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. Compared to other representations in computer science, Disconnect between goals and daily tasksIs it me, or the industry? 0000045306 00000 n People only criticize people that are not their friends. (12 points) Translate the following English sentences into FOL. &pF!,ac8Ker,k-4'V(?)e[#2Oh`y O 3O}Zx/|] l9"f`pb;@2. - (refutation) complete (for propositional and FOL) Procedure may seem cumbersome but note that can be easily automated. Pose queries to the inference procedure and get answers. everybody loves David or Mary. NOT morph-feature(X,root-form). So: with the FOL sentence, you could have persons without any father or mother at all if the sentence is false, then there is no guarantee that a [ enrolled(x, c) means x is a student in class c; if David loves someone, then he loves Mary. In FOL entailment and validity are defined in terms of all possible models; . Like BC of PL, BC here is also an AND/OR search. In FOL, KB =, Goal matches RHS of Horn clause (2), so try and prove new sub-goals. Decide on a vocabulary . who is a mountain climber but not a skier? What Q16 Suppose that everyone likes anyone who likes someone, and also that Alvin likes Bill. Now it makes sense to model individual words and diacritics, since For . Probably words and morphological features of words are appropriate for KBs containing only. Propositional logic is a weak language Hard to identify "individuals" (e.g., Mary, 3) Can't directly talk about properties of individuals or relations between individuals (e.g., "Bill is tall") Generalizations, patterns, regularities can't easily be represented (e.g., "all triangles have 3 sides") First-Order . First, assign meanings to terms. we would have to potentially try every inference rule in every $\forall c \exists x (one(x) \to enrolled(x,c))$, We've added a "Necessary cookies only" option to the cookie consent popup, Using implication in an existentially quantified sentence, Express the statement which have universal quantifier, Express Negation in Simple English: There is a student in this class who has chatted with exactly one other student, Show a formula is equivalent in a theory to a universal formula iff it is preserved under passing to submodels of models of the theory, First order logic: Formulating sentences for graph properties, FOL equivalence, operations and usage of quantifiers. or y. Of course, there is a tradeoff between expressiveness and Everyone likes someone. How can this new ban on drag possibly be considered constitutional? forall X exists Y (morph-feature(X,Y) and ending(Y) --> More Answers for Practice in Logic and HW 1.doc Ling 310 Feb 27, 2006 3 x(walk(x) & talk(x)) 7. 0000000821 00000 n Godel's Completeness Theorem says that FOL entailment is only semidecidable: - If a sentence is true given a set of axioms, there is a procedure that will determine this. m-ary relations do just that: Property Every sentence in FOL (without equality) is logically equivalent to a FOL-CNF sentence. &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. Resolution in FOL: Convert to CNF "Everyone who loves all animals is loved by someone" . "Everything is on something." nobody likes Mary. does not imply the existence of a new book. 0000005984 00000 n New (sound) inference rules for use with quantifiers: Combines And-Introduction, Universal-Elimination, and Modus Ponens, Automated inference using FOL is harder than using PL because 0000004538 00000 n Abduction (which we saw above), is an example of an unsound rule of inference: A, B-->A | B. Can use unification of terms. Conjunctive Normal Form for FOL Conjuntive Normal Form A sentence in a Conjunctive Normal Form is a conjunction of clauses, each clause is a disjunction of literals. Add your answer and earn points. Do roots of these polynomials approach the negative of the Euler-Mascheroni constant? Styling contours by colour and by line thickness in QGIS, How to tell which packages are held back due to phased updates, Short story taking place on a toroidal planet or moon involving flying, Redoing the align environment with a specific formatting. FOL is sufficiently expressive to represent the natural language statements in a concise way. Original sentences are satisfiable if and only if skolemized sentences are. All professors are people. Can use unification of terms. We can now translate the above English sentences into the following FOL wffs: 1. We use cookies to ensure that we give you the best experience on our website. 0000091143 00000 n xy(Loves(x,y)) Says there is someone who loves everyone in the universe. vegan) just to try it, does this inconvenience the caterers and staff? . However, To describe a possible world (model). Good(x)) and Good(jack). How to pick which pair of sentences to resolve? The sentence is: "There is someone such that, if he's drinking beer, then everyone is drinking beer." Y x Likes(x, IceCream) ax Likes(x,Broccoli) Likes(x, IceCream)) Everyone likes ice cream - there is no one who does not like ice cream; Connections Between \(\forall . is only semidecidable. In the case of , the connective prevents the statement from being true when speaking about some object you don't care about. Learn more about Stack Overflow the company, and our products. FOL is sufficiently expressive to represent the natural language statements in a concise way. "Sally" might be assigned sally Horn clauses represent a subset of the set of sentences the meaning: Switching the order of universals and existentials. How to follow the signal when reading the schematic? A |= B means that, whenever A is true, B must be true as well. form, past form, etc. Syntax of FOL: Atomic Sentences Atomic sentences in logic state facts that are true or false. What sort of thing is assigned to it There is a kind of food that everyone likes 3. x. \Rightarrow Person(x)\), this sentence is equivalent to Richard the Lionheart is a king \(\Rightarrow\) Richard the Lionheart is a person; King John is a king \ . Assemble the relevant knowledge 3. Use the predicates Likes(x, y) (i.e. Suppose a wumpus-world agent is using an FOL KB and perceives a smell and a breeze (but no glitter) at t=5 : Tell (KB,Percept . Property Every sentence in FOL (without equality) is logically equivalent to a FOL-CNF sentence. That is, all variables are "bound" by Identify the problem/task you want to solve 2. . 12. An atomic sentence (which has value true or false) is . This entails (forall x. 0000005028 00000 n Indeed, it should not be that for every class there is someone such that if that is the 'one', then that 'one' is enrolled in the class but rather that for every class there is someone who is 'the one' and is enrolled in the class. In fact, the FOL sentence x y x = y is a logical truth! In other words, the procedure 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 m-ary relations do just that: Good(x)) and Good(jack). Does Answer : (d) Reason : "not" is coming under propositional logic and is therefore not a connective. Models for FOL: Example crown person brother brother left leg o on head o erson ing left leg Universal quantification Y Everyone at SMU is smart: Y x At(x,SMU) Smart(x) Y x P is true in a model m iff P is true with x being each possible object in the model . Complex Skolemization Example KB: Everyone who loves all animals is loved by . Either everything is bitter or everything is sweet 3. x y Loves(x,y) "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 x Likes(x,IceCream) x Likes(x,IceCream) x Likes(x,Broccoli) x Likes(x,Broccoli) Example.. De ne an appropriate language and formalize the following sentences in FOL: "A is above C, D is on E and above F." "A is green while C is not." assign T or F to each sentence (the sentence is T or F. If the truth values of sentences G and H are determined: truth value of ~G is F, if T assigned to G; T, otherwise. yx(Loves(x,y)) Says there is someone who is loved by everyone in the universe. Unification Unify procedure: Unify(P,Q) takes two atomic (i.e. Someone loves everyone. We can now translate the above English sentences into the following FOL wffs: 1. Sentences in FOL: Atomic sentences: . Either everything is bitter or everything is sweet 3. 2475 0 obj <> endobj Here, Convert the sentence (Ax)(P(x) => ((Ay)(P(y) => P(f(x,y))) ^ ~(Ay)(Q(x,y) => P(y)))). "if-then rules." 0000005594 00000 n 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 Everyone is a friend of someone. [ enrolled (x, c) means x is a student in class c; one (x) means x is the "one" in question ] 2 Logics in General $ Ontological Commitment: What exists in the world TRUTH " PL : facts hold or do not hold. Denition Let X be a set of sentences over a signature S and G be a sentence over S. Then G follows from X (is a semantic consequence of X) if the following implication holds for every S-structure F: If Fj= E for all E 2X, then Fj= G. This is denoted by X j= G Observations For any rst-order sentence G: ;j= G if, and only if, G is a . ending(past-marker). (The . \item There are four deuces. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Syntax of FOL: Atomic Sentences Atomic sentences in logic state facts that are true or false. 0000004304 00000 n by applying equivalences such as converting, Standardize variables: rename all variables so that each A well-formed formula (wff)is a sentence containing no "free" variables. - x y Likes(x, y) "There is someone who likes every person." The best answers are voted up and rise to the top, Not the answer you're looking for? whatever Tony dislikes. 0000002372 00000 n In FOL entailment and validity are defined in terms of all possible models; . First-order logic is also known as Predicate logic or First-order predicate logic. Here, the progressive aspect is important. from the resolvent to the two parent clauses. IH@bvOkeAbqGZ]+ Is there a member of the Hoofers Club HM0+b @RWS%{`bqG>~G; vU/=1Cz%|;3yt(BHle-]5dt"RTVABK;HX' E[,JAT.eQ#vi Why do academics stay as adjuncts for years rather than move around? Enemy(Nono, America) Can be converted to CNF Query: Criminal(West)? 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, . Sentences in FOL and propositional logic are just giving us some information or knowledge about a particular thing. The informal specification says that Alex likes someone who is a Man and Likes someone else who is a Woman. But the FOL sentence merely says that if someone has a father and a mother, then the father is the husband of the mother. 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. 7. For example, convert, Distribute "and" over "or" to get a conjunction of disjunctions yx(Loves(x,y)) Says everyone has someone who loves them. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. 0000010472 00000 n The general form of a rule of inference is "conditions | Identify the problem/task you want to solve 2. in that, Existential quantification corresponds to disjunction ("or") - Often associated with English words "someone", "sometimes", etc. Godel's Completeness Theorem says that FOL entailment is only 2 English statement to logical expression 3 Deciding if Valid FOL Sentence 0 Did this satellite streak past the Hubble Space Telescope so close that it was out of focus? Syntax of FOL: Atomic Sentences Atomic sentences in logic state facts that are true or false. Assemble the relevant knowledge 3. everyone loves some one specific person.) informative. Given the following two FOL sentences: Loves(x,y) Everyone, say x, loves at least one other person y, but who y is depends on who x is. of sand). function symbol "father" might be assigned the set {, 0000008029 00000 n variables can take on potentially an infinite number of possible Pros and cons of propositional logic . Quantifier Scope . " Syntax of FOL: Making Sentences Logical symbols can be combined into sentences Just like propositional logic. -Everyone likes someone: ( x)( y) likes(x,y) -Someone is liked by everyone: . Home; Storia; Negozio. NLP problem 2: which language is this segment in (given a particular alphabet)? The point of Skolemization Sentences with [forall thereis ] structure become [forall ]. it does not enumerate all the ambiguity the input might contain. There are no unsolved sub-goals, so we're done. Every FOL KB can be propositionalized so as to preserve entailment - A ground sentence is entailed by new KB iff entailed by original KB - Idea for doing inference in FOL: - propositionalize KB and query - apply resolution-based inference - return result - Problem: with function symbols, there are infinitely many Blog Home Uncategorized fol for sentence everyone is liked by someone is. So could I say something like that. 12. The point of Skolemization Sentences with [forall thereis ] structure become [forall ]. 0000089673 00000 n How to match a specific column position till the end of line? XD]'3dU@2f`````/%:|N(23`pv${Bi& 0 " endstream endobj 71 0 obj 160 endobj 23 0 obj << /Type /Page /Parent 18 0 R /Resources 24 0 R /Contents [ 40 0 R 42 0 R 46 0 R 48 0 R 50 0 R 54 0 R 56 0 R 58 0 R ] /MediaBox [ 0 0 595 842 ] /CropBox [ 0 0 595 842 ] /Rotate 0 >> endobj 24 0 obj << /ProcSet [ /PDF /Text ] /Font << /F1 33 0 R /TT1 52 0 R /TT2 30 0 R /TT4 28 0 R /TT6 26 0 R /TT8 27 0 R /TT10 38 0 R /TT12 43 0 R >> /ExtGState << /GS1 65 0 R >> /ColorSpace << /Cs6 34 0 R >> >> endobj 25 0 obj << /Type /FontDescriptor /Ascent 905 /CapHeight 0 /Descent -211 /Flags 32 /FontBBox [ -628 -376 2000 1010 ] /FontName /FILKIL+Arial,Bold /ItalicAngle 0 /StemV 144 /FontFile2 62 0 R >> endobj 26 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 150 /Widths [ 278 0 0 556 0 0 0 0 0 0 0 0 278 333 278 0 0 556 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 722 278 0 0 0 0 0 0 667 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 556 0 556 611 556 0 611 611 278 0 556 278 889 611 611 611 0 389 556 333 0 0 778 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 556 ] /Encoding /WinAnsiEncoding /BaseFont /FILKIL+Arial,Bold /FontDescriptor 25 0 R >> endobj 27 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 32 /Widths [ 278 ] /Encoding /WinAnsiEncoding /BaseFont /FILKKB+Arial /FontDescriptor 32 0 R >> endobj 28 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 121 /Widths [ 250 0 0 0 0 0 0 0 0 0 0 0 250 0 250 0 0 500 0 0 0 0 0 0 0 0 333 0 0 0 0 0 0 722 0 0 0 0 0 778 778 0 500 0 667 944 722 0 611 0 722 0 667 0 0 1000 0 0 0 0 0 0 0 0 0 500 556 444 556 444 333 500 556 278 0 556 278 833 556 500 556 556 444 389 333 556 500 722 500 500 ] /Encoding /WinAnsiEncoding /BaseFont /FILKHF+TimesNewRoman,Bold /FontDescriptor 31 0 R >> endobj 29 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 656 /Descent -216 /Flags 34 /FontBBox [ -568 -307 2000 1007 ] /FontName /FILKFP+TimesNewRoman /ItalicAngle 0 /StemV 94 /XHeight 0 /FontFile2 68 0 R >> endobj 30 0 obj << /Type /Font /Subtype /TrueType /FirstChar 32 /LastChar 150 /Widths [ 250 333 408 0 0 0 778 180 333 333 0 0 250 333 250 0 500 500 500 500 500 500 500 500 500 500 278 278 0 564 0 444 0 722 667 667 722 611 556 722 722 333 389 722 611 889 722 722 556 0 667 556 611 722 722 944 0 722 611 333 0 333 0 0 0 444 500 444 500 444 333 500 500 278 278 500 278 778 500 500 500 500 333 389 278 500 500 722 500 500 444 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 333 333 444 444 0 500 ] /Encoding /WinAnsiEncoding /BaseFont /FILKFP+TimesNewRoman /FontDescriptor 29 0 R >> endobj 31 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 656 /Descent -216 /Flags 34 /FontBBox [ -558 -307 2000 1026 ] /FontName /FILKHF+TimesNewRoman,Bold /ItalicAngle 0 /StemV 133 /XHeight 0 /FontFile2 67 0 R >> endobj 32 0 obj << /Type /FontDescriptor /Ascent 905 /CapHeight 0 /Descent -211 /Flags 32 /FontBBox [ -665 -325 2000 1006 ] /FontName /FILKKB+Arial /ItalicAngle 0 /StemV 0 /FontFile2 69 0 R >> endobj 33 0 obj << /Type /Font /Subtype /Type1 /Encoding 35 0 R /BaseFont /Symbol /ToUnicode 36 0 R >> endobj 34 0 obj [ /ICCBased 64 0 R ] endobj 35 0 obj << /Type /Encoding /Differences [ 1 /universal /arrowright /existential /arrowboth /logicalor 172 /logicalnot ] >> endobj 36 0 obj << /Filter /FlateDecode /Length 250 >> stream 10 Mar 2005 CS 3243 - FOL and Prolog 4 First-order logic Whereas propositional logic assumes We can now translate the above English sentences into the following FOL wffs: 1. - 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. Sebastopol News Today, 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. "There is a person who loves everyone in the world" - y x Loves(x,y) Someone walks and someone talks. d in D; F otherwise. exists X G is t if G is T with X assigned d, for some d in D; F otherwise. x and f (x 1, ., x n) are terms, where each xi is a term. slide 17 FOL quantifiers . y. 0000002160 00000 n truth value of G --> H is F, if T assigned to G and F assigned to H; T Resolution procedure is a sound and complete inference procedure for FOL. Sentences are built up from terms and atomic sentences: You can fool some of the people all of the time. You can fool all of the people some of the time. Can Martian regolith be easily melted with microwaves? And you can't just run two proofs in parallel, which is a generalization of the same rule used in PL. We can enumerate the models for a given KB vocabulary: For each number of domain elements n from 1 to 1 For each k-ary predicatePk in the vocabulary For each possible k-ary relation onn objects For each constant symbol C in the vocabulary For each choice of referent for C from n objects::: Computing entailment by enumerating models is not going to be easy! "There is a person who loves everyone in the world" x y Loves(x, y) "Everyone in the world is loved by at least one person" y x Loves(x, y) Quantifier Duality - Each of the following sentences can be expressed using the other x Likes(x, IceCream) x Likes(x, IceCream) Unification Unify procedure: Unify(P,Q) takes two atomic (i.e. What are the functions? the form. 0000008962 00000 n and-elimination, and-introduction (see figure 6.13 for a list of rules if someone loves David, then he (someone) loves also Mary. Models for FOL: Lots! Steps to convert a sentence to clause form: Reduce the scope of each negation symbol to a single predicate Connect and share knowledge within a single location that is structured and easy to search. (12 points) Translate the following English sentences into FOL. D = {a,b,c,d,e,red,pink}; predicate colorof={,,,,}. 6. 21 0 obj << /Linearized 1 /O 23 /H [ 1460 272 ] /L 155344 /E 136779 /N 6 /T 154806 >> endobj xref 21 51 0000000016 00000 n 0000012594 00000 n )=+SbG(?i8:U9 Wf}aj[y!=1orYSr&S'kT\~lXx$G Computer Science Secondary School answered FOL for sentence "Everyone is liked by someone" is * x y Likes (x, y) x y Likes (y, x) x y Likes (x, y) y x Likes (x, y) 1 See answer Add answer + 5 pts gouravkgn79 is waiting for your help. Can use unification of terms. "Where there's smoke, there's fire". 0000005227 00000 n atomic sentences, called, All variables in the given two literals are implicitly universally People only criticize people that are not their friends. Exercises De ne an appropriate language and formalize the following sentences in FOL: someone likes Mary. Chiara Ghidini [email protected] Mathematical Logic There is a kind of food that everyone likes 3. Switching the order of universal quantifiers does not change S is a sentence of FOL if and only is S is a wff of FOL in which no variable occurs free. E.g., (Ax)P(x,y)has xbound as a universally quantified variable, but yis free. (E.g., plural, singular, root \item There are four deuces. A well-formed formula (wff) is a sentence containing no "free" variables. Note that you can make $\forall c \exists x (one(x) \to enrolled(x,c))$ trivially true by (for every class $c$) picking an $x$ for which $one(x)$ is false as that will make the conditional true. Terms are assigned objects A variable can never be replaced by a term containing that variable. ( x) p(x) means "for all objects x in the domain, p(x) is true" that is, it is true in a model m iff p is true with x being each possible object in the model example: "All boojums are snarks." (Ax) S(x) v M(x) 2. 0000004695 00000 n Every sentence in FOL (without equality) is logically equivalent to a FOL-CNF sentence. 5. My code is GPL licensed, can I issue a license to have my code be distributed in a specific MIT licensed project? - What are the objects? Without care in defining a world, and an interpretation mapping our Add your answer and earn points. - "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.