Second-hand information and imperfect information sources

Transcription

1 Second-hand information and imperfect information sources Laurence Cholvy ONERA Toulouse, France EUMASS 2013, december, Toulouse 1/10 Cholvy Second-hand information and imperfect information sources

2 Context: agents getting information reported by other agents which are not necessarly perfect. 2/10 Cholvy Second-hand information and imperfect information sources

3 Context: agents getting information reported by other agents which are not necessarly perfect. My friend, who lives in the mountains, tells me that the temperature has risen there? 2/10 Cholvy Second-hand information and imperfect information sources

4 Context: agents getting information reported by other agents which are not necessarly perfect. My friend, who lives in the mountains, tells me that the temperature has risen there? My neighbour tells me that his friend, who lives in the mountains, told him that the temperature has risen there. (second-hand information) 2/10 Cholvy Second-hand information and imperfect information sources

5 Context: agents getting information reported by other agents which are not necessarly perfect. My friend, who lives in the mountains, tells me that the temperature has risen there? My neighbour tells me that his friend, who lives in the mountains, told him that the temperature has risen there. (second-hand information) My neighbour s wife tells me that her husband told her that his friend, who lives in the moutains, told him that the temperature has risen there. (third-hand information) 2/10 Cholvy Second-hand information and imperfect information sources

6 First question: Q 1 : Can one accept as a new belief, a piece of information provided by agents who cite some other agents? 3/10 Cholvy Second-hand information and imperfect information sources

7 First question: Q 1 : Can one accept as a new belief, a piece of information provided by agents who cite some other agents? Can I conclude that the temperature has risen? 3/10 Cholvy Second-hand information and imperfect information sources

8 First question: Q 1 : Can one accept as a new belief, a piece of information provided by agents who cite some other agents? Can I conclude that the temperature has risen? Second question: Q 2 : What can I assume about the different agents so that I believe information they provide? 3/10 Cholvy Second-hand information and imperfect information sources

9 First question: Q 1 : Can one accept as a new belief, a piece of information provided by agents who cite some other agents? Can I conclude that the temperature has risen? Second question: Q 2 : What can I assume about the different agents so that I believe information they provide? On which conditions (about my neighbour, her wife, his friend) can I conclude that the temperature has risen? 3/10 Cholvy Second-hand information and imperfect information sources

10 A propositional modal logic with two families of operators: B i (beliefs) and R i (reporting). Axioms schemata of classical propositional logic (D B ) B i p B i p (K B ) B i p B i (p q) B i q (MP) p p q (Nec Bi ) (RE Ri ) q p B i p p q R i p R i q 4/10 Cholvy Second-hand information and imperfect information sources

11 Properties of agents [Demolombe 2004]: valid agents and complete agents valid(i, ϕ) R i ϕ ϕ complete(i, ϕ) ϕ R i ϕ 5/10 Cholvy Second-hand information and imperfect information sources

12 Properties of agents [Demolombe 2004]: valid agents and complete agents valid(i, ϕ) R i ϕ ϕ complete(i, ϕ) ϕ R i ϕ misinformer(i, ϕ) R i ϕ ϕ falsifier(i, ϕ) ϕ R i ϕ 5/10 Cholvy Second-hand information and imperfect information sources

13 Answering question (Q 1 ) B i R j R k ϕ B i valid(j, R k ϕ) B i valid(k, ϕ) B i ϕ B i R j R k ϕ B i valid(j, R k ϕ) B i misinformer(k, ϕ) B i ϕ B i R j R k ϕ B i misinformer(j, R k ϕ) B i complete(k, ϕ) B i ϕ B i R j R k ϕ B i misinformer(j, R k ϕ) B i falsifier(k, ϕ) B i ϕ My neighbour told me that his friend told him that the temperature has risen. I know that when my neighbour says such a sentence, it is false i.e, I can infer that his friend did not tell him that the temperature has risen. But suppose that I know that his friend always inform him when the temperature rises. I can conclude that the temperature had not risen. 6/10 Cholvy Second-hand information and imperfect information sources

14 Anwering question (Q 2 ) Let B i the set of explicit beliefs of i. A a formula. 7/10 Cholvy Second-hand information and imperfect information sources

15 Anwering question (Q 2 ) Let B i the set of explicit beliefs of i. A a formula. (Q 2 ) is modelled by: 7/10 Cholvy Second-hand information and imperfect information sources

16 Anwering question (Q 2 ) Let B i the set of explicit beliefs of i. A a formula. (Q 2 ) is modelled by: What are the formulas H: 1. B i {H} = B i A 2. H is composed with atomic formulas of the form B a valid(.,.), B a misinformer(.,.), B a complete(.,.) or B a falsifier(.,.). 3. B i {H} is consistent. 4. H is minimal, i.e, if H statisfies the two previous constraints and = H H then = H H. 7/10 Cholvy Second-hand information and imperfect information sources

17 Anwering question (Q 2 ) Let B i the set of explicit beliefs of i. A a formula. (Q 2 ) is modelled by: What are the formulas H: 1. B i {H} = B i A 2. H is composed with atomic formulas of the form B a valid(.,.), B a misinformer(.,.), B a complete(.,.) or B a falsifier(.,.). 3. B i {H} is consistent. 4. H is minimal, i.e, if H statisfies the two previous constraints and = H H then = H H. Find premisses which miss for deducing a given conclusion: abductive reasoning or equiv. logical consequence generation problem. 7/10 Cholvy Second-hand information and imperfect information sources

18 SOL-resolution (Skip Ordered Linear Resolution) is an inference rule sound and complete for generating 1. logical consequences of a set of clauses 4. minimal for the subsumption 2. belonging to a given language 3. satisfying some constraints When 2, 3 satisfy some condition (stable production field.) Inference rule defined in First Order Logic Reformulate Q 2 in First Order Logic 8/10 Cholvy Second-hand information and imperfect information sources

19 Def i = { B(i, valid(x, y)) (B(i, R(x, y)) B(i, y)), B(i, misinformer(x, y) (B(i, R(x, y)) B(i, not(y))), B(i, complete(x, y) (B(i, not(r(x, y))) B(i, not(y))), B(i, falsifier(x, y) (B(i, not(r(x, not(y)))) B(i, not(y)))} 9/10 Cholvy Second-hand information and imperfect information sources

20 Def i = { B(i, valid(x, y)) (B(i, R(x, y)) B(i, y)), B(i, misinformer(x, y) (B(i, R(x, y)) B(i, not(y))), B(i, complete(x, y) (B(i, not(r(x, y))) B(i, not(y))), B(i, falsifier(x, y) (B(i, not(r(x, not(y)))) B(i, not(y)))} (q 2 ): what are the clauses h: 1. Def i B i h = B(i, A) 2. h is composed of atomic formulas of the form B(i, valid(.,.)), B(i, misinformer(.,.)), B(i, complete(.,.)), B(i, falsifier(.,.)). 3. Def i B i h is consistent 4. h is minimal for subsumption 9/10 Cholvy Second-hand information and imperfect information sources

21 Def i = { B(i, valid(x, y)) (B(i, R(x, y)) B(i, y)), B(i, misinformer(x, y) (B(i, R(x, y)) B(i, not(y))), B(i, complete(x, y) (B(i, not(r(x, y))) B(i, not(y))), B(i, falsifier(x, y) (B(i, not(r(x, not(y)))) B(i, not(y)))} (q 2 ): what are the clauses h: 1. Def i B i h = B(i, A) 2. h is composed of atomic formulas of the form B(i, valid(.,.)), B(i, misinformer(.,.)), B(i, complete(.,.)), B(i, falsifier(.,.)). 3. Def i B i h is consistent 4. h is minimal for subsumption Conjecture: answering (q 2 ) is equivalent to answering (Q 2 ) 9/10 Cholvy Second-hand information and imperfect information sources

22 Def i = { B(i, valid(x, y)) (B(i, R(x, y)) B(i, y)), B(i, misinformer(x, y) (B(i, R(x, y)) B(i, not(y))), B(i, complete(x, y) (B(i, not(r(x, y))) B(i, not(y))), B(i, falsifier(x, y) (B(i, not(r(x, not(y)))) B(i, not(y)))} (q 2 ): what are the clauses h: 1. Def i B i h = B(i, A) 2. h is composed of atomic formulas of the form B(i, valid(.,.)), B(i, misinformer(.,.)), B(i, complete(.,.)), B(i, falsifier(.,.)). 3. Def i B i h is consistent 4. h is minimal for subsumption Conjecture: answering (q 2 ) is equivalent to answering (Q 2 ) We can apply SOL-resolution (constraints 2, 3 are OK)... but with ground(def i, n) instead of Def i. 9/10 Cholvy Second-hand information and imperfect information sources

23 Find the conditions under which the conjecture is OK Extend the model to topics Extend the model to uncertainty How can we decide that an agent is valid/misinformer/...? 10/10 Cholvy Second-hand information and imperfect information sources