. We also write v¯ |= ϕ or simply v |= ϕ. v¯ is a model of ϕ. A theory is a set of formulae: T ⊆ FmlL . v satisfies T if v(ϕ) ¯ = true for all ϕ ⊆ T . We write v |= T . A L -formula ϕ is called L -tautology if for all possible valuations v in L v |= ϕ holds.

From now on we suppress the language L , because it is obvious from context. Nevertheless it needs to be carefully defined. Jürgen Dix and Wojtek Jamroga ·


1. Classical Logic


Lemma 1.5 (Properties of Cn(T )) The semantic consequence operator has the following properties: 1 T -extension: T ⊆ Cn(T ), 0 0 2 Monotony: T ⊆ T ⇒ Cn(T ) ⊆ Cn(T ), 3 Closure: Cn(Cn(T )) = Cn(T ). Lemma 1.6 (ϕ 6∈ Cn(T)) ϕ 6∈ Cn(T ) if and only if


there is a model v with v |= T and v(ϕ) ¯ = false.


1. Classical Logic

1. Classical Logic



Definition 1.7 (MOD(T), Cn(U )) Definition 1.8 (Completeness of a Theory T )

If T ⊆ FmlL then we denote with MOD(T ) the set of all L -structures A which are models of T :

T is called complete if for each formula ϕ ∈ Fml: T |= ϕ or T |= ¬ϕ holds.

MOD(T ) =def {A : A |= T }. If U is a set of models, we consider all those sentences, which are valid in all models of U . We call this set Cn(U ):

Attention: Do not mix up this last condition with the property of a valuation (model) v: each model is complete in the above sense.

Cn(U ) =def {ϕ ∈ FmlL : ∀v ∈ U : v(ϕ) ¯ = true}. MOD is obviously dual to Cn: Cn(MOD(T )) = Cn(T ), MOD(Cn(T )) = MOD(T ).


1. Classical Logic


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1. Classical Logic

3. Sudoku

1.3 Sudoku Definition 1.9 (Consistency of a Theory) T is called consistent if there is a valuation (model) v with v(ϕ) ¯ = true for all ϕ ∈ T . Lemma 1.10 (Ex Falso Quodlibet) T is consistent if and only if Cn(T ) 6= FmlL .





1. Classical Logic

3. Sudoku

1. Classical Logic

3. Sudoku

Can they be solved with sentential logic?

Sudoku Since some time, Sudoku puzzles are becoming quite famous.

Idea: Given a Sudoku-Puzzle S, construct a language LSudoku and a theory TS ⊆ FmlLSudoku such that MOD(TS ) = Solutions of the puzzle S Solution

In fact, we construct a theory TSudoku and for each (partial) instance of a 9 × 9 puzzle S a particular theory TS such that MOD(TSudoku ∪ TS ) = {S : S is a solution of S} Jürgen Dix and Wojtek Jamroga ·




Table 1: A simple Sudoku (S1 )

1. Classical Logic

3. Sudoku

We introduce the following language LSudoku : 1 2 3 4 5 6 7 8 9

einsi, j , 1 ≤ i, j ≤ 9, zweii, j , 1 ≤ i, j ≤ 9, dreii, j , 1 ≤ i, j ≤ 9, vieri, j , 1 ≤ i, j ≤ 9, fuenfi, j , 1 ≤ i, j ≤ 9, sechsi, j , 1 ≤ i, j ≤ 9, siebeni, j , 1 ≤ i, j ≤ 9, achti, j , 1 ≤ i, j ≤ 9, neuni, j , 1 ≤ i, j ≤ 9.

1. Classical Logic

3. Sudoku

We distinguished between the puzzle S and a solution S of it. What is a model (or valuation) in the sense of Definition 1.2?

This completes the language, the syntax. How many symbols are these? Table 2: How to construct a model S ? Jürgen Dix and Wojtek Jamroga ·




1. Classical Logic

3. Sudoku

1. Classical Logic

We have to give our symbols a meaning: the semantics! einsi, j means i, j contains a 1 zweii, j means i, j contains a 2 .. . neuni, j means i, j contains a 9 To be precise: given a 9 × 9 square that is completely filled out, we define our valuation v as follows (for all 1 ≤ i, j ≤ 9). true, if 1 is at position (i, j), v(einsi, j ) = false, else . Jürgen Dix and Wojtek Jamroga ·


1. Classical Logic How does

TS

3. Sudoku

true, if 2 is at position (i, j), false, else .

v(zweii, j ) =



v(dreii, j ) = v(vieri, j ) = etc.

v(neuni, j ) =


Therefore any 9 × 9 square can be seen as a model or valuation with respect to the language LSudoku . Jürgen Dix and Wojtek Jamroga ·

1. Classical Logic


3. Sudoku

How should the theory TSudoku look like (s.t.

look like?

TS = { eins1,4 , eins5,8 , eins6,6 , zwei2,2 , zwei4,8 , drei6,8 , drei8,3 , drei9,4 , vier1,7 , vier2,5 , vier3,1 , vier4,3 , vier8,2 , vier9,8 , .. . neun3,4 , neun5,2 , neun6,9 , }

models of TSudoku ∪ TS correspond to solutions of the puzzle)?

First square: T1 1 2 3 4 5 6 7 8 9


3. Sudoku


eins1,1 ∨ . . . ∨ eins3,3 zwei1,1 ∨ . . . ∨ zwei3,3 drei1,1 ∨ . . . ∨ drei3,3 vier1,1 ∨ . . . ∨ vier3,3 fuenf1,1 ∨ . . . ∨ fuenf3,3 sechs1,1 ∨ . . . ∨ sechs3,3 sieben1,1 ∨ . . . ∨ sieben3,3 acht1,1 ∨ . . . ∨ acht3,3 neun1,1 ∨ . . . ∨ neun3,3



1. Classical Logic

3. Sudoku

The formulae on the last slide are saying, that 1 The number 1 must appear somewhere in the first square. 2 The number 2 must appear somewhere in the first square. 3 The number 3 must appear somewhere in the first square. 4 etc Does that mean, that each number 1, . . . , 9

1. Classical Logic

3. Sudoku

No! We have to say, that each number occurs only once:

T10 : 1 2 3 4 5 6 7 8

¬(einsi, j ∧ zweii, j ), 1 ≤ i, j ≤ 3, ¬(einsi, j ∧ dreii, j ), 1 ≤ i, j ≤ 3, ¬(einsi, j ∧ vieri, j ), 1 ≤ i, j ≤ 3, etc ¬(zweii, j ∧ dreii, j ), 1 ≤ i, j ≤ 3, ¬(zweii, j ∧ vieri, j ), 1 ≤ i, j ≤ 3, ¬(zweii, j ∧ fuenfi, j ), 1 ≤ i, j ≤ 3, etc

How many formulae are these?

occurs exactly once in the first square?


1. Classical Logic


3. Sudoku


1. Classical Logic


3. Sudoku

Second square: T2 1 2 3 4 5 6 7 8 9

eins1,4 ∨ . . . ∨ eins3,6 zwei1,4 ∨ . . . ∨ zwei3,6 drei1,4 ∨ . . . ∨ drei3,6 vier1,4 ∨ . . . ∨ vier3,6 fuenf1,4 ∨ . . . ∨ fuenf3,6 sechs1,4 ∨ . . . ∨ sechs3,6 sieben1,4 ∨ . . . ∨ sieben3,6 acht1,4 ∨ . . . ∨ acht3,6 neun1,4 ∨ . . . ∨ neun3,6

The same has to be done for all 9 squares.

And all the other formulae from the previous slides (adapted to this case): T20





1. Classical Logic

3. Sudoku

1. Classical Logic

3. Sudoku

First Row: TRow 1 What is still missing:

1

Rows: Each row should contain exactly the numbers from 1 to 9 (no number twice). Columns: Each column should contain exactly the numbers from 1 to 9 (no number twice).

2 3 4 5 6 7 8 9



1. Classical Logic

3. Sudoku

Analogously for all other rows, eg. Ninth Row: TRow 9 1 2 3 4 5 6 7 8 9

eins9,1 ∨ eins9,2 ∨ . . . ∨ eins9,9 zwei9,1 ∨ zwei9,2 ∨ . . . ∨ zwei9,9 drei9,1 ∨ drei9,2 ∨ . . . ∨ drei9,9 vier9,1 ∨ vier9,2 ∨ . . . ∨ vier9,9 fuenf9,1 ∨ fuenf9,2 ∨ . . . ∨ fuenf9,9 sechs9,1 ∨ sechs9,2 ∨ . . . ∨ sechs9,9 sieben9,1 ∨ sieben9,2 ∨ . . . ∨ sieben9,9 acht9,1 ∨ acht9,2 ∨ . . . ∨ acht9,9 neun9,1 ∨ neun9,2 ∨ . . . ∨ neun9,9

Is that sufficient? What if a row contains several

1's?




1. Classical Logic

3. Sudoku

First Column: TColumn 1 1 2 3 4 5 6 7 8 9


Analogously for all other columns. Is that sufficient? What if a column contains several



1's?



1. Classical Logic

3. Sudoku

1. Classical Logic

3. Sudoku

Here is a more difficult one. All put together: TSudoku = T1 ∪ T10 ∪ . . . ∪ T9 ∪ T90 TRow 1 ∪ . . . ∪ TRow 9 TColumn 1 ∪ . . . ∪ TColumn 9

Table 3: A difficult Sudoku Sdifficult


1. Classical Logic


3. Sudoku

The above formulation is strictly formulated in propositional logic. Theorem provers, even if they consider only propositional theories, often use predicates, variables etc. smodels uses a predicate logic formulation, including variables. But as there are no function symbols, such an input can be seen as a compact representation. It allows to use a few rules as a shorthand for thousands of rules using propositional constants. Jürgen Dix and Wojtek Jamroga ·



1. Classical Logic


3. Sudoku

For example smodels uses the following constructs: 1 row(0..8) is a shorthand for row(0), row(1), ..., row(8). 2 val(1..9) is a shorthand for val(1), val(2), ..., val(9). 3 The constants 1, ..., 9 will be treated as numbers (so there are operations available to add, subtract or divide them).



1. Classical Logic 1

2

3. Sudoku

Statements in smodels are written as head :- body This corresponds to an implication body → head. An important feature in smodels is that all atoms that do not occur in any head, are automatically false.

For example the theory p(X, Y, 5) :- row(X), col(Y) means that the whole grid is filled with 5’s and only with 5's: eg. ¬p(X,Y, 1) is true for all X,Y , as well as ¬p(X,Y, 2) etc. Jürgen Dix and Wojtek Jamroga ·


1. Classical Logic

3. Sudoku

1. Classical Logic

3. Sudoku

More constructs in smodels 1 { p(X,Y,A) : val(A) } 1 :row(X), col(Y) this makes sure that in all entries of the grid, exactly one number (val()) is contained. 1 { p(X,Y,A) : row(X) : col(Y) : eq(div(X,3), div(R,3)) : eq(div(Y,3), div(C,3) } 1 :- val(A), row(R), col(C) this rule ensures that in each of the 9 squares each number from 1 to 9 occurs only once.



1. Classical Logic

Sudoku formalization

4. Calculi for SL

1.4 Calculi for SL

row(0..8). col(0..8). val(1..9).

1 { p(X,Y,A) : val(A) } 1 :- row(X), col(Y). 1 { p(X,Y,A) : row(X) } 1 :- val(A), col(Y). 1 { p(X,Y,A) : col(Y) } 1 :- row(X), val(A). 1 { p(X,Y,A) : row(X) : col(Y) : eq(div(X,3),div(R,3)) : eq(div(Y,3),div(C,3)) } 1 :- row(R), col(C), val(A).





1. Classical Logic

4. Calculi for SL

1. Classical Logic

4. Calculi for SL

A general notion of a certain sort of calculi.

We now define a particular instance of our general notion.

Definition 1.11 (Hilbert-Type Calculi)

Definition 1.12 (Calculus for Sentential Logic SL)

A Hilbert-Type calculus over a language L is a pair hAx, Infi where Ax: is a subset of FmlL , the set of well-formed formulae in L : they are called axioms, Inf: is a set of pairs written in the form φ1 , φ2 , . . . , φn ψ where φ1 , φ2 , . . . , φn , ψ are L -formulae: they are called inference rules. Intuitively, one can assume all axioms as “true formulae” (tautologies) and then use the inference rules to derive even more new formulae.

SL We define HilbertSL L = hAxL , {MP}i, the Hilbert-Type calculus, as follows. The underlying language is L ⊆ LSL with the wellformed formulae FmlL as defined in Definition 1.1. Axioms in SL (AxSL L ) are the following formulae: 1 φ → >, → φ, ¬> → , → ¬>, 2 (φ ∧ ψ) → φ, (φ ∧ ψ) → ψ, 3 φ → (φ ∨ ψ), ψ → (φ ∨ ψ), 4 ¬¬φ → φ, (φ → ψ) → ((φ → ¬ψ) → ¬φ), 5 φ → (ψ → φ), φ → (ψ → (φ ∧ ψ)).



1. Classical Logic

4. Calculi for SL

The only inference rule in SL is modus ponens: MP : Fml × Fml → Fml : (ϕ, ϕ → ψ) 7→ ψ. or short ϕ, ϕ → ψ . ψ

(ϕ, ψ are arbitrarily complex formulae).



1. Classical Logic


4. Calculi for SL

Definition 1.13 (Proof)

Definition (continued)

(MP)

φ, ψ stand for arbitrarily complex formulae (not just constants). They represent schemata, rather than formulae in the language.


A proof of a formula ϕ from a theory T ⊆ FmlL is a sequence ϕ1 , . . . , ϕn of formulae such that ϕn = ϕ and for all i with 1 ≤ i ≤ n one of the following conditions holds: ϕi is substitution instance of an axiom, ϕi ∈ T , there is ϕl , ϕk = (ϕl → ϕi ) with l, k < i. Then ϕi is the result of the application of modus ponens on the predecessor-formulae of ϕi . We write: T ` ϕ (ϕ can be derived from T ). Jürgen Dix and Wojtek Jamroga ·


1. Classical Logic

4. Calculi for SL

We have now introduced two important notions: Syntactic derivability `: the notion that certain formulae can be derived from other formulae using a certain calculus, Semantic validity |=: the notion that certain formulae follow from other formulae based on the semantic notion of a model.


1. Classical Logic


4. Calculi for SL

1. Classical Logic

4. Calculi for SL

Definition 1.14 (Correct-, Completeness for a calculus) Given an arbitrary calculus (which defines a notion `) and a semantics based on certain models (which defines a relation |=), we say that Correctness: The calculus is correct with respect to the semantics, if the following holds: Φ ` φ implies Φ |= φ. Completeness: The calculus is complete with respect to the semantics, if the following holds: Φ |= φ implies Φ ` φ.



1. Classical Logic

4. Calculi for SL

Theorem 1.15 (Correct-, Completeness for HilbertSL L )

Theorem 1.16 (Compactness for HilbertSL L )

A formula follows semantically from a theory T if and only if it can be derived:

A formula follows from a theory T if and only if it follows from a finite subset of T : Cn(T ) =

T |= ϕ if and only if T ` ϕ




{Cn(T 0 ) : T 0 ⊆ T, T 0 finite}.

[


1. Classical Logic

4. Calculi for SL

It is well-known, that any formula φ can be written as a conjunction of disjunctions mi n _ ^

φi, j

i=1 j=1

The φi, j are just constants or negated constants. W i φi, j are called clauses of The n disjunctions mj=1 φ. Normalform

Instead of working on arbitrary formulae, it is often easier to work on finite sets of clauses.


1. Classical Logic


4. Calculi for SL

Set-notation of clauses

A disjunction A ∨ ¬B ∨C ∨ . . . ∨ ¬E is often written as a set {A, ¬B,C, . . . , ¬E}. Thus the set-theoretic union of such sets corresponds again to a clause: {A, ¬B} ∪ {A, ¬C} represents A ∨ ¬B ∨ ¬C. Note that the empty set 0/ is identified with .



1. Classical Logic

4. Calculi for SL

A resolution calculus for SL The resolution calculus is defined over the language L res ⊆ LSL where the set of well-formed formulae FmlLRes res consists of all disjunctions of the following form A ∨ ¬B ∨C ∨ . . . ∨ ¬E, i.e. the disjuncts are only constants or their negations. No implications or conjunctions are allowed. These formulae are called clauses. is also a clause: the empty disjunction.


1. Classical Logic


4. Calculi for SL

We define the following inference rule on FmlLRes res : Definition 1.17 (SL resolution)

Let C1 ,C2 be clauses (disjunctions). Deduce the clause C1 ∨C2 from C1 ∨ A and C2 ∨ ¬A: C1 ∨ A, C2 ∨ ¬A (Res) C1 ∨C2 / then C1 ∨C2 = . If C1 = C2 = 0,



1. Classical Logic

4. Calculi for SL

If we use the set-notation for clauses, we can formulate the inference rule as follows: Definition 1.18 (SL resolution (Set notation))

Deduce the clause C1 ∪C2 from C1 ∪ {A} and C2 ∪ {¬A}: (Res)

C1 ∪ {A}, C2 ∪ {¬A} C1 ∪C2

Again, we identify the empty set 0/ with .

1. Classical Logic

4. Calculi for SL

Definition 1.19 (Resolution Calculus for SL)

We define the resolution calculus / {Res}i as follows. The RobinsonSL L res = h0, underlying language is L res ⊆ LSL defined on Slide 57 together with the well-formed formulae FmlLRes res . Thus there are no axioms and only one inference rule. The well-formed formulae are just clauses. Question:

Is this calculus correct and complete?



1. Classical Logic

4. Calculi for SL



1. Classical Logic

Answer: It is correct, but it is not complete!

5. Wumpus in SL

1.5 Wumpus in SL

But every problem of the kind “T |= φ” is equivalent to “T ∪ {¬φ} is unsatisfiable” or rather to T ∪ {¬φ} ` (` stands for the calculus introduced above). Theorem 1.20 (Completeness of Resolution Refutation) If M is an unsatisfiable set of clauses then the empty clause can be derived in RobinsonSL L res . We also say that resolution is refutation complete.





1. Classical Logic Breeze

Stench

4

1. Classical Logic

5. Wumpus in SL

Breeze

3

Stench

PIT

1,4

Breeze

1,3

2,4

3,4

4,4

5. Wumpus in SL

2,3

3,3

4,3

PIT

Gold

1,2

= Agent = Breeze = Glitter, Gold = Safe square = Pit = Stench = Visited = Wumpus

1,4

2,4

3,4

4,4

1,3

2,3

3,3

4,3

2,2

3,2

4,2

2,2

3,2

4,2

1,2

2,1

3,1

4,1

1,1

Breeze

Stench

2

A B G OK P S V W

A Breeze

1

OK

2,1 V OK

Breeze

PIT

P?

OK

OK 1,1

OK

3,1

A B OK

START

1

2


3


1,3

W!

2,4

3,4

4,4

2,3

3,3

4,3

5. Wumpus in SL

1,2

A

2,2

S OK 1,1

4,2

= Agent = Breeze = Glitter, Gold = Safe square = Pit = Stench = Visited = Wumpus

1,4

2,4

1,3 W!

2,3

1,2

2,2

S V OK

OK 3,1

B V OK

P!

4,1

1,1

(a)

P?

3,4

4,4

3,3 P?

4,3

3,2

4,2


(b)


1. Classical Logic

5. Wumpus in SL

Language definition: Si, j stench Bi, j breeze Piti, j is a pit Gli, j glitters Wi, j contains Wumpus

V OK B V OK

3,1

P!

4,1

(b)

General knowledge: ¬S1,1 −→ (¬W1,1 ∧ ¬W1,2 ∧ ¬W2,1 ) ¬S2,1 −→ (¬W1,1 ∧ ¬W2,1 ∧ ¬W2,2 ∧ ¬W3,1 ) ¬S1,2 −→ (¬W1,1 ∧ ¬W1,2 ∧ ¬W2,2 ∧ ¬W1,3 ) S1,2


4,1

2,1 V OK

A S G B

2,1 V OK

3,2

A B G OK P S V W

P?

4

1. Classical Logic 1,4

(a)


−→ (W1,3 ∨W1,2 ∨W2,2 ∨W1,1 )



1. Classical Logic

5. Wumpus in SL

1. Classical Logic

5. Wumpus in SL

Knowledge after the 3rd move: ¬S1,1 ∧ ¬S2,1 ∧ S1,2 ∧ ¬B1,1 ∧ ¬B2,1 ∧ ¬B1,2 Question: Can we deduce that the wumpus is located at (1,3)? Answer: Yes. Either via resolution or using our Hilbert-calculus.



2. Modal Logic

Programming versus knowledge engineering programming knowledge engineering choose language choose logic write program define knowledge base write compiler implement calculus run program derive new facts



2. Modal Logic

Chapter 2. Modal Logic In this chapter:

2.1 2.2 2.3 2.4 2.5

Modal Logic Reasoning about Knowledge Kripke Semantics Axioms for Modal Logics Normative Reasoning References



We extend propositional logic by modality operators. They can express beliefs, knowledge, etc. We introduce the semantics of modal logics, based on Kripke structures. Finally we consider variants of modal logics representing obligations and permissions.



2. Modal Logic

2. Modal Logic

is an extension of classical logic by new connectives and ♦ : necessity and possibility. Modal logic

p is true” means p is necessarily true, i.e. “ true in every possible scenario, ♦ p is true” means p is possibly true, i.e. “♦ true in at least one possible scenario. Independently of the precise definition, the following holds

Various modal logics: knowledge → epistemic logic, beliefs → doxastic logic, obligations → deontic logic, actions → dynamic logic, time → temporal logic, and combinations of the above: most famous multimodal logics: BDI logics of beliefs, desires, intentions (and time).

♦p ↔ ¬¬p Jürgen Dix and Wojtek Jamroga ·


2. Modal Logic


2. Modal Logic

Most of it can be translated to classical logic; . . . but it looks horribly UGLY there; . . . and in most cases it’s not automatizable any more. Remember Frege’s notation of predicate logic:




Good to know: MSPASS is a theorem prover implementing many modal logics, also description logics, relational calculus, etc built upon SPASS, a resolution prover for first-order logic with equality, check out http: //www.cs.man.ac.uk/~schmidt/mspass/.



2. Modal Logic

1. Reasoning about Knowledge

2.1 Reasoning about Knowledge

2. Modal Logic


Example 2.1 (Muddy children – Shoham’s version) A group of n children enters the house after having played in the mud outside. They are greeted by their father, who notices that k of them have mud on their foreheads (no kid can see whether she herself has a muddy forehead, but they can see all other foreheads). Can the kids find out whether they have a muddy forehead?

The father announces: At least one of you has mud on her forehead. He also says If you know (can prove) that your forehead is muddy, then raise your hands now. Nothing happens. The father keeps repeating the question. After exactly k rounds, all the children with muddy foreheads raise their hands. Jürgen Dix and Wojtek Jamroga ·

2. Modal Logic




2. Modal Logic



Definition 2.2 (Partition Model) An n-agent partition model over a language L is a tuple (W , I1 , . . . , In ), where

How is that possible? The announcement of the father does not reveal anything new, or does-it?

W : is a set of possible worlds, i.e. a set of L -structures (each L -sentence ϕ is either true in each of w ∈ W or not). Ii : each Ii is a partition of W : Ii = {Wi1 ,Wi2 , . . . ,Wir } with Wi j ∩Wik = 0/ for j 6= k S and 1≤ j≤r Wi j = W . We also use the notation (w ∈ W )

Ii (w) := {w0 : w ∈ Wi j and w0 ∈ Wi j } all worlds in the partition containing world w, according to agent i’s view of the universe. Jürgen Dix and Wojtek Jamroga ·




2. Modal Logic


The worlds in W can be propositional valuations or even first-order structures. In this lecture, we will mainly consider the propositional version.

3

1

cc c

3 mc c

3

1

cc c

3 mc c

3 mc 1c

cc c

mc 1c

mc m 1

cc m

3 cc c

mc c

mud1 ∈ π(w) iff w = mxx mud2 ∈ π(w) iff w = xmx mud3 ∈ π(w) iff w = xxm




mm 1c

mm c

3

1

3

2

mc m

3

3 mc c

1 cm c

m m

2

cc m

1

mm c

cm c

2

1 cm c

mm 1c

3

3

2

mc m

cm c

cm m

1 cm m

2

mm c

3

2

cc m

1

mm 1c

mm m 1

mm m

3

2

mm c

cm m

3

2

mc m

mm m 1

3

3

2

1

cm c

2

cc m

1

mm c


mm m

3

2

2

3

2

2

mc m

2

2

1

cm c

cm m

1

2

mm m

3

3 mc c

cm m

1

3 mm c

2. Modal Logic


mm m


2

2. Modal Logic


2


2

How can we formalise the notion of one agent knowing something? And reason about what agents know and draw conclusions?

2

The language L in our example consists just of the propositional constants mud1 , . . . , mudn .



We introduce new operators Ki with the intuitive meaning Ki ϕ “agent i knows that ϕ holds”. We can now formulate statements in a language L (K1 , . . . , Kn ) containing these operators. It remains to define a precise semantics for this notion.

2

2. Modal Logic


mc c

m m

2. Modal Logic


2. Modal Logic

2. Kripke Semantics

2.2 Kripke Semantics

Definition 2.3 (Semantics for partition models)

Let an n-agent partition model A = (W , I1 , . . . , In ) over a language L be given. Then we define the relation |= for L (K1 , . . . , Kn ) formulae as follows: for ϕ ∈ L : A , w |= ϕ if and only if w |= ϕ, A , w |= Ki ϕ if and only if for all worlds w0 , if w0 ∈ Ii (w) then A , w0 |= ϕ.

2. Modal Logic


2. Kripke Semantics

A slight generalisation of partition models leads to modal logic.


2. Modal Logic



2. Kripke Semantics

An Example

How can one look at a partition? It is like a set of equivalence classes: for agent i, all worlds in one partition are equivalent. They are all possible scenarios. So let’s generalise and introduce a binary relation R on all worlds: w1 Rw2 meaning that world w2 can be accessed (is reachable) from world w1 .


q0

x=0

q0

x=0

s s

q0

q2 x=2


q1

q2

x=1x=2

s s


q1

q2

x=1x=2


2. Modal Logic

2. Kripke Semantics

2. Modal Logic

2. Kripke Semantics

Definition 2.4 (Modal Logic with n modalities)

The language Lmodaln of modal logic with n modal operators 1 , . . . , n is the smallest set containing the propositional constants of L , and with formulae ϕ, ψ also the formulae i ϕ, ¬ϕ, ϕ ∧ ψ. We treat ∨, →, ↔, ♦ as macros (defined as usual). Note that the operators can be nested:

Definition 2.5 (Semantics of Modal Logic) The truth of a Lmodal1 -formula is evaluated relative to a world w and a Kripke structure hW , Ri, where W is a set of (propositional) models (possible worlds) and R is a binary relation on them, the accessibility relation. The relation |= is defined as follows: hW , Ri |=w p if p is primitive and w |= p, hW , Ri |=w ϕ ∧ ψ if hW , Ri |=w ϕ and hW , Ri |=w ψ. hW , Ri |=w ¬ϕ if not hW , Ri |=w ϕ hW , Ri |=w ϕ if for any w0 ∈ W with wRw0 : hW , Ri |=w0 ϕ

(1 2 1 p) ∨ 3 ¬p


2. Modal Logic


3. Axioms for Modal Logics

2.3 Axioms for Modal Logics


2. Modal Logic



As in classical logic, one can ask about a complete axiom system. Is there a calculus that allows to derive all sentences true in all Kripke models? Definition 2.6 (System K)

The system K is an extension of the propositional calculus by the axiom Axiom K (ϕ ∧ (ϕ → ψ)) → ψ and the inference rule ϕ (Necessitation) . ϕ Jürgen Dix and Wojtek Jamroga ·




2. Modal Logic


Theorem 2.7 (Soundness/completeness of System K)

System K is sound and complete with respect to arbitrary Kripke models. If we allow n modalities, the theorem as well as the definitions extend in an obvious way. The calculus is then called System Kn to account for the n modalities.


2. Modal Logic



To which axioms do the following properties lead? Reflexivity: xRx for all x. Transitivity: xRy and yRz implies xRz for all x, y, z. Euclidean: xRy and xRz implies yRz for all for all x, y, z. Serial: for all x there is a x0 such that xRx0 .



2. Modal Logic


Note that we have not assumed any properties of the accessibility relation R: it is just any binary relation. How does our partition model fit in this framework? A partition is nothing else than an equivalence relation! Assuming that R is an equivalence relation, what additional statements (axioms) are true in all Kripke models?


2. Modal Logic



Lemma 2.8

A binary relation is an equivalence relation if and only if it is reflexive, transitive and euclidean.



2. Modal Logic


2. Modal Logic


Definition 2.9 (Extending K by Axioms D, T, 4,5)

Theorem 2.10 (Sound/complete Subsystems of KDT45)

The system K is often extended by (a subset of) the following axioms:

Let X be any subset of {D, T, 4, 5} and let X be any subset of {serial, re f lexive,transitive, euclidean} corresponding to X. Then K ∪ X is sound and complete with respect to Kripke models the accessibility relation of which satisfies X .

K D T 4 5

(Ki ϕ ∧ Ki (ϕ → ψ)) → Ki ψ ¬Ki (ϕ ∧ ¬ϕ) Ki ϕ → ϕ Ki ϕ → Ki Ki ϕ ¬Ki ϕ → Ki ¬Ki ϕ

called as above for historical reasons. The system consisting of KT45 is also called S5.


2. Modal Logic



Exercise: Show that 1 The axiom D follows from KT45. 2 Show that KD45 is not equivalent to K45: axiom D does not follow from K45.



Corollary 2.11 (Sound-, completeness of KT45 (S5))

System KT45 is sound and complete with respect to Kripke models the accessibility relation of which is reflexive, transitive and euclidean. Jürgen Dix and Wojtek Jamroga ·

2. Modal Logic



Up to now we were thinking of Ki ϕ (the box operator i ) as agent i knows that ϕ. What if we interpret the operator as belief? Under such an interpretation axiom T has to be dropped. But all other axioms make sense. Axiom system KD45 is called the standard logic of beliefs. Axiom K is called logical omniscience, axiom D is called consistency, axiom 4 (resp. axiom 5) is called positive (resp. negative) introspection.



2. Modal Logic

4. Normative Reasoning

2.4 Normative Reasoning

2. Modal Logic


Deontic Logic is concerned with normative law and normative reasoning in law. The operators used denote obligation, permission, prohibition etc. We describe two systems: system OS (Von Wright, 1951) and KD (Åqvist, 1984).


2. Modal Logic




2. Modal Logic



Definition 2.12 (System OS)

The system OS is based on two modalities (deontic operators): O, meaning obligation, and P meaning permission. It is an extension of the propositional calculus by the axioms

It can be shown that this system can be seen as a modal logic with O as primitive modality. But then, because of necessitation, one has to accept the derivable sentence (O>) O(p ∨ ¬p) (stating the existence of an empty normative system): von Wright originally rejected this as an axiom.

OS1 Op ↔ ¬P¬p OS2 Pp ∨ P¬p OS3 P(p ∨ q) ↔ Pp ∨ Pq and the inference rule ϕ↔ψ (OS4) . Pϕ ↔ Pψ Jürgen Dix and Wojtek Jamroga ·




2. Modal Logic


The KD System The KD System is a von Wright-type system including the F (forbidden) operator and consisting of the following additional axioms and rules (again the classical axioms and modus ponens are included). (KD1) (KD2) (KD3) (KD4) (KD5)

O(p → q) → (Op → Oq) Op → Pp Pp ≡ ¬O¬p Fp ≡ ¬Pp p O-necessitation: Op



2. Modal Logic

5. References

2.5 References

2. Modal Logic


Axiom (KD1) is the so called K-axiom; (KD2) is the D-axiom, stating that obligatory implies permitted; (KD3) states that permission is the dual of obligation and (KD4) says that forbidden is not permitted. (KD1) holds for any modal necessity operator. O is the basic operator (all others are definable with it). The axioms reflect the idea that something is obligated if it holds in all perfect (ideal) worlds (relative to the world where one is).



2. Modal Logic

5. References

[aqvist84] Åqvist, L. (1984). Deontic logic. In D. M. Gabbay and F. Guenther (Eds.), Handbook of Philosophical Logic, Vol II, pp. 605–714. Reidel. [Fagin et al. 1995] Fagin, R., Halpern, J. Y., Moses, Y. & Vardi, M. Y. Reasoning about Knowledge. MIT Press: Cambridge, MA. [fipa] Foundation for Intelligent Physical Agents. FIPA home page. http://www.fipa.org/.





2. Modal Logic

5. References

[fipa-acl] Foundation for Intelligent Physical Agents (2002). FIPA ACL message structure specification. Approved for standard, 2002-12-06. http://www.fipa.org/specs/fipa00061/. [HiDBoHoMe98] Hindriks, K. V., F. S. de Boer, H. Hoek, and J.-J. C. Meyer (1998). Formal semantics of the core of AGENT-0. In ECAI’98 Workshop on Practical Reasoning and Rationality, pp. 20–29.


2. Modal Logic


5. References

[mascardi02] Mascardi, V. (2002). Logic-Based Specification Environments for Multi-Agent Systems. Ph. D. thesis, Computer Science Department of Genova University, Genova, Italy. DISI-TH-2002-04. [MascardiMS04] Mascardi, V., M. Martelli, and L. Sterling (2004). Logic-based specification languages for intelligent software agents. TPLP 4(4), 429–494.



2. Modal Logic

5. References

[kripke63b] Kripke, S. (1963a). Semantical analysis of modal logic I. Normal propositional calculi. Zeitschrift fur math. Logik und Grundlagen der Mathematik 9, 67–96. [kripke63a] Kripke, S. (1963b). Semantical considerations on modal logic. Acta Philosophica Fennica 16, 83–94. [kripke65] Kripke, S. (1965). Semantical analysis of modal logic II. Non-normal modal propositional calculi. In Addison, Henkin, and Tarski (Eds.), The theory of models, Amsterdam, North-Holland, pp. 206–220.


2. Modal Logic


5. References

[Parunak-odell] Odell, J., H. V. D. Parunak, and B. Bauer (2000a). Extending UML for agents. In G. Wagner, Y. Lespérance, and E. Yu (Eds.), Proceedings of the 2nd Workshop on Agent-Oriented Information Systems (AOIS’00). Held in conjunction with 17th National conference on Artificial Intelligence (AAAI’00), pp. 3–17. [aose54] Odell, J., H. V. D. Parunak, and B. Bauer (2000b). Representing agent interaction protocols in UML. In P. Ciancarini and M. Wooldridge (Eds.), Proceedings of the 1st International Workshop on Agent-Oriented Software Engineering (AOSE’00), pp. 121–140. Springer-Verlag. LNCS 1957.



2. Modal Logic

5. References

3. Action and Time

Chapter 3. Action and Time [vonWright51] von Wright, G. H. (1951). Deontic logic. Mind 60, 1–15. Reprinted in G. H. von Wright, Logical Studies, pp. 58–74. Routledge and Kegan Paul, 1957.



3. Action and Time



3. Action and Time

In this chapter:

Up to now: Several operators Ki , each defines an accessibility relation on worlds. Description of static systems: no possibility of change But: MAS are dynamic!


3.1 3.2 3.3 3.4

Action and Time Dynamic Logic Temporal Logic ATL References


We add program/action modalities [α], hαi, and define dynamic logic, where we can talk about actions and their outcomes. We discuss two basic kinds of temporal logic, i.e. linear and branching time temporal logic. We end with introducing ATL, in which temporal logic is combined with elements of game theory.



3. Action and Time

1. Dynamic Logic

3.1 Dynamic Logic

3. Action and Time

1. Dynamic Logic

1st idea: Consider actions or programs α. Each such α defines a transition (accessibility relation) from worlds into worlds. 2nd idea: We need statements about the outcome of actions: [α]ϕ: “after every execution of α, ϕ holds, hαiϕ: “after some executions of α, ϕ holds. As usual, hαiϕ ≡ ¬[α]¬ϕ.


3. Action and Time


1. Dynamic Logic



3. Action and Time

1. Dynamic Logic

Definition 3.1 (Labelled Transition System)

3rd idea: Programs/actions can be combined (sequentially, nondeterministically, iteratively), e.g.: [α; β]ϕ would mean “after every execution of α and then β, formula ϕ holds”.

A labelled transition system is a pair α

hQ, {−→: α ∈ L}i where Q is a non-empty set of states and L is a non-empty set of labels and for each α ∈ L: α −→⊆ Q × Q. Definition 3.2 (Dynamic Logic: Models)

A model of propositional dynamic logic is given by a labelled transition systems and an evaluation of propositions. Jürgen Dix and Wojtek Jamroga ·




3. Action and Time

1. Dynamic Logic

3. Action and Time

wait Definition 3.3 (Semantics of DL) α

M , s |= [α]ϕ iff for every t such that s −→ t, we have M ,t |= ϕ.

q0

1. Dynamic Logic

wait

try

q1

start

wait

q0

wait

try

q1

halt start

halt

start → htryihalt start → ¬[try]halt start → htryi[wait]halt


3. Action and Time


1. Dynamic Logic

Definition 3.4 (Composite labels)

We require that the set of labels forms a ∪,∗∗i. We also assume that Kleene algebra hL, ;,∪ the set of labels contains constructs of the form “ϕ?”, whenever ϕ is a formula not involving any modalities.



3. Action and Time

1. Dynamic Logic

“;” means sequential composition, ∪” means nondeterministic choice, “∪ “∗∗” means finite iteration (regular expr.), “ϕ?” means test. Thus we assume that the labels obey the following conditions: α;β

α

α∪β

α

β

s −→ t iff s −→ s0 and s0 −→ t, β

s −→ t iff s −→ t or s −→ t, α∗ s −→ t is the reflexive and transitive closure of α s −→ t, ϕ?

s −→ s iff s |=M ϕ. Jürgen Dix and Wojtek Jamroga ·




3. Action and Time

1. Dynamic Logic

3. Action and Time

1. Dynamic Logic

fuel

What has this to do with programs? roL 2 fuelOK caL

roL nofuel 1 caL

if ϕ then a else b

(ϕ?;a) ∪ (¬ϕ?;b)

5

9



3. Action and Time

1. Dynamic Logic

roL nofuel caP

9

roL nofuel caL

roL 2 fuelOK caL

roL nofuel caR

roL 6 fuelOK caR

roL nofuel caP

roL fuelOK caP

10

7

roP nofuel caR

roP fuelOK caR 8

roP nofuel caP 11

roP fuelOK caP 12


3. Action and Time

1. Dynamic Logic

(nofuel?;fuel)È(fuelOK?;load)

roP nofuel caL 3

roP fuelOK caL 4

roP nofuel caR

roP fuelOK caR 8

roP nofuel caP 11

roL fuelOK caP

10


load

5

7

roP fuelOK caL 4

(ϕ?;a)∗ ; (¬ϕ?)

while ϕ do a

1

roL 6 fuelOK caR

roL nofuel caR

roP nofuel caL 3

roL nofuel 1 caL

5

roP fuelOK caP 12

9

roL nofuel caR

roL nofuel caP

roL 2 fuelOK caL

roL 6 fuelOK caR

roL fuelOK caP

10

7

roP nofuel caL 3

roP fuelOK caL 4

roP nofuel caR

roP fuelOK caR 8

roP nofuel caP 11

roP fuelOK caP 12

caR → [(no f uel?; f uel) ∪ ( f uelOK?; load)]caR Jürgen Dix and Wojtek Jamroga ·




3. Action and Time

2. Temporal Logic

3.2 Temporal Logic

3. Action and Time

2. Temporal Logic

Reasoning about Time: CTL Ideas: The accessibility relation represents time. time: linear vs. branching start

start



3. Action and Time

2. Temporal Logic

ϕ ♦ϕ ϕU ψ

ϕ is true in the next moment in time ϕ is true in all future moments ϕ is true in some future moment ϕ is true until the moment when ψ becomes true

((¬passport ∨ ¬ticket) → ¬board _ f light) send(msg, rcvr) → ♦ receive(msg, rcvr)


3. Action and Time


2. Temporal Logic

Temporal logic was originally developed in order to represent tense in natural language.

Typical temporal operators

ϕ



Within Computer Science, it has achieved a significant role in the formal specification and verification of concurrent and distributed systems. Much of this popularity has been achieved because a number of useful concepts can be formally, and concisely, specified using temporal logics, e.g. safety properties liveness properties fairness properties



3. Action and Time

2. Temporal Logic

Safety / maintenance: “something bad will not happen” “something good will always hold”

3. Action and Time

2. Temporal Logic

Liveness: “something good will happen”

Typical examples: Typical examples:

♦ rich roL → ♦ roP and so on . . .

¬bankrupt ( f uelOK ∨ f uelOK) and so on . . . Usually: ¬....


Usually: ♦ ....


3. Action and Time

2. Temporal Logic


3. Action and Time


2. Temporal Logic

Strong fairness

Combinations of safety and liveness possible: ♦ roP (roL → ♦ roP)


fairness


“if something is attempted/requested, then it will be successful/allocated” Typical examples: (attempt → ♦ success) ♦attempt → ♦success



3. Action and Time

2. Temporal Logic

3. Action and Time

2. Temporal Logic

Linear Time: LTL Fairness: Useful when scheduling processes, responding to messages, etc. Good for specifying properties of the environment.


3. Action and Time


2. Temporal Logic

: Linear Time Logic Reasoning about a particular computation of a system Time is linear: just one possible future path is included! Models: paths LTL


3. Action and Time

2. Temporal Logic

Important: computational vs. behavioral structure

Definition 3.5 (Models of LTL)

A model of LTL is a sequence of time moments (states). We call such models paths, and denote them by λ. Evaluation of atomic propositions at particular time moments is also needed.

1

Computational str.

2

pos0

q0

2 1

pos2

pos1 2


System

pos0

1

Notation: λ[i]: ith time moment λ[i . . . j]: all time moments between i and j λ[i . . . ∞]: all timepoints from i on



q2

pos2


q1 pos1


3. Action and Time

2. Temporal Logic

3. Action and Time

2. Temporal Logic

Important: computational vs. behavioral structure Computational str.

Behavioral str. pos0

q0

q0

pos0

LTL models are defined as behavioral structures!

q0q0 pos0

pos2

pos1


q0q0q1 pos1

....

q1

q2


3. Action and Time

2. Temporal Logic



3. Action and Time

2. Temporal Logic

Definition 3.6 (Semantics of LTL)

iff p is true at moment λ[0]; iff λ[1..∞] |= ϕ; iff λ[i..∞] |= ϕ for some i ≥ 0; iff λ[i..∞] |= ϕ for all i ≥ 0; iff λ[i..∞] |= ψ for some i ≥ 0, and λ[ j..∞] |= ϕ for all 0 ≤ j ≤ i.

l pos0

q0

l' pos1

q1

pos2

q2

pos0

q0

pos1

pos2

q1

q2

....

λ |= p λ |= ϕ λ |= ♦ϕ λ |= ϕ λ |= ϕ U ψ

pos0

q0

pos1

q1

λ |= ♦pos1 λ0 = λ[1..∞] |= pos1 pos1 ∈ π(λ0 [0])

Note that: ϕ ≡ ¬♦¬ϕ ♦ϕ ≡ ¬¬ϕ ♦ϕ ≡ > U ϕ Jürgen Dix and Wojtek Jamroga ·




pos

q2

3. Action and Time

3. Action and Time

2. Temporal Logic

2. Temporal Logic

Branching Time: CTL

l q1

pos2

q2

pos0

q0

pos1

pos2

q1

q2

....

q0

pos1

pos0

q0

pos1

q1

λ |= ♦pos1 λ[0..∞] |= ♦pos1 λ[1..∞] |= ♦pos1 λ[2..∞] |= ♦pos1 ...



3. Action and Time

2. Temporal Logic

l[0..¥] pos2

q2

pos0

q0

pos1 Computational pos2 pos pos pos2 pos0 CTL: Tree 0Logic. 1 q1 q2 q q1 q2 q0 Reasoning about possible0 computations of a system Time is branching: we want all alternative paths included! Models include: states (time points, situations), transitions (changes) Paths: courses of action, computations.

....

pos0

l[0..¥]


3. Action and Time

pos1


2. Temporal Logic

CTL models: transition systems

Definition 3.8 (Paths)

Definition 3.7 (Transition System)

A path λ is an infinite sequence of states that can be affected by subsequent transitions. A path must be full, i.e. either infinite, or ending in a state with no outgoing transition.

A transition system is a pair hQ, −→i where Q is a non-empty set of states −→⊆ Q × Q is a transition relation. Note that, formally, a transition relation is just a modal accessibility relation.



q1

Usually, we assume that the transition relation is serial (time flows forever). Then, all paths are infinite.



pos2

q2

3. Action and Time

2. Temporal Logic

Reasoning about Time: CTL

“Vanilla” CTL: every temporal operator must be immediately preceded by exactly one path quantifier; CTL*: no syntactic restrictions; Reasoning in “vanilla” CTL can be automatized. European Agent Systems Summer School (Durham 2007) 145/253

3. Action and Time

2. Temporal Logic

Example: Rocket and Cargo

1

roL nofuel caL

roL 2 fuelOK caL

roP nofuel caL 3

roL nofuel caR

roL 6 fuelOK caR

7

roP nofuel caR

roP fuelOK caL 4

roP fuelOK caR 8

A(roL ∨ roP) roL → A (roP → nofuel)

9

roL nofuel caP

roL fuelOK caP

10


roP nofuel caP 11

A rocket and a cargo, The rocket can be moved between London (proposition roL) and Paris (proposition roP), The cargo can be in London (caL), Paris (caP), or inside the rocket (caR), The rocket can be moved only if it has its fuel tank full (fuelOK), When it moves, it consumes fuel, and nofuel holds after each flight.


3. Action and Time


2. Temporal Logic

Definition 3.9 (Semantics of CTL*: state formulae)

roL → E♦roP 5

2. Temporal Logic


Path quantifiers: A (for all paths), E (there is a path); Temporal operators: (nexttime), ♦ (sometime), (always) and U (until);


3. Action and Time

roP fuelOK caP 12


M, q |= Eϕ if, by definition, there is a path λ, starting from q, such that M, λ |= ϕ; M, q |= Aϕ if, by definition, for all paths λ, starting from q, we have M, λ |= ϕ. Definition 3.10 (Semantics of CTL*: path formulae)

Exactly like for LTL! M, λ |= ϕ iff M, λ[1...∞] |= ϕ; M, λ |= ϕ U ψ iff M, λ[i...∞] |= ϕ for some i ≥ 0, and M, λ[ j...∞] |= ϕ for all 0 ≤ j ≤ i. Jürgen Dix and Wojtek Jamroga ·


3. Action and Time

2. Temporal Logic

3. Action and Time

2. Temporal Logic


1

5

9

roL nofuel caL

roL 2 fuelOK caL

roL nofuel caR

roL 6 fuelOK caR

roL nofuel caP

roL fuelOK caP

10

7

roP nofuel caL 3

roP fuelOK caL 4

roP nofuel caR

roP fuelOK caR 8

roP nofuel caP 11

roP fuelOK caP 12


Exercise:

How to express that there is no possibility of a deadlock? E♦caP


3. Action and Time

3. ATL

3.3 ATL



3. Action and Time

3. ATL

ATL: What Agents Can Achieve ATL: Agent Temporal Logic [Alur et al. 1997] Temporal logic meets game theory Main idea: cooperation modalities hhAiiΦ: coalition A has a collective strategy to enforce Φ





3. Action and Time

3. ATL

hh jamesbondii♦win: “James Bond has an infallible plan to eventually win” hh jamesbond, bondsgirliifun U shot: “James Bond and his girlfriend are able to have fun until someone shoots at them”

Agents, actions, transitions, atomic propositions Atomic propositions + interpretation Actions are abstract


3. Action and Time

3. ATL

Definition 3.11 (Concurrent Game Structure)

A concurrent game structure is a tuple M = hAgt, Q, π, Act, d, oi, where: Agt: a finite set of all agents Q: a set of states π: a valuation of propositions Act: a finite set of (atomic) actions d : Agt × Q → P (Act) defines actions available to an agent in a state o: a deterministic transition function that assigns outcome states q0 = o(q, α1 , . . . , αk ) to states and tuples of actions Jürgen Dix and Wojtek Jamroga ·

3. ATL

ATL Models: Concurrent Game Structures

“Vanilla” ATL: every temporal operator preceded by exactly one cooperation modality; ATL*: no syntactic restrictions; Jürgen Dix and Wojtek Jamroga ·

3. Action and Time




3. Action and Time

3. ATL

Rocket Example: Simultaneous Actions

1 2

1

5

9


roL nofuel caL

roL 2 fuelOK caL

roL nofuel caR

roL 6 fuelOK caR

roL nofuel caP

roL fuelOK caP

10

7

roP nofuel caL 3

roP fuelOK caL 4

roP nofuel caR

roP fuelOK caR 8

roP nofuel caP 11

roP fuelOK caP 12


3. Action and Time

3. Action and Time

3. ATL

3. ATL

Definition 3.12 (Strategy)

A strategy is a conditional plan. We represent strategies with functions sa : Q → Act.

Definition 3.13 (Semantics of ATL)

M, q |= hhAiiΦ if, by definition, there is a collective strategy SA such that, for every path λ ∈ out(q, SA ), we have M, λ |= Φ.

Function out(q, SA ) returns the set of all paths that may result from agents A executing strategy SA from state q onward.




3. Action and Time


3. Action and Time

3. ATL

3. ATL

Example: Robots and Carriage Note:

h

us

ait

it,p

sh ,w

wa

us h

ait

,w

it,p

sh

wa

pu

us h

ait ,w

it,p

sh

wa

pu

us h

,w

it,p

sh

wa

pu

pu

pos2

h


pos1

it

pos2

push,wait

us

pos1

wait,wait push,push is exactly the topic of the next section. push,push wait,push wait,push q1 wait q q2

wa


q1

it,p

pos2

push,wait

sh, pu

pos1

q2

wait,wait

pos0 wa

q1

wait,push

wait,wait wait,wait Which push,push push,push 2 push

it wa sh, sh pu ,pu


push,wait

pos0

it wa

pos2

wait,push

wait,wait wait,wait push,push push,push

q0

it wa sh, sh pu ,pu

q2

pos0

When we combine time (actions, strategies...) wait,wait wait wait,wait push,push push,push with knowledge desires, intentions, wait pos0 q0(beliefs, q0 obligations...), we finally obtain a fairly realistic model of MAS.

it wa

wait,wait push,push

q0

wait,wait push,push

it wa sh, h pu us it,p wa

ait

wait,wait push,push

push,wait


q1

w p

pos1

3. Action and Time

3. ATL

3. Action and Time

Practical importance of temporal and strategic logics:

4. References

3.4 References

Automatic verification in principle possible (model checking): good complexity results for “vanilla” CTL and “vanilla” ATL model checkers exist for LTL and “vanilla” CTL. Can be used for automated planning. Executable specifications can be used for programming. More: Jürgen Dix and Wojtek Jamroga ·

3. Action and Time

Chapter 5. European Agent Systems Summer School (Durham 2007) 161/253

4. References

[Alur et al. 2002] R. Alur, T. A. Henzinger, and O. Kupferman. Alternating-time Temporal Logic. Journal of the ACM, 49:672–713, 2002. [Emerson 1990] E. A. Emerson. Temporal and modal logic. Handbook of Theoretical Computer Science, volume B, 995–1072. Elsevier, 1990.


3. Action and Time


4. References

[Huth00] Huth, M. & Ryan, M. Logic in Computer Science: Modeling and reasoning about systems. Cambridge University Press.

[Fisher 2006] Fisher, M. (2006). Temporal Logics. Kluwer.





4. Combining Modalities


Chapter 4. Combining Modalities

4.1 4.2 4.3 4.4

We have seen how to model belief, knowledge, time and action using modal logic. How about combining them?

Combining Modalities BDI (Cohen et al.) BDI (Rao et al.) Beware! References




1. BDI (Cohen et al.)

4.1 BDI (Cohen et al.)





Beliefs, Desires, Intentions BDI according to Cohen and Levesque: Mental primitives: beliefs and goals, Separate operators and relations for each agent Time and action: LTL and DL. Altogether:




multi-modal logic



Operator Bi ϕ Gi ϕ

α Done α


Meaning agent i believes ϕ agent i has goal of ϕ action α will happen next action α has just happened

Additionally: Action constructors “;” and “?”, as in DL; Derived operators: ♦α (sometime α), α (always α), (Later ϕ): strict sometime, (Before ϕ , ψ): ϕ holds before ψ.







The primitive constructs above allow us to define the following operators: Derived Operator Definition Meaning

x;α?) sometime α ♦α ∃x( ♦¬α α ¬♦ always α ♦ (Later ϕ) ¬ϕ ∧ ϕ strict sometime (Before ϕ , ψ) ϕ → ψ ϕ holds before ψ Examples: Gcitizen safea G police Bcitizen safecitizen




2. BDI (Rao et al.)

All goals have to be dropped eventually:

4.2 BDI (Rao et al.)

♦ ¬(Gi (Later ϕ)) How to define a persistent goal? P-Goal i ϕ = (G i (Later ϕ)) ∧ Bi ¬ϕ ∧ Before ((Bi ϕ) ∨ (Bi ¬ϕ) (¬(Gi Later ϕ)) How to define intention?

ϕ))? ; ϕ] Intend i ϕ = (P-Goal i [Donei (Bi (






2. BDI (Rao et al.)


2. BDI (Rao et al.)

Example: Card Play BDI according to Rao and Georgeff: Mental primitives: beliefs, desires and intentions Time: CTL Sophisticated semantic structure

W1

q0

hasA



2. BDI (Rao et al.)

Bela E win : Agent A believes that there is a way to win in one step Desa A win : the agent desires that every path leads to a victory, so he does not have to worry about his decisions However, he does not believe it is possible: ¬Bela A win On the other hand, he believes that a real optimist would believe that winning forever is inevitable: Bela Belopt A win.



AK1

W3

q0

AK2

q0

Ba

Da

hasA win

W4

q0

AK1

Da hasA

hasA

Bopt hasA win


W2

AQ1

hasA

AQ1

Bopt AK2

QK2



hasA

AQ2

KQ2 win

hasA win

KQ2


2. BDI (Rao et al.)

Of course, it is possible to extend BDI: horizontally: with other modal dimensions (e.g., BOID); vertically: to a language of higher order (e.g., LORA).




2. BDI (Rao et al.)


2. BDI (Rao et al.)

What do we use these frameworks for?

What do we use these frameworks for?

Analysis & Design

Automatic Generation of Behaviours

Modeling systems (the frameworks provide intuitive conceptual structures, and a systematic approach); Specifying desireable properties of systems.

Reasoning about concrete systems; Correctness testing.



Philosophy of Mind and Agency

Characterization of mental attitudes; Discussion of rational agents; Testing rationality assumptions.

Verification & Exploration


Programming with executable specifications; Automatic planning.

3. Beware!




3. Beware!

4.3 Beware! Let’s put ATL and epistemic logic in one box. Alternating-time Temporal Epistemic Logic

Problems!





4. Combining Modalities (-,-)

3. Beware!

q0


3. Beware!

start

Problem: The Agent Knows too Much

keep

(K,A)

trade

q3

keep

(K,Q)

trade

q4

keep

(Q,A)

trade

q5

keep

(Q,K)

trade

q6

keep

trade

win

win

win

win

win

(A,K )

(Q,K )

(K,A )

(Q,A )

(A,Q )

(K,Q )

(Q,A )

q9 q10 q11 q12 q13 q14 q15 q16 q17 q18 (K,A )

q8 (Q,K )

(A,K )

q7

q2

(K,Q )

keep

(A,Q)

trade

(A,Q )

(A,K)

q1

Agent a cannot really enforce ♦win ! Incomplete information, deterministic choices: if q is indistinguishable from q00 then a shouldn’t be allowed to plan one action in q, and another in q00 .

win

start → hhaii♦win start → Ka hhaii♦win Jürgen Dix and Wojtek Jamroga ·



3. Beware!




4. References

4.4 References Beware:

Strategic and epistemic abilities are not independent!






4. References


4. References

[Broersen et al., 2001a] Broersen, J., Dastani, M., Huang, Z., and van der Torre, L. (2001a). The BOID architecture: conflicts between beliefs, obligations, intentions and desires. In Müller, J., Andre, E., Sen, S., and Frasson, C., editors, Proceedings of the Fifth International Conference on Autonomous Agents, pages 9–16. ACM Press.

[Rao and Georgeff1991] Rao, A. S. and M. Georgeff (1991). Modeling Rational Agents within a BDI-Architecture. In J. F. Allen, R. Fikes, and E. Sandewall (Eds.), Proceedings of the International Conference on Knowledge Representation and Reasoning, Cambridge, MA, pp. 473–484. Morgan Kaufmann.

[Cohen and Levesque, 1990] Cohen, P. and Levesque, H. (1990). Intention is choice with commitment. Artificial Intelligence, 42:213–261.

[Wooldridge, 2000] Wooldridge, M. (2000). Reasoning about Rational Agents. MIT Press : Cambridge, Mass.

[Fagin et al. 1995] Fagin, R., Halpern, J. Y., Moses, Y. & Vardi, M. Y. Reasoning about Knowledge. MIT Press: Cambridge, MA.




Modalities in Action Verification and Planning Temporal Programming Agent Programming References




Chapter 5. Modalities in Action

5.1 5.2 5.3 5.4


In this chapter: We suggest that model checking formulas of temporal and strategic logics can be used for verification and planning in Multi-Agent Systems. We present several complexity results for CTL and ATL model checking. We mention briefly two ideas of programming agents with specifications based on modal logics: MetaTeM (programming in temporal logics) Agent-0 (programming with beliefs, capabilities and commitments), the very first agent logic developed by Shoham in the early 90ies.





1. Verification and Planning

5.1 Verification and Planning



Model Checking Formulae of CTL and ATL Model checking: Does ϕ hold in model M and state q? Global model checking: Return the exact set of states q in M such that ϕ holds in M , q Natural for verification of existing systems; also during design (“prototyping”)





Verification example: we want to make sure that the cargo can be always moved to the other location. A


caL → E♦caP ∧ caP → E♦caL ∧ caR → (E♦caL ∨ E♦caP)






function mcheck(M , ϕ). Model checking formulae of CTL. Returns the exact subset of Q for which formula ϕ holds. case ϕ ≡ p : return {q ∈ Q | p ∈ π(q)} case ϕ ≡ ¬ψ : return Q \ mcheck(M , ψ) case ϕ ≡ ψ1 ∧ ψ2 : return mcheck(M , ψ1 ) ∩ mcheck(M , ψ2 ) case ϕ ≡ E ψ : return pre(mcheck(M , ψ)) case ϕ ≡ Eψ : Q1 := Q; Q2 := Q3 := mcheck(M , ψ); while Q1 6⊆ Q2 do Q1 := Q1 ∩ Q2 ; Q2 := pre(Q1 ) ∩ Q3 od; return Q1 case ϕ ≡ Eψ1 U ψ2 : / Q1 := 0; Q2 := mcheck(M , ψ2 ); Q3 := mcheck(M , ψ1 ); while Q2 6⊆ Q1 do Q1 := Q1 ∪ Q2 ; Q2 := pre(Q1 ) ∩ Q3 od; return Q1 end case





Example: Modified Rocket Domain,

E♦caL



Two perspectives to model checking MAS: Verification

1

5

9

2

roL nofuel caL

roL fuelOK caL

roL nofuel caR

roL 6 fuelOK caR

roL nofuel caP

roL fuelOK caP

10



7

roP nofuel caL 3

roP fuelOK caL 4

roP nofuel caR

roP fuelOK caR 8

roP nofuel caP 11

roP fuelOK caP 12



Agent Planning with ATL Automated verification of ATL properties: Can the agents do it? Planning: How can we do it? Planning as model checking: CTL (Giunchiglia and Traverso, 1999). Agent planning with ATL: even more natural.



Model represents the view of an objective observer Formula: specification to be met Planning

Model represents the subjective view of an agent Formula: goal to be achieved





function plan(ϕ). Returns a subset of Q for which formula ϕ holds, together with a (conditional) plan to achieve ϕ. The plan is sought within the context of concurrent game structure S = hAgt, Q, Π, π, oi. case ϕ ∈ Π : return {hq, −i | ϕ ∈ π(q)} case ϕ = ¬ψ : P1 := plan(ψ); return {hq, −i | q ∈ / states(P1 )} case ϕ = ψ1 ∨ ψ2 : P1 := plan(ψ1 ); P2 := plan(ψ2 ); return {hq, −i | q ∈ states(P1 ) ∪ states(P2 )} case ϕ = hhAii ψ : return pre(A, states(plan(ψ))) case ϕ = hhAiiψ : P1 := plan(true); P2 := plan(ψ); Q3 := states(P2 ); while states(P1 ) 6⊆ states(P2 ) do P1 := P2 |states(P1 ) ; P2 := pre(A, states(P1 ))|Q3 od; return P2 |states(P1 ) case ϕ = hhAiiψ1 U ψ2 : / P1 := 0; Q3 := states(plan(ψ1 )); P2 := plan(true)|states(plan(ψ2 )) ; while states(P2 ) 6⊆ states(P1 ) do P1 := P1 ⊕ P2 ; P2 := pre(A, states(P1 ))|Q3 od; return P1 end case





Auxiliary functions: Weakest precondition: pre(A, Q1 ) = {hq, σA i | ∀σΣ\A δ(q, σA , σΣ\A ) ∈ Q1 } States for which plan P is defined: states(P) = {q ∈ Q | ∃σ hq, σi ∈ P} Extending plan P1 with P2 : P1 ⊕ P2 = P1 ∪ {hq, σi ∈ P2 | q ∈ / states(P1 )} P restricted to the states from Q1 : P|Q1 = {hq, σi ∈ P | q ∈ Q1 }


1 2

1

5

9





plan(hh1ii♦caP)={ h9, −i, h10, −i, h11, −i, h12, −i } plan(hh1, 2ii♦caP)={ h2, 1 : load·2 : nopi, h6, 1 : move·2 : nopi, h7, 1 : unload·2 : unloadi, h8, 1 : unload·2 : unloadi, h9, −i, h10, −i, h11, −i, h12, −i } plan(hh1, 2, 3ii♦caP)={ h1, 1 : load·3 : loadi, h2, 1 : load·3 : loadi, h3, 1 : nop·3 : f ueli, h4, 1 : move·3 : nopi, h5, 1 : load·3 : f ueli, h6, 1 : move·3 : nopi, h7, 1 : unload·3 : nopi, h8, 1 : unload·3 : nopi, h9, −i, h10, −i, h11, −i, h12, −i }




roL nofuel caL

roL 2 fuelOK caL

roL nofuel caR

roL 6 fuelOK caR

roL nofuel caP

roL fuelOK caP

10



7

roP nofuel caL 3

roP fuelOK caL 4

roP nofuel caR

roP fuelOK caR 8

roP nofuel caP 11

roP fuelOK caP 12



Nice results: model checking CTL and ATL is tractable! Theorem (Clarke, Emerson & Sistla 1986)

CTL model checking is P-complete, and can be done in time linear in the size of the model and the length of the formula. Theorem (Alur, Kupferman & Henzinger 1998)

ATL model checking is P-complete, and can be done in time linear in the size of the model and the length of the formula. Jürgen Dix and Wojtek Jamroga ·






Complexity of Model Checking CTL and ATL Nice results: model checking CTL and ATL is tractable. But: the result is relative to the size of the model and the formula Well known catch (CTL): size of models is exponential wrt a higher-level description Another problem (ATL): transitions are labeled So: the number of transitions can be

So, let's model-check!

Not as easy as it seems.

exponential in the number of agents.









3 agents/attributes, 12 states, 216 transitions

1 2 roL nofuel 1 caL

5

9 Jürgen Dix and Wojtek Jamroga ·

roL nofuel caR

roL nofuel caP

roL 2 fuelOK caL

roL 6 fuelOK caR

roL fuelOK caP

10

Proposition (Jamroga & Dix 2005), (Laroussinie et al. 2006)

7

roP nofuel caL 3

roP fuelOK caL 4

roP nofuel caR

roP fuelOK caR 8

roP nofuel caP 11

roP fuelOK caP 12


ATL model checking (wrt CGS) is ∆P3 -complete with respect to the number of states and agents. For positive ATL model checking (wrt CGS) is even ΣP2 -complete.





Some Complexity Classes



Complexity Class ΣP2

P: problems solvable in polynomial time by a deterministic Turing machine NP: problems solvable in polynomial time by a non-deterministic Turing machine PSPACE: problems solvable in polynomial space EXPTIME: problems solvable in exponential time

ΣPi : problems solvable in polynomial time by a non-deterministic Turing machine making adaptive queries to a ΣPi−1 oracle ΣP2 = NPNP : problems solvable in polynomial time by a non-deterministic Turing machine making adaptive queries to an NP oracle

What is this about? Scalability! Jürgen Dix and Wojtek Jamroga ·




Complexity of Temporal and Strategic Logics m, l

n, k, l

nlocal , k, l

CTL

P-complete [1]

P-complete [1]

PSPACE-complete [2]

ATL

P-complete [3]

ATLir ∆P2 -complete [4,7] [1] [2] [3] [4] [5] [6] [7] [8] [9]

∆P3 -complete [5,6] EXPTIME-complete [8,9] ∆P3 -complete [7]

PSPACE-complete [9]

Clarke, Emerson & Sistla (1986). Automatic verification of finite-state .... ACM Prog. Lang. Syst Kupferman, Vardi & Wolper (2000). An automata-theoretic approach to .... J. ACM Alur, Henzinger & Kupferman (2002). Alternating-time Temporal Logic. J. ACM Schobbens (2004). Alternating-time logic with imperfect recall. ENTCS Jamroga & Dix (CEEMAS 2005). Do agents make model checking explode? Laroussinie, Markey & Oreiby (FORMATS 2006). Model-Checking Timed. Jamroga & Dix (2007). Model checking abilities of agents. Theory of Computing Systems Hoek, Lomuscio & Wooldridge (AAMAS 2006). On the complexity of practical ATL model checking. Jamroga & Ågotnes (AAMAS 2007). Modular Interpreted Systems







Some variants: Information vs. recall ATLIr : Perfect information, no memory (imperfect recall). This is classical ATL, see [3]. ATLIR : Perfect information, full memory (perfect recall). Same complexity as ATL: polynomial in m, l. See [3]. ATLir : Imperfect information, imperfect recall: see last slide. ATLiR : Imperfect information, perfect recall: undecidable (currently working on a proof).







Even if model checking appears very easy, it can be very hard.

Main message: Complexity is very sensitive to the context! In particular, the way we define the input, and measure its size, is crucial.

Still, people do automatic model checking! LTL: SPIN CTL/ATL: MOCHA, MCMAS, VeriCS Even if model checking is theoretically hard, it can be feasible in practice.




2. Temporal Programming





5.2 Temporal Programming METATEM is based on a first-order temporal logic based on discrete, linear models with finite past and infinite future, called FML.







The formulae for FML are generated as usual, starting from a set L p of predicate symbols, a set Lv of variable symbols, a set Lc of constant symbols, the quantifiers ∀ and ∃, and the set Lt of terms (constants and variables). The set Fml is defined by: If t1 , ...,tn are in Lt and p is a predicate symbol of arity n, then p(t1 , ...,tn ) is in Fml. > (true) and ⊥ (false) are in Fml. If A and B are in Fml, then so are ¬A, A ∧ B, AU B, AS B, and (A). If A is in Fml and v is in Lv , then ∃v.A and ∀v.A are both in Fml.




ϕU ψ ϕS ψ

ϕ

−ϕ

primitive primitive [⊥U ϕ]

ϕ holds until ψ ϕ since ψ ϕ is true in the next state the current state is not the initial state, and ϕ was true in the previous state

•ϕ

if the current state is not the initial state, then ϕ was true in the previous state

♦ϕ ϕ ϕ ϕ

[⊥S ϕ]

ϕ will be true in some future state ϕ was true in some past state ϕ will be true in all future states ϕ was true in all past states



[¬ − ¬ϕ] [>U ϕ] [>S ϕ] [¬ ♦¬ϕ] [¬¬ϕ]



Definition 5.1 (FML model)

Semantics of FML. The models for FML formulae are given by a structure consisting of a sequence of worlds (also called states), together with an assignment of truth values to atomic sentences within states, a domain D which is assumed to be constant for every state, and mappings from elements of the language into denotations. Jürgen Dix and Wojtek Jamroga ·


Temporal operators:

Syntax of FML.




A FML model is a tuple M = hσ, D , hc , h p i where σ is the ordered set of states s0 , s1 , s2 , ..., hc is a map from the constants into D , and h p is a map from N × L p into D n → { true, false } (the first argument of h p is the index i of the state si ). Thus, for a particular state s, and a particular predicate p of arity n, h(s, p) gives truth values to atoms constructed from n-tuples of elements of D . A variable assignment hv is a mapping from the variables into elements of D . Given a variable and the valuation function hc , a term assignment τvh is a mapping from terms into D defined in the usual way. Jürgen Dix and Wojtek Jamroga ·




The semantics of FML is given by the |= relation that gives the truth value of a formula in a model M at a particular moment in time i and with respect to a variable assignment. hM , i, hv i |= > hM , i, hv i 6|= ⊥ hM , i, hv i |= p(x1 , ..., xn ) hM , i, hv i |= ¬ϕ hM , i, hv i |= ϕ ∨ ψ hM , i, hv i |= ϕ U ψ hM , i, hv i |= ϕ S ψ hM , i, hv i |= ∀x.ϕ hM , i, hv i |= ∃x.ϕ

h p (i, p)(τvh (x1 ), ..., τvh (xn )) = true hM , i, hv i 6|= ϕ hM , i, hv i |= ϕ or hM , i, hv i |= ψ for some k s.t. i < k, hM , k, hv i |= ψ and for all j, if i < j < k then hM , j, hv i |= ϕ iff for some k s.t. 0 ≤ k < i, hM , k, hv i |= ψ and for all j, if k < j < i then hM , j, hv i |= ϕ iff for all d ∈ D , hM , i, hv [d/x]i |= ϕ iff there exists d ∈ D s.t. hM , i, hv [d/x]i |= ϕ iff iff iff iff





An agent interface consists of three components: a unique agent identifier which names the agent a set of predicates defining what messages will be accepted by the agent – they are called environment predicates; a set of predicates defining messages that the agent may send—these are called component predicates. Besides environment and component predicates, an agent has a set of internal predicates with no external effect. The computational engine of an object is based on the METATEM paradigm of executable temporal logics. The idea behind this approach is to directly execute a declarative agent specification given as a set of program rules.





Concurrent METATEM is a programming language for distributed AI based on FML. A system contains a number of concurrently executing agents which are able to communicate through message passing. Each agent executes a specification of its desired behavior. Each agent has two main components: an interface which defines how the agent may interact with its environment (i.e., other agents); a computational engine, defining how the agent may act.





Program rules are temporal logic formulae of the form: antecedent: past → consequent: future The intuitive interpretation of such a rule is: on the basis of the past, do the future





Contract Proposal from [MascardiMS04] Seller

Buyer contractProposal

accept

refuse contractProposal acknowledge

accept



Seller agent may receive a contractProposal message from a buyer agent. According to the amount of merchandise required and the price proposed by the buyer, the seller may accept the proposal, refuse it or try to negotiate a new price by sending a contractProposal message back to the buyer. The buyer agent can do the same (accept, refuse or negotiate) when it receives a contractProposal message back from the seller.

refuse

contractProposal acknowledge









Behaviour of seller if the received message is contractProposal(merchandise, amount, proposed price) then if there is enough merchandise in the warehouse and the price is greater or equal than a max value, the seller accepts by sending an accept message to the buyer and concurrently ships the required merchandise to the buyer (if no concurrent actions are available, answering and shipping merchandise will be executed sequentially);



if there is not enough merchandise in the warehouse or the price is lower or equal than a min value, the seller agent refuses by sending a refuse message to the buyer; if there is enough merchandise in the warehouse and the price is between min and max, the seller sends a contractProposal to the buyer with a proposed price evaluated as the mean of the price proposed by the buyer and max.







The Concurrent METATEM program for the seller agent may be as follows. The interface of the seller agent is the following

The merchandise to be exchanged are oranges, with minimum and maximum price 1 and 2 euro respectively. The initial amount of oranges that the seller possesses is 1000.





The internal knowledge base of the seller agent contains the following rigid predicates (predicates whose value never changes): min-price(orange, 1). max-price(orange, 2). The internal knowledge base of the seller agent contains the following flexible predicates (predicates whose value changes over time): storing(orange, 1000).



seller(contractProposal)[accept, re f use, contractProposal, ship] meaning that: the seller agent, identified by the seller identifier, is able to recognize a contractProposal message with its arguments, not specified in the interface; the messages that the seller agent is able to broadcast to the environment, including both communicative acts and actions on the environment, are accept, refuse, contractProposal, ship.



5. Modalities in Action 2. Temporal Programming The program rules of the seller agent are the following ones (lowercase symbols = constants, uppercase = variables): ∀ Buyer, Merchandise, Req_Amnt, Price

− [contractProposal(Buyer, seller, Merchandise, Req_Amnt, Price) ∧ storing(Merchandise, Old_Amount) ∧ Old_Amount ≥ Req_Amnt ∧ max − price(Merchandise, Max) ∧ Price ≥ Max] =⇒ [ship(Buyer, Merchandise, Req_Amnt, Price) ∧ accept(seller, Buyer, Merchandise, Req_Amnt, Price)]

If there was a previous state where Buyer sent a contractProposal message to seller, and in that previous state all the conditions were met to accept the proposal, then accept the Buyer’s proposal and ship the required merchandise.





∀Buyer, Merchandise, Req_Amnt, Price

− [contractProposal(Buyer, seller, Merchandise, Req_Amnt, Price)∧ storing(Merchandise, Old_Amount)∧ min − price(Merchandise, Min)∧ Old_Amount < Req_Amnt ∨ Price ≥ Min] =⇒ re f use(seller, Buyer, Merchandise, Req_Amnt, Price)

If there was a previous state where Buyer sent a contractProposal message to seller, and in that previous state the conditions were not met to accept the Buyer’s proposal, then send a refuse message to Buyer.




3. Agent Programming

5.3 Agent Programming





∀Buyer, Merchandise, Req_Amnt, Price

− [contractProposal(Buyer, seller, Merchandise, Req_Amnt, Price)∧ storing(Merchandise, Old_Amount) ∧ min − price(Merchandise, Min)∧ max − price(Merchandise, Max) ∧ Old_Amount ≥ Req_Amnt∧ Price > Min ∧ Price < Max ∧ New_Price = (Max + Price)/2] =⇒ contractProposal(seller, Buyer, Merchandise, Req_Amnt, New_Price)

If there was a previous state where Buyer sent a contractProposal message to seller, and in that previous state the conditions were met to send a contractProposal back to Buyer, then send a contractProposal message to Buyer with a new proposed price.





Shoham suggests the following three components of an Agent Oriented Programming system: A formal language with clear syntax for describing the mental state. A programming language for defining agents. A method of transforming legacy code into an agent.





Shoham: an agent has commitments (or obligations). One such commitment can be to perform an action. No need for action selection. But the language specification must have a mechanism for adopting commitments or not. All rules must be created at compile time: huge amount of possible scenarios.





Atoms: Sentences in a point-based temporal framework: is_ f riend(aa,bb) (facts), turn(I, le f t)t (instantaneous actions).


AGENT-0 Programming Language

AGENT-0 is based on a quantified multi-modal logic, with direct reference to time. Three modalities: Beliefs, Commitments, Capabilities.






Beliefs: BInow go_swimming(II ,CLZ)t : I believe now that I am going to swim in CLZ at time t. BInow ontable(block_b)t . Ba3 Bb4 is_ f riend(aa,bb).

No distinction between actions and facts.







Obligations: (also called commitments) are the beliefs that one agent will create the truth of a statement (for another agent) t+1 OBLIt ,you , you go_swimming(II ,CLZ) (at time t, I am obligated to you to go swimming in CLZ at time t + 1). The argument does not need to be you an action: OBLt(II ,you you)t+1 . you you) in_lecture(you



Capabilities: An agent is said to be capable of a statement if it has the ability to see that that statement hold at the specified time: CANInow in_lecture(you)t (I am capable of seeing to it that you are in a lecture at time t.) Note: capabilities may change.

A decision is an obligation to oneself: DECat ϕ := OBLt(aa,aa) ϕ. Jürgen Dix and Wojtek Jamroga ·




AGENT-0 Programs

An agent in AGENT-0 consists of (1) a set of initial beliefs, (2) a set of capabilities, (3) a set of initial commitments, and (4) a set of commitment rules of the form agent (COMMITmsgcond mntlcond (agent agent, t, action)), “Commit to perform action for agent at time t (if msgcond holds of the new incoming messages, if mntlcond holds in the mental state, if the agent is currently capable of doing action).”







agent agent: this is the name of an agent; action: a private or communicative action. Only primitive actions are allowed! Action “Find the gold and bring it home” is, most likely, not primitive. It requires a plan (a sequence of primitive actions). msgcond: a message condition of the form (Sender Type Content). mntlcond: this describes a condition about the own mental state of the agent (i.e., the agent’s beliefs and commitments); Jürgen Dix and Wojtek Jamroga ·





Cycle

initialize

AGENT-0 follows the following simple control loop when executing a program: At each time step 1 gather incoming messages and update the mental state accordingly, 2 execute commitments (using capabilities).

3. Agent Programming messages in

beliefs update beliefs commitments update commitments abilities

More recent agent-oriented programming languages follow similar ideas!

EXECUTE

messages out internal actions Jürgen Dix and Wojtek Jamroga ·







4. References

5.4 References What makes an agent do something


?





4. References

[BFG89] Barringer, H., M. Fisher, D. Gabbay, G. Gough, and R. Owens (1990). METATEM: A framework for programming in temporal logic. In J. W. de Bakker, W. P. de Roever, and G. Rozenberg (Eds.), Proceedings of the 1989 REX Workshop on Stepwise Refinement of Distributed Systems: Models, Formalisms, Correctness, pp. 94–129. Springer-Verlag. LNCS 430.




4. References

[concMTM13] Fisher, M. (1993). Concurrent METATEM – A language for modeling reactive systems. In A. Bode, M. Reeve, and G. Wolf (Eds.), Proceedings of the 5th International Conference on Parallel Architectures and Language, Europe (PARLE’93), pp. 185–196. Springer-Verlag. LNCS 694. [concMTM14] Fisher, M. and H. Barringer (1991). Concurrent METATEM processes – A language for distributed AI. In E. Mosekilde (Ed.), Proceedings of the 1991 European Simulation Multiconference. SCS Press.




4. References

[finger-fisher-owens] Finger, M., M. Fisher, and R. Owens (1993). METATEM at work: Modelling reactive systems using executable temporal logic. In P. Chung, G. L. Lovegrove, and M. Ali (Eds.), Proceedings of the 6th International Conference on Industrial and Engineering Applications of Artificial Intelligence and Expert Systems (IEA/AIE’93), pp. 209–218. Gordon and Breach Publishing. [fisherNormalForm] Fisher, M. (1992). A normal form for first-order temporal formulae. In D. Kapur (Ed.), Proceedings of the 11th International Conference on Automated Deduction (CADE’92), pp. 370–384. Springer-Verlag. LNCS 607.




4. References

[concMTM16] Fisher, M. and M. Wooldridge (1993). Executable temporal logic for distributed AI. In K. Sycara (Ed.), Proceedings of the 12th International Workshop of Distributed AI (IWDAI’93), pp. 131–142. [GiunchigliaTraverso00] Fausto Giunchiglia and Paolo Traverso. In Susanne Biundo and Maria Fox (Eds.) Recent Advances in AI Planning, 5th European Conference on Planning, ECP’99, Durham, UK, September 8-10, 1999, Proceedings, 1–20, LNCS 1809, Springer 2000.




4. References

[HiDBoHoMe98] Hindriks, K. V., F. S. de Boer, H. Hoek, and J.-J. C. Meyer (1998). Formal semantics of the core of AGENT-0. In ECAI’98 Workshop on Practical Reasoning and Rationality, pp. 20–29. [Jamroga, 2004] Jamroga, W. (2004). Strategic planning through model checking of ATL formulae. In et al., L. R., editor, Artificial Intelligence and Soft Computing: Proceedings of ICAISC’04, volume 3070 of Lecture Notes in Computer Science, pages 879–884. Springer Verlag.




4. References

[mascardi02] Mascardi, V. (2002). Logic-Based Specification Environments for Multi-Agent Systems. Ph. D. thesis, Computer Science Department of Genova University, Genova, Italy. DISI-TH-2002-04. [MascardiMS04] Mascardi, V., M. Martelli, and L. Sterling (2004). Logic-based specification languages for intelligent software agents. TPLP 4(4), 429–494.




4. References

[Jamroga/Dix1] Wojtek Jamroga and Jürgen Dix. Model Checking Strategic Abilities of Agents under Incomplete Information. In Mario Coppo, Elena Lodi and G. Michele Pinna (Eds.), Proceedings of the Italian Conference on Theoretical Computer Science (ICTCS ’05), pages 295–308. LNCS 3701. Springer, 2005. [Jamroga/Dix2] Wojtek Jamroga and Jürgen Dix. Do Agents Make Model Checking Explode (Computationally)? M. Pe(chouc(ek and P. Petta and L.Z. Varga (Eds.), Proceedings of the 4th International Central and Eastern European Conference on Multi-Agent Systems (CEEMAS ’05), pages 398-407. LNCS 3690. Springer, 2005.




4. References

[Parunak-odell] Odell, J., H. V. D. Parunak, and B. Bauer (2000a). Extending UML for agents. In G. Wagner, Y. Lespérance, and E. Yu (Eds.), Proceedings of the 2nd Workshop on Agent-Oriented Information Systems (AOIS’00). Held in conjunction with 17th National conference on Artificial Intelligence (AAAI’00), pp. 3–17. [aose54] Odell, J., H. V. D. Parunak, and B. Bauer (2000b). Representing agent interaction protocols in UML. In P. Ciancarini and M. Wooldridge (Eds.), Proceedings of the 1st International Workshop on Agent-Oriented Software Engineering (AOSE’00), pp. 121–140. Springer-Verlag. LNCS 1957.




4. References

[Pistore and Traverso, 2001] Pistore, M. and Traverso, P. (2001). Planning as model checking for extended goals in non-deterministic domains. In Proceedings of IJCAI, pages 479–486. [Shoham90] Shoham, Y. (1993, March). Agent-oriented programming. Artificial Intelligence 60, 51–92. [placa] Thomas, S. R. (1995). The PLACA agent programming language. In M. Wooldridge and N. R. Jennings (Eds.), Proceedings of the 1st International Workshop on Agent Theories, Architectures, and Languages (ATAL’94), pp. 355–370. Springer-Verlag. LNCS 890.

