One Method of Constructing a Formal System Khimuri Rukhaia and Lali Tibua Institute of Applied...

One Method of Constructing a Formal System

Khimuri Rukhaia and Lali Tibua

Institute of Applied Mathematics, TSU Email: [email protected]

27-29 August,2007 1Intas,Moscow

Talk outline

IntroductionNotation Theory The SR TheoryOpen Problem


Abstract:Research in the field of automated theorem proving mainly has been conducted in two directions:

(a)Simple representation of the input problem through the improvement if the logical language

and

(b) Search for effective proof methods and their implementation.


Results of this research are essentially based on the first-order theory. In this theory the operator sign of Bourbaki does not occur in the basic symbols, or is it possible to introduce it through Pkhakadze's rational system of rules for defining contracting symbols. The absence of the operator sign in a theory in some sense restricts its expressive power. In this paper the SR logic is constructed, whose language, as its basic symbols, includes the operator sign and S and R operator signs of substitution. In this theory the existential and universal quantifiers are defined by the rational system of the defining rules. The same system is used to deductively extend and develop the language of SR theory and, therefore, it has sufficient expressive power.


The SR-Logic languageThe language of SR logic consists of the following symbols: I. Fundamental symbols: 1) Logical connectives: (of the weight 1), , (each of the weight 2).

2) Logical operational sign of the weight (1,1).

3) Substantive special substitution operator S of the weight (1,2). 4) Relational logico-special substitution operator R of the weight (1,2) and with the logicality indicator 2. 5) Object letters:

,...,,210

XXX27-29 August,2007 5Intas,Moscow

6)Predicate symbols = and (each of the weight 2)

Predicate letters:

and Expressive strings of uppercase letters.

7) Functional symbol that has the weight 2, and functional letters:

and expressive strings of lowercase letters.8) [ and ] (left and right brackets).

II. Signs, introduced by the definitions of the types I, II and II‘.

,...}2,1,0{,...,,,1100

nQPQP nnnn

,...}2,1,0{,...,,,1100

ngfgf nnnn


Finite sequence of signs of SR are called a word

of SR- logic. The words are the SR- logic operators with the weight 1. The words

and are the SR logic operators with the weight 2. Besides, the operators are substantive partial quantifiers with the binding indicator 2, and the operators are logico-special partial quantifiers with logicality and binding indicator 2.


,...,10

xx ,...,

10SxSx

,...,10

RxRx

,...,10

SxSx

,...,10

RxRx

Operators

Simple operators

Composed operators

Quantifiers Partial quantifiers

,...,, xxx ,..., RxSx


,...,, complement

The operator is called :

1) A logical relational if its operands are formulas,and results is formulae. For examples: 2) A logical substantive if its operands are formulas,and results is term. For examples: 3) A special relacional if its operands are terms,and resu lts is formulae. For example: <represent>x,…

4) A special substantive if its operands are termsand results is term. For examples: 5) A logico-special relacional if its operands are terms,and resu lts is formulae. Rx, <root>x,…

6) A logico-special substantive if its operands are terms and results is term. <subset>x,…

,...,,, xx

,..., xx

,...sin,, Sx


Formulas and terms of SR logic are defined in following way: (1) Subject letters are simple terms. (2) If is an n-ary relational logical (resp. special) operator, then (resp. ) is either a formula or a term depending whether is relational or substantive.(3) Let be a sequence of formulas. If is an n-ary logico-special operator whose logicality indicator is , and is the maximal subsequence of the sequence consisting of formulas only, then is either a formula or a term depending whether is relational or substantive.(4) C is formula or term if and only if it is derived by the three rules above.

n

AA ...1

n

TT ...1

ncc ,...,

1

),...,(1 k

nnknn

cc ,...,1

ncc ,...,

1

ncc ,...,

1


The expression is a term that denotes a privileged subject that has the property A, if such a subject exists. Otherwise it is a term that denotes an arbitrary element in the interpretation domain. For example, is a term that denotes a privileged number whose squaring gives 1. For instance, ``1'' can be such a term. The expression

is a term that denotes some number from the interpretation domain of real numbers, e.g., ``0''.

27-29 August,2007 Intas,Moscow 11

Ax

)1( 2 xx

)1( 2 xx

Definitions:A form of SR- logic is called a fundamental forms(a form of level 0), if it is constructed from fundamental symbols. The symbols of level n of the

SR- logic are defined as follows:

1) The fundamental symbols of the SR- logic are symbols of level 0.2) A symbol of level n (n=1,2,…) is a contracting

symbol of SR- logic introduced by a definition such that each symbol in its right hand side has the level less than n, and there is at least one symbol with the level n-1.


Definitions:Below we label definitions of the types I, II, and II' with the , where k is the number of the definition, i denotes the type of the definition, and j denotes the level of the operator obtained by the definition.

Reads: ``There exists such that A''. The operator is logical relational.

Reads:,, For all ''. The operator is logical relational.

],[ jiDk

xAARxxAIID ],[1

x x

AxxAIIID ],[2

x xA


where the variable is different from and does not occur in . Reads:A set of object of properties A.

The operator is logical substantive. Reads: ``If A then B’’. Reads: `` A and B’’

]],[[],[ '

3AyxxyxAsetIIIIID

y xA

xset

BABAIID ],[4

][],[5

BABAIID


where the variable x does not occur in the terms T and U. Reads: ``The complements of the set U with respect of the set T’’. The operator <complement> is special substantive.

where the variable y is different from x and does not occur in the terms T and U.

Reads: ``U can be represented as T with respect of the x’’. The operator <represent> is special relational partial quantifier with the binding indicator (2).

),(],[7

SxyTUyxUTrepresentIIID

][],[6

UxTxxsetUTcomplementIVIID


Reads: ``T is a solution of the formula A with respect to x ''. The operator <root> x is a logico-special relational partial quantifier with the logical and binding indicator (2).

where x and y are distinct variables , x do not occur in T and y do not occur in T and A .

Reads: ``The set of all the elements of T with the property A’’. The operator <subset> x is a logico-special substantial partial quantifier with the logical and binding indicator(2).

RxTAxTArootIID ],[8

]]],[[[],[9

ATxyxxyxTAsubsetIIIIID


The inference rules of SR logic are the following:

a. if A and , then B;b. if A and , then B;c. If A and A is congruence of B, then B;

BA BA


The axiom schemes of SR logic are the following:

1. []2. []3. [][]4. []5. 6. 7. Note that in the axiom schemas HA6 and HA7 the substitution (T/x) does not bind free variables in T.

HA8.

HA9.

HA10.

]][][[11

BAAA

xARxTA AxTRxTA )/(

UxTSxTU )/(

][][ xBxABAx ][]][][[ RxUBRxTAUTBAx ][]][][[

1111USxUTSxTUTUTx


Last, assume is a definition, then (respectively is an axiom schema if C is a formula (resp. C is a term).

Examples of the axiom schema are:

where y does not occur in A.

.m

HAD1

CC ,...)2,1( mDm

1CC

1CC

mHAD

xAARxxAHAD .1

],[.2

AyxxyxAsetHAD


The axioms and inference rules of SR logic imply validity of the counterparts of all the deductive criteria from [1] and the following theorems:

Theorem 1. If A is a main formula of SR and B is a contracted form of A, then | B, if and only if | A.

Theorem 2. Let the type of the operator be I, II, or II'. If all the operators in the right side of the definition of are invariant then is invariant.

Theorem 3. All operators of SR are invariant.

,0,...1

mxxm

mxx ...

1

mxx ...

1


Open Problem:We have a similar situation in artificial languages. Here computer (machine) languages play the r\^{o}le of the main language, and programming languages can be considered as languages extended by contracting symbols. Each programming language is obtained by introducing a new operator (these operators can be called contracting symbols). Programming languages are closer to natural languages, than computer languages. When writing programs, a programmer uses general, yet unproved, laws that connect computer languages and programming languages. The use of these laws is based on intuition. This is why program testing becomes necessary, but testing can not guarantee program correctness. If these laws can be proved for programming languages, then program testing will not be necessary. To prove them, it will be necessary to find a mathematical notion of contracting symbol (i.e., an operator of a programming language).


Hence, a general theory of programming should be created in the same method that was used for creating the Notation Theory. To achieve this goal, operators of modern programming languages should be studied, a rational notion of contracting symbol should be introduced, and the theory of contracting symbols for artificial languages should be constructed. Creating such a theory essentially means to create a general theory of programming. It would imply reliability of programs written in the language of this theory. Moreover, such a theory would help to come up with recommendations on how machine languages should be, to which direction computers should develop, what kind of devices should be created for processing mathematical texts, etc.


The logical form of the adjudge`s operator

the logical form is

The operator is substantive partial quantifier with the binding indicator 2.

The logical form of the recursive definition

1:ancestor(x,y) parent(x,y)

N:ancestor(x,y) ( parent(x,z) N-1:ancestor (z,y)).


1: xx xxSx )1( Sx

z


.~][~][~.15 ABAandBAC MMM

|— M~ ,

|— ][~M A , ,5.,2...,.., CBASBABBB

,1.CA —| ][~M A

,5.,3. SAAAAAACAA

,5.

,11.

,5.

CAA

CAA

CAAAA

.

,5.

,1.

CA

SAAA

—| M~ .

Towards One Logical Method of Program Verification

For the past yers a lot of attention has been paid to the studies of mathematical methods of asserting the correctness of program.One of such metods has been described in the monograph (Chang Ch;Lee R; ,,Symbolic Logic and Mechanical Theorem Proving’’). This method helps in the analysis of programs whose edge describing control predicate requires that it should be a quantifier-free formula. This presens rather great restriction and is caused by the fact that the application of the considered method requires reducing the program describing formula A to the Skolem standard form B. It is evident that when formuli A contains quantifiers, as a rule, B is not equivalent A, i.e. formula B cannot be a describing formula of the given program.


Theorem 1. Let be clause set of describing formula of every -program P then is satisfiable.

Theorem 2. Given a –program P ,let S denote the set of clauses representing . Then S is satisfiable.

Theorem 3. Let for given – program P a formula be obtained from describing formula by rubbing out all such literals that include than P will finish working if and only if is inconsistent.

Theorem 4. Let P be –program and S a clause set that represents a formula: . The program P will finish working if and only if deduction(answer) of halting clause from a set S (by resolution rule) exists.


pA pA

ISP AAA

TA

PA

HQ

IsT AAA

ISP AAA

References:[1] N. Bourbaki. Elements de mathematique. Theorie des ensembles. Chapitres 1 et 2. Hermann, Paris, 1966.

[2] A. Church. Introduction to Mathematical Logic. Princeton University Press,Princeton, N.J., 1956.

[3] D. Hilbert and P. Bernays. Grundlagen der Mathematik, volume I,II. Springer,Berlin, 1934, 1939.

[4] Sh. Pkhakadze. Some problems of the notation theory. Tbilisi University Press,Tbilisi, 1977. In Russian.

[5] Kh. Rukhaia. On one variant of -theory extended with contracting symbols.In Jenaer Frege-Konferenz, pages 365-381, Friedrich-Schiller UniversitÄat, Jena,1979.


One Method of Constructing a Formal System Khimuri Rukhaia and Lali Tibua Institute of Applied...

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