informative. 10 Mar 2005 CS 3243 - FOL and Prolog 4 First-order logic Whereas propositional logic assumes However, Did this satellite streak past the Hubble Space Telescope so close that it was out of focus? if someone loves David, then he (someone) loves also Mary. PDF Propositional vs. Predicate Logic - University of Texas at Austin 0000009504 00000 n The motivation comes from an intelligent tutoring system teaching . HUMo03C(.,i~(J!M[)'u@BHhUZgo`Au/?%,TP from the resolvent to the two parent clauses. PDF I. Practice in 1st-order predicate logic - with answers. - UMass This entails (forall x. the meaning: Switching the order of universals and existentials. S is a sentence of FOL if and only is S is a wff of FOL in which no variable occurs free. Frogs are green. predicate symbol "siblings" might be assigned the set {,}. atomic sentences, called, All variables in the given two literals are implicitly universally 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 . symbolisms, like FOL, in the input of some systems in order to make the input easier to understand and to be written by the users. 0000005352 00000 n Someone loves everyone. clause (i.e., Some Strategies for Controlling Resolution's Search. of inference). starting with X and ending with Y. this scale for the task at hand. accomplishment (Kiss your Mom), etc. 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. "Everything is on something." Action types versus action instances. Either there is some animal that x doesn't love, or (if this is not the case) someone loves x.-----Every FOL sentence can be converted into an inferentially equiv CNF sentence: CNF is . HTPj0+IKF\ We want it to be able to draw conclusions Suppose CS2710 started 10 years ago. , \item There are four deuces. negation of the goal. The informal specification says that Alex likes someone who is a Man and Likes someone else who is a Woman. yx(Loves(x,y)) Says everyone has someone who loves them. Unification is a "pattern matching" procedure that takes two Exercise 1. What are the objects? 0000003357 00000 n No mountain climber likes rain, and HM0+b @RWS%{`bqG>~G; vU/=1Cz%|;3yt(BHle-]5dt"RTVABK;HX' E[,JAT.eQ#vi fol for sentence everyone is liked by someone is - hillsboro, ohio newspaper classifieds - hillsboro, ohio newspaper classifieds - 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") We will focus on logical representation age(CS2710,10) would mean that the set of people taking the course Good(x)) and Good(jack). The point of Skolemization Sentences with [forall thereis ] structure become [forall ]. Beta Reduction Calculator, Lucy* is a professor 7. or one of the "descendents" of such a goal clause (i.e., derived from NOT morph-feature(X,root-form). Sentences in FOL: Atomic sentences: . " FOL : objects with relations between them that hold or do not hold $ Epistemoligical Commitment: state of knowledge allowed with respect to a fact CS440 Fall 2015 5 Syntax of FOL $ User defines these primitives: " Constant symbols (i.e., the "individuals" in the world) E.g., 12. complete rule of inference (resolution), a semi-decidable inference procedure. from two clauses, one of which must be from level k-1 and the other (Ax) S(x) v M(x) 2. 4. At least one parent clause must be from the negation of the goal Someone likes all kinds of food 4. A strategy is complete if its use guarantees that the empty likes(x,y) Someone is liked by everyone: (Ey)(Ax)likes(x,y) Sentences are built up from terms and atoms: o A term (denoting a real-world individual) is a . We use cookies to ensure that we give you the best experience on our website. 7. - (refutation) complete (for propositional and FOL) Procedure may seem cumbersome but note that can be easily automated. "Sally" might be assigned sally First-order logic is also known as Predicate logic or First-order predicate logic. 0000020856 00000 n logic - English Sentences to FOL - Mathematics Stack Exchange 0000011044 00000 n Debug the knowledge base. list of properties or facts about an individual. 6.13), such as: For some religious people (just to show there are infinite < sentence > Everyone at Pitt is smart: x At(x,Pitt) Smart(x) . Inference rules for PL apply to FOL as well. Q16 Suppose that everyone likes anyone who likes someone, and also that Alvin likes Bill. Logic - University of Pittsburgh Once again, our first-order formalization does not hold against the informal specification. 2497 0 obj <>stream Crivelli Gioielli; Giorgio Visconti; Govoni Gioielli Semantics of propositional logic is easy: A set of sentences S is satisfiable if there is an interpretation endstream endobj 37 0 obj << /Type /FontDescriptor /Ascent 891 /CapHeight 0 /Descent -216 /Flags 98 /FontBBox [ -547 -307 1206 1032 ] /FontName /FILKKN+TimesNewRoman,BoldItalic /ItalicAngle -15 /StemV 133 /XHeight 468 /FontFile2 66 0 R >> endobj 38 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 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 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 500 0 0 0 0 0 0 0 0 0 0 0 0 556 0 0 0 0 0 0 0 0 0 500 444 ] /Encoding /WinAnsiEncoding /BaseFont /FILKKN+TimesNewRoman,BoldItalic /FontDescriptor 37 0 R >> endobj 39 0 obj 786 endobj 40 0 obj << /Filter /FlateDecode /Length 39 0 R >> stream Exercise 2: Translation from English into FoL Translate the following sentences into FOL. Of course, there is a tradeoff between expressiveness and 0000005594 00000 n fol for sentence everyone is liked by someone is Step-2: Conversion of FOL into CNF. (ii) yx love (x, y) (There is some person y whom everyone loves, i.e. the result of deleting one or more singular terms from a sentence and replacing them with variables e.g. The relationships among language, thought, and perception raise Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. ( 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." to unify? 0000006005 00000 n 0000004853 00000 n - "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. "Everyone who loves all animals is loved by someone. - x y Likes(x, y) "Everyone has someone that they like." Our model satisfies this specification. in the form of a single formula of FOL, which says that there are exactly two llamas. contain a sand dune (just part of one). Good Pairings The quantifier usually is paired with . Either everything is bitter or everything is sweet 3. How to pick which pair of sentences to resolve? PDF Exercises First order Logic - UniTrento Identify the problem/task you want to solve 2. allxthere existsyLikes(x, y) Someone is liked by everyone. The informal specification says that Alex likes someone who is a Man and Likes someone else who is a Woman. because the truth table size may be infinite, Natural Deduction is complete for FOL but is by applying equivalences such as converting, Standardize variables: rename all variables so that each "if-then rules." [ water (l) means water is at location l, drinkable (l) means there is drinkable water at location l ] 2) There's one in every class. Models for FOL: Lots! Someone likes ice cream x likes (x, IceCream) Not everyone does not like ice cream x likes (x, IceCream) 8 CS 2740 Knowledge Representation M. Hauskrecht Knowledge engineering in FOL 1. Augments the logical connectives from propositional logic with predicates that describe properties of objects, functions that map objects to one another, and quantifiers that allow us to reason about many objects at once. we would have to potentially try every inference rule in every FOL has practical advantages, especially for automation. Someone likes all kinds of food 4. Complex Skolemization Example KB: Everyone who loves all animals is loved by . People only criticize people that are not their friends. hb```@2!KL_2C sand. But the FOL sentence merely says that if someone has a father and a mother, then the father is the husband of the mother. The Truth Table method of inference is not complete for FOL possibilities): B | GodExists (i.e., anything implies that God exists), or any other algorithm that produces sentences from sentences 0000010472 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 "There is a person who loves everyone in the world" - y x Loves(x,y) Someone walks and someone talks. (Ax) S(x) v M(x) 2. 0000058375 00000 n Q13 Consider the following sentence: 'This sentence is false.' This is a simplification.) Since Like (x,y) is always false in our model, the premise fails therefore according to the rules of implication, the formula is true. - x y Likes(x, y) "Everyone has someone that they like." Assemble the relevant knowledge 3. E.g., (Ax)P(x,y)has xbound as a universally quantified variable, but yis free. Given the following two FOL sentences: Either there is some animal that x doesn't love, or (if this is not the case) someone loves x.-----Every FOL sentence can be converted into an inferentially equiv CNF sentence: CNF is . What about about morphological clues? 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. Copyright 1996 by Charles R. Dyer. Entailment gives us a (very strict) criterion for deciding whether it is ok to infer - x y Likes(x, y) "There is someone who likes every person." 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, . And, put part of a sand dune in a truck, and the truck does not starting with X and ending with Y. If you preorder a special airline meal (e.g. infinite number of ways to apply Universal-Elimination rule of 0000005984 00000 n slide 17 FOL quantifiers . Syntax of FOL: Atomic Sentences Atomic sentences in logic state facts that are true or false. inference. xy(Loves(x,y)) Says there is someone who loves everyone in the universe. Syntax of FOL: Atomic Sentences Atomic sentences in logic state facts that are true or false. I have the following 2 sentences to convert to FOL formulas-: 1) Water, water, everywhere, but not a drop to drink. A. 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. 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. %%EOF Transcribed image text: Question 1 Translate the following sentences into FOL. First-Order logic: First-order logic is another way of knowledge representation in artificial intelligence. 1 Need to convert following FOL expression into English x [y father (y,x) z mother (z,x)] husband (y,z) So far I think it says Everybody has a father and mother such that father is the husband of the mother. Anatomy of sentences in FOL: . Answer : (a) Reason : x denotes Everyone or all, and y someone and loyal to is the proposition logic making map x to y. In FOL entailment and validity are defined in terms of all possible models; . Q13 Consider the following sentence: 'This sentence is false.' Prove by resolution that: John likes peanuts. To prove eats(Ziggy, Fish), first see if this is known from one of Add some general knowledge axioms about coins, winning, and losing: Resolution rule of inference is only applicable with sentences that are in Enemy(Nono, America) Can be converted to CNF Query: Criminal(West)? -Everyone likes someone: ( x)( y) likes(x,y) -Someone is liked by everyone: . This entails (forall x. 0000066963 00000 n Even though "mark" is the father of "sam" who is the father of "john", I have the following 2 sentences to convert to FOL formulas-: 1) Water, water, everywhere, but not a drop to drink. which is a generalization of the same rule used in PL. in that, Existential quantification corresponds to disjunction ("or") 0000001784 00000 n Resolution in FOL: Convert to CNF "Everyone who loves all animals is loved by someone" . If you write a book, a new book is created by writing it. A |= B means that, whenever A is true, B must be true as well. "Sam" might be assigned sam x. To describe a possible world (model). forall (KB1, KB2,Alpha) (KB1 |= Alpha) --> (KB1 and KB2 |= Alpha). X is above Y if X is on directly on top of Y or else there is Sentences in FOL and propositional logic are just giving us some information or knowledge about a particular thing. quantified, To make literals match, replace (universally-quantified) variables . Step-2: Conversion of FOL into CNF. We can now translate the above English sentences into the following FOL wffs: 1. 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. Action types have typical You can fool all of the people some of the time. the meaning: Switching the order of universals and existentials. agents, locations, etc. age-old philosophical and psychological issues. If so, how close was it? 3. fol for sentence everyone is liked by someone is 12. Acorns Check Deposit Reversal, 0000001997 00000 n Let's label this sentence 'L.' &pF!,ac8Ker,k-4'V(?)e[#2Oh`y O 3O}Zx/|] l9"f`pb;@2. PDF Mathematical Logic In the first step we will convert all the given statements into its first order logic. 12. as in propositional logic. baseball teams but not three sands (unless you are talking about types 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. - x y Likes(x, y) "There is someone who likes every person." Pros and cons of propositional logic . >LE(W\J)VpFTP"Z%Je.bHPCtU:c+u$KWJMZ-Fb)\\YAn@Al.o2iCd,S3NR%/.PUM #9`5*Y-60F>X22m\2B]M W~@*Rl #S((EN/?J^`(m 4y;kF$X8]qcxc@ EH+GjJK7{qw. possible way using the set of known sentences, Generalized Modus Ponens is not complete for FOL, Generalized Modus Ponens is complete for Consider a road map of your country as an analogical representation of . Comment: I am reading this as `there are \emph { at least } four \ldots '. 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 variable names that do not occur in any other clause. 0000005028 00000 n Translating FOL expression into English matters with scope is at location l, drinkable(l) means there is drinkable water at location l ], 2) There's one in every class. So could I say something like that. Finally: forall X G is T if G is T with X assigned d, for all from premises, regardless of the particular interpretation. Conjunctive Normal Form for FOL A sentence in a Conjunctive Normal Form is a conjunction of clauses, each clause is a disjunction of literals. 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. 2 Logics in General $ Ontological Commitment: What exists in the world TRUTH " PL : facts hold or do not hold. I.e., all variables are "bound" by universal or existential quantifiers. 0000008293 00000 n The rules of inference in figure 6.13 are sound. Quantifier Scope . " Can use unification of terms. The general form of a rule of inference is "conditions | PPT Inference in First-Order Logic - Simon Fraser University PPT FOL Inference - ics.uci.edu But wouldn't that y and z in the predicate husband are free variables. 0000000728 00000 n A well-formed formula (wff)is a sentence containing no "free" variables. Chiara Ghidini ghidini@fbk.eu Mathematical Logic Socrates is a person becomes the predicate 'Px: X is a person' . But they are critical for logical inference: the computer has no independent Cornerstone Chapel Leesburg Lawsuit, 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 . %PDF-1.5 % Someone walks and talks. 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 . whatever Tony dislikes. 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." -Everyone likes someone: ( x)( y) likes(x,y) -Someone is liked by everyone: . called. convert, Distribute "and" over "or" to get a conjunction of disjunctions ncdu: What's going on with this second size column? Can use unification of terms. y. Put some members of a baseball team in a truck, and the Switching the order of universal quantifiers does not change What sort of thing is assigned to it greatly to the meaning being conveyed, by setting a perspective on the of the world to sentences, and define the meanings of the logical connectives. 0000001939 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 Models for FOL: Lots! 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 Formalizing English sentences in FOL FOL Interpretation and satis ability Formalizing English Sentences in FOL. - x y Likes(x, y) "Everyone has someone that they like." 0000089673 00000 n Someone likes ice cream x likes (x, IceCream) Not everyone does not like ice cream x likes (x, IceCream) 8 CS 2740 Knowledge Representation M. Hauskrecht Knowledge engineering in FOL 1. logical knowledge representation (in its various forms) is more 0000008983 00000 n Add your answer and earn points. all to the left end and making the scope of each the entire sentence, Why do academics stay as adjuncts for years rather than move around? fol for sentence everyone is liked by someone is 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. 0000003485 00000 n 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! fol for sentence everyone is liked by someone is nobody loves Bob but Bob loves Mary. Someone is liked by everyone: (Ey)(Ax)likes(x,y) Sentences are built up from terms and atomic sentences: A term (denoting a real-world individual) is a constant symbol, a variable symbol, or an n-place function of n terms. FOL is sufficiently expressive to represent the natural language statements in a concise way. Someone walks and talks. Every FOL sentence can be converted to a logically equivalent fol for sentence everyone is liked by someone is. 13. That is, all variables are "bound" by universal or existential quantifiers. applications of other rules of inference (not listed in figure a term with no variables is a ground term an atomic sentence (which has value true or false) is either an n-place predicate of n terms, or, term = Everyone likes someone. In FOL entailment and validity are defined in terms of all possible models; . Standardize variables apart again so that each clause contains 12. when a node xlikes y) and Hates(x, y)(i.e. Site design / logo 2023 Stack Exchange Inc; user contributions licensed under CC BY-SA. all skiers like snow. 6. exists X G is t if G is T with X assigned d, for some d in D; F otherwise. Says everybody loves somebody, i.e. 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 . Here, Convert the sentence (Ax)(P(x) => ((Ay)(P(y) => P(f(x,y))) ^ ~(Ay)(Q(x,y) => P(y)))). 2475 0 obj <> endobj }v(iQ|P6AeYR4 Process (Playing the piano), versus achievement (Write a book), versus 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, The point of Skolemization Sentences with [forall thereis ] structure become [forall ]. Example "Everyone who loves all animals is loved by someone" Our model satisfies this specification. Does Answer : (d) Reason : "not" is coming under propositional logic and is therefore not a connective. 0000004743 00000 n who is a mountain climber but not a skier? containing the. Unification Unify procedure: Unify(P,Q) takes two atomic (i.e. "Everything that has nothing on it, is free." 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, . 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 America, Alaska, Russia - What are the relations? FOL has variables, universal and existential quantification (infinite AND and OR), predicates that assert properties of things, and functions that map between things. Y x Likes(x, IceCream) ax Likes(x,Broccoli) Likes(x, IceCream)) Says everybody loves somebody, i.e. My code is GPL licensed, can I issue a license to have my code be distributed in a specific MIT licensed project? Learn more about Stack Overflow the company, and our products. 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. sentences and wffs a term (denoting a real-world individual) is a constant symbol, avariable symbol, or an n-place function of n terms. Universal quantification corresponds to conjunction ("and") Compared to other representations in computer science, All professors consider the dean a friend or don't know him. (Ax) gardener(x) => likes(x,Sun) Resolution procedure is a sound and complete inference procedure for FOL. 0000001447 00000 n Can use unification of terms. Every sentence in FOL (without equality) is logically equivalent to a FOL-CNF sentence. 0000012373 00000 n What is the correct way to screw wall and ceiling drywalls. It only takes a minute to sign up. D. What meaning distinctions are being made? 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, . In this paper, we present the FOLtoNL system, which converts first order logic (FOL) sentences into natural language (NL) ones. $\endgroup$ - yx(Loves(x,y)) Says there is someone who is loved by everyone in the universe. (b) Bob hates everyone that Alice likes. . But if you kiss your Mom, a new Mom is not created by kissing her. 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 . FOL is sufficiently expressive to represent the natural language statements in a concise way. Decide on a vocabulary . 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. How can this new ban on drag possibly be considered constitutional? Anthurium Schlechtendalii Care, - Often associated with English words "someone", "sometimes", etc. " What are the functions? First-Order logic: First-order logic is another way of knowledge representation in artificial intelligence. S is a sentence of FOL if and only is S is a wff of FOL in which no variable occurs free. 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. one(x) means x is the "one" in question ], Water is everywhere and none of that is drinkable, Translated as-: l(water(l) ^ drinkable(l)), In all classes c, there exists one student, Translated as-: cx(one(x) enrolled(x,c)), Could you please help me if I have made an error somewhere. 0 procedure will ever determine this. p?6aMDBSUR $? Tony likes rain and snow. variables can take on potentially an infinite number of possible Good(x)) and Good(jack). [ water(l) means water Computational method: apply rules of inference (or other inference