Fun with Henry. Dirk Pattinson Department of Computing Imperial College London

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1 Fun with Henry Dirk Pattinson Department of Computing Imperial College London 1

2 Example: The Tudors Henry VIII Henry Carey Mary Boleyn There have been speculation that Mary s two children, Catherine and Henry, were fathered by Henry, but this has never been proven Dirk Pattinson (Imperial College London) Fun with Henry 2

and Henry, were fathered by Henry, but this has never been

3 What do we know? 20% Henry VIII Henry Carey Mary Boleyn Mary Boleyn was Henry s Mistress some time between 1520 and 1526 Suppose that Henry Carey is a child of Henry VIII with probability 0.2 Dirk Pattinson (Imperial College London) Fun with Henry 3

4 What are good Models? Probabilistic Information about offspring {,,,...} {,,,. {,,...} } 0.8 Models are probability distributions on sets: W D(P(W )) Dirk Pattinson (Imperial College London) Fun with Henry 4

5 What are good models? Certainty about amorous affairs: Madge Shelton Elisabeth Blount Models are Relations: Combination of both facets: Dirk Pattinson (Imperial College London) W P(W ) W D(P(W )) P(W ) Fun with Henry 5

6 Semantic Framework Models map worlds to structured successors: Coalgebras: W TW where T : Set Set is a construction on sets (a functor) Example: The reign of Henry VIII W D(P(W )) P(W ) In general: T is a parameter graded modal logics, probabilistic logics, neighbourhood frames etc. closed under modular combinations 6

Henry VIII W D(P(W )) P(W ) In general: T is a parameter graded modal logics,

7 Coalgebraic Logics Extension of (classical) propositional logic with modal operators: Example: Henry had an affair with Mary Henry is father of Henry with likelihood of at least 20% In general: DL Syntax: Henry had affair.mary Henry father of 0.2.HenryC Modal operators talk about structured succesors Lifting of world predicates to successor predicates Semantically: given model (W, γ : W TW) and w W : w = A γ(w) (A) where : P(W ) P(TW) determines the interpretation of 7

8 Given: a set Λ Semantic Framework of modal operators Interpretation via -structure: Λ Endofunctor T : Set Set (defining the model class) Interpretations ( W : P(W ) P(TW)) W Set (for operators) Models: Tuples (W, γ, π) where W set of worlds and π : V P(W ) valuation of variables γ : W TW structure map Formal syntax: F(Λ) A, B ::= p A B A B A A ( Λ,p V ) Formal semantics over M =(W, γ, π) : A M = γ 1 ( W )(A M ) 8

Tuples (W, γ, π) where W set of worlds and π : V P(W ) valuation of variables γ : W TW structure map

9 Questions Satisfiability over empty TBox: Given A F(Λ), is there a model M such that A M =? Satisfiability over general TBox (knowledge base): Given A F(Λ) and T F(Λ), is there a model M such that M = B for all B T and A M =? Interpretation: T contains background knowledge, satisfied by model A is a hypothesis, consistent relative to T Extensions: New constructs (individuals, fixpoints) Other Questions (query answering) Task: Decision procedures parametric in the notion of model 9

10 Syntactical Approach: Tableaux Example: Satisfiability A F(Λ) unsatisfiable A has closed tableau Two components: fixed propositional rules, e.g. Γ,A B Γ, A, B Γ,A B Γ,A Γ,B plus plus instance specific modal rules, e.g. graded modal logic: l 1 A 1,...,l m A m, k 1 B 1,..., k n B n i r ib i s j A j < 0 Rule format in general: rule premise describes property of state (purely modal) rule conclusion describes property of successors (Examples: coalition logic, conditional logics, probabilistic logics,...) 10

11 Example: Probabilistic Modal Logic Models: coalgebras γ : W D(W ) over probability distributions D(W )={µ : W [0, 1] supp(µ) finite, w W µ(w) =1} Operators: formal linear inequalities over formulas l c written as c µ( A) c k µ(a k ) c k=1 Interpretation: instantiation with local distribution w = l c k µ(a k ) c l c k γ(w)(a k ) c k=1 k=1 11

linear inequalities over formulas l c written as c µ( A) c k µ(a k ) c k=1

12 The Satisfiability Problem Given: coalgebraic semantics plus coherence conditions Theorem. The tableau calculus characterises satisfiability, i.e. A F(Λ) unsatisfiable A has closed tableau Proof Strategy: model construction using coherence conditions no reference to underlying semantics Applications. Satisfiability is (mostly) PSPACE analysis of (polynomially deep) tableaux Extensions. Tableaux for background knowledge, individuals, fixpoints... precisely identical coherence conditions extended to cater for specific constructs 12

using coherence conditions no reference to underlying semantics Applications.

13 Exploting Uniformity Syntax: modal operators, Semantics: coalgebras W TW coherence conditions Generic Results: Completeness, Complexity Generic Algorithms: Automated deduction concrete instances Concrete results: e.g. complexity of majority logic Implemented tools: e.g. consistency checking 13

Algorithms: Automated deduction concrete instances Concrete results: e.g. complexity of majority logic Implemented tools: e.

14 Fun with Henry... (this is where things will start to go wrong...) 14

15 Applications Example: cell cycle of budding yeast automated hypotheses checking testing mutants in silico needs quantitative information Example: insurance policy nonmonotonic reasoning no rule without exceptions quick classification 15

needs quantitative information Example: insurance policy

16 Thanks for your attention! Any Questions? 16

CHAPTER 7 GENERAL PROOF SYSTEMS

CHAPTER 7 GENERAL PROOF SYSTEMS 1 Introduction Proof systems are built to prove statements. They can be thought as an inference machine with special statements, called provable statements, or sometimes