Combinations of Normal and Non-normal Modal

Outline

Overview

Background

Modalization/Temporalization

Computing Trust

Indep. Comb.

Further Work

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Combinations of Normal and Non-normal Modal Logics for Normative MAS Clara Smith1

Agust´ın Ambrossio1 Leandro Mendoza1 Antonino Rotolo2

(1) Facultad de Inform´ atica, UNLP, Argentina (2) CIRSFID, University of Bologna, Italy

Presentation at IRIT, Universit`e Toulouse 1 Capitole June 27, 2011

C. Smith, A. Ambrossio, L. Mendoza, A. Rotolo

Comb. of Normal and Non-Normal Modal Logics for ...

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Outline

Overview

Background


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General overview Why to Combine Logics? A Case: Normative MAS

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Background Logical Framework Involved Modalization/Temporalization Independent Combination

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Modalization/Temporalization Syntax and Semantics

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Computing Trust Decidability Model Checking

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Indep. Comb. Syntax and Semantics

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Further Work Further Work

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References References C. Smith, A. Ambrossio, L. Mendoza, A. Rotolo

Computing Trust

Indep. Comb.

Further Work

References


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Outline

Overview

Background


Computing Trust

Indep. Comb.

Further Work

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Why to Combine Logics Our agents are seen as autonomous goal-directed entities, so we are taking a BDI approach to model the system.



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Overview

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Computing Trust

Indep. Comb.

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Why to Combine Logics Our agents are seen as autonomous goal-directed entities, so we are taking a BDI approach to model the system. Some aims of combining logics: 1 2 3

Modularity Re-use of independent proofs Different kinds of Combinations lead to Different Expressiveness



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Outline

Overview

Background


Computing Trust

Indep. Comb.

Further Work

References

Why to Combine Logics Our agents are seen as autonomous goal-directed entities, so we are taking a BDI approach to model the system. Some aims of combining logics: Modularity 2 Re-use of independent proofs 3 Different kinds of Combinations lead to Different Expressiveness Some issues of combining logics: 1

1

The “Transfer Problem” arises... Which properties of the base logics remain valid/holds in the combined system. Completeness / Soundness Decidability / FMP

All these necessary for a correct implementation of the combined system. C. Smith, A. Ambrossio, L. Mendoza, A. Rotolo


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Outline

Overview

Background


Computing Trust

Indep. Comb.

Further Work

References

Normality vs. Non-normality

Normal modal logics systems are called normal because they satisfy the axiom K : K : (ϕ → ψ) → (ϕ → ψ) Some systems need weaker assumptions to work, such as the Does modality, that does not complies with K , (B. Chellas, 1980)



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Our Case Example 1

Internal Representation of an Agent Beliefs Belx A “agent x has the belief that A” Intentions Intx A “agent x has the intention to make A true” Goals Goalx A “agent x has the goal A true”



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Our Case Example 1


2

Internal Representation of a Group Common Beliefs C -BelG A “agents in the group i has the common belief that A” Common Intentions C -IntG A “agents in the group i has the common intention to make A true” All the above as depicted in [DV2002] with normal Kripke Models



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Overview

Background


Computing Trust

Indep. Comb.

Further Work

References

Our Case Example 1


2

Internal Representation of a Group Common Beliefs C -BelG A “agents in the group i has the common belief that A” Common Intentions C -IntG A “agents in the group i has the common intention to make A true” All the above as depicted in [DV2002] with normal Kripke Models

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Visible Behavior of an Agent Agency: Doesx A “agent x brings it about that A ” The above is Elgesem’s E modality with Scott-Montague semantics [Elg1997] (Governatori and Rotolo, 2004). C. Smith, A. Ambrossio, L. Mendoza, A. Rotolo


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Overview

Background


Computing Trust

Indep. Comb.

Further Work

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Adding Trust Modeling Trust. Trust can be modelled by means of delegation, weak or strong, depending on agreements, deal or promises (or the lack of). (Falcone and Castelfranchi, 2001). These can be illustrated as follows:



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Overview

Background


Computing Trust

Indep. Comb.

Further Work

References

Adding Trust Modeling Trust. Trust can be modelled by means of delegation, weak or strong, depending on agreements, deal or promises (or the lack of). (Falcone and Castelfranchi, 2001). These can be illustrated as follows: Levels of Trust: Joint Trust. Suppose agent y is in the bus stop, there’s a group G close to y , expecting that y will rise her hand. Reliance. It’s Mary’s birthday. Her co-workers give money to y , relying on y to purchase the gift. Everyone trusts that y will do so. Collective Trust. A gangster group G entrusts member y to sabotage other gangster association.

(Smith and Rotolo, 2010)



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Formal Definitions for Trust Levels:

Trustxy ϕ ≡ Goalx ϕ ∧ Belx Doesy ϕ ∧ Intx (Doesy ϕ ∧ ¬Doesx ϕ) ∧ ≡ ∧ Goalx Inty ϕ ∧ Belx Inty ϕ



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Computing Trust

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Formal Definitions for Trust Levels:

Trustxy ϕ ≡ Goalx ϕ ∧ Belx Doesy ϕ ∧ Intx (Doesy ϕ ∧ ¬Doesx ϕ) ∧ ≡ ∧ Goalx Inty ϕ ∧ Belx Inty ϕ

Joint Trust. JTrustyG A ≡ (

V

i∈G

Trustyi A )

Reliance. RelyG ≡ JTrustyG A ∧ MIntG (JTrustyG A ) Collective Trust. CTrustxG ≡ RelsG A ∧ CBelG (RelsG A ) All the above is depicted in the prior work “Collective Trust and Normative Agents” (Smith and Rotolo 2010) and also (Falcone and Castelfranchi, 2001). Both MInt G (Mutual Intention) and CBel G (Common Belief) as depicted in [DV2002] . C. Smith, A. Ambrossio, L. Mendoza, A. Rotolo


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Overview

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Computing Trust

Indep. Comb.

Further Work

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Logical Framework Structure of the MAS: (Smith, Rotolo, 2010) F = hA, W , {Bi }i∈A , {Gi }i∈A , {Ii }i∈A , {Di }i∈A i where: A is the finite set of agents; W is a set of situations, or points, or possible worlds; {Bi }i∈A is a set of accessibility relations wrt Bel, which are transitive, euclidean and serial; {Gi }i∈A is a set of accessibility relations wrt Goal, (standard Kn semantics); {Ii }i∈A is a set of accessibility relations wrt Int, which are serial {Di }i∈A is a family of sets of accessibility relations Di wrt Does, witch are pointwise closed under intersection, reflexive and serial.

Just for clearness and for keeping the system manageable, the modality Obl is put apart in the following presentation C. Smith, A. Ambrossio, L. Mendoza, A. Rotolo


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Outline

Overview

Background


Computing Trust

Indep. Comb.

Further Work

References

Combination of Logics Combination Methods - Modalization/Temporalization: Modalization/Temporalization Method (Finger and Gabbay 1992). Intuitively this method arranges two logics one “on top of” the other, that is, if we are modeling/temporalizing two logics L and M, the resulting combined logic L(M) is described graphically below: models for L

L dj(wi)

dj(wi+1)

Model Mi

Model Mi+1

dj(wi+2)

dj(wi+3)

Model Mi+2

Model Mi+3

M

models for M C. Smith, A. Ambrossio, L. Mendoza, A. Rotolo


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Outline

Overview

Background


Computing Trust

Indep. Comb.

Further Work

References

Combination of Logics - Cont. Combination Methods - Independent Combination: Independent Combination (Finger and Gabbay 1996) Intuitively, this method arranges two logics in a bi-dimensional “mesh like” structure, that is, if we are independently combining two logics L and M, the resulting combined logic L ⊕ M is: logic

M logic


L


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Overview

Background


Computing Trust

Indep. Comb.

Further Work

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Syntax: Recall: F = hA, W , {Bi }i∈A , {Gi }i∈A , {Ii }i∈A , {Di }i∈A i Call N to the restriction of F to its normal part, and call Does to restriction of F to its non-normal part.



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Outline

Overview

Background


Computing Trust

Indep. Comb.

Further Work

References


Syntax: Recall: F = hA, W , {Bi }i∈A , {Gi }i∈A , {Ii }i∈A , {Di }i∈A i Call N to the restriction of F to its normal part, and call Does to restriction of F to its non-normal part. We partition Does in two sets (as in Franceschet 2004): BDoes: Boolean formulas (outermost connective is boolean); eg. (Doesi A ∧ Doesi B) and, MDoes: Monolithic formulas (otherwise); eg. Doesi A .



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Outline

Overview

Background


Computing Trust

Indep. Comb.

Further Work

References


Syntax: Recall: F = hA, W , {Bi }i∈A , {Gi }i∈A , {Ii }i∈A , {Di }i∈A i Call N to the restriction of F to its normal part, and call Does to restriction of F to its non-normal part. We partition Does in two sets (as in Franceschet 2004): BDoes: Boolean formulas (outermost connective is boolean); eg. (Doesi A ∧ Doesi B) and, MDoes: Monolithic formulas (otherwise); eg. Doesi A .

Call N(Does) the modalization of Does by means of N.



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Outline

Overview

Background


Computing Trust

Indep. Comb.

Further Work

References

Modalization/Temporalization - Syntax - Cont.

Language: Let LDoes denote the language of agency.



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Language: Let LDoes denote the language of agency. Let LN denote the language of N.



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Overview

Background


Computing Trust

Indep. Comb.

Further Work

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Language: Let LDoes denote the language of agency. Let LN denote the language of N. The Language LN(Does) of N(Does) is obtained by replacing propositional letters by “monolithic formulas in LDoes ”. This process is known as “fuzzling” (Finger, Gabbay, 1996).



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Overview

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Computing Trust

Indep. Comb.

Further Work

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A Modalized model for N(Does) has the structure: M = hA, W , {Bi }i∈A , {Gi }i∈A , {Ii }i∈A , V 0 , {di }i A, W , {Bi }i∈A , {Gi }i∈A , {Ii }i∈A defined as before. V 0 is the valuation function V restricted to normal operators, as follows: standard boolean conditions; V 0 (w , Beli , A ) = 1 iff ∀v ∈ W (if wBi v then V 0 (v , A ) = 1); V 0 (w , Goali , A ) = 1 iff ∀v ∈ W (if wGi v then V 0 (v , A ) = 1); V 0 (w , Inti , A ) = 1 iff ∀v ∈ W (if wIi v then V 0 (v , A ) = 1); and



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Modalization/Temporalization - Syntax - Cont. models for L

L dj(wi)

dj(wi+1)

Model Mi

Model Mi+1

dj(wi+2)

Model Mi+2

dj(wi+3)

Model Mi+3

M

models for M



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Overview

Background


Computing Trust

Indep. Comb.

Further Work

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Modalization/Temporalization - Syntax - Cont. each di is a total function mapping, for each world w ∈ W , for each agent i, into a multi-relational model of the form: η = hW , Di , vi where: W is the (same original) set of worlds Di is a family of sets of accessibility relations Di wrt agency regarding agent i which are pointwise closed under intersection, reflexive and serial. v is V restricted to the non-normal operators. Doesi A holds in w iff A is true in the neighborhoods of w . Formally: standard Boolean conditions; v(w , Doesi A ) = 1 iff ∃Di ∈ Di such that ∀u(w Di u iff v(u, A ) = 1)



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Overview

Background


Computing Trust

Indep. Comb.

Further Work

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Modalization/Temporalization - Semantics

Given a model M, given a w ∈ W , given V 0 , and given functions di , the semantics are given by replacing the clause: M, w |= p iff p ∈ V 0 (w ), whenever p ∈ P with the clause: M, w |= A iff di (w ) |= A , whenever A ∈ MDoes.



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Computing Trust

Indep. Comb.

Further Work

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Computing Collective Trust - Decidability N(Does) is Complete due Modalization/Temporalization transfer properties if both logics are complete.



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Computing Trust

Indep. Comb.

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Computing Collective Trust - Decidability N(Does) is Complete due Modalization/Temporalization transfer properties if both logics are complete. N(Does) is Decidable due Modalization/Temporalization transfer properties if both logics are decidable.



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Computing Collective Trust - Decidability N(Does) is Complete due Modalization/Temporalization transfer properties if both logics are complete. N(Does) is Decidable due Modalization/Temporalization transfer properties if both logics are decidable. This required results for Does are proven in (Rotolo and Governatori, 2004), for N these results were proven in (Ambrossio and Mendoza, 2011) as follows: Completeness: using canonical models (every normal modal logic is complete respect to its canonical model) (Blackburn 2001, Chap. 4). Decidability: through finite model property, found “via filtration” (closing the formulas in a manageable structure)(Blackburn 2001, Chap 6).



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Computing Collective Trust - Model Checking Using the transfer properties of the Modalization/Temporalization method, we obtained a model checker for the logic N(Does) combining the model checkers for N and Does with the “fuzzling” technique.



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Overview

Background


Computing Trust

Indep. Comb.

Further Work

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Computing Collective Trust - Model Checking Using the transfer properties of the Modalization/Temporalization method, we obtained a model checker for the logic N(Does) combining the model checkers for N and Does with the “fuzzling” technique. Function MCN(Does) ((A, W , Bi , Gi , Ii , V 0 , {di }), ϕ) input: a modalized model M and a formula ϕ ∈ LN(Does) compute MMLDoes (ϕ) for every α ∈ MMLDoes (ϕ) i := identify the agent involved in α for every w ∈ W if (MCDoes (di (w ), α) = true) S then V 0 (w ) := V 0 (w ) {pα } /*fuzzling */ build up ϕ0 /* systematically replace variables generated above */ return MCN ((A, W , Bi , Gi , Ii , V 0 , {di }), ϕ0 );/*calls to the normal checker */



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Independent Combination of Mental States and Actions

Using the Modalization method, we are unable to write a formula such as Doesi (Goalj A ). (i.e. a form of persuasion) Thus, now we consider F as splitted into two sub-structures again, one capturing the internal (mental) motivational and informational aspects of the agents; and another one capturing the external, visible, behavioral side of agents. For doing this, first, for simplicity, we duplicate and subscript the elements in W to get a set of situations WN , and another set WD . Now we can build an ontology WN × WD of pairs (wN , wD ).



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Independent Combination - Syntax and Semantics

Syntax: Again, let LN denote the language of N, and LDoes denote the language of agency. The language LN×Does is obtained by taking the union of the formation rules for the combination of both languages.

Semantics: Assume that we have two separate structures: (A, WN , {Bi }, {Gi }, {Ii }, V 0 ), and (A, WD , {Di }, v). Interpret LN×Does over a combined model: C = hA, WN × WD , {Bi }i∈A , {Gi }i∈A , {Ii }i∈A , {Di }i∈A , Vi



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Independent Combination - Semantics - Cont.

C = hA, WN × WD , {Bi }i∈A , {Gi }i∈A , {Ii }i∈A , {Di }i∈A , Vi where: A is a set of agents. WN × WD is a set of pairs of situations. {Bi }i∈A , {Gi }i∈A , {Ii }i∈A are the accessibility relations for the normal operators. {Di }i∈A are the accessibility relations wrt agency; and V : WN × WD → Pow (P) is a function assigning to each pair (wN , wD ) ∈ WN × WD the set of proposition letters in P which are true.



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Independent Combination - Semantics - Cont.

The definition of a formula in LN×Does being satisfied in a model C at state (wN , wD ) amounts to: C, (wN , wD ) |= Beli A iff ∀vN ∈ WN ( if wN BvN then C, (vN , wD ) |= A ). C, (wN , wD ) |= Goali A iff ∀vN ∈ WN ( if wN GvN then C, (vN , wD ) |= A ). C, (wN , wD ) |= Inti A iff ∀vN ∈ WN ( if wN IvN then C, (vN , wD ) |= A ). C, (wN , wD ) |= Doesi A iff there exists a neighborhood n of wD such that ∀v ∈ n (C, (wN , v ) |= A ).



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Independent Combination - Syntax - Cont.

logic

M logic


L


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Computing Trust

Indep. Comb.

Further Work

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Further Work

The Obl modality: we have to check the transfer of properties when including Obl to the MAS. Our hypothesis is that there is transfer. Complexity: The proposed logic (modalized/temporalized), though decidable, is EXP-TIME complete. Dunnin-Keplicz and Verbrugge (2007) propose some methods for reducing this complexity. Further Combinations: We can use the idea of reasoning about time for tracking the evolution of the system.



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Overview

Background


Computing Trust

Indep. Comb.

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References

Ambrossio, A., Mendoza, L.: Completitud e implementaci´ on de modalidades en MAS. Thesis report, Facultad de Informatica UNLP (2011) Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic, Cambridge Tracts in Theoretical Computer Science, vol. 53. Cambridge University Press, Cambridge (2001). Chellas, B.: Modal Logic: An Introduction, Cambridge University Press, Cambridge, 1980. Dunin-Keplicz, B., Verbrugge, R.: Collective intentions. Fundam. Inform. pp. 271-295 (2002) Elgesem, D.: The modal logic of agency. Nordic Journal of Philosophical Logic 2, 1-46 (1997) M. Franceschet, A. Montanari, and M. De Rijke. Model checking for combined logics with an application to mobile systems. Automated Software Eng., 11:289321, June 2004. M. Finger and D. Gabbay. Combining temporal logic systems. Notre Dame Journal of Formal Logic, 37, 1996. Governatori, G., Rotolo, A.: On the axiomatization of Elgesem’s logic of agency. In: AiML 2004 Advances in Modal Logic. pp. 130-144. Department of Computer Science, University of Manchester (2004), http://www.cs.man.ac.uk/cstechrep/index.html Halpern, J., Moses, Y.: A guide to completeness and complexity for modal logics of knowledge and belief. Artificial Intelligence 54, 311-379 (1992) Smith, C., Rotolo, A.: Collective trust and normative agents. Logic Journal of IGPL 18(1), 195-213 (2010), http://jigpal.oxfordjournals.org/content/18/1/195.abstract



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Overview

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Computing Trust

Indep. Comb.

Further Work

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Thanks!!



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Computing Trust

Indep. Comb.

Further Work

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Thanks!! Any questions?



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