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Ｈｉｌｂｅｒｔ ＩＩ Presentation of Formal Correct Mathematical Knowledge Logical Language ════════════════════════════════════════════════════════════════════════════════ ═══════ Michael Meyling The source for this document can be found here: http://www.qedeq.org/0_04_00/doc/project/qedeq_logic_language.xml Copyright by the authors. All rights reserved. If you have any questions, suggestions or want to add something to the list of modules that use this one, please send an email to the address mime@qedeq.org The authors of this document are: Michael Meyling ___________________________________________________ Description ═══════════ The project Ｈｉｌｂｅｒｔ ＩＩ includes formal correct mathematical knowledge. Here we introduce the underlying formal language for the mathematical formulas. This is done in an informal way. Important theorems (e.g.: universal decomposition, and any proofs) are left out. All we will do is manipulate symbols. We build lists of symbol strings and use certain simple rules to get new lists. So by starting with a few basic lists we create a whole universe of derived symbol lists. It turns out that these lists could be interpreted as a view to the incredible world of mathematics. ___________________________________________________ Chapter 1 Entities ════════ To describe the logical language we firstly deal with a more basic notation. This notation enables us to formulate the syntax of formulas and terms later on. 1.1 Elements, Atoms and Lists ――――――――――――――――――――――――――――― The basic structure we have to deal with is an element. An element is either an atom or a list. An atom carries textual data, atoms are just strings. Each list has an operator and can contain elements again. An operator is also nothing more than a simple string. A list has a size: the number of elements it contains. Their elements can be accessed by their position number. An atom has no operator, no size and no subelements in the previous sense. 1.2 List Notation ――――――――――――――――― Lists and atoms can be written in the following manner. We write down string atoms quoted with " and the lists as the contents of the operator string followed by ( and a comma separated list of elements and an closing ). 1.3 Examples ―――――――――――― In this syntax we can write down the following element examples. "I am a string atom" EMPTY_LIST() THIS_LIST("contains", "three", "atoms") OPERATOR("argument 1", "argument 2") FUNCTION_A(FUNCTION_B("1", "2"), "3") In the last example we have a list that has the operator #FUNCTION_A# and contains two elements. The first element is #FUNCTION_B("1", "2")# which is a list too. The second element is the atom #"3"#. ___________________________________________________ Chapter 2 Logical Language ════════════════ There are different basic things we have to do with. These are predicates, functions, subject variables and logical connectives. In the following all of them are named and described. 2.1 Logical Operator Overview ――――――――――――――――――――――――――――― Lists are categorized according to their operators. Before we introduce the formal language in detail the used operators are briefly listed. 2l@ l o g i c a l A N D logical conjunction operator ∧ O R logical disjunction operator ∨ I M P L logical implication operator → E Q U I logical biconditional operator ↔ N O T logical negation operator ¬ 2l@ l o g i c a l q u a n t i f i e r s F O R L L universal quantifier ∀ E X I S T S existential quantifier ∃ E X I S T S U unique existential quantifier ∃ ! 2l@ v a r i a b l e s V A R subject variables x, y, z, ... P R E D V A R predicate variables A, B, R, ... F U N C V A R function variables f, g, h, ... 2l@ c o n s t a n t s P R E D C O N predicate constants =, ∈, ⊆, ... F U N C C O N function constants ∅, ℘, ... C L A S S class term { x | φ(x) } 2.2 Terms and Formulas ―――――――――――――――――――――― Now we define recursivly our formal language. We call some elements s u b j e c t v a r i a b l e s , t e r m s and some other f o r m u l a s . We also define the relations a subject variable i s f r e e i n and i s b o u n d i n a term or a formula. If something is not according to the formal rules errors occur. The error codes are also described. 2.2.1 General Error Codes The atoms and lists that build up a formula or term are subject to restrictions. The following errors occur if an atom has no content or has content with length of 0 or an list has no operator or one of its sub-elements does not exist. These are mainly technical error codes, only the error code 30470 shows an semantical error. 30400 no element an element doesn’t exist - it is null 30410 no atom an atom doesn’t exist - it is null 30420 no list a list doesn’t exist - it is null 30430 no atom content an atom has no content - it is null 30440 atom content empty an atom has content with 0 length 30450 no operator a list has no operator - it is null 30460 operator empty a list has an operator with 0 length 30470 list expected list element expected but not found 2.2.2 Subject Variable We call an element s u b j e c t v a r i a b l e iff it has the operator V A R and its list size is 1 with an atom as its only argument. Each subject variable is also called a t e r m . Only the subject variable itself is free in itself. No subject variable is bound in a subject variable. 30710 not exactly one argument list has not exactly one element 30730 atom element expected the first and only list element must be an atom 2.2.3 Function Term If an element has the operator F U N V A R or F U N C O N and its list size is greater than or equal to 1 with an atom as its first argument and the remaining arguments are all terms then it is called a term too. Iff a subject variable is free in any sub-element it is also free in the new term. No other subject variables are free. Analogous for bound subject variables. 30720 argument(s) missing if operator is F U N C O N the list must have at least one element 30730 atom element expected the first list element must be an atom 30740 argument(s) missing if operator is F U N V A R the list must have more than one element 30770 free bound mixed found a bound subject variable that is already free in a previous list element 30780 free bound mixed found a free subject variable that is already bound in a previous list element 30690 undefined constant the operator is F U N C O N and this function constant has not been defined for this argument number Any other error for term checks may occur due to the fact that all (but the first) sub-elements must be terms too. 2.2.4 Predicate Formula If an element has the operator P R E D V A R or P R E D C O N and its list size is greater than or equal to 1 with an atom as its first argument and the remaining arguments are all terms and no errors occur then it is called a f o r m u l a . Iff a subject variable is free in any sub-element it is also free in the new formula. No other subject variables are free. Analogous for bound subject variables. 30720 argument(s) missing list must have at least one element 30730 atom element expected the first list element must be an atom 30770 free bound mixed found a bound subject variable that is already free in a previous list element 30780 free bound mixed found a free subject variable that is already bound in a previous list element 30590 undefined constant the operator is P R E D C O N and this predicate constant has not been defined for this argument number Any other error for formula checks may occur due to the fact that all (but the first) sub-elements must be terms. 2.2.5 Logical Connectives If an element has the operator A N D , O R , I M P L or E Q U I and its list size is greater than or equal to 2 and the remaining arguments are all formulas and no errors occur then it is called a formula too. Iff a subject variable is free in any sub-element it is also free in the new formula. No other subject variables are free. Analogous for bound subject variables. 30740 argument(s) missing list must have more than one element 30760 exactly 2 elements expected the operator is I M P L and this list size is not equal to 2 30770 free bound mixed found a bound subject variable that is already free in a previous list element 30780 free bound mixed found a free subject variable that is already bound in a previous list element Any other error for formula checks may occur due to the fact that all sub-elements must be formulas. 2.2.6 Negation If an element has the operator N O T , its list size is exactly 1 and its only sub-element arguments is a formula then it is called a formula too. Iff a subject variable is free in the sub-element it is also free in the new formula. No other subject variables are free. Analogous for bound subject variables. 30710 exactly 1 argument expected list must have exactly than one element Any other error for formula checks may occur due to the fact that the sub-element must be a formula. 2.2.7 Quantifiers If an element has the operator F O R A L L , E X I S T S or E X I S T S U its first sub-element is a subject variable and its second and perhaps its third sub-element is a formula then the element is called a f o r m u l a too. Iff a subject variable is free in the sub-element it is also free in the new formula. No other subject variables are free. Analogous for bound subject variables. 30760 2 or 3 arguments expected list must have exactly 2 or 3 elements 30540 subject variable expected first sub-element must be a subject variable 30550 already bound subject variable already bound in second or third sub-element 30770 free bound mixed found a bound subject variable that is already free in a previous list element 30780 free bound mixed found a free subject variable that is already bound in a previous list element Any other error for formula checks may occur due to the fact that the sub-element must be a formula. 2.2.8 Class Term An list element with the operator C L A S S , containing an subject variable and an formula is a term. Iff a subject variable is free in the formula and is not equal to the first sub-element (which is a subject variable) it is also free in the new term. No other subject variables are free. If a subject variable is bound in the formula it is bound in the new term. Also the first sub-element is bound. No other subject variables are bound. 30760 2 arguments expected the list must contain exactly two arguments 30540 subject variable expected the first sub-element must be a subject variable 30550 already bound the subject variable is already bound in the formula 30680 undefined class operator the class operator is still unknown Any other error for formula checks may occur due to the fact that the second sub-element must be a formula. 2.2.9 Term When checking an element for beeing a term the element must have the operator for a S u b j e c t V a r i a b l e , F u n c t i o n T e r m or C l a s s T e r m . 30620 unknown term operator element has no operator that is known as a term operator Any other error for the accordant operator checks may occur. 2.2.10 Formula When checking an element for beeing a formul the element must have the operator for a P r e d i c a t e F o r m u l a , L o g i c a l C o n n e c t i v e , N e g a t i o n or Q u a n t i f i e r . 30530 unknown logical operator element has no known logical operator Any other error for the accordant operator checks may occur. ___________________________________________________ Chapter 3 Representations ═══════════════ The representation of elements differ according to the viewpoint. Lets take the following formula for example. y = { x | φ(x) } ↔ ∀ z (z ∈ y ↔ z ∈ { x | φ(x) } ) The predicate constant ∈ must have been defined in previous sections. 3.1 List Notation ――――――――――――――――― In list notation (see chapter1_section1) the above formula looks like the following. EQUI( PREDCON( "equal", VAR("y"), CLASS( VAR("x"), PREDVAR( "φ", VAR("x") ) ) ), FORALL( VAR("z"), EQUI( PREDCON( "in", VAR("z"), VAR("y") ), PREDCON( "in", VAR("z"), CLASS( VAR("x"), PREDVAR( "φ", VAR("x") ) ) ) ) ) ) Due to XSD restrictions for the XML document some error codes listed in Chapter chapter0 will not occur. Instead the XML will be classified as invalid. 3.2 Java ―――――――― The list notation leads directly to the following Java code. Element el = new ElementListImpl("EQUI", new Element[] new ElementListImpl("PREDCON", new Element[] new AtomImpl("equal"), new ElementListImpl("VAR", new Element[] new AtomImpl("y"),), new ElementListImpl("CLASS", new Element[] new ElementListImpl("VAR", new Element[] new AtomImpl("x"),), new ElementListImpl("PREDVAR", new Element[] new AtomImpl(" phi"), new ElementListImpl("VAR", new Element[] new AtomImpl("x"),)))), new ElementListImpl("FORALL", new Element[] new ElementListImpl("VAR", new Element[] new AtomImpl("z"),), new ElementListImpl("EQUI", new Element[] new ElementListImpl("PREDCON", new Element[] new AtomImpl("in"), new ElementListImpl("VAR", new Element[] new AtomImpl("z"),), new ElementListImpl("VAR", new Element[] new AtomImpl("y"),)), new ElementListImpl("PREDCON", new Element[] new AtomImpl("in"), new ElementListImpl("VAR", new Element[] new AtomImpl("z"),), new ElementListImpl("CLASS", new Element[] new ElementListImpl("VAR", new Element[] new AtomImpl("x"),), new ElementListImpl("PREDVAR", new Element[] new AtomImpl(" phi"), new ElementListImpl("VAR", new Element[] new AtomImpl("x"),))))))); 3.3 XML ――――――― The XML representation within an QEDEQ module looks a little bit different. Here all first list atoms are represented as the attribute ref or id. So the above formula may look like the following. Due to XSD restrictions for the XML document some error codes listed in Chapter chapter0 will not occur. Instead the XML will be classified as invalid. ___________________________________________________