A Short Course in Belief Revision Pavlos Peppas University of Technology Sydney and University of Patras Two parts: - Walk through the main developents in AI and Logic - Focus on Belief Revision This talk focuses more on motivation rather than technicalities

An Example John’s car is a BMW BMWs are made in Germany Germany is part of the EU All cars made in EU take unleaded petrol John’s car takes unleaded petrol Rational Belief Revision John’s car takes leaded petrol BMWs are made in Germany Germany is part of the EU All cars made in the EU take unleaded petrol John’s car takes leaded petrol There is life on Mars. Η νέα πληροφορία έρχεται σε αντίθεση με κάποια από τα συμπεράσματα των πρωταρχικών αντιλήψεων. Το ζητούμενο είναι ο προσδιορισμός της ορθολογικής αναθεώρησης αντιλήψεων. Εισαγωγή νέας πληροφορίας + concistency δεν αρκούν για να εξασφαλίσουν ορθολογικότητα. Πολλαπλές επιλογές για ορθολογικότητα. Διαφοροποίηση των αντιλήψεων ως προς την σπουδαιότητά τους.

Some History (Carlos Alchourron and David Makinson) Old Legal Code The children may watch TV only if they eat their dinner. The children may eat their dinner only if they do their homework ? New Legal Code Amendment: On Fridays the children may watch TV without doing their homework . Παράδειγμα διαβάθμισης προτεραιοτήτων σε αυτόν το χώρο είναι το - Κοινοτικό Δίκαιο - Σύνταγμα - Υπουργικές αποφάσεις

History II: Semantics for Counterfactuals (Peter Gardernfors) A B A  B T T T T F F F T T F F T ├ If Napoleon had not invaded Russia, then he would had conquered Europe. Κ Ramsey Test: The sentence "If A, then B" is true at belief state K, iff B is true at the state that results from the revision of K by A. K ⊨ A > B iff (K*A) ⊨ B A K*A K * ¬A B Ramsey Test  Monotonicity (i.e. K  K’  (K*A)  (K’ * A) ) TRIVIALITY RESULT: (L is trivial iff it contains no more than three propos. variables) Note that the connective > is in the object language.

The Birth of Belief Revision (1985) C. Alchourron, P. Gardernfors, and D. Makinson, “On the logic of theory change: Partial meet functions for contraction and revision”, Journal of Symbolic Logic, 1985 The AGM Framework for Theory Change Ramsey Test  Monotonicity (i.e. K  K’  (K*A)  (K’ * A) ) TRIVIALITY RESULT: (L is trivial iff it contains no more than three propos. variables) Note that the connective > is in the object language.

Rational Belief Revision Using Logic to Model Belief Revision ab ab a (ab)  d (de)  f d a b (ab)  d (de)  f Rational Belief Revision * Κ Κ*Α Beliefs are modeled as sentences of propositional logic. Belief States are modeled as sets of sentences closed under logical implication.

Models for Belief Revision Axiomatic Models AGM Revision Functions AGM Contraction Functions Levi / Harper Identity Grove, 1988 Alchourron, Gardenfors & Makinson, 1985, 1988 Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα Preorders on possible worlds Epistemic Entrenchments Selection Functions Constructive Models

The AGM Postulates for Belief Revision * A Κ Κ*Α The new belief state differs as little as possible from the old belief states, in light of the new information Principle of Minimal Change: (K*1) Κ*Α is a theory (Κ*2) Α (K*A) Πρόβλημα με την Επιλογή της Αναπαράστασης: - Υπεραπλουστευμένη (probabilities may need to be attached to senteces) - Closure ========================================================= K+A = Cn(K{A}) (K*3) + (K*4) : Limiting case where A  K (K*7) + (K*8) : If B  K*A then K*(AB) = (K*A)+B (K*1) – (K*8)  [ K*(AB) = K*A or K*B or (K*A)  (K*B) ] (K*3) K*A  K+A (K*4) If A  K then K+A  K*A (K*5) If A is consistent then K*A is consistent (K*6) If AB then K*A = K*B (K*7) K*(AB)  (K*A)+B (K*8) If B  (K*A) then (K*A)+B  K*(AB)

The Plurality of AGM Revision Functions * Κ Κ*Α For a given belief state K and new information A, the AGM postulates (K*1) - (K*8) do not specify uniquely the new belief state K*A. * If B  K then ( B  K*A )  ( B  K*A ) If A  K then K*A is complete. Proposition: (M)  (N) (K*M) If K  H then K*A  H*A Proposition: (K*1) – (K*8), (K*M)  Contradiction for non-trivial languages. Proof: K = Cn( B ), Λ = Cn( C ), and H = Cn(BC). K*(B  C) = Cn(B  C). Λ*(B  C) = Cn( B  C). (K*M)  K*(B  C)  H*(B  C) and Λ*(B  C)  H*(B  C) . Consequently, H*(B  C) is inconsistent, which however contradicts (K*5). * * * * * * * * * * * * Functions that satisfy the postulates (Κ*1) - (Κ*8)

Some Additional Conditions * A Κ Κ*Α (K*M) If K  H then K*A  H*A. Theorem: Condition (K*M) is inconsistent with (K*1) – (K*8). Proof for (K*R): ------------------ Consider any sentence B. Since AK, (AB) K. Assume that (AB) K*A. Then clearly BK*A. Assume therefore that (AB) K*A. Then, by (K*R), (AB) K*A, and consequently, (A  B)  K*A. Hence, in either case K*A contains B or  B. Proof for (K*M): -------------------- K = Cn( B ), Λ = Cn( C ), and H = Cn(BC). K*(B  C) = Cn(B  C). Λ*(B  C) = Cn( B  C). (K*M)  K*(B  C)  H*(B  C) and Λ*(B  C)  H*(B  C) . Consequently, H*(B  C) is inconsistent, which however contradicts (K*5). (K*R) If B  K and B  K*A then B  K*A. Theorem: If * satisfies (K*1) – (K*8) and (K*R), then K*A is complete whenever A  K.

Models for Belief Revision Axiomatic Models AGM Revision Functions AGM Contraction Functions Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα Preorders over possible worlds Epistemic Entrenchments Selection Functions Constructive Models

A Nice Possible World Germany Australia Greece

Possible Worlds vs Sentences All academics are rich All academics are nice A possible world is a consistent complete theory of the object language. Expressibility Problems!!! Peppas, Foo, and Williams “On the Expressibility of Propositions”, Logique et Analyse, 1992. All academics are rich and nice

Belief Revision with Worlds John’s car is a BMW BMWs are made in Germany Germany is part of the EU All cars made in EU take unleaded petrol John’s car takes leaded petrol A possible world is a consistent complete theory of the object language. Expressibility Problems!!! Peppas, Foo, and Williams “On the Expressibility of Propositions”, Logique et Analyse, 1992.

Plausibility Rankings John’s car is a BMW BMWs are made in Germany Germany is part of the EU All cars made in EU take unleaded petrol John’s car takes leaded petrol ≤ ≤ ≤ A possible world is a consistent complete theory of the object language. Expressibility Problems!!! Peppas, Foo, and Williams “On the Expressibility of Propositions”, Logique et Analyse, 1992. (S*) [K*A] = min([A], ≤)

Representation Result  Revision Functions (K*1) - (K*8) * (S*) Preorders on Possible Worlds Proof: U is a sphere iff - for all u  U, there is an A  L, s.t. u  [K*A] - if [A]  U   then [K*A]  U.

Models for Belief Revision Axiomatic Models AGM Revision Functions AGM Contraction Functions Preorders on possible worlds Epistemic Entrenchments Selection Functions Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα Constructive Models

The AGM Postulates for Belief Contraction - A Κ Κ-Α The new belief state differs as little of possible from the old belief state, in view of the sentence A that needs to be removed. Principle of Minimal Change: (K-1) Κ-Α is a theory (Κ-2) K-A  K (K-3) If A  K then K - A = K Contraction alone occurs mainly in the course of deliberation. Example: What if I hadn’t married Victoria… or two disputants disagree on the status of A, so they remove both A (the first disputant) and A the second disputant and see what follows from the remaining beliefs. Problem: Removing A alone is not enough since other beliefs may entail A. Moreover, more than one choices. (K-1) – (K-4)  If A  K then (K-A)+A  K (K-5) : RECOVERY : K-A should be large enough to give back K. ( Problimatic especially for finite bases or in a probabilistic setting). (K-1)-(K-4)  (K-5) iff K-A = K((K-A)+(A)) (K-7) + (K-8): Compare K – (AB) with K-A. When contracting with AB there seems to be a choice between giving up A or giving up B. Thus K – (AB) is in a sense less “dramatic” than giving up A or B. More generarly, (K-D) if C ⊢ B then K-B  K-C The condition (K-D) seems reasonable but is in fact too strong (see example below). Moreover (K-D)  K-(AB) = (K-A)  (K-B). Thus we weaken it to (K-7) and (K-8). Example: A: The morning train will be on time. B: My car will start this morning. C: I will be able to catch the train. A,B.C  K. C  K-A, but C  K-(AB) (since the train is more reliable than my car). Hence C  K-A but C  K-(AB). (K-7) + (K-8)  K-(AB) = K-A or K-B or (K-A)  (K-B). Minimal Change: A = “John always wears his hat when it rains”, B = “Today it rains”, C = “Today John wears his hat”. Causal links should count as well in measuring change! (K-4) If ⊭ A then A  K-A (K-5) If A K then K  (K-A)+A (K-6) If A≣B then K-A = K-B (K-7) (K-A)(K-B)  K-(A  B) (K-8) If A  K-(A  B) then K-(A  B)  K-A

Levi Identity A * Κ (Κ - (A)) + A - (A) + A K - (A) (LI) K*A = (Κ - (A)) + A Contraction and Revision were defined in terms of independently motivated postulates. Isaac Levi argued about the Levi Identity much before the postulates (1977). In fact he claims that there are only two primitive types of change – expansion and contraction. Revision is built from these two. The Levi Identity is motivated by the Principle of Minimal Change. Note that for the Levi Identity, (K-5) is not required. The theorem about the Levi identity can be refined to include only the first six revision/contraction postulates, and then can be enhanced gradually with the remaining two. (LI) - * - * THEOREM : - * - - * * (K-1) - (K-8) (K*1) - (K*8)

Harper Identity A - (Κ * (A))  K Κ *(A)  K K * (A) (HI) K-A = (Κ * (A))  K Contraction and Revision were defined in terms of independently motivated postulates. Isaac Levi argued about the Levi Identity much before the postulates (1977). In fact he claims that there are only two primitive types of change – expansion and contraction. Revision is built from these two. The Levi Identity is motivated by the Principle of Minimal Change. Note that for the Levi Identity, (K-5) is not required. The theorem about the Levi identity can be refined to include only the first six revision/contraction postulates, and then can be enhanced gradually with the remaining two. (HI) * - * - THEOREM : * - * - * - (K*1) - (K*8) (K-1) - (K-8)

Inter-definability * - - * * - * - * - (HI) (LI) (K*1) - (K*8) The recovery postulate can be withdrawn with no effect on the corresponding revision functions. Withdrawal functions: (K-1) – (K-4), (K-6). Theorem: For each withdrawal function -, there is a unique contraction function –’ that is revision equivalent to -. Moreover, -’ is the maximal withdrawal function that Is revision equivalent to -. (K*1) - (K*8) (K-1) - (K-8)

Models for Belief Revision Axiomatic Models AGM Revision Functions AGM Contraction Functions Preorders on possible worlds Epistemic Entrenchments Selection Functions Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα Constructive Models

Epistemic Entrenchment John’s car does not take unleaded petrol Germany belongs to Ε.U. All BMW are made in Germany All cars made in E.U. take unleaded petrol John’s car is a BMW    Intuition Behind Entrenchment: Minimize Loss of Epistemic Value Paradigm Approach: Kuhn and Lakatos - Informational Economy Approach. (Philosophy of Science). Η κωδικοποίηση των pragmatic factors μέσα από μια διάταξη Entrenchment  Probability. Entrenchment: Επεξηγηματική Ισχύς, Το θέμα προς συζήτηση, κλπ. (eg, in modern chemistry, knowledge about the combining weights is much more important for chemical experiments that knowledge about the color or taste of the substances. Thus, if a chemist for some reason needs to change her opinion about chemistry she would rather give up the part that concerns taste than combining weights. Epistemic entrenchment can be determined independently of the sentence by which we are revising/contracting. Epistemic entrenchment is more fundamental than contraction/revision. John’s car does not take unleaded petrol John’s car is not a BMW All BMW are made in Germany Germany belongs to E.U. All cars made in E.U. take unleaded petrol

Epistemic Entrenchment Α (E-)  Κ  Κ-Α  (EE1) If Α  B and B  C, then A  C. (ΕΕ2) If Α ⊨ Β then Α  B. (ΕΕ3) Α  AB or B  AB. (ΕΕ4) If Κ  L, then Α  K iff A  B, for all BL. (C) B  A iff B  K-(AB). (EE3): To give up AB, one needs to give up either A or B, or both. Therefore, the entrenchment of AB is equal to the either the entrenchment of A or to the entrenchment of B. (EE1) – (EE3)  Totality (Ε*) Β  (K*A) iff (A  B) < (A  B) or ⊢ A (ΕΕ5) If Α  B for all A L, then ⊨ B (C-) A  Β iff A  K-(AB) (Ε-) Β  (K-A) iff B∈K and A < A  B or ⊨ A

Representation Result  - (E-) Epistemic Entrenchments Axioms (EE1) - (EE5) Contraction Functions Axioms (K-1) - (K-8)

Models for Belief Revision Axiomatic Models AGM Revision Functions AGM Contraction Functions Preorders on possible worlds Epistemic Entrenchments Selection Functions Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα Constructive Models

(Semi-) Open Problems Relevance-Sensitive Revision Iterated-Revision Revision over Weaker Logics Implementations - Representational Cost Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα

Relevance-Sensitive Revision Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα

Relevance-Sensitive Belief Revision * K K*Α An non-intuitive AGM revision function: A K*A = K+A, if A  K Cn(A), otherwise * Κ Κ*Α

Parikh’s Notion of Relevance A = (abe)  (abe) = ae a  c d¬a ge y y * K K*Α (P) If K = Cn(X,Y), LXLY = and A  LX, then (K*A)LY = KLY.

Distance Between Worlds Diff(w,r) = the set of variable that have different values in w and r. e.g. , Diff(abc, abc) = {a, b} (SP) If Diff(w,r)  Diff(w,z) then r < z. e.g. abc abc abc abc abc abc abc abc ≤ (P) If K = Cn(X,Y), LXLY = and A  LX, then (K*A)LY = KLY.

Distance Between a Theory and a World a  c d¬a ge bf r = { a, b, c, d, g, e, f } ⇒ Diff(K, r) = {a, c, d} (SP) If Diff(w,r)  Diff(w,z) then r < z. (Q1) If Diff(K,r)  Diff(K,z) and Diff(K,r)∩Diff(r,z) = ∅ then r < z. K (Q2) If Diff(K,r) = Diff(K,z) and Diff(K,r)∩Diff(r,z) = ∅ then r ≈ z. abc abc abc ≤ ≤ abc ≤ abc ≤ abc abc abc K = Cn(a≣b, c)

Representation Result Peppas, Williams, Chopra, and Foo (2015) Revision Functions (K*1) – (K*8) Preorders in Possible Worlds *   * *    * * (S*) *   *  * (P)  (Q1) - (Q2) * *   * * 

Strong (P) A * X K K*A Y Y A * X H H*A Z Z (P) If K = Cn(X,Y), LXLY = and A  LX, then (K*A)LY = KLY. (sP) If K = Cn(X,Y), H = Cn(X,Z), LXLY =, LXLZ =, and A  LX, then (K*A)LX = (H*A)LX.

Representation Result Peppas, Williams, Chopra, and Foo (2015) Revision Functions (K*1) – (K*8) Preorders in Possible Worlds *   * *    * * (S*) *  (sP) (Q3)  *  * (P)  (Q1) – (Q2) * *   * * 

Future Work on Relevance * K K*A A * K ? K*A

Iterated Revision Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα

Iterative Belief Revision ≤ ? * ≤ K K*A A possible world is a consistent complete theory of the object language. Expressibility Problems!!! Peppas, Foo, and Williams “On the Expressibility of Propositions”, Logique et Analyse, 1992.

Iterative Belief Revision ≤ ≤ * * ≤ ≤ ≤ K K*A K*A*B A possible world is a consistent complete theory of the object language. Expressibility Problems!!! Peppas, Foo, and Williams “On the Expressibility of Propositions”, Logique et Analyse, 1992.

Darwiche and Pearl’s Postulates for Iteration B ≤ ≤ * * ≤ ≤ ≤ K K*A K*A*B (DP1) If B ⊨ A then K*A*B = K*B. (IS1) If w,r  [A] then w  r iff w ’ r. A possible world is a consistent complete theory of the object language. Expressibility Problems!!! Peppas, Foo, and Williams “On the Expressibility of Propositions”, Logique et Analyse, 1992. (DP2) If B ⊨ A then K*A*B = K*B. (IS2) If w,r  [A] then w  r iff w ’ r. (DP3) If A  K*B then A  K*A*B. (IS3) If w  [A] and r  [A] then w < r entails w <’ r. (DP4) If A  K*B then A  K*A*B. (IS4) If w  [A] and r  [A] then w  r entails w ’ r. Jin and Thielscher (Ind) If A  K*B then A  K*A*B. (IndR) If w  [A] and r  [A] then w  r entails w <’ r.

Conflicts between Iteration and Relevance Peppas, et. al. (2008) * * K K*A K*A*B Theorem: (DP1) – (DP4) and (P) are inconsistent. A possible world is a consistent complete theory of the object language. Expressibility Problems!!! Peppas, Foo, and Williams “On the Expressibility of Propositions”, Logique et Analyse, 1992.

Revision over Weaker Logics Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα

Belief Revision over Horn Theories A * Κ Κ*Α Horn theory Horn theory Horn clause: a1∧ . . . ∧an ⇒ a Any Horn clause is a Horn sentence. If φ, ψ are Horn sentences then so is φ∧ψ. A Horn theory K, is any set of Horn sentences closed under ⊨; i.e. if φ is a Horn sentence and K⊨φ then φ∈K.

Problems with the “Naive” AGM Horn Revision *   * *   * * *    * Revision Functions (K*1) - (K*8) Preorders on Possible Worlds The correspondence between Horn revision functions and preorders over worlds breaks down: There exists Horn revision functions that can not be constructed by any preorder over worlds. There exist preorders over worlds that induce non-Horn revision functions.

The Solution to AGM Horn Revision (Delgrande and Peppas, 2015) *   * *   * * *    * Revision Functions (K*1) - (K*8) Preorders over Possible Worlds (HC) If r ≈ r’, then r∩r’ <r. (Acyc) If (K*a1)+a0 ⊭⊥, . . . , (K*an)+an-1 ⊭⊥, and (K*a0)+an ⊭⊥, then (K*a0)+an ⊭⊥ Delgrande, Peppas, and Woltran, “General Belief Revision”, 2017.

Implementations - Representational Cost Η επιχειρηματολογία είναι ανάλογη με αυτήν σε computational models της δεκαετίας του ’30. ( Turing machines  Recursive Functions  Grammers ) Δομή της Ομιλίας * Μεταφορά του διαγράμματος στον πίνακα

Specifying a Revision Function *   *  * *   * Revision Functions Preorders For a language with n propositional elements, there are 2n worlds that need to be ordered. Even worse, there are 22n theories, and for each one we need to specify an ordering over worlds.

Solution I: Use an Off-the-Shelf Operator (Dalal) abc abc abc abc abc abc abc ≤ ≤ ≤ abc r ≤ r’ iff for some w∈[K], |Diff(w,r)| ≤ |Diff(w’,r’)|, for all w’ ∈[K]. Pros: No information about the revision function needs to be specified. Cons: Restricted range of applicability.

Solution II: Parameterised Difference Operators (Peppas and Williams, 2016) abc abc abc abc abc abc abc ≤ ≤ ≤ abc Domain Specific : Info c < a ≈ b abc abc abc ≤ abc ≤ ≤ ≤ abc ≤ abc abc abc abc abc abc abc ≤ ≤ ≤ abc abc abc abc

Properties of Parameterised Difference Operators *   PD revisions * PD preorders *   * * *    * AGM Revision Functions Preorders A single preorder over the n variables of the language suffices to specify the preorders for all theories. A natural generalization of Dalal’s operator. At the same level in the polynomial hierarchy as Dalal’s operator (2nd level). PD revisions satisfy Parikh’s relevance axiom (P). A built-in solution to the iterated revision problem. An axiomatic characterization of PD revisions.

Conclusion A * Κ Κ*Α The AGM approach is a very elegant framework for studying Belief Revision. More work is needed in (at least): Relevance-Sensitive Revision Iterated Revision Easy to use Implementations