sentences and wffs a term (denoting a real-world individual) is a constant symbol, avariable symbol, or an n-place function of n terms. First-order logic is a logical system for reasoning about properties of objects. _t\xUh`p+rF\8 <1 endstream endobj 41 0 obj 603 endobj 42 0 obj << /Filter /FlateDecode /Length 41 0 R >> stream it does not enumerate all the ambiguity the input might contain. We can now translate the above English sentences into the following event or state. N-ary function symbol from the resolvent to the two parent clauses. So our sentence is also true in a model where it should not hold. Since Like (x,y) is always false in our model, the premise fails therefore according to the rules of implication, the formula is true. The truth values of sentences with logical connectives are determined First-order logicalso known as predicate logic, quantificational logic, and first-order predicate calculusis a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a . sentence that is in a "normal form" called. or one of the "descendents" of such a goal clause (i.e., derived from convert, Eliminate existential quantification by introducing, Remove universal quantification symbols by first moving them We will focus on logical representation Propositionalization 26 Every FOL KB and query can be propositionalized Algorithms for deciding PL entailment can be used Problem:infinitely large set of sentences Infinite set of possible ground-term substitution due to function symbols e.g., ( ( ( ))) Solution: Theorem (Herbrand,1930):If a sentence is entailed by an FOL KB, Exercise 2: Translation from English into FoL Translate the following sentences into FOL. Step-2: Conversion of FOL into CNF. - What are the objects? 12. complete rule of inference (resolution), a semi-decidable inference procedure. 2497 0 obj <>stream Translation into FOL Sentences Let S(x) mean x is a skier, M(x) mean x is a mountain climber, and L(x,y) mean x likes y, where the domain of the first variable is Hoofers Club members, and the domain of the second variable is snow and rain. I have the following 2 sentences to convert to FOL formulas-: 1) Water, water, everywhere, but not a drop to drink. For example, Resolution procedure can be used to establish that a given sentence, Resolution procedure won't always give an answer since entailment If the suggestion was that there are \emph { exactly } two, then a different FOL sentence would be required, namely: \\. mapping from D^N to D The informal specification says that Alex likes someone who is a Man and Likes someone else who is a Woman. My code is GPL licensed, can I issue a license to have my code be distributed in a specific MIT licensed project? First-order logic is also known as Predicate logic or First-order predicate logic. variable names that do not occur in any other clause. Our model satisfies this specification. rev2023.3.3.43278. 0000003317 00000 n the negation of the goal. quantifier on a variable C at the front and infer from it the formula obtained by dropping the quantifier and if you like replacing the occurence of X by any variable or . Now consider the following statement taken from the OP: AxEy(Likes( man(x), woman(y) ) -> Likes(alex, man(x) )) This statement is from a different language. Can use unification of terms. What - "There is a person who loves everyone in the world" y x Loves(x,y) - "Everyone in the world is loved by at least one person" Quantifier duality: each can be expressed using the other xLikes(x,IceCream) x Likes(x,IceCream) x Likes(x,Broccoli) x Likes(x,Broccoli) But wouldn't that y and z in the predicate husband are free variables. Try to rebuild your world so that all the sentences come out true. 0000061209 00000 n Complex Skolemization Example KB: Everyone who loves all animals is loved by . - A common mistake is to represent this English sentence as the FOLsentence: ( x) student (x) => smart (x) It also holds if there no student exists in the domain because student (x) => smart (x) holds for any individual who is not astudent. This entails (forall x. expressed by ( x) [boojum(x) snark(x)]. whatever Tony dislikes. because if A is derived from B using a sound rule of inference, then hbbd``b`y$ R zH0O QHpEb id100Ma which is a generalization of the same rule used in PL. A common mistake is to represent this English sentence as the FOL sentence: (Ex) cs170-student(x) => smart(x) But consider what happens when there is a person who is NOT a cs170-student. Every sentence in FOL (without equality) is logically equivalent to a FOL-CNF sentence. Syntax of FOL: Atomic Sentences Atomic sentences in logic state facts that are true or false. clause (i.e., Some Strategies for Controlling Resolution's Search. (The . Syntax of FOL: Atomic Sentences Atomic sentences in logic state facts that are true or false. 0000002898 00000 n xy(Loves(x,y)) Says there is someone who loves everyone in the universe. xhates y) (a) Alice likes everyone that hates Bob. - A common mistake is to represent this English sentence as the FOLsentence: ( x) student (x) => smart (x) It also holds if there no student exists in the domain because student (x) => smart (x) holds for any individual who is not astudent. The first one is correct, the second is not. hVo7W8`{q`i]3pun~h. is at location l, drinkable(l) means there is drinkable water at location l ], 2) There's one in every class. and then just dropping the "prefix" part. &kdswhuv )luvw 2ughu /rjlf 'u 'dlv\ 7dqj,q zklfk zh qrwlfh wkdw wkh zruog lv eohvvhg zlwk remhfwv vrph ri zklfk duh uhodwhg wr rwkhu remhfwv dqg lq zklfk zh hqghdyru wr uhdvrq derxw wkhp (b) Bob hates everyone that Alice likes. If so, how close was it? - If the sentence is false, then there is no guarantee that a procedure will ever determine this-i.e., it may never halt. "Everyone loves somebody": Either x. iff the sentences in S are all true under I, A set of sentences that is not satisfiable is inconsistent, A sentence is valid if it is true under every interpretation, Example of an inconsistent sentence? Formalizing English sentences in FOL FOL Interpretation and satis ability Formalizing English Sentences in FOL. Just like in PL, restrictions on sentence types allows simple inference Find rules that are "triggered" by known facts PL: A ^ B => X FOL: King(x) ^ Greedy(x) => Evil(x) Use Unify() to match terms Keep matching/generating new facts until fixed point: we only derive facts we already know. FOL is sufficiently expressive to represent the natural language statements in a concise way. We can now translate the above English sentences into the following FOL wffs: 1. if the sentence is false, then there is no guarantee that a (d) There is someone who likes everyone that Alice hates. 1.Everything is bitter or sweet 2.Either everything is bitter or everything is sweet 3.There is somebody who is loved by everyone 4.Nobody is loved by no one 5.If someone is noisy, everybody is annoyed 1 "There is a person who loves everyone in the world" yx Loves(x,y) "Everyone in the world is loved by at least one person" Quantifier duality: each can be expressed using the other x Likes(x,IceCream) . Decide on a vocabulary . 1. So: $\forall c \exists x (one(x) \land enrolled(x,c))$, In all classes c, there exists one student who is 'the one'. Is it possible to create a concave light? Quantifier Scope FOL sentences have structure, like programs In particular, the variables in a sentence have a scope For example, suppose we want to say "everyone who is alive loves someone" ( x) alive(x) ( y) loves(x,y) Here's how we scope the variables ( x) alive(x) ( y) . Deb, Lynn, Jim, and Steve went together to APT. Horn clause that has the consequent (i.e., right-hand side) of the The motivation comes from an intelligent tutoring system teaching. Resolution procedure can be thought of as the bottom-up construction of a we know that B logically entails A. trailer << /Size 72 /Info 19 0 R /Root 22 0 R /Prev 154796 /ID[<4685cf29f86cb98308caab2a26bcb12a>] >> startxref 0 %%EOF 22 0 obj << /Type /Catalog /Pages 18 0 R /Metadata 20 0 R /PageLabels 17 0 R >> endobj 70 0 obj << /S 69 /L 193 /Filter /FlateDecode /Length 71 0 R >> stream Answer : (d) Reason : "not" is coming under propositional logic and is therefore not a connective. What are the predicates? 10 Mar 2005 CS 3243 - FOL and Prolog 4 First-order logic Whereas propositional logic assumes the world contains facts, first-order logic (like natural language) assumes the world contains {Objects: people, houses, numbers, colors, baseball games, wars, {Relations: red, round, prime, brother of, bigger than, part of, comes between, FOL syntax Sentence: T/F expression Atom Complex sentence using connectives: . Hb```"S 8 8a All men are mortal, Logical level: Forall X (man(X) --> mortal(X)), Implementation level: (forall (X) (ant (man X)(cons (mortal X))). Says everybody loves somebody, i.e. Note: G --> H is logically equivalent to ~G or H, G = H means that G and H are assigned the same truth value under the interpretation, Universal quantification corresponds to conjunction ("and") FOL has practical advantages, especially for automation. 1.All dogs don't like cats No dog likes cats 2.Not all dogs bark There is a dog that doesn't bark 3.All dogs sleep There is no dog that doesn't sleep 4.There is a dog that talks Not all dogs can't talk Notational differences Different symbolsfor and, or, not, implies, . "There is a person who loves everyone in the world" yx Loves(x,y) "Everyone in the world is loved by at least one person" Quantifier duality: each can be expressed using the other x Likes(x,IceCream) . in the form of a single formula of FOL, which says that there are exactly two llamas. Typical and fine English sentence: "People only vote against issues they hate". the meaning: Switching the order of universals and existentials. x and f (x 1, ., x n) are terms, where each xi is a term. For example, informative. This is a simplification.) the axioms directly. yx(Loves(x,y)) Says everyone has someone who loves them. Everyone likes someone: (Ax)(Ey)likes(x,y) Someone is liked by everyone: (Ey)(Ax)likes(x,y) Just like in PL, restrictions on sentence types allows simple inference Find rules that are "triggered" by known facts PL: A ^ B => X FOL: King(x) ^ Greedy(x) => Evil(x) Use Unify() to match terms Keep matching/generating new facts until fixed point: we only derive facts we already know. forall (KB1, KB2,Alpha) (KB1 |= Alpha) --> (KB1 and KB2 |= Alpha). 12. An atomic sentence (which has value true or false) is . Every member of the Hoofers Club is either a skier >;bh[0OdkrA`1ld%bLcfX5 cc^#dX9Ty1z,wyWI-T)0{+`(4U-d uzgImF]@vsUPT/3D4 l vcsOC*)FLi ]n]=zh=digPlqUC1/e`-g[gfKYoYktrz^C5kxpMAoe3B]r[|mkI1[ q3Fgh At least one parent clause must be from the negation of the goal Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Acorns Check Deposit Reversal, First-order logicalso known as predicate logic, quantificational logic, and first-order predicate calculusis a collection of formal systems used in mathematics, philosophy, linguistics, and computer science.First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables, so that rather than propositions such as "Socrates is a . Either everything is bitter or everything is sweet 3. In this paper, we present the FOLtoNL system, which converts first order logic (FOL) sentences into natural language (NL) ones. 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