Abductive Logic Programming (ALP) and its Application in Agents and Multi-agent Systems

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Abductive Logic Programming (ALP) and its Application in Agents and Multi-agent Systems. Fariba Sadri Imperial College London ICCL Summer School Dresden August 2008. Contents. ALP recap ALP and agents Abductive planning via abductive event calculus (AEC) Dynamic planning with AEC

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### Abductive Logic Programming(ALP) and its Application in Agents and Multi-agent Systems

Imperial College London

ICCL Summer School Dresden

August 2008

Contents
• ALP recap
• ALP and agents
• Abductive planning via abductive event calculus (AEC)
• Dynamic planning with AEC
• Abductive proof procedures – IFF and CIFF
• Hierarchical Planning
• Reactivity
• Plan repair
• Dialogues and Negotiation

Abductive Logic Programs <P, A, IC> have three components:

• P is a normal logic program.
• A is a set of abducible predicates.
• IC, the set of integrity constraints.

Often, ICs are expressed as conditionals:

If A1 &...& An then B A1 &...& An  B

or as denials:

not (A1 &...& An & not B) A1 &...& An & not B  false

Normally, P is not allowed to contain any clauses whose conclusion contains an abducible predicate.

(This restriction can be made without loss of generality.)

ALP Semantics Recap from Bob Kowalski’s Slides

Semantics:

Given an abductive logic program,

< P,A,IC >, an abductive answer for a goal G is a set Δ of ground atoms in terms of the abducible predicates such that:

G holds in P  Δ

IC holds in P  Δ or P  Δ  IC is consistent.

Abduction is normally used to explain observationsRecap from Bob Kowalski’s Slides

Program P: Grass is wet if it rained

Grass is wet if the sprinkler was on

The sun was shining

Abducible predicates A: it rained, the sprinkler was on

Integrity constraint:

it rained & the sun was shining  false

Observation: Grass is wet

Two potential explanations: it rained, the sprinkler was on

The onlyexplanation that satisfies the integrity constraint is

the sprinkler was on.

ALP and Agents

Some references: Two ALP-based agent models:

R.A. Kowalski, F. Sadri, From logic programming towards multi-agent systems. In Annals of Mathematics and Artificial Intelligence Volume 25, pages 391-419  (1999)

Later developments of this model in later papers by Bob Kowalski

A. Kakas. P. Mancarella, F. Sadri, K. Stathis, F. Toni. Computational logic foundations of KGP agents, Journal of Artificial Intelligence Research. To appear

ALP and Agents

Agents can be seen as ALPs

• Logic Programs represent beliefs

(more elaborate beliefs than Agent0 or AgentSpeak)

• Abducibles represent observations and actions
• Integrity Constraints represent
• Condition-action rules for reactivity
• Plan repair rules
• Communication policies
• Obligations and prohibitions

ALP and Agents

Abductive Logic Programs used for

• Planning
• Reactivity
• Plan repair
• Negotiation

ALP and AgentsA small planning example

have(X) if borrow(X)

IC: buy(X) & no-money  false

Goal: have(tv)

Δ1: buy(tv) & register(tv) (Plan 1)

Δ2: borrow(pc) (Plan 2)

If P also includes no-money then

The only solution (only plan) is Δ2:borrow(pc).

ALP for PlanningAbductive Event Calculus (AEC)

Some References

Original Event Calculus

R.A. Kowalski and M.J. Sergot (1986)

A logic-based calculus of events. New Generation Computing, 4(1), 67-95.

Abductive Event calculus

P. Mancarella, F. Sadri, G. Terreni, F. Toni (2004) Planning partially for situated agents. In Leite, Torroni (eds.), Computational Logic in Multi-agent Systems, CLIMA V, Lecture Notes in Computer Science, Springer, 230-248.

Also papers by M. Shanahan.

ALP for PlanningAbductive Event Calculus (AEC)

Domain Independent Rules

holds_at(F,T2) ¬ happens(A,T1), T1<T2, initiates(A, T1, F),

¬clipped(T1, F, T2)

holds_at(F,T) ¬ initially(F), 0T,

¬clipped(0,F,T)

clipped(T1,F,T2) ¬ happens(A,T), terminates(A,T,F), T1T<T2

ALP for PlanningAbductive Event Calculus (AEC)

holds_at(¬F,T2) ¬ happens(A,T1), T1<T2, terminates(A, T1, F),

¬declipped(T1, F, T2)

holds_at(¬F,T) ¬ initially(¬F), 0T,

¬declipped(0,F,T)

declipped(T1,F,T2) ¬ happens(A,T), initiates(A,T,F), T1T<T2

AECDomain dependent rules

Example:

initially(¬have(money))

initiates(buy(X), T, have(X) )  ¬X=money

initiates(borrow(X), T, have(X))

AECDomain dependent rules

Another Example

Actions can be communicative actions : tell(Ag1, Ag2, Content, D)

initially(no-info(tr-ag, arrival(tr101))

initiates(tell(X, tr-ag, inform(Q,I), D),T,have-info(tr-ag,Q,I)) ¬ holds_at(trustworthy(X),T)

terminates(tell(X, tr-ag, inform(Q,I), D),T,no-info(tr-ag,Q)) ¬ holds_at(trustworthy(X),T)

precondition(tell(tr-ag,X, inform(Q,I), D), have-info(tr-ag,Q,I))

AEC
• Abducible happens
• Domain Independent Integrity Constraints

holds_at(F,T) & holds_at(F,T)  false

happens(A,T)& precondition(A,P)holds_at(P,T)

• and any Domain Dependent Integrity Constraints

For example:

holds_at(open-shop, T)  T≥9 & T≤18

happens(tell(a, Ag, accept(R), D), T) &

happens(tell(a, Ag, refuse(R), D), T)  false

ALP with Constraints
• Notice that AEC has times and constraint predicates, T1<T2, T1T2, T1=T2, etc.
• We can extend the notion of abductive answer to cater for these, in the spirit of Constraint Logic Programming (CLP).
• Structure R consisting of
• a domain D(R) and
• a set of constraint predicates and
• an assignment of relations on D(R) for each such constraint predicate.

ALP with ConstraintsSemantics

Given an ALP with constraints,

< P,A,IC,R>, an abductive answer for a goal G is a Δ=(D, C) such that

D is in terms of the abducible predicates, and

for all groundings  of the variables in G, D, C such that satisfies C (according to R)

G holds in P  D

IC holds in P  D or P  D IC is consistent.

Back to AEC

Given AEC and a goal G

holds_at(g1, T1) &holds_at(g2, T2) & … & holds_at(gn, Tn)

an answer for G is a parially ordered plan for achieving G.

That is an answer for G is a

 = (As, TC)

As is a set of happens atoms, and

TC is a set of temporal constraints, and

 is an abductive answer to G wrt AEC with constraints.

Example

Domain dependent part :

initially(¬have(money))

initiates(buy(X), T, have(X) )  ¬X=money

initiates(borrow(X), T, have(X))

happens(buy(X), T)  T≥9 & T≤18

Example cntd.

Goal: holds_at(have(pc), T) & T<12

1= (happens(borrow(pc), T1), T1<T<12)

2= (happens(borrow(money),T1),

T1<T2, T29, T2<T<12)

Example cntd.

Goal: holds_at(have(pc), T) & holds_at(have(tv), T)

How many answers (plans) are there ????

1=(happens(borrow(pc),T1), happens(borrow(tv),T2), T1<T, T2<T)

2=(happens(borrow(money),T1),

happens(borrow(money),T3),

T1<T2, T29, T2 ≤18, T2<T3, T3<T4, T49, T4 ≤18, T4<T)

Act

Observe

Reason

Agent Cycle

Messages Goals + ALP

Environment + Observations

Effects of actions

An ALP agent cycle with explicit time

to cycle at time T,

record any observations at time T,

resume thinking,

giving priority to forward reasoning with the new observations,

evaluate to false any alternatives containing sub-goals that are not marked as observations but are atomic actions to be performed at an earlier time,

select sub-goals, that are not marked as observations, from among those that are atomic actions to be performed at times consistent with the current time,

attempt to perform the selected actions,

record the success or failure of the performed actions and mark the records as observations,

cycleat time T+ n, where n is small.

Note selecting an action involves both selecting an alternative branch of the search space and prioritising conjoint subgoals.

Agent Cycle

Now consider using AEC in an agent cycle.

We will have:

• Observations
• Plan execution
• Interleaved planning and plan execution

So we have to extend the AEC theory for this dynamic setting.

(Dynamic) AECBridge Rules

Bridge rules for connecting the AEC theory to observations:

holds_at(F,T2) ¬ observed(F,T1), T1 T2,

¬clipped(T1,F,T2)

holds_at(¬F,T2) ¬ observed(¬F,T1), T1 T2,

¬declipped(T1,F,T2)

happens(A,T) ¬ executed(A,T)

happens(A,T) ¬ observed(_, A[T], _)

happens(A, T) ¬ assume_happens(A, T)

clipped(T1,F,T2) ¬ observed(¬F,T), T1T<T2

declipped(T1,F,T2) ¬ observed(F,T), T1T<T2

(Dynamic) AEC

Abducible assume_happens

As well asabducibleswe can also have a set of observables. These are fluents (or fluent literals) that can only be proved by being observed. The agent cannot plan to achieve them.

In the KGP architecture they are called sensing goals.

E.g. shop is open

it is raining.

Observations

In general observations can involve:

• Observable predicates
• Abducible predicates
• Defined predicates – in which case the observation may be explained through abductions.

(Dynamic) AEC

Modify the set of domain independent integrity constraints:

holds_at(F,T) & holds_at(F,T)  false

assume_happens(A,T) & precondition(A,P)  holds_at(P,T)

assume_happens(A,T) &

¬executed(A,T), time_now(T’)  T>T’

Example

Domain dependent part :

initially(¬have(money))

initiates(buy(X), T, have(X) )  ¬X=money

initiates(borrow(X), T, have(X))

observable

Example cntd.

Goal: holds_at(have(pc), T) & T<12

The agent can consider two alternative plans, to borrow pc

Then an observation ¬open-shop

(maybe a notice saying the shop is closed all day or until further notice) makes the agent focus on the first plan.

Some references:

CIFF:

Endriss U., Mancarella P., Sadri F., Terreni G., Toni F., Abductive logic programming with CIFF: system description.

The CIFF proof procedure for abductive logic programming with constraints.

Both in Proceedings of Jelia 2004.

Proof Procedures

Endriss U., Mancarella P., Sadri F., Terreni G., Toni F., The CIFF proof procedure for abductive logic programming with constraints: theory, implementation and experiments. Forthcoming.

A-System:

Kakas A., Van Nuffelen B., Denecker M., A-system: problem solving through abduction. In Proceedings of the 17th Internationla Joint Conference on Artificial Intelligence, 2001, 591-596.

Proof Procedure : CIFF
• Builds on earlier work by Fung and Kowalski: The IFF proof procedure for abductive logic programming. Journal of Logic Programming, 1997.
• CIFF adds constraint satisfaction and a few other features for efficiency and extended applicability.
• Here we review the IFF proof procedure.

IFF

Roughly speaking :

Given ALP <P,A,IC> :

IFF reasons forwards with IC and backwards with the selective completion of P (wrt non-abducibles).

IFF – Selective Completion

Example of selective completion wrt non-abducibles):

have(X) if borrow(X)

IC: buy(X) & no-money  false

Selective completion of P is:

no-money iff false

i.e. the abducibles are open predicates, the rest are completed.

We can also designate the observables as open predicates.

IFF – Backward reasoning (Unfolding)

Backward reasoning (Unfolding) uses iff-definitions to reduce atomic goals

(and observations) to disjunctions of conjunctions of sub-goals.

E.G.

A goal have(pc) will be unfolded to

IFF –Forward reasoning (Propagation)

Forward reasoning(Propagation) tests and actively maintains consistency and the integrity constraints

by matching a new observation or new atomic goal p

with a condition of an implicational goal p & q  r

to derive the new implicational goal q  r.

q can be reduced to subgoals by backward reasoning (unfolding)

or can be removed by forward reasoning (propagation).

r is added as a new goal (after p &q has been removed).

r can then trigger forward reasoning

or can be reduced to sub-goals.

r is (in general) a disjunction of conjunctions of sub-goals.

IFF –Forward reasoning (Propagation)

In the example propagation will give:

[buy(pc) & (no-money  false)] or borrow(pc).

Suppose now we observe no-money.

Another propagation step will give

IFF – Some Other Inference Rules
• Logical equivalences replace a subformula by another

formula which is both logically equivalent and simpler.

These include the following equivalences used as rewrite rules:

G & true iff G G & false iff false

D or true iff true D or false iff D.

[true  D] iff D [false  D] iff true

• Splitting uses distributivity to replace a formula of the form

(D or D') & G by (D & G) or (D'& G)

• Factoring“unifies two atomic subgoals P(t) & P(s)

replacing them by the equivalent formula

[P(t) & s=t] or [P(t) & P(s) & s t]

IFF - Negation

Two versions:

• Negation re-writing:

¬P is re-written as p false.

• Combining negation re-writing and negation as failure:

Some negative literals are marked These are evaluated using special inference rules that provide the effect of NAF.

IFF - Search Space

The search space is represented by a logical formula,

e. g.

[happens(borrow(money), T1) & happens(buy(pc), T2) & T1<T2 & happens(borrow(tv), T3) & precondition(buy(pc),P)  holds_at(P,T2)]

or

[happens(borrow(pc),T)] & happens(happens(borrow(tv),T’)]

Each disjunct is analogous to an alternative branch of a prolog-like search space.

Notice - 1
• Propagation together with unfolding allows ECA or condition-action rule type of behaviour.

E & C  A E & C  G

trigger via propagation

evaluate via propagation or unfolding

fires the action or goal

Notice - 2

Factoring allows repeating actions at a later stage if an earlier attempt has not been effective.

Scenario: e-shopping – logged on with one credit card, choose item, then card fails because there are not enough funds.

Given :

happens(logon(Card), T1) & happens(logon(visa), 5)& Rest(Card)

planned action recorded observation

factoring obtains:

[happens(logon(Card), T) & Card=visa &T=5 & Rest(Card)] or

[happens(logon(Card), T) & happens(logon(visa), 5) & (T5 or Cardvisa) & Rest(Card)]

Notice – 2 cntd.
• So if Rest(Card) fails with Card=visa &T=5 the 2nd disjunct allows further attempts:
• either to logon with a different card at a later time
• or to logon with visa at a later time
• Notice also that any work done in Rest(Card) in the first attempt is saved and available in the 2nd disjunct, for example choice of item to buy.

Semantics
• LetD be a conjunction of definitionsin iff-form
• Let G be a goal, i.e.conjunction of literals
• Let IC be a set on integrity constraints
• Let O be a conjunction of positive and negative observations, including actions successfully or unsuccessfully

performed by the agent.

G & Ois an answer iff all the below hold:

• GandOare both conjunctions of formulae in the abducible and constraint (e.g.=) (or observable) predicates
• D & G & O |= G
• D & O |= O
• D & G & O |= IC.

AEC for Hierarchical Planning
• So far you have seen AEC formalised for planning from 1st principles.
• This may not be ideal in an agent setting.
• AEC lends itself to formalising plan libraries, macro-actions and hierarchical and progressive planning.
• We will look at some of these through examples.

AEC for Hierarchical Planning

A Reference:

M. Shanahan, An abductive event calculus planner. The Journal of Logic Programming, Vol 44, 2000, 207-239

Robot mail delivery domain

Primitive actions

initiates(pickup(P), T, got(P))  holds_at(in(R),T), holds_at(in(P,R),T),

initiates(putdown(P),T, in(P,R))  holds_at(in(R),T), holds_at(got(P),T)

initiates(gothrough(D), T, in(R1))  holds_at(in(R2),T), connects(D,R2,R1)

terminates(gothrough(D), T, in(R))  holds_at(in(R),T)

AEC for Hierarchical PlanningExample cntd.

Compound action

happens(shiftPack(P,R1,R2,R3),T1,T6) happens(goToRoom(R1,R2),T1,T2), ¬clipped(T2,in(R2),T3), ¬clipped(T1,in(P,R2),T3), happens(pickup(P),T3), happens(goToRoom(R2,R3),T4,T5), ¬clipped(T3,got(P),T6), ¬clipped(T5,in(R3),T6), happened(putdown(P),T6),

T2<T3<T4, T5<T6

AEC for Hierarchical PlanningExample cntd.

goToRoom(R1,R2)

maintain effect

pickup(P)

maintain effect

goToRoom(R2,R3)

maintain effect

putdown(P)

shiftPack

(P,

R1,

R2,

R3)

AEC for Hierarchical PlanningExample cntd.

initiates(shiftPack(P,R1,R2,R3),T, in(P,R3))  holds_at(in(R1), T), holds_at(in(P,R2),T)

Verifiable:

The effects of compound actions should follow from the effects of their sub-actions. This can be verified formally, or, in this case by inspection.

AEC for Hierarchical PlanningExample cntd.

happens(goToRoom(R,R),T,T)

happens(goToRoom(R1,R3),T1,T3)  connects(D,R1,R2), happens(gothrough(D),T1), ¬clipped(T1,in(R2),T2), T1<T2

happens(goToRoom(R2,R3),T2,T3)

initiates(goToRoom(R1,R2),T,in(R2))  holds_at(in(R1),T)

Conditional and recursive compound action

AEC for Hierarchical Planningin agent model

Hierarchical planning in the agent model allows :

• Building heuristics and expertise into planning
• Generating actions in progressive order – first action first (as opposed to regressive order – last action first).

Progressive planning fits well within an agent cycle:

• A partial plan can be executed and give useful results.
• Observe effect of actions and the state of the environment to decide whether it is worth continuing with that plan.

ALP for Reactivity
• To specify reactions to the changes/events in the environment, similar to condition-action rules
• To specify plan repair steps
• To specify interaction policies

specify reactions to the environment

ALP allows and extends the active behaviour provided by ECA/Condition-Action rules and gives it semantics.

ALP for Reactivity

Domain Dependent Integrity Constraints in the ALP can be specified and used to achieve reactive behaviour.

Triggers, Other-Conditions  Reaction

Conjunction of:Conjunction of:

Observations holds_at(F,T)

Executed actions happens(A,T)

Planned actions temporal constraints

Examples

Informal syntax

in room R at time T & alarm sounds in R at T  leave R at T+1

(Smart Home Ambient Intelligence AmI- See for example work by Augusto et al on agent-based ECA-based AmI)

P leaves kitchen at T & gas is on at T &

¬ X enters kitchen by T+15 

raise alarm at T+15

Plan Repair

Most agent models that have plan repair use (ad hoc) ECA/Condition-Action rules for this purpose (see, for example 2APL/3APL).

A reference for 3APL:

M. Dastani, B, van Riemsdijk, F. Dignum, J-J. Meyer, A programming language for cognitive agents goal directed 3APL. Proceedings of the First Workshop on Programming Multiagent Systems: Languages, frameworks, techniques, and tools (ProMAS03) to be held at AAMAS'03, Melbourne, July 2003.

Plan Repair

An example from 2APL:

Goto(R2,R1);X <- not pos(R2) and pos(R3)| {goto(R3,R1);X}

An example from 3APL:

G(on(X,Y))<- B(tooheavy(X) and ¬heavy(Z)) | G(on(Z,Y)

Plan Repair

Using ALP we can

• Formalise and use such ad hoc (active) rules for plan repair – examples seen earlier
• Achieve the behaviour of such rules just by executing AEC – example on next slide
• Derive such specialised plan repair rules from the general AEC theory and then use them explicitly – example on slide after next
• Simulate TR-Program type of behaviour by manipulating the search space and search strategy – work in progress

Plan Repair

Achieve the behaviour of such rules just by executing AEC:

Example:

Goto(R2,R1);X <- not pos(R2) and pos(R3)| {goto(R3,R1);X}

In AEC there will be a goal, e.g pos(r1) requiring an action goto(Y,r1) with a precondition pos(Y).

Plan Repair

pos(r1)

goto(Y,r1)

pos(Y), goto(Y,r1)

Y=r2, ¬clipped(pos(r2)), goto(Y,r1)

observe(pos(r3))

This entails clipped(pos(r2)) and pos(r3)

Y=r3, ¬clipped(pos(r3)), goto(Y,r1)

Plan Repair

Deriving specialised plan repair rules from the general AEC theory

E-shopping Example using simplified notation :

logged on with one credit card, choose item, then card fails because there are not enough funds. We want to use another card, but first we have to logout the first card.

Specialised plan repair rule

logon(Card1,T1) & logged-on(Card2, T2) & T2<T1 logout(Card2, T3) & T2<T3<T1

Such a rule is derivable from AEC and domain-dependent rules

precondition(logon(Card), ¬logged-on(Card’))

terminates(logout(Card), T, logged-on(Card))

Plan Repair

How? Briefly : Given

happens(logon(Card1),T1)

holds(¬logged-on(Card2), T1)

using the precondition IC

holds(logged-on(Card2),T1)  false

using the F and ¬F cannot hold together IC

happens(logon(Card2),T2) & T2<T1 & ¬clipped(T2, logged-on(Card2), T1)  false

using definition of holds

happens(logon(Card2),T2) & T2<T1clipped(T2,logged-on(Card2),T1)

using negation re-writing

happens(logon(Card2),T2) & T2<T1 happens(logoout(Card2),T3) & T2<T3<T1

using definition of clipped and terminates

So:

happens(logon(Card1),T1) & happens(logon(Card2),T2) & T2<T1  happens(logoout(Card2),T3) & T2<T3<T1

ALP for negotiation

Some references:

Sadri F., Toni F., Torroni P.:

An abductive logic programming architecture for negotiating agents, Jelia 02.

Minimally intrusive negotiating agents for resource sharing, IJCAI 03.

Dialogues for negotation: agent varieties and dialogue sequences, ATAL 01.

Endriss U., Maudet N., Sadri F., Toni F.: Protocol conformance for logic-based agents, IJCAI 03.

F. Sadri: Multi-agent Cooperative Planning and Information Gathering, 11th International Workshop CIA 2007 on Cooperative Information Agents, September 2007, LNAI series by Springer Verlag.

ALP for negotiation
• Dialogues between agents as a means of interaction
• Often based on fixed protocols (rules of interaction)
• Negotiation is one form of dialogue
• Others include (Classification of dialogues [Walton & Krabbe, 1995])
• Persuation
• Information seeking

ALP for negotiation

Negotiation:

• Negotiation is “the process by which a group of agents communicate with each other to try and come to a mutually acceptable agreement on some matter”.

[Bussman & Muller 1992]

• One reason agents may negotiate is for resource sharing and allocation.

ALP for negotiationGeneral Idea 1

Agent1 has a plan requiring resources A,B. It has A,E, and is missing B.

Agent2 has a plan requiring resources D,E. It has B,C,D, and is missing E.

Can you give me B?

Yes, if you give me E.

ALP for negotiationGeneral Idea 2

Can you give me B?

No, why do you want B?

I have a goal G and a plan

bla bla and I need B for it.

Well, You can solve G with plan bla’ bla’ which needs C but not B. I can give you C if you give me E.

ALP for negotiationGeneral Idea 3

Can you give me B

from 9 to 5?

No, I can give it to you at 9

but I need it back at 3.

ALP for negotiationCommunication language

Communication language

tell(Ag1,Ag2,Content,D)

Content can be :

request request(give(R)) request(give(R),(Ts,Te)

accept

refuse

promise promise(R, (T1,T2), (T3,T4))

Change change(promise(R, (T1,T2), (T3,T4)))

Challenge challenge(request(give(R)))

....

ALP for negotiationInteraction Policies

Expressed as integrity constraints of the form

Pi & C  Pi+1

Dialogue move

Intended meaning:

If the agent receives a move pi and the conditions C are satisfied in its KB then it generates a move Pi+1.

ALP for negotiationInteraction Policies Examples

observed(C, tell(C,a,request(R),D,T1),T) &

holds_at(have(R),T1) & holds_at(¬need(R),T1) 

happens(tell(a,C,accept(request(R)),D,T1),T2) & T+5>T2>T

observed(C, tell(C,a,request(R),D,T1),T) &

holds_at(need(R),T1) 

happens(tell(a,C,refuse(request(R)),D,T1),T2) & T+5>T2>T

observed(C, tell(C,a,request(R),D,T1),T) &

holds_at(¬have(R),T1) 

happens(tell(a,C,refuse(request(R)),D,T1),T2) & T+5>T2>T

Interaction Protocols

These policies conform to the following simple protocol

request accept

refuse

Interaction Protocols

request refuse

accept

promise

change promise

accept promise