Models of Distributed Reasoning - 0.5ex-3ex - Laboratoire …meghyn/papers/slides-mdr.pdf · 2010-11-30 · Models of Distributed Reasoning Meghyn Bienvenu ... What new reasoning

Models of Distributed ReasoningMeghyn Bienvenu (CNRS – Univ. Paris Sud)

November 22, 2010

1. About this course

General course information.

• Time and place: Mondays 14h-17h in room FIRTECH

• Duration: 8 sessions, ending January 31st

• Instructors (in order of appearance):Meghyn Bienvenu, François Goasdoué, Philippe Chatalic,Laurent Simon, Philippe Dague, Pascal Poizat, Fatiha Zaidi

• Evaluation: 1/3 coursework + 2/3 final exam

1. About this course 3/44

Course plan.

1 Nov. 22 Introduction to the course2 Nov. 29 Propositional reasoning in P2P systems3 Dec. 6 Inconsistency-handling in P2P systems4 Dec. 13 RDFS querying in P2P setting5 Jan. 10 Distributed satisfiability algorithms6 Jan. 17 Distributed diagnosis algorithms7 Jan. 24 Web service composition8 Jan. 31 Web service verification


Format of coursework.

Coursework will be done in groups of two students.

Each pair of students will have two research articles (in English).

Each student will be responsible for one of the articles.

The group will deliver a five-page report consisting of:

• two-page summary of each article (individually prepared)

• a single page comparing and contrasting the two articles(jointly prepared)


Format of coursework, cont..

Each pair of articles relates to one session of the course.

Research reports will be handed in to the instructor associated withthe session.

A short interview will also be scheduled with the instructor.

Timing: reports due two sessions after corresponding session

Note: individual grades, based on the report and interview.


Distribution of articles.

Groups and articles will be decided today.

A handout with descriptions of the available articles will beprovided.

During the break: please try to find a partner and decide whichpairs of articles you are interested in.

At the end of the lecture, I will ask for your preferences and usethem to assign the articles.


2. Introduction to Distributed Reasoning

What is a distributed system?.

Group of computing devices which communicate via a network.

Different possible architectures:

Client-server

Example: mail server

Peer-to-peer

Example: file-sharing

2. Introduction to Distributed Reasoning 9/44

Reasoning and distributed systems.

Aim of this course:Investigate relationship between reasoning and distributed systems

Three perspectives on the interaction of these two areas:

1. How to extend traditional reasoning algorithms to adistributed context?

2. How to exploit distributed computing to improve theefficiency of reasoning algorithms?

3. What new reasoning tasks are relevant to distributed systems,and how can we solve them?









We will study various types of reasoning in distributed systems.

1 Nov. 22 Introduction to the course2 Nov. 29 Propositional reasoning in P2P systems3 Dec. 6 Inconsistency-handling in P2P systems4 Dec. 13 RDFS querying in P2P setting5 Jan. 10 Distributed satisfiability algorithms6 Jan. 17 Distributed diagnosis7 Jan. 24 Web service composition8 Jan. 31 Web service verification


Reasoning in P2P systems.Peer data management systems

set of independent peers, each with its own knowledge base (KB),which may be expressed using its own vocabulary

______________________________________________________________________________________________________________

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>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>


Advantages + disadvantages.Advantages over centralized data management systems:• flexibility: peers can enter and leave system• robustness: system does not depend on a central authority• personalization: each peer describes its data as it wishes• privacy: each peer controls its own data

Possible disadvantages (or rather challenges ):• since no peer has full view of the system,

can be hard to adapt centralized reasoning algorithms• peers may use different vocabularies, so need way of

translating information between peers• peers may disagree, so inconsistency-handling very important


Advantages + disadvantages.Advantages over centralized data management systems:• flexibility: peers can enter and leave system• robustness: system does not depend on a central authority• personalization: each peer describes its data as it wishes• privacy: each peer controls its own data

Possible disadvantages (or rather challenges ):• since no peer has full view of the system,

can be hard to adapt centralized reasoning algorithms• peers may use different vocabularies, so need way of

translating information between peers• peers may disagree, so inconsistency-handling very important


Propositional peer inference systems.

In the next two courses, we suppose peers have propositional KBs.

Why use propositional logic?

• simple yet powerful logic

• basis for study of other languages (e.g. RDF)

Reasoning task we will study: consequence finding (CF)


Very brief review of propositional logic.Key concepts you should know from the “tronc commun”:

Syntax:conjunction (∧), disjunction (∨), negation (¬), implication (→),bi-implication (↔), literal, clause, term, conjunctive normal form(CNF), disjunctive normal form (DNF)

Semantics:model = assignment of truth value to each variable/atom,notions of satisfiability & logical consequence (entailment)

Proof systems:resolution rule (α ∨ x + β ∨ ¬x⇒ α ∨ β),principle of refutation, correctness, completeness


Deduction vs. consequence finding.Deduction: test whether formula entailed

ϕψ

ϕ |= ψ ? yes

no

Consequence finding: generate entailed formulas

ϕ ϕ |= .... ? α

β

ω

δζ

Note: consequence finding more general than satisfiability / deduction2. Introduction to Distributed Reasoning 16/44

Consequence finding defined.

Of course, we don’t want to produce all consequences,since most are redundant or irrelevant.

So we generate a representative subset of them. Specifically:

prime implicates = logically strongest entailed clauses

Formally: a prime implicate of φ is a clause λ such that(a) φ |= λ

(b) there is no clause λ′ such that φ |= λ′ |= λ but λ ̸|= λ′

Consequence finding = generate formula’s prime implicates


Consequence finding using resolution.We can use the resolution rule to do consequence finding.

1. Put input formula into CNF2. Apply resolution rule to get new clause3. Remove any subsumed clauses in the clause set4. Repeat steps 2 and 3 until no new clause can be produced

Important property: above procedure is complete(i.e. guaranteed to produce all prime implicates)

More sophisticated resolution algorithms can be used to moreefficiently generate only those prime implicates built from a givenvocabulary.


Example: CF using resolution.

a ∨ b ¬e ∨ ¬b ∨ cd ∨ e ¬c ∨ d

e ∨ ¬d



a ∨ b ¬e ∨ ¬b ∨ cd ∨ e ¬c ∨ d

e ∨ ¬d

d ∨ ¬b ∨ c



a ∨ b ¬e ∨ ¬b ∨ cd ∨ e ¬c ∨ d

e ∨ ¬d

d ∨ ¬b ∨ c



a ∨ b ¬e ∨ ¬b ∨ cd ∨ e ¬c ∨ d

e ∨ ¬d

d ∨ ¬b ∨ c

a ∨ d ∨ c



a ∨ b ¬e ∨ ¬b ∨ cd ∨ e ¬c ∨ d

e ∨ ¬d

d ∨ ¬b ∨ c

a ∨ d ∨ c



a ∨ b ¬e ∨ ¬b ∨ cd ∨ e ¬c ∨ d

e ∨ ¬d

d ∨ ¬b ∨ c

a ∨ d ∨ c

e



a ∨ b ¬e ∨ ¬b ∨ cd ∨ e ¬c ∨ d

e ∨ ¬d

d ∨ ¬b ∨ c

a ∨ d ∨ c

e



a ∨ b ¬e ∨ ¬b ∨ cd ∨ e ¬c ∨ d

e ∨ ¬d

d ∨ ¬b ∨ c

a ∨ d ∨ c

e



a ∨ b ¬e ∨ ¬b ∨ c¬c ∨ d

d ∨ ¬b ∨ c

a ∨ d ∨ c

e



a ∨ b ¬e ∨ ¬b ∨ c¬c ∨ d

d ∨ ¬b ∨ c

a ∨ d ∨ c

e

¬b ∨ c



¬b ∨ c

a ∨ b ¬e ∨ ¬b ∨ c¬c ∨ d

d ∨ ¬b ∨ c

a ∨ d ∨ c

e



¬b ∨ c

a ∨ b ¬e ∨ ¬b ∨ c¬c ∨ d

d ∨ ¬b ∨ c

a ∨ d ∨ c

e



¬b ∨ c

a ∨ b¬c ∨ d

a ∨ d ∨ c

e

a ∨ c



a ∨ c

¬b ∨ c

a ∨ b¬c ∨ d

a ∨ d ∨ c

e



a ∨ c

¬b ∨ c

a ∨ b¬c ∨ d

e¬b ∨ d



¬b ∨ da ∨ c

¬b ∨ c

a ∨ b¬c ∨ d

e



¬b ∨ da ∨ c

¬b ∨ c

a ∨ b¬c ∨ d

ea ∨ d



a ∨ d

¬b ∨ da ∨ c

¬b ∨ c

a ∨ b¬c ∨ d

e



a ∨ d

¬b ∨ da ∨ c

¬b ∨ c

a ∨ b¬c ∨ d

e

No further resolutions or deletions possible, so we’re done!

Remaining clauses are exactly the prime implicates.


Back to propositional P2P systems.

Basic setup:• each peer P has its own propositional KB,

using its own vocabulary VPI each peer’s theory is set of clausesI can assume variables of peer i are of the form i : v

• a peer can establish connections with other peers by declaringmappings, which relate its vocabulary to those of other peers

I formally: clauses using variables from different peersI e.g. pair of mappings ¬2 : b ∨ 1 : a and ¬1 : a ∨ 2 : b states

that variables 1 : a and 2 : b are equivalent


Distributed consequence finding.DECA: fully decentralised message-passing CF algorithm• input: a propositional clause, in selected peer’s vocabulary• output: all proper prime implicates of input clause and global

theory, which are formulated using given target variables

******************************************************************************************************************************************************

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########################################################################

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

1 : a

¬1 : a ∨ 1 : b ∨ ¬1 : c




******************************************************************************************************************************************************

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########################################################################

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

1 : a 1 : b

1 : c

¬1 : a ∨ 1 : b ∨ ¬1 : c1 : b ∨ ¬1 : c




******************************************************************************************************************************************************

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ##################

########################################################################

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

1 : a 1 : b

1 : c

¬1 : a ∨ 1 : b ∨ ¬1 : c1 : b ∨ ¬1 : c




******************************************************************************************************************************************************

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ##################

########################################################################

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

1 : a 1 : b

1 : c

¬1 : a ∨ 1 : b ∨ ¬1 : c1 : b ∨ ¬1 : c




******************************************************************************************************************************************************

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ##################

########################################################################

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

1 : a 1 : b

1 : c

¬1 : a ∨ 1 : b ∨ ¬1 : c1 : b ∨ ¬1 : c

4 : e ∨ 5 : f

¬3 : h17 : p




******************************************************************************************************************************************************

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% ##################

########################################################################

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

1 : a 1 : b

1 : c

¬1 : a ∨ 1 : b ∨ ¬1 : c1 : b ∨ ¬1 : c

4 : e ∨ 5 : f

¬3 : h17 : p

4 : e ∨ 5 : f ∨ 17 : p


Handling inconsistency.Reasonable to suppose each peer has consistent KB.

Not reasonable to whole system consistent!!

Well-founded consequence: supported by consistent subset ofglobal theory

1 : a1 : b ∨ ¬1 : c

¬1 : a ∨ 2 : b

¬2 : b ∨ ¬2 : c

1

2

3

2 : c ∨ ¬2 : d

3 : d3 : c

¬3 : c ∨ 2 : c¬3 : d ∨ ¬2 : e

m1m2m3

¬2 : e is well-founded consequence(need to use both m1 and m2, which induce contradiction)


Handling inconsistency.Reasonable to suppose each peer has consistent KB.

Not reasonable to whole system consistent!!

Well-founded consequence: supported by consistent subset ofglobal theory

1 : a1 : b ∨ ¬1 : c

¬1 : a ∨ 2 : b

¬2 : b ∨ ¬2 : c

1

2

3

2 : c ∨ ¬2 : d

3 : d3 : c

¬3 : c ∨ 2 : c¬3 : d ∨ ¬2 : e

m1m2m3

¬3 : c NOT well-founded consequence(need to use both m1 and m2, which induce contradiction)


RDF reasoning in P2P setting.

In the fourth course, we consider a somewhat different setup.

Now peers have RDF data (triples) and ontologies (schema).

Relationship between peer vocabularies also via RDF ontologies.

Reasoning task: conjunctive query answering

Technique we employ: reduction to propositional reasoning

Result of CF = rewriting of input conjunctive query

Domain typing: domain of property P is A ; ¬P ∨ Adom

Class inclusion: A subclass of B ; ¬Adom∨Bdom,¬Arange∨Brange


We will study various types of reasoning in distributed systems.



What is abductive reasoning?.Abductive reasoning aims at generating explanations.

Formalization as a logical reasoning problem:

• Input: background theory B + effects / observations E +hypothesis vocabulary V (set of variables)

• Output: hypotheses / causes / explanations H satisfying:I B ∧ H is consistent (i.e. H is possible)I B ∧ H |= E (i.e. H explains E)I H is a cube (= conjunction of literals) over V

Important: Abductive reasoning can be rephrased in termsof consequence finding, using the following equivalence.

B ∧ H |= E if and only if B ∧ ¬E |= ¬H2. Introduction to Distributed Reasoning 25/44

Example: abductive diagnosis.Consider two inverters I and J in series:

I J0 1

Assume each inverter can behave according to 4 distinct andexhaustive behavioural modes:• correct (hypotheses IC and JC): IC → (outputI ↔ ¬inputI)

• faulty as followers (hyp. IF and JF): IF → (outputI ↔ inputI)

• faulty stuck-at-0 (hyp. I0 and J0): I0 → ¬outputI

• faulty stuck-at-1 (hyp. I1 and J1): I1 → outputI

Here outputI means I outputs 1 (so ¬outputI means I outputs 0).


Example, cont..Our background theory B consists of the 8 formulas above, plus:

inputJ ↔ outputI ¬inputI

The observation E to explain is the faulty output: outputJ

Our hypothesis vocabulary is: {IC, IF, I0, I1, JC, JF, J0, J1}

We compute the prime implicates of B ∧ ¬E on vocabulary V.

¬J1,¬IF ∨ ¬JC,¬IC ∨ ¬JF,¬I0 ∨ ¬JC,¬I1 ∨ ¬JF

Taking negation yields 8 possible explanations (4 single faults; 4double faults):

J1(+IC/IF/I0/I1) IF ∧ JC IC ∧ JF I0 ∧ JC I1 ∧ JF


Consistency-based reasoning.

What happens if the causes are not all known (incompleteness)?

Or, if the dependencies between causes and effects are not known?(e.g. in diagnosis: no fault model available, only correct model)

Then we should consider a weaker form of reasoning thanabduction, where we require only consistency with the effects(rather than their entailment):

• Input: background theory B; effects E; vocabulary V

• Output: hypotheses H (= valuation over V) such thatB ∧ H ∧ E is consistent


Example: consistency-based diagnosis.Same example, but with only correct behavioural modes:

IC → (outputI ↔ ¬inputI) JC → (outputJ ↔ ¬inputJ)

Hypothesis vocabulary: V = {IC, JC}

Want to find hypotheses that consistently extend B ∧ E.

First compute prime implicates on V: B ∧ E |= ¬IC ∨ ¬JC

Next, prime implicants of the prime implicate(s): {¬IC,¬JC}

These code concisely the consistent models

The minimal consistent hypotheses are thus (single faults):

IC ∧ ¬JC ¬IC ∧ JC


Distributed diagnosis.Want to move from centralized to distributed case (e.g. P2P).

General ideas:

• Distributed abductive diagnosis: do distributed CF, i.e.distributed computation of prime implicates (cf. session 2)

• Distributed consistency-based diagnosis:I avoid using intermediate step of prime implicates computationI find directly the consistent models restricted to VI much easier if local theories are in DNF instead of CNF:

it will be assumed that peer theories are small enough for theinitial transformation CNF –> DNF to be tractable

I do distributed computation of prime implicants









Extending traditional reasoning algorithms to P2P networks.



Propositional satisfiability.

Satisfiability = does a satisfying truth assignment exist?

Many practical problems can be reduced to satisfiability (SAT):

• planning (generate a sequence of actions to achieve goal)

• verification (finding bugs in programs)

• cryptography (decryption, finding keys)

• biology (identify common genes in population)

• ...

For this reason, lots of work on improving SAT solvers.


Simple SAT algorithm.

Algorithm SAT

Input: a set Σ of propositional clauses, a partial valuation vOutput: true if v can be extended to a sat. valuation for Σ, else false

remove from Σ all clauses which are satisfied by vif Σ is now empty, then return trueremove from all clauses in Σ all literals made false by vif Σ contains an empty clause, then return falsechoose some variable x which does not appear in vlet v′ (resp. v′′) be v extended by x← true (resp. x← false)if SAT(Σ,v′)=true then return trueelse return SAT(Σ,v′′)


Example of SAT algorithm.

¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ =

∅



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

{a}

t



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

{a}

t



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

{a}

t



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

{a, b}

t

t



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

{a, b}

t

t

t



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

{a, b}

empty clause!t

t

t

false



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

{a,¬b}

t

t f

false



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

{a,¬b}

t

t f

false



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

{a,¬b}

t

t f

false



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

t

t f

false

t{a,¬b, c}

c



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

t

t f

false

t{a,¬b, c}

c

empty clauses!

false2. Introduction to Distributed Reasoning 35/44


¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

t

t f

false

t

c

false

f{a,¬b,¬c}



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

t

t f

false

t

c

false

f{a,¬b,¬c}



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

t

t f

false

t

c

false

f

true

no clauses left!

{a,¬b,¬c}



¬b ∨ a ∨ c

¬c ∨ b

¬a ∨ ¬c

¬b ∨ ¬a

v =

Σ = a

b

t

t f

false

t

c

false

f

true

SAT!!

{a,¬b,¬c}


Exploiting distribution to improve efficiency.Two ideas:• split original problem into sub-problems, assigned to different

computers• give whole problem to each computer, but use different

techniques on each

ϕ

ψ1 ψ2 ψ3 ψ4

ϕ











Web services: basic scenario.

Web provides not justinformation, also services.

Often need to combine multipleservices to achieve a task(e.g. travel planning).

New reasoning task:provide support for(automatically) composing webservices to satisfy user needs

login(...)logout(...)get_credit(...)ask_bill(...)

<< service >>paypal

order(...)ship(...)charge_cc(...)charge_pp(...)bill(...)finalize(...)

<< service >>ebay

order(...)cancel(...)ship(...)bill(...)charge(...)gift(...)ack(...)

<< service >>pear_store

I need a tracking number

I want to buy something

I agree to give the product and

personal info

Find me service(s) so that I can

call select, purchase, ship, ...


Components of service descriptions.Set of capabilities, which are functionalities requested by user orprovided by a service.

Set of (semantic) data types, which can be used as input or output.

Data semantic structure (DSS) = relations between data types

addressshipping addr billing addr

user addr

product

pear product

ePadePhone

user nameuser info

CC info

CC number

CC holder

amazon login

amazon pwd

paypal login

paypal pwd

product price

order amount

product specs

pear product info

PIM wallet

amazon info

paypal info


Describing web services.

Service w = (Operations O, WFO)where for each operation o = (in,out,k)

• in: inputs(what o needs)

• out: outputs(what o provides)

• k: capability(what o does)

order ship bill

cancel

charge ack

gift_wrapper

order ship

charge_pp

finalizecharge_cc bill

paypal

ebay

pear_store

login get_credit

logout

logoutask_bill

order pear product pear product info, sidPS product selection

cancel sidPS

ship shipping addr, sidPS shipping setup

bill billing addr, sidPS billing setup

charge CC info, sidPS payment

gift wrapper giftcode, sidPS payment

ack sidPS tracking number order finalization

PS

SS

SS

$

$

F


Describing user needs.

I need a tracking number

I want to buy something

I agree to give the product and

personal info

Din = {product, user info}

Dout = {tracking number}

product_selection

shipping_setup

billing_setup

payment

order_finalization

Input Din + Output Dout + Capability workflow


Web service composition via planning.Technique used: reduction to propositional planning

Propositional planning problem:

• set S ⊆ 2V of possible states• an initial state s0 ∈ S• goal g ⊆ V – variables we want to make true• a set A of actions, with each action being defined by:

I its preconditions, i.e. atoms required for action to be possibleI positive effects, i.e. atoms made true by performing actionI negative effects, i.e. atoms made false by performing action

Plan = sequence of actions transforming initial state to a goal state


Verifying and testing web service compositions.Different but related reasoning task:Check whether a given service composition satisfies user’s needs.

NeedsNeeds Conceptual Conceptual diagram.diagram.

formal modelformal modelformal modelformal modelModel Model

specificationspecification

specificationspecificationabstract BPEL, abstract BPEL,

BPMN, UMLBPMN, UML

implementationimplementationexecutable executable

BPEL, Java, BPEL, Java, .net, ....net, ...

test casetest case

!"#$"

static analysis

verify

1. translate

2. generate

conform?

3. test

test suite

%"&'(')*$'+, -.*)/0-+#1$"2$',3

CodeCodeModelModel

SourceSourcecodecode

verifyverify

$&*,2.*$"

%"&'(4


Documents

Models of Distributed Reasoning - 0.5ex-3ex - Laboratoire …meghyn/papers/slides-mdr.pdf · 2010-11-30 · Models of Distributed Reasoning Meghyn Bienvenu ... What new reasoning