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First-Order Logic

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Conceptualization The formalization of knowledge begins with a conceptualization. This includes the objects presumed or hypothesized to exist in the world and their relationships. An object can be anything about we want to say something. –Objects can be concrete (book, earth, sun, etc) –Objects can be abstract (the number 2, the set of all integers, the justice etc) –Objects can be primitive or composite (a circuit that contains many circuits) –Objects can be fictional (Sherlock Holmes, Miss Right) A knowledge-representation task doesn’t require that we consider all the objects in the world; only those objects in a particular set are relevant. The set of objects about which knowledge is being expressed is called a universe of discourse.

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Example – Blocks World Here we conceptualize the blocks. Some could conceptualize also the table, but we ignore it. The universe of discourse corresponding to this conceptualization is {A,B,C,D,E} Although in this example the universe of discourse is finite, this is not always the case. E.g. it is common in math to consider as the universe of discourse the set of all integers, the set of all real numbers etc. A B C D E

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More about Conceptualization I A function is a kind of interrelationship among the objects in a universe of discourse. Although we can define many functions for a given set of objects, in conceptualizing a portion of the world we usually emphasize some functions and ignore others. The set of functions emphasized in a conceptualization is called the functional basis set. E.g. in our blocks world, it would make sense to define a (partial) function Hat that maps a block into the block on top of it, if such exists. The tuples corresponding to this function are: {,, }

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More about Conceptualization II A relation is the second kind of interrelationship among the objects in a universe of dicourse. The set of relations emphasized in a conceptualization is called the relational basis set. E.g. in our blocks world, we can consider the On relation {,, } Also, we can consider the Above relation {,,, } More, we can consider the unary Clear relation {, } Or, the unary Table relation {, }

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Formally… A conceptualization is a triple consisting of: a universe of discourse, a functional basis set for that universe of discourse and a relational basis set. E.g. No matter how we choose to conceptualize the world, there are other conceptualizations as well. What makes a conceptualization more appropriate than another for knowledge formalization?

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Reification E.g. consider a Blocks World conceptualization in which there are five blocks, no functions, and there are three unary relations This conceptualization allows us to consider the color of blocks but not the properties of colors. In order to overcome this, we reify the relations and have instead the conceptualization The advantage of this is that it allows us to consider properties of of properties.

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First-Order Logic Given a conceptualization of the world, we begin to formalize knowledge in a language appropriate to that conceptualization. E.g. we can express the fact that the block A is above block B by taking a relation symbol such as above and object symbols a and b and writing above(a,b) We can combine sentences by using logical operators ( , , , , , ) in order to create more complex sentences. above(a,b) above(b,a) The names we use, such as “above”, “a”, “b” are irrelevant. We could have said: abnwyrcn(pjkj, oiuy) abnwyrcn(oiuy, pjkj) What really matters is that they should should have identifiable referents in the real world, (better saying in the conceptualization that we choose)

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Syntax

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Constants Constants refer to objects, functions and relationships. joe, mary, loves, happy, Simple sentences express relationships among objects. loves(joe,mary) They are called atoms. Compound sentences capture relationships among relations. loves(x,y) loves(y,x) loves(x,y) loves(y,x) happy(x) Relations can be unary as well. tall(amy)

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Let’s blur the distinction… Constants refer to objects, functions and relationships. joe, mary, loves, happy, hat Ok, in order to be formal we should use “object constant,” “function constant,” and “relation constant,” but this is painful. So, let’s say for: –Object constantobject –Function constantfunction –Relation constantrelation It should be clear that the ones on the left are different from ones on the right; We are doing this for the ease of presentation.

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Example with Functions E.g. How about saying that Jimmy Durante has a big nose? durante is an object and nose_of (durante) is a function that constructs an object from the argument object. Then, we can write: big(nose_of (durante)) E.g. Quincy loves his dog. loves(quincy, dog_of (quincy)) Note: We are allowed to relate sentences only. So, we can say: loves(quincy, dog_of (quincy)) loves(quincy, cat_of (quincy)) But not, loves(quincy, dog_of (quincy) cat_of (quincy)) because dog_of (quincy), cat_of (quincy) aren’t sentences, they are objects.

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Full-blown First-Order Logic: , The language that we have described so far, consisting of atoms and the connectives ( , , , , , ) is typically called predicate logic. To extend it to first-order logic, we need to add quantifiers. The purpose of quantifiers is to allow us to say things about sets of objects. To say that Quincy loves everything we write: x. loves (quincy, x) We can think of as a big conjunction. For example, if there are only three objects quincy, dog, and cat, what the above asserts is: loves (quincy, dog) loves (quincy, cat) loves (quincy, quincy) To say that Quincy loves something we write: x. loves (quincy, x) We can think of as a big disjunction. For example, if there are only three objects as above, then what we are asserting is: loves (quincy, dog) loves (quincy, cat) loves (quincy, quincy)

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Mushrooms The universe of discourse is the set of all plants. There is a unary relation for being mushroom, another one for being purple, and a third for being poisonous. We designate these relations with the unary relation symbols mushroom, purple, and poisonous. All purple mushrooms are poisonous. x (purple(x) mushroom(x) poisonous(x)) A mushroom is poisonous only if it is purple. x (mushroom(x) poisonous(x) purple(x)) No purple mushroom is poisonous. x (purple(x) mushroom(x) poisonous(x)) There is exactly one mushroom. x mushroom(x) ( y y x mushroom(y))

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Reverse translation Translate the following into English. x hesitates(x) lost(x) He who hesitates is lost. x business(x) like(x,Showbusiness) There is no business like show business. x glitters(x) gold(x) Not everything that glitters is gold. x t person(x) time(t) canfool(x,t) You can fool some of the people all the time.

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Semantics

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Models This is all well and good, but there isn’t much value to saying what sentences belong to a language unless we also say what those sentences mean. Simply said, we are interested in the truth value of given (possibly compound) sentences. Suppose that our first-order language had no logical connectives at all, just objects, functions, and relations. –In that case, the sentences in our language would be simply what we called atoms. Let, A be the set of all atoms in our impoverished language. A model M is a subset of the set of atoms A. By a model M A, we will mean the state of affairs in which all of the atoms in M are true and all of the atoms not in M are false. Given a model it is obviously straightforward to decide whether or not some particular atom a is true in M: it’s true if a M and false if a M. What makes first-order logic powerful, is that we can also determine whether compound sentences are true or false in M.

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Example Suppose that Tom is our only object, and that we have two relations, rich and poor. Now, A = {rich(tom), poor(tom)} is the set of all atoms. So, we have 4 possible models. In M 1 = , Tom is neither rich nor poor. In M 2 ={rich(tom)}, Tom is rich and not poor. In M 3 ={poor(tom)}, tom is poor and not rich. In M 4 ={rich(tom), poor(tom)}, Tom is both rich and poor. Now, consider the sentence poor(tom) rich(tom) Using the truth table for the implication, the above sentence will be true in M 1, M 2, M 3 and false only in M 4. (try it)

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Entailment Definition. Given two sentences and , we will say that entails , writing |= , if holds in every model that holds. Example: Consider ( ) |= For any sentences and , any model M in which ( ) holds is model in which holds and also a model in which holds. But since holds in M, it must be the case that holds in M as well. By the definition of entailment, the claim follows.

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Entailment II A set of premises logically entails a conclusion if and only if every interpretation that satisfies the premises also satisfies conclusion. In the case of Propositional Logic, the number of interpretations is finite, and so it is possible to check logical entailment directly in finite time. In the case of Relational Logic, the number of models is infinite, and so a direct check of logical entailment is not feasible.

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Example {} {p(a)} {q(a)} {p(a),q(a)} {p(b)} {q(b)} {p(b),q(b)} {p(a), p(b)} {p(a), p(b), q(a)} {p(a), p(b), q(b)} {p(a), p(b), q(a), q(b)}... Infinitely many models.

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Good News Given any set of sentences, there is a specially defined subset of interpretations called Herbrand interpretations. Under certain conditions, checking just the Herbrand models suffices to determine logical entailment. Checking just the Herbrand interpretations is less work than checking all interpretations.

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Herbrand The Herbrand universe for a set of sentences is the set of all ground terms that can be formed from just the constants used in those sentences. –Ground terms are either object constants or functions applied to object constants. The Herbrand base for a set of sentences is the set of all ground atomic sentences that can be formed using just the elements in the Herbrand universe. A Herbrand model for a set of sentences is any subset of the Herbrand base for those sentences.

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Example Sentences x. (r(a,x) r(x,b)) x. y. z. (r(x,y) r(x,y) r(x,z)) Herbrand Universe (constants used in sentences only) {a,b} Herbrand Base {r(a,a), r(a,b), r(b,a), r(b,b)}

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Herbrand Models {} {r(a,a)} {r(a,b)} {r(b,a)} {r(b,b)} {r(a,a), r(a,b)} {r(a,a), r(b,a)} {r(a,a), r(b,b)} {r(a,b), r(b,a)} {r(a,b), r(b,b)} {r(b,a), r(b,b)} {r(a,a), r(a,b), r(b,a)} {r(a,a), r(a,b), r(b,b)} {r(a,a), r(b,a), r(b,b)} {r(a,b), r(b,a), r(b,b)} {r(a,a), r(a,b), r(b,a), r(b,b)} 16 Herbrand interpretations in all. Note: 16< .

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Herbrand Theorem Herbrand Theorem: A set of (pure) -free sentences is satisfiable if and only if it has a Herbrand model that satisfies it. Proof. –If a set of -free sentences is satisfiable, then there is a model that satisfies it. –Take the intersection of this model with the Herbrand base. This is a Herbrand model. –Moreover, it is easy to see that it satisfies the sentences. If the sentences contain variables, the instances must all be true, including those in which the variables are replaced only by elements in the Herbrand universe. Note. With -free we mean pure -free. E.g. p(x), is nothing else but p(x). So, pure -free: no negations preceding .

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Example Premises: x. p(x) q(x) p(a) q(a) Conclusion: q(a) Herbrand Models: {} {p(a)} {q(a)} {p(a),q(a)}

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Herbrand Method Add negation of conclusion to the premises to form the satisfaction set. Loop over Herbrand interpretations. Cross out each interpretation that does not satisfy the sentences in the satisfaction set. If all Herbrand interpretations are crossed out, by the Herbrand Theorem, the set is unsatisfiable. Sound and Complete: Negating the conclusion leads to a contradiction; therefore, the premises logically entail the conclusion. Termination: Since there are only finitely many Herbrand interpretations, the process halts.

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Example Premises: x. p(x) q(x) p(a) q(a) Conclusion: q(a) Satisfaction Set: x. p(x) q(x) p(a) q(a) q(a) Herbrand Models: {} {p(a)} {q(a)} {p(a),q(a)} By considering as a big conjunction we can verify that the satisfaction set is unsatisfiable in each Herbrand model.

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Herbrand for the rest of FOL… In the presence of function constants, all ground functional terms are included. –E.g. if we find in our sentences a,b,f,g then the Herbrand universe is: {a, b, f(a), f(b), g(a), g(b), f(f(a)), f(f(b)), f(g(a)), f(g(b)),...} Sad: The size of the Herbrand universe for a functional language is infinite. Upshot: Checking the Herbrand models for a language to determine logical entailment is not feasible in finite time. Solution: Use formal proofs!

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