This dpANS specifies the syntax and semantics of Knowledge Interchange Format (KIF) and a syntactic variant of KIF in "infix" form.
Knowledge Interchange Format (KIF) is a language designed for use in the interchange of knowledge among disparate computer systems (created by different programmers, at different times, in different languages, and so forth).
KIF is not intended as a primary language for interaction with human users (though it can be used for this purpose). Different computer systems can interact with their users in whatever forms are most appropriate to their applications (for example Prolog, conceptual graphs, natural language, and so forth).
KIF is also not intended as an internal representation for knowledge within computer systems or within closely related sets of computer systems (though the language can be used for this purpose as well). Typically, when a computer system reads a knowledge base in KIF, it converts the data into its own internal form (specialized pointer structures, arrays, etc.). All computation is done using these internal forms. When the computer system needs to communicate with another computer system, it maps its internal data structures into KIF.
The purpose of KIF is roughly analogous to that of Postscript. Postscript is commonly used by text and graphics formatting systems in communicating information about documents to printers. Although it is not as efficient as a specialized representation for documents and not as perspicuous as a specialized wysiwyg display, Postscript is a programmer-readable representation that facilitates the independent development of formatting programs and printers. While KIF is not as efficient as a specialized representation for knowledge nor as perspicuous as a specialized display (when printed in its list form), it too is a programmer-readable language and thereby facilitates the independent development of knowledge-manipulation programs.
The following categorical features are essential to the design of KIF.
ISO/IEC 10646-1:1993, Information Technology (IT) - Universal Multiple-Octet Coded Character Set (UCS).
ISO/IEC 14481:1998, Information Technology (IT) - Conceptual Schema Modeling Facilities (CSMF).
For the purpose of this dpANS, the terms and definitions given in ISO/IEC 10646-1:1993 and ISO/IEC 14481:1998 apply.
In this chapter, the syntax of KIF is presented using a modified BNF notation. All nonterminals and BNF punctuation are written in boldface, while characters in KIF are expressed in plain font. The notation {x1,...,xn} means the set of terminals x1,...,xn. The notation [nonterminal] means zero or one instances of nonterminal; nonterminal* means zero or more occurrences; nonterminal+ means one or more occurrences; nonterminal^n means n occurrences. The notation nonterminal1 - nonterminal2 refers to all of the members of nonterminal1 except for those in nonterminal2. The notation int(n) denotes the decimal representation of integer n. The nonterminals space, tab, return, linefeed, and page refer to the characters corresponding to ascii codes 32, 9, 13, 10, and 12, respectively. The nonterminal character denotes the set of all 128 ascii characters. The nonterminal empty denotes the empty string.
KIF characters are classified as upper case letters, lower case letters, digits, alpha characters (non-alphabetic characters that are used in the same way that letters are used), special characters, white space, and other characters (every ascii character that is not in one of the other categories).
upper ::= A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z lower ::= a | b | c | d | e | f | g | h | i | j | k | l | m | n | o | p | q | r | s | t | u | v | w | x | y | z digit ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 alpha ::= ! | $ | % | & | * | + | - | . | / | < | = | > | ? | @ | _ | ~ | special ::= " | # | ' | ( | ) | , | \ | ^ | ` white ::= space | tab | return | linefeed | pageA normal character is either an upper case character, a lower case character, a digit, or an alpha character.
normal ::= upper | lower | digit | alpha
The function of a lexical analyzer is cyclic. It reads characters from the input string until it encounters a character that cannot be combined with previous characters to form a legal lexeme. When this happens, it outputs the lexeme corresponding to the previously read characters. It then starts the process over again with the new character. Whitespace causes a break in the lexical analysis process but otherwise is discarded.
There are five types of lexemes in KIF -- special lexemes, words, character references, character strings, and character blocks.
Each special character forms its own lexeme. It cannot be combined with other characters to form more complex lexemes, except through the ``escape'' syntax described below.
A word is a contiguous sequence of (1) normal characters or (2) other characters preceded by the escape character \.
word ::= normal | word normal | word\character
It is possible to include the character \ in a word by preceding it by another occurrence of \, i.e. two contiguous occurrences of \ are interpreted as a single occurrence. For example, the string A\\\'B corresponds to a word consisting of the four characters A, \, ', and B.
Except for characters following \, the lexical analysis of words is case insensitive. The output lexeme for any word corresponds to the lexeme obtained by converting all letters not following \ to their upper case equivalents. For example, the word abc and the word ABC map into the same lexeme. The word a\bc maps into the same lexeme as the word A\bC, which is not the same as the lexeme for the word ABC, since the second character is lower case.
A character reference consists of the characters #, \, and any character. Character references allow us to refer to characters as characters and differentiate them from one-character symbols, which may refer to other objects.
charref ::= #\characterA character string is a series of characters enclosed in quotation marks. The escape character \ is used to permit the inclusion of quotation marks and the \ character itself within such strings.
string ::= "quotable" quotable ::= empty | quotable strchar | quotable\character strchar ::= character - {",\}Sometimes it is desirable to group together a sequence of arbitrary bits or characters without imposing escape characters, e.g. to encode images, audio, or video in special formats. Character blocks permit this sort of grouping through the use of a prefix that specifies how many of the following characters are to grouped together in this way. A character block consists of the character # followed by the decimal encoding of a positive integer n, the character q or Q, and then n arbitrary characters.
block ::= # int(n) q character^n | # int(n) Q character^nFor the purpose of grammatical analysis, it is useful to subdivide the class of words a little further, viz. as variables, operators, and constants.
A variable is a word in which the first character is ? or @. A variable that begins with ? is called an individual variable. A variable that begins with an @ is called a sequence variable.
variable ::= indvar | seqvar indvar ::= ?word seqvar ::= @wordOperators are used in forming complex expressions of various sorts. There are three types of operators in KIF -- term operators, sentence operators, and definition operators. Term operators are used in forming complex terms. Sentence operators and user operators are used in forming complex sentences. Definition operators are used in forming definitions.
operator ::= termop | sentop | defop termop ::= value | listof | quote | if sentop ::= holds | = | /= | not | and | or | => | <= | <=> | forall | exists defop ::= defobject | defunction | defrelation | deflogical | := | :-> | :<= | :=>All other words are called constants.
constant ::= word - variable - operatorSemantically, there are four categories of constants in KIF -- object constants, function constants, relation constants, and logical constants. Object constants are used to denote individual objects. Function constants denote functions on those objects. Relation constants denote relations. Logical constants express conditions about the world and are either true or false. KIF is unusual among logical languages in that there is no syntactic distinction among these four types of constants; any constant can be used where any other constant can be used. The differences between these categories of constants is entirely semantic.
There are nine types of terms in KIF -- individual variables, constants, character references, character strings, character blocks, functional terms, list terms, quotations, and logical terms. Individual variables, constants, character references, strings, and blocks were discussed earlier.
term ::= indvar | constant | charref | string | block | funterm | listterm | quoterm | logterm
A implicit functional term consists of a constant and an arbitrary number of argument terms, terminated by an optional sequence variable and surrounded by matching parentheses. Note that there is no syntactic restriction on the number of argument terms; arity restrictions in KIF are treated semantically.
funterm ::= (constant term* [seqvar])
A explicit functional term consists of the operator value and one or more argument terms, terminated by an optional sequence variable and surrounded by matching parentheses.
funterm ::= (value term term* [seqvar])A list term consists of the listof operator and a finite list of terms, terminated by an optional sequence variable and enclosed in matching parentheses.
listterm ::= (listof term* [seqvar])Quotations involve the quote operator and an arbitrary list expression. A list expression is either an atom or a sequence of list expressions surrounded by parentheses. An atom is either a word or a character reference or a character string or a character block. Note that the list expression embedded within a quotation need not be a legal expression in KIF.
quoterm ::= (quote listexpr) | 'listexpr listexpr ::= atom | (listexpr*) atom ::= word | charref | string | blockLogical terms involve the if and cond operators. The if form allows for the testing of a single condition or multiple conditions. An optional term at the end allows for the specification of a default value when all of the conditions are false. The cond form is similar but groups the pairs of sentences and terms within parentheses and has no optional term at the end.
logterm ::= (if logpair+ [term]) logpair ::= sentence term logterm ::= (cond logitem*) logitem ::= (sentence term)The following BNF defines the set of legal sentences in KIF. There are six types of sentences. We have already mentioned logical constants.
sentence ::= constant | equation | inequality | relsent | logsent | quantsentAn equation consists of the = operator and two terms. An inequality consist of the /= operator and two terms.
equation ::= (= term term) inequality ::= (/= term term)An implicit relational sentence consists of a constant and an arbitrary number of argument terms, terminated by an optional sequence variable. As with functional terms, there is no syntactic restriction on the number of argument terms in a relation sentence.
relsent ::= (constant term* [seqvar])
A explicit relational sentence consists of the operator holds and one or more argument terms, terminated by an optional sequence variable and surrounded by matching parentheses.
relsent ::= (holds term term* [seqvar])
It is noteworthy that the syntax of implicit relational sentences is the same as that of implicit functional terms. On the other hand, their meanings are different. Fortunately, the context of each such expression determines its type (as an embedded term in one case or as a top-level sentence or argument to some sentential operator in the other case); and so this slight ambiguity causes no problems.
The syntax of logical sentences depends on the logical operator involved. A sentence involving the not operator is called a negation. A sentence involving the and operator is called a conjunction, and the arguments are called conjuncts. A sentence involving the or operator is called a disjunction, and the arguments are called disjuncts. A sentence involving the => operator is called an implication; all of its arguments but the last are called antecedents; and the last argument is called the consequent. A sentence involving the <= operator is called a reverse implication; its first argument is called the consequent; and the remaining arguments are called the antecedents. A sentence involving the <=> operator is called an equivalence.
logsent ::= (not sentence) | (and sentence*) | (or sentence*) | (=> sentence* sentence) | (<= sentence sentence*) | (<=> sentence sentence)There are two types of quantified sentences -- a universally quantified sentence is signalled by the use of the forall operator, and an existentially quantified sentence is signalled by the use of the exists operator. The first argument in each case is a list of variable specifications. A variable specification is either a variable or a list consisting of a variable and a term denoting a relation that restricts the domain of the specified variable.
quantsent ::= (forall (varspec+) sentence) | (exists (varspec+) sentence) varspec ::= variable | (variable constant)Note that, according to these rules, it is permissible to write sentences with free variables, i.e. variables that do not occur within the scope of any enclosing quantifiers. The significance of the free variables in a sentence depends on the use of the sentence. When we assert the truth of a sentence with free variables, we are, in effect, saying that the sentence is true for all values of the free variables, i.e. the variables are universally quantified. When we ask whether a sentence with free variables is true, we are, in effect, asking whether there are any values for the free variables for which the sentence is true, i.e. the variables are existentially quantified.
The following BNF defines the set of legal KIF definitions. There are three types of definitions -- unrestricted, complete, and partial. Within each type, there are four cases, one for each category of constant. Object constants are defined using the defobject operator. Function constants are defined using the deffunction operator. Relation constants are defined using the defrelation operator. Logical constants are defined using the deflogical operator.
definition ::= unrestricted | complete | partial unrestricted ::= (defobject constant [string] sentence*) | (deffunction constant [string] sentence*) | (defrelation constant [string] sentence*) | (deflogical constant [string] sentence*) complete ::= (defobject constant [string] := term) | (deffunction constant (indvar* [seqvar]) [string] := term) | (defrelation constant (indvar* [seqvar]) [string] := sentence) | (deflogical constant [string] := sentence) partial ::= (defobject constant [string] :-> indvar :<= sentence) | (defobject constant [string] :-> indvar :=> sentence) | (deffunction constant (indvar* [seqvar]) [string] :-> indvar :<= sentence) | (deffunction constant (indvar* [seqvar]) [string] :-> indvar :=> sentence) | (defrelation constant (indvar* [seqvar]) [string] :<= sentence) | (defrelation constant (indvar* [seqvar]) [string] :=> sentence) | (deflogical constant [string] :<= sentence) (deflogical constant [string] :=> sentence)A form in KIF is either a sentence or a definition.
form ::= sentence | definitionIt is important to note that definitions are top level constructs. While definitions contain sentences, they are not themselves sentences and, therefore, cannot be written as constituent parts of sentences or other definitions (unless they occur inside of a quotation.
A knowledge base is a finite set of forms. It is important to keep in mind that a knowledge base is a set of sentences, not a sequence; and, therefore, the order of forms within a knowledge base is unimportant. Order may have heuristic value to deductive programs by suggesting an order in which to use those sentences; however, this implicit approach to knowledge exchange lies outside of the definition of KIF.
A universe of discourse is the set of all objects presumed or hypothesized to exist in the world. The notion of object used here is quite broad. Objects can be concrete (e.g. a specific carbon atom, Confucius, the Sun) or abstract (e.g. the number 2, the set of all integers, the concept of justice). Objects can be primitive or composite (e.g. a circuit that consists of many subcircuits). Objects can even be fictional (e.g. a unicorn, Sherlock Holmes).
Different users of a declarative representation language, like KIF, are likely to have different universes of discourse. KIF is conceptually promiscuous in that it does not require every user to share the same universe of discourse. On the other hand, KIF is conceptually grounded in that every universe of discourse is required to include certain basic objects.
The following basic objects must occur in every universe of discourse.
In KIF, relationships among objects take the form of relations. Formally, a relation is defined as an arbitrary set of finite lists of objects (of possibly varying lengths). Each list is a selection of objects that jointly satisfy the relation. For example, the < relation on numbers contains the list <2,3>, indicating that 2 is less than 3.
A function is a special kind of relation. For every finite sequence of objects (called the arguments), a function associates a unique object (called the value). More formally, a function is defined as a set of finite lists of objects, one for each combination of possible arguments. In each list, the initial elements are the arguments, and the final element is the value. For example, the 1+ function contains the list <2,3>, indicating that integer successor of 2 is 3.
Note that both functions and relations are defined as sets of lists. In fact, every function is a relation. However, not every relation is a function. In a function, there cannot be two lists that disagree on only the last element. This would be tantamount to the function having two values for one combination of arguments. By contrast, in a relation, there can be any number of lists that agree on all but the last element. For example, the list <2,3> is a member of the 1+ function, and there is no other list of length 2 with 2 as its first argument, i.e. there is only one successor for 2. By contrast, the < relation contains the lists <2,3>, <2,4>, <2,5>, and so forth, indicating that 2 is less than 3, 4, 5, and so forth.
Many mathematicians require that functions and relations have fixed arity, i.e they require that all of the lists comprising a relation have the same length. The definitions here allow for relations with variable arity, i.e. it is perfectly acceptable for a function or a relation to contain lists of different lengths. For example, the relation < contains the lists <2,3> and <2,3,4>, reflecting the fact that 2 is less than 3 and the fact that 2 is less than 3 and 3 is less than 4. This flexibility is not essential, but it is extremely convenient and poses no significant theoretical problems.
The value of a functional term without a terminating sequence variable is obtained by applying the function denoted by the function constant in the term to the objects denoted by the arguments.
For example, the value of the term (+ 2 3) is obtained by applying the addition function (the function denoted by +) to the numbers 2 and 3 (the objects denoted by the object constants 2 and 3) to obtain the value 5, which is the value of the object constant 5.
If a functional term has a terminating sequence variable, the value is obtained by applying the function to the sequence of arguments formed from the values of the terms that precede the sequence variable and the values in the sequence denoted by the sequence variable.
Assume, for example, that the sequence variable @l has as value the sequence 2, 3, 4. Then, the value of the term (+ 1 @l) is obtained by applying the addition function to the numbers 1, 2, 3, and 4 to obtain the value 10, which is the value of the object constant 10.
A simple relational sentence without a terminating sequence variable is true if and only if the relation denoted by the relation constant in the sentence is true of the objects denoted by the arguments. Equivalently, viewing a relation as a set of tuples, we say that the relational sentence is true if and only if the tuple of objects formed from the values of the arguments is a member of the set of tuples denoted by the relation constant.
If a relational sentence terminates in a sequence variable, the sentence is true if and only if the relation contains the tuple consisting of the values of the terms that precede the sequence variable together with the objects in the sequence denoted by the variable.
An equation is true if and only if the terms in the equation refer to the same object in the universe of discourse.
An inequality is true if and only if the terms in the equation refer to distinct objects in the universe of discourse.
The truth value of true is true, and the truth value of false is false.
The value of a logical term involving the if operator is the value of the term following the first true sentence in the argument list. For example, the term (if (> 1 2) 1 (> 2 1) 2 0) is equivalent to 2.
If none of the embedded sentences of a logical term involving the if operator is true and there is an isolated term at the end, the value of the conditional term is the value of that isolated term. For example, if the object constant a denotes a number, then the term (if (> a 0) a (- a)) denotes the absolute value of that number.
If none of the embedded sentences is true and there is no isolated term at the end, the value is undefined (i.e. bottom). In other words, the term (if (p a) a) is equivalent to (if (p a) a bottom).
The value of a logical term involving the cond operator is the value of the term following the first true sentence in the argument list. For example, the term (cond ((> 1 2) 1) ((> 2 1) 2)) is equivalent to 2.
If none of the embedded sentences is true, the value is undefined (i.e. bottom). In other words, the term (cond ((p a) a)) is equivalent to (cond ((p a) a) (true bottom)).
A negation is true if and only if the negated sentence is false.
A conjunction is true if and only if every conjunct is true.
A disjunction is true if and only if at least one of the disjuncts is true.
If every antecedent in an implication is true, then the implication as a whole is true if and only if the the consequent is true. If any of the antecedents is false, then the implication as a whole is true, regardless of the truth value of the consequent.
A reverse implication is just an implication with the consequent and antecedents reversed.
An equivalence is equivalent to the conjunction of an implication and a reverse implication.
A simple existentially quantified sentence (one in which the first argument is a list of variables) is true if and only if the embedded sentence is true for some value of the variables mentioned in the first argument.
A simple universally quantified sentence (one in which the first argument is a list of variables) is true if and only if the embedded sentence is true for every value of the variables mentioned in the first argument.
Quantified sentences with complicated variables specifications can be converted into simple quantified sentences by replacing each complicated variable specification by the variable in the specification and adding an appropriate condition into the body of the sentence. Note that, in the case of a set restriction, it may be necessary to rename variables to avoid conflicts. The following pairs of sentences show the transformation from complex quantified sentences to simple quantified sentences.
(forall (... (?x r) ...) s) (forall (... ?x ...) (=> (r ?x) s)) (exists (... (?x r) ...) s) (exists (... ?x ...) (and (r ?x) s))
Note that the significance of free variables in quantifier-free sentences depends on context. Free variables in an assertion are assumed to be universally quantified. Free variables in a query are assumed to be existentially quantified. In other words, the meaning of free variables is determined by the way in which KIF is used. It cannot be unambiguously defined within KIF itself. To be certain of the usage in all contexts, use explicit quantifiers.
The definitional operators in KIF allow us to state sentences that are true ``by definition'' in a way that distinguishes them from sentences that express contingent properties of the world. Definitions have no truth values in the usual sense; they are so because we say that they are so.
On the other hand, definitions have content -- sentences that allow us to derive other sentences as conclusions. In KIF, every definition has a corresponding set of sentences, called the content of the definition.
The defobject operator is used to define objects. The legal forms are shown below, together with their content. In the first case, the content is the equation involving the object constant in the definition with the defining term. In the second case, the content is the conjunction of the constituent sentences.
(defobject s := t) (= s t) (defobject s p1 ... pn) (and p1 ... pn) (defobject s :-> v :=> p) (=> (= s v) p) (defobject s :-> v :<= p) (<= (= s v) p)
The deffunction operator is used to define functions. Again, the legal forms are shown below, together with their defining axioms. In the first case, the content is the equation involving (1) the term formed from the function constant in the definition and the variables in its argument list and (2) the defining term. In the second case, as with object definitions, the content is the conjunction of the constituent sentences.
(deffunction f (v1 ...vn) := t) (= (f v1 ...vn) t) (deffunction f p1 ...pn) (and p1 ...pn) (deffunction f (v1 ... vn) :-> v :=> p) (=> (= (f v1 ... vn) v) p) (deffunction f (v1 ... vn) :-> v :<= p) (<= (= (f v1 ... vn) v) p)
The defrelation operator is used to define relations. The legal forms are shown below, together with their defining axioms. In the first case, the content is the equivalence relating (1) the relational sentence formed from the relation constant in the definition and the variables in its argument list and (2) the defining sentence. In the second case, as with object and function definitions, the content is the conjunction of the constituent sentences.
(defrelation r (v1 ...vn) := p) (<=> (r v1 ...vn) p) (defrelation r p1 ...pn) (and p1 ...pn) (defrelation r (v1 ... vn) :=> p) (=> (r v1 ... vn) p)) (defrelation r (v1 ... vn) :<= p) (<= (r v1 ... vn) p))
The referent of every numerical constant in KIF is assumed to be the number for which that constant is the base 10 representation. Among other things, this means that we can infer inequality of all distinct numerical constants, i.e. for every t1 and distinct t2 the following sentence is true.
(/= t1 t2)
We use the intended meaning of numerical constants in defining the numerical functions and relations in this chapter. In particular, we require that these functions and relations behave correctly on all numbers represented in this way.
Note that this does mean that it is incorrect to apply these functions and relations to terms other than numbers. For example, a nonnumerical term may refer to a number, e.g. the term two may be defined to be the same as the number 2 in which case it is perfectly proper to write (+ two two).
The user may also want to extend these functions and relations to apply to objects other than numbers, e.g. sets and lists.
In KIF, we use the term (listof t1 ... tk) to denote the list of objects denoted by t1, ..., tk. For example, the following expression denotes the list of an object named mary, a list of objects named tom, dick, and harry, and an object named sally.
(defrelation list (?x) := (exists (@l) (= ?x (listof @l))))The object constant nil denotes the empty list. null tests whether or not an object is the empty list. The relation constants single, double, and triple allow us to assert the length of lists containing one, two, and three elements, respectively.
The first method is use of charref syntax, i.e. the characters # and \, followed by the character to be represented.
While this method works for all 128 characters, it is less than ideal for documents like this one, because of the difficulty of writing out non-printing characters. Using this method, it is also difficult to assert properties of some classes of characters. For this reason, KIF supports an alternative method of specification, viz. the use of the 7 bit code corresponding to the character. The relationship between characters and their numerical codes is given via the functions char-code and code-char. The former maps the nth character cn into the corresponding 7-bit integer n, and the latter maps a 7-bit integer n into the corresponding character cn. The values of these functions on all other arguments are undefined.
(= (char-code #\cn) n) (= (code-char n) #\cn)The relation constant character is true of the characters of KIF and no other objects.
One way of referring to strings is through the use of the string syntax described in chapter 4. In this method, we refer to the string abc by enclosing it in double quotes, i.e. "abc".
A second way is through the use of character blocks, the block syntax described in chapter 4. In this method, we refer to the string abc by prefixing with the character #, a positive integer indicating the length, the letter q, and the characters of the string, i.e. #3qabc
A third way of referring to strings is to use the listof function. For example, we can denote the string abc by a term of the form (listof #\a #\b #\c).
The advantage of the listof representation over the preceding representations is that it allows us to quantify over characters within strings. For example, the following sentence says that all 3 character strings beginning with a and ending with a are nice.
For example, we conceptualize the sentence (not (p (+ a b c) d)) as a list consisting of the operator not and the sentence (p (+ a b c) d). This sentence is treated as a list consisting of the relation constant p and the terms (+ a b c) and d. The first of these terms is a list consisting of the function constant + and the object constants a, b, and c.
For Lisp programmers, this conceptualization is relatively obvious, but it departs from the usual conceptualization of formal languages taken in the mathematical theory of logic. It has the disadvantage that we cannot describe certain details of syntax such as parenthesization and spacing (unless we augment the conceptualization to include string representations of expressions as well). However, it is far more convenient for expressing properties of knowledge and inference than string-based conceptualizations.
In order to assert properties of expressions in the language, we need a way of referring to those expressions. There are two ways of doing this in KIF.
One way is to use the quote operator in front of an expression. To refer to the symbol john, we use the term 'john or, equivalently, (quote john). To refer to the expression (p a b), we use the term '(p a b) or, equivalently, (quote (p a b)).
With a way of referring to expressions, we can assert their properties. For example, the following sentence ascribes to the individual named john the belief that the moon is made of a particular kind of blue cheese.
(believes john '(material moon stilton))Note that, by nesting quotes within quotes, we can talk about quoted expressions. In fact, we can write towers of sentences of arbitrary heights, in which the sentences at each level talk about the sentences at the lower levels.
Since expressions are first-order objects, we can quantify over them, thereby asserting properties of whole classes of sentences. For example, we could say that Mary believes everything that John believes. This fact together with the preceding fact allows us to conclude that Mary also believes the moon to be made of blue cheese.
(=> (believes john ?p) (believes mary ?p))The second way of referring to expressions is KIF is to use the listof function. For example, we can denote a complex expression like (p a b) by a term of the form (listof 'p 'a 'b), as well as '(p a b).
The advantage of the listof representation over the quote representation is that it allows us to quantify over parts of expressions. For example, let us say that Lisa is more skeptical than Mary. She agrees with John, but only on the composition of things. The first sentence below asserts this fact without specifically mentioning moon or stilton. Thus, if we were to later discover that John thought the sun to be made of chili peppers, then Lisa would be constrained to believe this as well.
(=> (believes john (listof 'material ?x ?y)) (believes lisa (listof 'material ?x ?y)))While the use of listof allows us to describe the structure of expressions in arbitrary detail, it is somewhat awkward. For example, the term (listof 'material ?x ?y) is somewhat awkward. Fortunately, we can eliminate this difficulty using uparrow and comma. Rather than using the listof function constant as described above, we write the expression preceded by the uparrow character ^ and add a comma character , in front of any subexpression that is not to be taken literally. For example, we would rewrite the preceding sentence as follows.
(=> (believes john ^(material ,?x ,?y)) (believes lisa ^(material ,?x ,?y)))
For every individual variable v, there is an axiom asserting that it is indeed an individual variable. Each such axiom is a defining axiom for the indvar relation.
(indvar (quote v))For every sequence variable s, there is an axiom asserting that it is a sequence variable. Each such axiom is a defining axiom for the seqvar relation.
(indvar (quote s))For every word w, there is an axiom asserting that it is a word. Each such axiom is a defining axiom for the word relation.
(word (quote w))Using this basic vocabulary and our vocabulary for lists, it is possible to define type relations for all types of syntactic expressions in KIF.
(=> (and (r 0) (forall (?n) (=> (r ?n) (r (1+ ?n))))) (forall (?n) (r ?n)))This schema encodes infinitely many sentences, the principle of mathematical induction for named relations. The following sentences are instances.
(=> (and (p 0) (forall (?n) (=> (p ?n) (p (1+ ?n))))) (forall (?n) (p ?n))) (=> (and (q 0) (forall (?n) (=> (q ?n) (q (1+ ?n))))) (forall (?n) (q ?n)))Axiom schemata are differentiated from axioms due to the presence of metavariables or other metalinguistic notation (such as dots or star notation), together with conditions on the variables. They describe sentences in a language, but they are not themselves sentences in the language. As a result, they cannot be manipulated by procedures designed to process the language (presentation, storage, communication, deduction, and so forth) but instead must be hard coded into those procedures.
As we have seen, it is possible in KIF to write expressions that describe KIF sentences. As it turns out, there is also a way to write sentences that assert the truth of the sentences so described. The effect of adding such metalevel sentences to a knowledge base is the same as directly including the (potentially infinite) set of described sentences in the knowledge base.
The use of such a language simplifies the construction of knowledge-based systems, since it obviates the need for building axiom schemata into deductive procedures. It also makes it possible for systems to exchange axiom schemata with each other and thereby promotes knowledge sharing.
The KIF truth predicate is called wtr (which stands for ``weakly true''). For example, we can say that a sentence of the form (=> (p ?x) (q ?x)) is true by writing the following sentence.
(wtr '(=> (p ?x) (q ?x)))This may seem of limited utility, since we can just write the sentence denoted by the argument as a sentence in its own right. The advantage of the metanotation becomes clear when we need to quantify over sentences, as in the encoding of axiom schemata. For example, we can say that every sentence of the form (=> p p) is true with the following sentence. (The relation sentence can easily be defined in terms of quote, listof, indvar, seqvar, and word.
(=> (sentence ?p) (wtr ^(=> ,?p ,?p)))Semantically, we would like to say that a sentence of the form (wtr 'p) is true if and only if the sentence p is true. Unfortunately, this causes serious problems. Equating a truth function with the meaning it ascribes to wtr quickly leads to paradoxes. The English sentence ``This sentence is false.'' illustrates the paradox. We can write this sentence in KIF as shown below. The sentence, in effect, asserts its own negation.
(wtr (subst (name ^(subst (name x) ^x ^(truth ,x))) ^x ^(not (wtr (subst (name x) ^x ^(not (wtr ,x)))))))No matter how we interpret this sentence, we get a contradiction. If we assume the sentence is true, then we have a problem because the sentence asserts its own falsity. If we assume the sentence is false, we also have a problem because the sentence then is necessarily true.
Fortunately, we can circumvent such paradoxes by slightly modifying the proposed definition of wtr. In particular, we have the following axiom schema for all p that do not contain any occurrences of wtr. For all p that do contain occurrences, wtr is false.
(<=> (wtr 'p) p)With this modified definition, the paradox described above disappears, yet we retain the ability to write virtually all useful axiom schemata as metalevel axioms.
From the point of view of formalizing truth, wtr is a cheat, since it fails to cover those interesting cases where sentences contain the truth predicate. However, from the point of view of capturing axiom schemata not involving the truth predicate, it works just fine. Furthermore, unlike the solutions to the problem of formalizing truth, the framework presented here is easy for users to understand, and it is easy to implement.
Two other constants round out KIF's level-crossing vocabulary.
The term (denotation t) denotes the object denoted by the object denoted by t. A quotation denotes the quoted expression; the denotation of any other object is bottom. As with wtr, the dentotation of a quoted expression is the embedded expression, provided that the expression does not contain any occurrences of denotation. Otherwise, the value is undefined.
(= (denotation 't) t)The term (name t) denotes the standard name for the object denoted by the term t. The standard name for an expression t is (quote t); the standard name for a non-expression is at the discretion of the user. (Note that there are only a countable number of terms in KIF, but there can be worlds with uncountable cardinality; consequently, it is not always possible for every object to have a unique name.)
Infix KIF is a syntactic variant of prefix KIF using infix syntax. For every expression in Infix KIF, there is a logically equivalent expression in KIF. However, the reverse is not true; some KIF expressions have been dropped for simplicity, eg. complex conditionals, complex variable specifications in quantified sentences, and documentation strings in definitions.
This chapter gives the full grammar for Infix KIF using nonterminals similar to those in the specification of Prefix KIF. (A few extra nonterminals are used here to define the more complicated syntax of Infix KIF.) This parallel treatment of nonterminals conveys the relationship between the two syntaxes.
Note that Infix KIF is NOT intended for use in the exchange of knowledge between computers. It is provided solely for consumption by humans, in articles about KIF and possibly in knowledge base editors. Prefix KIF is preferred for exchange of knowledge between machines. Its simpler syntax makes it easier to write parsers and makes it possible to write parsers that are more efficient.
upper ::= A | B | C | D | E | F | G | H | I | J | K | L | M | N | O | P | Q | R | S | T | U | V | W | X | Y | Z lower ::= a | b | c | d | e | f | g | h | i | j | k | l | m | n | o | p | q | r | s | t | u | v | w | x | y | z digit ::= 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 alpha ::= $ | % | ? | @ | _ special ::= " | # | ' | ( | ) | , | \ | ^ | ` | | ! | & | * | + | - | . | / | < | = | > | ~ white ::= space | tab | return | linefeed | page normal ::= upper | lower | digit | alpha
word ::= normal | word normal | word\character charref ::= #\character string ::= "quotable" quotable ::= empty | quotable strchar | quotable\character strchar ::= character - {",\} block ::= # int(n) q character^n | # int(n) Q character^n variable ::= upper word operator ::= termop | sentop | defop termop ::= listof | quote | if sentop ::= = | /= | not | and | or | => | <= | <=> | forall | exists defop ::= defobject | defunction | defrelation | deflogical | := | :-> | :<= | :=> constant ::= word - variable - operator atom ::= word | charref | string | block
The following BNF grammar defines the legal expressions in Infix KIF. Note that parentheses are specified only where essential. It is permissible to add parentheses, in pairs, around any term or sentence, in order to specify a particular association of expressions. In the absence of parentheses, parsers are required to associate to the right.
term ::= indvar | constant | charref | string | block | funterm | listterm | quoterm | logterm terms ::= empty | termlist | termlist ; seqvar termlist ::= term | term , termlist funterm ::= constant(terms) listterm ::= [terms] quoterm ::= 'expr' expr ::= atom | [] | [exprlist] exprlist ::= expr | expr , exprlist logterm ::= if(loglist) | if(loglist,term) loglist ::= sentence , term | sentence , term , loglist sentence ::= constant | equation | inequality | relsent | logsent | quantsent sentencelist ::= sentence | sentence , sentencelist equation ::= term=term inequality ::= term/=term relsent ::= constant(terms) logsent ::= ~sentence sentence & sentence | sentence | sentence | sentence => sentence | sentence <= sentence | sentence <=> sentence quantsent ::= Avariable).sentence | Evariable).sentence vars ::= empty | varlist | varlist ; seqvar varlist ::= indvar | indvar , varlist definition ::= unrestricted | complete | partial unrestricted ::= defobject(constant,sentencelist) | deffunction(constant,sentencelist) | defrelation(constant,sentencelist) | deflogical(constant,sentencelist) complete ::= defobject(constant := term) | deffunction(constant(vars) := term) | defrelation(constant(vars) := sentence) | deflogical(constant := sentence) partial ::= defobject(constant :-> indvar :<= sentence) | defobject(constant :-> indvar :=> sentence) | deffunction(constant(vars) :-> indvar :<= sentence) | deffunction(constant(vars) :-> indvar :=> sentence) | defrelation(constant(vars) :<= sentence) | defrelation(constant(vars) :=> sentence) | deflogical(constant :<= sentence) deflogical(constant :=> sentence) form ::= sentence | definition
As with prefix KIF, a knowledge base in Infix KIF is a finite set of forms, not a sequence, though, as with Prefix KIF, there may be heursitic value in the order.
KIF is a highly expressive language. For many, this is a desirable feature; but there are disadvantages. One disadvantage is that it complicates the job of building fully conforming systems. Another disdvantage is that the resulting systems tend to be "heavyweight" (i.e. they are larger and in some cases less efficient than systems that employ more restricted languages).
In order to deal with these problems, the KIF committee in the Fall of 1997 voted to augment the basic language specification with a set of "conformance dimensions". These dimensions are not the same as the "conformance levels" of other languages. Rather, each conformance dimension has a variety of levels within that dimension.
A "conformance profile" is a selection of alternatives from each conformance dimension. System builders are expected to make choices for each dimension and and then ensure that their systems adhere to the resulting comformance profile. Systems are expected to use the terminology defined here to share information about their conformance profile with other systems (in a protocol-specific manner).
Although this conformance profile scheme is more complex than one based on conformance levels, it accommodates varying capabilities and/or computational constraints while providing a migration path from more restrictive to more expressive.
A conformance dimension is a classification of KIF sentences into conformance categories on the basis of a single syntactic criterion. (For example, the quantification dimension provides two categories, quantified KIF and unquantified KIF, based on whether or not a conforming knowledge base contans quantifiers.)
The first conformance dimension concerns logical form. There are five basic categories: atomic, conjunctive, positive, logical, and rule-like. Rule-like knowledge bases are further categorized as Horn or non-Horn and recursive or non-recursive.
A knowledge base is atomic if and only if it contains no logical operators.
A knowledge base is conjunctive if and only if it contains no logical operators except for conjunction.
A knowledge base is positive if and only if it contains no logical operators except for conjunction and disjunction.
A knowledge base is logical if and only if it contains no logical operators except for conjunction, disjunction, and negation.
A knowledge base is rule-like if and only if every sentence is either atomic or an implication or reverse implication in which all subexpressions are atomic sentences or negations of atomic sentences. A rule system is a rule-like knowledge base.
A rule system is Horn if and only if every constituent of every rule is atomic (i.e. no negations allowed). Otherwise, the rule system is said to be non-Horn.
The dependency graph for a rule system is a graph whose nodes are the constants in relational position. There is an edge from the node for a given relation constant p to the node of relation constant q if and only if p appears in the body of a rule whose head predicate is p.
A rule system is recursive if there is a cycle in its dependency graph. Otherwise, the rule system is said to be non-recursive.
The nature of terms defines a second conformance dimension. There are two categories: simple and complex.
A knowledge base is simple if and only if the only terms occurring the knowledge base are constants and variables.
A knowledge base is complex if and only if it contains terms other than constants or variables, e.g. functional terms or logical terms.
The third conformance dimension concerns the presence or absence of variables.
A knowledge base is ground, or zeroth-order, if and only if it contains no variables. Otherwise, a knowledge base in nonground.
A knowledge base is first-order if and only if there are no variables in the first argument of any explicit functional term or explicit relational sentence.
A knowledge base is higher-order otherwise.
For nonground knowledge bases, there are two alternatives -- quantified and unquantified.
A nonground knowledge base is quantified if and only if it contains at least one explicit quantifier.
A nonground knowledge base is unquantified if and only if it contains no explicit quantifiers.
The final conformance dimension concerns the ability to express metaknowledge, e.g. to write sentences about sentences.
A knowledge base is baselevel if and only if it contains no occurrences of the quote operator or the wtr relation.
Otherwise, the knowledge base is metalevel.
A conformance profile is a selection of alternatives for each conformance dimension. Given the dimensions and categories defined in the preceding section, it is possible to define a large number of profiles. A single system may use different profiles in different types of communication. In particular, it is common to use one profile for assertions and another for queries. The following paragraphs define a few common types of systems with their corresponding profiles.
A database system is one in which (1) all assertions are atomic, simple, ground, and baselevel and (2) all queries are positive, simple, unquantified, and baselevel.
A Horn system (e.g. pure Datalog) is one in which (1) all assertions are rules that are Horn, unquantified, and baselevel and (2) all queries are positive, non-recursive, unquantified, and baselevel.
A relational system is one in which (1) all assertions are rules that are simple, unquantified (but may be non-Horn and non-recursive), and baselevel and (2) all queries are logical, non-recursive, unquantified, and baselevel.
A first-order system is one that allows the broadest categories within each conformance dimension except that only first-order expressions are accommodated.
A full KIF system is one that accepts the broadest categories within each conformance dimension, i.e. any KIF knowledge base is acceptable in any context.
The existence of multiple conformance profiles raises the question of what happens when systems with different profles must communicate.
Whenever the conformance profile of a receiver is known, a sender should avoid sending expressions that fall outside the receiver's conformance profile.
Unfortunately, this rule cannot be enforced in all situations. In some cases, conformance information about receivers is unavailable; and, even when conformance information is available, it may be desirable to send a message that falls outside a receiver's profile, e.g. it may be most efficient for a sender to broadcast a single knowledge base to a large number of receivers with differing conformance profiles rather than sending different knowledge bases to each receiver.
Whenever a receiver receives a non-conforming expression, it is free to ignore the expression, even though it may be able to make sense of portions of that expression. If the receiver ignores a non-conforming expression and the sender requests a reply, the receiver should report a failure.