Help Taking a step back
Logic query Languages and expressiveness Propositional logic Specification of propositional logic data Learning model Inference Inference algorithms for propositional logic Inference in propositional logic with only definite clauses Inference in full propositional logic predictions Firstorder logic Specification of firstorder logic Models describe how the world works (relevant to some task) Inference algorithms for firstorder logic Inference in firstorder logic with only definite clauses What type of models? Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang CS221: Artificial Intelligence (Autumn 2012) Percy Liang 1
Some modeling paradigms Outline
State space models: search problems, MDPs, games Languages and expressiveness Applications: route finding, game playing, etc. Propositional logic Specification of propositional logic Think in terms of states, actions, and costs Inference algorithms for propositional logic Variablebased models: CSPs, Markov networks, Bayesian Inference in propositional logic with only definite networks clauses Applications: scheduling, object tracking, medical diagnosis, etc. Inference in full propositional logic Think in terms of variables and factors Firstorder logic Specification of firstorder logic Logical models: propositional logic, firstorder logic Inference algorithms for firstorder logic Applications: proving theorems, program verification, reasoning Inference in firstorder logic with only definite clauses Think in terms of logical formulas and inference rules Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 2 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 3
Language
Language is a mechanism for expressing models/knowledge/ideas.
Natural languages: Desiderata: want a language than can represent complex facts English: even numbers about the world and allows for sophisticated reasoning about those German: geraden Zahlen facts...
Programming languages:
Python: def even(x): return x % 2 == 0 C++: bool even(int x) { return x % 2 == 0; }
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Procedural languages represent knowledge using data structures, Variablebased models are a declarative language: define algorithms manipulates data structures. constraints/factors over variables; ask any query over variables. Implicit representation? def f(positions): All students work hard. distances = [dist(pos, me) for pos in positions] John is a student. minDistance = min(distances) Therefore, John works hard. return minDistance Variablebased models would explictly represent all the students — intuitively shouldn't be necessary. Easy query: Given positions, what is minDistance? Higherorder reasoning? Hard query: Given minDistance, what are positions? John believes it will rain. Will it rain? Need a declarative language ( ) Does John believe it will not rain (assuming John is logical)? Need something more expressive to represent...
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Natural languages A puzzle
A dime is better than a nickel. If John likes probability, then John likes logic. A nickel is better than a penny. Therefore, a dime is better than penny. If it is Thursday, then John likes probability or John likes logic.
General rule (transitivity): if and , then . It is Thursday. A penny is better than nothing. Nothing is better than world peace. Does John like probability? Therefore, a penny is better than world peace??? Does John like logic? Natural language is not formal, can be slippery...
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Notes A puzzle Propositional logic chat demo
You get extra credit if you write a paper and you solve the Tell me some something or ask me something. I will try to convert your utterance into propositional logic and apply resolution to carry out your request. I have no personality. problems. Example: "If it rained, then the ground is wet.", "It rained.", "Is the ground wet?" Example: "(implies (or rain snow) wet)", "rain", "(or wet cold)?"
You didn't get extra credit.
You solve the problems.
Did you write a paper?
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Syntax: defines a set of valid formulas ( ) Syntax: what are valid expressions in the language? Semantics: Semantics: what do these expressions mean? A model describes a possible situation in the world An interpretation function mapping each Different syntax, same semantics: and model to a truth value x + y y + x Inference rules: what new formulas can be added without changing semantics ( )? Same syntax, different semantics: Inference algorithm: apply inference rules in some clever order to 3 / 2 (Python) 3 / 2 (Javascript) answer queries (is true?)
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Outline Outline
Languages and expressiveness Languages and expressiveness Propositional logic Propositional logic Specification of propositional logic Specification of propositional logic Inference algorithms for propositional logic Inference algorithms for propositional logic Inference in propositional logic with only definite Inference in propositional logic with only definite clauses clauses Inference in full propositional logic Inference in full propositional logic Firstorder logic Firstorder logic Specification of firstorder logic Specification of firstorder logic Inference algorithms for firstorder logic Inference algorithms for firstorder logic Inference in firstorder logic with only definite clauses Inference in firstorder logic with only definite clauses Inference in full firstorder logic Inference in full firstorder logic Other logics Other logics Logic and probability Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 14 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 15
Syntax of propositional logic Syntax of propositional logic
Propositional symbols: (think variables in CSPs); these are Formula: formulas Nonformula: Logical connectives:
Build up formulas recursively—if and are formulas, so are the following: Negation: Note: formulas are just symbols — no meaning yet! Conjunction: Disjunction: Implication: Biconditional:
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A model in propositional logic is an assignment of truth values Given a formula and a model , interpretation function to propositional symbols returns either true (1) or false (0). Example: Base case: 3 propositional symbols: For a propositional symbol (e.g., ): possible models : Recursive case: For any two formulas and :
0 0 1 0 0 1 1 0 1 1 0 1 1 0 1 0 0 0 1 0 0 1 1 0 1 1 1 1
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Example Formulas represent sets of models
Formula: Fach formula and model has an interpretation Model: Formula: Interpretation: Models with true interpretation :
Point: formula is symbols that compactly represent a set of models. Think of formula as putting constraints on the world.
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Types of formulas Knowledge base
Validity: for all models (tautologies that provide no Let . information, e.g., )
Unsatisfiability: for all models (contradictions, e.g., ) Definition: Knowledge base A knowledge base is a set of formulas with . Contingent: neither valid nor unsatisfiable (provides information, e.g., ) Note: think conjunction:
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Intuition: contradicts what we know (captured in ). Intuition: added no information/constraints (it was already known). Example: contradicts Definition: Entailment entails (written ) iff . Relationship between entailment and contradiction:
Example: entails ( ) iff is unsatisfiable
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Contingency Interacting with a knowledge base
or response Tell: It rained. ( ) Possible responses: Already knew that: entailment ( ) Don't believe that: contradiction ( ) Intuition: adds nontrivial information to Learned something new (update knowledge base): contingent Ask: Did it rain? ( ) Possible responses: Yes: entailment ( ) No: contradiction ( ) Example: and I don't know: contingent Everything boils down to entailment (checking satisfiability)...
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Model checking Outline
Checking satisfiability (SAT) is special case of solving CSPs Languages and expressiveness propositional symbol variable Propositional logic formula constraint Specification of propositional logic model assignment Inference algorithms for propositional logic Inference in propositional logic with only definite A B C clauses Solving CSPs is model checking (operate over models). Inference in full propositional logic Popular algorithms: Firstorder logic DPLL (backtracking search) Specification of firstorder logic WalkSat (Gibbs sampling) Inference algorithms for firstorder logic But logic allows us to peer inside factors and exploit structure... Inference in firstorder logic with only definite clauses theorem proving operates on formulas. Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 28 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 29 Inference rules Inference framework
Example of making an inference: In general, have a set of rules with the following form: It rained. ( ) If it rained, then the ground is wet. ( ) Therefore, the ground is wet. ( ) Forward inference algorithm Repeat until no changes to : Choose set of formulas . Modus Ponens inference rule Find matching rule . For any formulas and : Add to . Say that ( derives/proves ) if there exists sequence Key: operate on syntax, not semantics (can be more efficient). of rule applications that eventually adds to .
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Soundness and completeness Soundness
What properties does a set of inference rules have? Is (Modus ponens) sound? Definition: Soundness (only prove entailed formulas) A set of rules is sound if: ? implies 0 1 0 1 0 1 Definition: Completeness (prove all entailed formulas) 0 0 0 A set of rules is complete if: 1 1 1 implies Yes! Models represented by and is subset of Point: These properties link syntax (inference rules) and semantics those represented by . (entailment).
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Soundness Soundness
Is (Modus tollens) sound? Is sound?
? ?
0 1 0 1 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 1 1 1
Yes! No!
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Example: Is (resolution rule) sound?
? Can verify that all is sound, but not complete...for example, given , can't infer . 0 1 0 1 0 1 0,0 0,0 0,0 What set of rules is complete? This is tricker than soundness... 0,1 0,1 0,1 Plan: 1,0 1,0 1,0 Propositional logic with only definite clauses: only need Modus 1,1 1,1 1,1 ponens Yes! Propositional logic: only need resolution rule (+ preprocessing)
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Outline A restriction of propositional logic
Languages and expressiveness Assume knowledge base (KB) contains only definite clauses: Propositional logic Definition: Definite clause Specification of propositional logic A definite clause has the following form: Inference algorithms for propositional logic Inference in propositional logic with only definite clauses for propositional symbols . Inference in full propositional logic Intuition: if premises hold, then conclusion holds. Firstorder logic Example: Specification of firstorder logic Inference algorithms for firstorder logic Nonexample: Inference in firstorder logic with only definite clauses Nonexample: Inference in full firstorder logic Allowed queries to the KB: Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 38 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 39
Example scenario Inference rule
Suppose we have the following knowledge base: The single rule is sound and complete for propositional logic with only definite clauses: Definition: Modus ponens
Proof tree:
Queries: ? ? Each node is a formula derived from children using modus ponens, leaves are original formulas in .
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KB: Languages and expressiveness Query: Propositional logic Forward chaining: Specification of propositional logic From known propositions, iteratively apply rules to derive new Inference algorithms for propositional logic propositions. Inference in propositional logic with only definite Proactively make new inferences when information comes in. clauses Time: linear in size of knowledge base. Inference in full propositional logic Backward chaining: Firstorder logic Start from query and recursively derive premises that conclude Specification of firstorder logic the query. Inference algorithms for firstorder logic Make inferences tailored towards answering a particular query. Inference in firstorder logic with only definite clauses Time: often much less than linear in size of knowledge base. Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 42 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 43
Highlevel strategy Resolution algorithm
Goal: determine whether Goal: determine whether is satisfiable. Example: Example: Algorithm: Algorithm (performs proof by contradiction): Convert all formulas in into conjunctive normal form. Set . Example: Example: Repeatedly apply resolution rule. Example: Run inference algorithm to check satisfiability of . Conclude iff is unsatisfiable. Example: unsatisfiable Return unsatisfiable iff derive false (0).
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Conjunctive normal form Conversion to CNF
Definition: Conjunctive normal form (CNF) Goal: convert arbitrary propositional formula into CNF formula A CNF formula is a conjunction of disjunctions of optional Steps (exercise: verify semantic equivalence): negations of propositional symbols. Eliminate : Eliminate : Eliminate double negation: Example: Move inwards: Move inwards: Distribute over :
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Resolution rule A model describes a possible state of the world (e.g., ). Each formulas (e.g., ) describes a set of models (think of providing information or imposing constraints). Example: Inference rules (e.g., ) allow one to derive new formulas from old ones. Soundness/completeness links syntax and semantics. Definite clauses only: ( ) forward/backward chaining yields linear time inference. Propositional logic: ( ) resolution yields exponential time inference.
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Outline Limitations of propositional logic
Languages and expressiveness Alice and Bob both know arithmetic. Propositional logic Specification of propositional logic Inference algorithms for propositional logic All students know arithmetic. Inference in propositional logic with only definite clauses Inference in full propositional logic Firstorder logic Specification of firstorder logic Every even integer greater than 2 is the sum of two primes. Inference algorithms for firstorder logic Inference in firstorder logic with only definite clauses ??? Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 50 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 51
Limitations of propositional logic Outline
All students know arithmetic. Languages and expressiveness Propositional logic Specification of propositional logic Inference algorithms for propositional logic Inference in propositional logic with only definite Propositional logic is very clunky. What's missing? clauses Objects and relations: propositions (e.g., Inference in full propositional logic ) has more internal structure ( , Firstorder logic , ) Specification of firstorder logic Inference algorithms for firstorder logic Quantifiers and variables: all is a quantifier which references all Inference in firstorder logic with only definite clauses people, don't want to enumerate them all... Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 52 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 53 Some examples of firstorder logic Syntax of firstorder logic
Alice and Bob both know arithmetic. Ingredients: Connectives from propositional logic: Constant symbols (e.g., ): refer to objects Predicate symbols (e.g., ): relate multiple objects Function symbols (e.g., ): map objects to single object All students know arithmetic. Variables (e.g., ): refers to objects Quantifiers (e.g., ): aggregate results from different assignments to variables
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Syntax of firstorder logic Models in firstorder logic
Terms (refer to objects): constant symbol (e.g., ), Recall a model represents a possible state of affairs (mapping from variable (e.g., ), or function applied to terms (e.g., ) symbols to their interpretation). Formulas (refer to truth values): Propositional logic: Model maps propositional symbols to truth Atomic formulas: Predicate applied to terms (e.g., values. ); analogue of propositional symbol in e.g., propositional logic Firstorder logic: Connectives applied to formulas (e.g., Model maps constant symbols to objects ); same as e.g., propositional logic Model maps predicate symbols to tuples of objects Quantifiers applied to formulas (e.g., ) e.g.,
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Notes Graph representation Database semantics (alternative)
A model as be represented a directed graph (if only have binary There are two students, John and Bob. predicates):
Definition: Unique names assumption Each object has at most one constant symbol. This rules out . Definition: Domain closure Each object has at least one constant symbol. This rules out . Nodes are objects, labeled with constant symbols Definition: Closedworld assumption Directed edges are relations, labeled with predicate symbols All atomic formulas not known (labels not present) are false.
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Universal quantification ( ): Universal quantification ( ):
Every student knows arithmetic. Think conjunction: is like
Existential quantification ( ):
Think disjunction: is like Existential quantification ( ): Some properties: Some student knows arithmetic. equivalent to different from
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Notes Some examples of firstorder logic Outline
There is some course that every student has taken. Languages and expressiveness Propositional logic Specification of propositional logic Inference algorithms for propositional logic Every even integer greater than 2 is the sum of two primes. Inference in propositional logic with only definite clauses Inference in full propositional logic Firstorder logic If a student takes a course and the course covers some concept, then Specification of firstorder logic the student knows that concept. Inference algorithms for firstorder logic Inference in firstorder logic with only definite clauses Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 62 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 63
Outline Definite clauses
Languages and expressiveness Assume knowledge base (KB) contains only definite clauses: Propositional logic Definition: Definite clause Specification of propositional logic A definite clause has the following form: Inference algorithms for propositional logic Inference in propositional logic with only definite clauses for atomic formulas and variables that Inference in full propositional logic appear in the atomic formulas. Firstorder logic Example: Specification of firstorder logic Inference algorithms for firstorder logic Inference in firstorder logic with only definite Intuition: think of firstorder definite clause compactly representing clauses all instantiations of the variables (for all objects). Inference in full firstorder logic Other logics CS221: ALrtioficgiali Icnt elalignendce p(Arutoumbn a20b12i)l i Pteyrcy Liang 64 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 65 Substitution and unification Substitution
Goal: define inference rules that work on formulas with quantifiers Definition: Substitution A substitution maps variables to constant symbols or variables. Example: returns the result of performing substitution on . Given and .
Infer ? Examples: Problem: and don't match exactly. Two concepts: Substitution: morph a formula into another Unification: make two formulas the same
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Unification Inference rule
Definition: Unification Generalized modus ponens Unification takes two formulas and and returns a substitution , which is the most general unifier: such that where is the most general unifier . or fail if no exists. Example inputs: Examples:
Example result:
Derive
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Forward/backward chaining Time/space complexity
If there are no function symbols, then bounded by number of Inference algorithms analogous to those for propositional logic. domain elements to the maximum arity of a predicate (3 in this case). Forward chaining: starting from known atomic formulas (e.g., If there are function symbols (e.g., ), then infinite... ), find rules whose premises unify with them, and derive conclusion. Theorem: Semidecidability Firstorder logic (even restricted to only definite clauses) is semi Backward chaining: starting from query atomic formula (e.g., decidable. ), find rules whose conclusion unifies with If , forward/backward chaining will prove in finite time. it, and recursive on premises. If , no algorithm can show this in finite time.
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Languages and expressiveness Goal: given a knowledge base, can we derive a contradiction Propositional logic (unsatisfiable)? Specification of propositional logic Inference algorithms for propositional logic Recall: Firstorder logic can include formulas like this (not a Inference in propositional logic with only definite definite clause) clauses Inference in full propositional logic Firstorder logic Specification of firstorder logic Highlevel strategy (same as in propositional logic): Inference algorithms for firstorder logic Convert all formulas to CNF Inference in firstorder logic with only definite clauses Repeatedly apply resolution rule Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 72 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 73
Conversion to CNF Conversion to CNF
Input: Input: Eliminate implications (old):
Push inwards (old): Output: Standardize variables (new):
Replace existentially quantified variables with Skolem functions (new): New to firstorder logic: Distribute over (old): All variables (e.g., ) have universal quantifiers by default Remove universal quantifiers (new): Introduce Skolem functions (e.g., ) to represent existential quantified variables Interpretation: represents animal that doesn't like, represents person who likes
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Resolution Outline
Generalized resolution rule Languages and expressiveness Propositional logic Specification of propositional logic where . Inference algorithms for propositional logic Inference in propositional logic with only definite clauses Example: Inference in full propositional logic Firstorder logic Specification of firstorder logic Inference algorithms for firstorder logic with substitution . Inference in firstorder logic with only definite clauses Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 76 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 77 Motivation Temporal logic
Goal: represent knowledge and perform inferences Barack Obama is the US president.
Why use anything besides propositional or firstorder logic?
Expressiveness: George Washington was the US president. Temporal logic: express time Modal logic: express alternative worlds Higherorder logic: fancier quantifiers Some woman will be the US president. Notational convenience, computational efficiency: Description logic
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Temporal logic Modal logic for propositional attitudes
Point: all formulas interpreted at a current time. Alice believes one plus one is two. The following operators change the current time and quantify over ?? it (think of as ): Alice believes Boston is a city. : held at some point in the past ?? : will hold at some point in the future Problem: is true, is true, : held at every point in the past but two are not interchangeable in this context. : will hold at every point in the future Solution: every formula interpreted with respect to a possible world, operator interprets according to Alice's world Every student will at some point never be a student again.
Model: map from time points to models in firstorder logic Model: map from possible worlds to models in firstorder logic
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Higherorder logic: lambda calculus Description logic
Simple: People with at least three sons who are all unemployed and married to doctors, and at most two daughters who are professors...are weird. Alice has visited some museum. Lambda calculus:
More complex: Description logic: Alice has visited at least 10 museums. : boolean function representing set of museums Alice has visited Advantages: Generalized quantifiers without variables: notationally more compact Higherorder logic allows us to model these generalized quantifiers. Firstorder is semidecidable, description logic is decidable
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Propositional logic: Languages and expressiveness Firstorder logic: Propositional logic Specification of propositional logic Temporal / modal logic: Inference algorithms for propositional logic Description logic: Inference in propositional logic with only definite clauses Higherorder logic (lambda calculus): Inference in full propositional logic Firstorder logic Specification of firstorder logic Inference algorithms for firstorder logic Inference in firstorder logic with only definite clauses Inference in full firstorder logic Other logics Logic and probability CS221: Artificial Intelligence (Autumn 2012) Percy Liang 84 CS221: Artificial Intelligence (Autumn 2012) Percy Liang 85
Limitations Markov logic
In logic, every formula is true or false. In reality, there is Assume database semantics. Defines Markov network: uncertainty. Random variables: Firstorder formula: Probability used to define joint distributions: Think of as propositional symbols A model is Equivalent propositional logic formulas: We are placing a distribution over possible worlds . Probability theory: Pro: allows us to manage uncertainty in a coherent way Con: captures propositional logic
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Markov logic Summary
One parameter for each firstorder formula (e.g., ) Logic is a language for expressing facts in a knowledge base is the interpretation of in Markov logic defines a Markov network: Considerations: expressiveness, notational convenience, inferential complexity
Defines distribution over possible worlds (models) Propositional logic with definite clauses, propostional logic, description logic, firstorder logic, temporal logic, modal logic, All grounded instances of a formula have same parameter higherorder logic weight Can do lifted probabilistic inference for efficiency (important Markov logic: marry logic (abstract reasoning by working on for learning) formulas) and probability (maintaining uncertainty) As , get ordinary logic
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