1. CMSC 691M Agent Architectures & Multi-Agent Systems
Spring 2002 – February 26
Class #9 – Formal Methods for MAS
Prof. Marie desJardins

2. Today’s overview
Reading: Weiss Chap. 8, “Formal Methods in DAI” (Munindar P. Singh, Anand S. Rao, and Michael P. Georgeff)
Why use formal methods?
Classes of logics
Beliefs, desires, and intentions
Implementing BDI models
Coordinating BDI agents
Communicating BDI agents
Societies of BDI agents

3. Why use formal methods? Specify properties of agents declaratively
Provide reasoning mechanisms for agents
Disadvantages
Intractable in general case
Limiting precisely because of formalism and abstract representation
Benefits
Specify complex behavior at an abstract level
Validate agent behaviors

4. What’s to be modeled?
First-order logic
Propositional logic
To design and implement intelligent agents, we may need to be able to reason about the truth of propositions and relations between objects in the world, to reason about what may or must be true, to reason about how the agent’s actions affect the state of the world, and to reason about how other agents and external agents change the world over time.
Modal
logic
Temporal logic
Dynamic logic

5. Classes of logics Wffs (syntax) Proof theory (inference)
Model theory (semantics)
Propositional
Atoms closed under  and ¬
Implication and rules for  and ¬
“Meaning” of propositions
Predicate (first-order)
Adds quantifiers  and , relations, variables
Inference rules for quantifiers and variable binding
“Meaning” of relations
Modal
Adds possibility  and necessity 
Inference rules for  and 
Possible worlds
Dynamic
Adds sequencing, branching, testing
Inference rules for outcomes
Possible worlds with transitions
Temporal
Adds notion of time points or intervals, ordering
Inference for propositions w/ temporal extent
Possible worlds with multiple transitions

6. Propositional logic L = true atomic propositions
P is entailed iff P  L
P  Q is entailed iff P and Q are entailed
P is entailed iff P is not entailed

7. Predicate (first-order) logic
x (Q(x)) is entailed iff Q(l) holds for every object l
x (Q(x)) is entailed iff Q(l) holds for some object l

8. Modal logic Possible worlds semantics
Accessibility relation R(W1, W2):
Possibility: P is entailed in world w iff P is true in some possible world ( w’: R(w,w’)  P is entailed in w’)
Necessity: □P is entailed in w iff P is true in every possible world ( w’: R(w,w’) → P is entailed in w’)

9. Dynamic logic (“modal logic of action”)
Sequencing: a;b – do a, then do b
Choice: a+b – do either a or b nondeterministically
Testing: p? –TRUE if p, FALSE if p
((q?;a) + ((q?;b)) ≡ if q then a else b
Accessibility relation RA – reachability of worlds via (composite) action A

10. Dynamic logic – modeling outcomes
Possible outcomes
<A>P is entailed in w iff P is entailed in some world reachable by applying action A
<A>P  w’: RA(w,w’)  P entailed in w’
Necessary outcomes
[A]P is entailed in w iff P is entailed in all worlds reachable by applying action A
[A]P w’: RA(w,w’) → P entailed in w’

11. Temporal logic – variations
Linear vs. branching: modeling a single sequence of events/outcomes vs. modeling a branching series of alternative possible worlds
Discrete vs. dense (continuous): time treated as discrete intervals vs. continuously flowing
Moment-based (point) vs. period-based (interval): units of time treated as points or intervals

12. Discrete moment-based branching temporal logic
Moment in time are partially ordered
Each moment is associated with a possible world
The actions of multiple agents can influence which moment (possible world) occurs next

13. Linear temporal logic
p  q at moment t means that p holds from t until t’ and q holds at t’
X p at moment t means that p holds in the moment immediately following t
P p at moment t means that p was true at t’ where t’ is before t
F p at moment t means that p is true at some moment t’ after t
G p at moment t means that p is true at every moment t’ after t

14. Branching temporal logic
“The present moment”
“Reality”
A p means that p is true in all paths at the present moment (i.e., no matter what may have gone before or will happen in the future, p is true now) – temporal equivalent of the necessity operator of modal logic
E p means that p is true in some path at the present moment – temporal equivalent of possibility operator

15. Branching temporal logic II
X<a>p is true (at a particular moment, on a particular path) iff p is a possible outcome of agent x performing action a
X[a]p is true (at a particular moment, on a particular path) iff p is a necessary outcome of agent x performing action a
V a : p is true (at a moment and on a path) iff there is some action a under which p becomes true

16. Brief commentary on logics
Branching temporal logic is very powerful, and has been used to develop planners and other agent architectures
Many researchers use ideas from some or all of these logics in their agent designs and representations
Very few researchers use the full formal specification of these logics in building systems (though it isn’t uncommon to see them in conference and journal papers)

17. Beliefs, desires, and intentions
Use modal logic to model agent’s cognitive attitudes: beliefs, desires, goals, know-how, and intentions

18. Beliefs
X Bel p iff p is entailed in every possible world the agent believes it can be in (modeled by the B accessibility relation)
Interestingly, although a proposition q may be believed by this definition, an agent may not believe that it believes q
Limited rationality / limited computational resources means that the agent can’t derive everything that it “believes”

19. Desires
x Des p iff p holds in all possible worlds reachability by the D accessibility relation
The agent might not know how to reach the states it desires to be in
An agent can desire to be in conflicting states
Goals are the subset of the agent’s desires that are achievable and consistent

20. Intentions
x Int p iff p is true along all paths that are reachable by the I accessibility relation
According to this definition, an agent can “intend” something it doesn’t desire
An agent can also have an unsatisfiable intention (if the set of reachable paths is empty)
An agent can intend something, and yet fail to make it come true (if it proceeds along a path that isn’t in its set of intended paths)
Know-how models when an agent can guarantee the success of its actions
More useful might be to model when an agent might be able to guarantee the success of its actions

21. Commitments
Agents that persist with their intentions (as long as they are satisfiable) are said to be committed to those intentions
The concept of a commitment is very useful in modeling societies of agents

22. Basic interpreter basic-interpreter initialize-state(); do until quit.
options := option-generator (event-queue, S);
selected-options := deliberate (options, S);
update-state (selected-options, S);
execute (S);
event-queue := get-new-events();
until quit.
internal state
percepts
“intentions”

23. BDI interpreter BDI-interpreter initialize-state(); do until quit.
Beliefs, desires (goals),
and intentions
BDI-interpreter
initialize-state(); do
options := option-gen (event-queue, B, G, I);
selected-options := deliberate (options, B, G, I);
update-intentions (selected-options, I);
execute (I);
event-queue := get-new-events();
drop-successful-attitudes (B, G, I);
drop-impossible-attitudes (B, G, I);
until quit.
satisfied or
unrealizable
beliefs, goals, and
intentions

24. Issues in implementation
Updating the BDI structures is intractable in the general case
Use only explicit beliefs and goals
Represent beliefs, goals and intentions as plan structures that are followed by the agent
Support means-ends reasoning
Hierarchically structured

25. Coordinating BDI agents
Model actions of the agents in terms of how they can be affected by other agents’ preferences
Flexible actions can be delayed or omitted
Inevitable actions can be delayed but not omitted
Immediate actions can be neither delayed nor omitted
Triggerable actions can be performed at the request of another agent
Use a finite state automaton (skeleton) to model the state transitions of the agent

26. Coordination relationships
Model the relationships between two agents’ events
Is-required-by
Disables
Enables
Conditionally enables
(guaranteeing enablement)
Initiates
Jointly-required-by
Compensates-for-failure

27. Communicating BDI agents
Performative: speech act that is itself an action
Informing
Requesting
Promising
Permitting
Prohibiting
Declaring
Expressing

28. Communicating: Ontologies
Ontology – representation of objects and relationships in the world
Not quite the same as a knowledge base
An ontology is typically the “representational” part of a knowledge base…
…but sometimes axioms and rules are in an ontology

29. Societies of BDI agents
Groups of agents interact in some way
Agents may have different roles within the group
The agents may be heterogeneous or homogeneous
Teams of agents share (some) common goals

30. Mutual BDI Mutual beliefs Joint intentions
Everyone believes p, believes that the others believe p, believes that the others believe …
Impossible to achieve perfect mutual information in environments where communication can fail: “We attack at dawn”
Joint intentions
Everyone intends p; everyone will persist with p until achieved or impossible
Shared plans: intending-to and intending-that
Social commitments: promises and persistence

31. A few notes on grammar “Punctuation always goes inside a quote.”
“That” is used to define; “which” is used to clarify or extend
The system that Weiss describes is … (“that” tells which system I’m talking about)
The PRS system, which Georgeff et al. developed, … (“which” tells more about the only system in question)
Useful references:
Strunk and White, Elements of Style
DuPre, Bugs in Writing
Chicago Manual of Style