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Some remarks on the possibility of extending resolution proof procedures to intuitionistic logic
Information ProcessingLetters 22 (1986) 87-90 North-Holland
18 January 1986
S O M E REMARKS ON T H E POSSIBILITY OF EXTENDING R E S O L U T I O N PROOF P R O C E D U R E S T O I N T U I T I O N I S T I C LOGIC
Marta C I A L D E A Langages et Systbmes Informatiques, Laboratoire Associb au CNRS, i~quipe Cornprkhension du Raisonnement Naturel, Universitb Paul Sabatier, 118 route de Narbonne, 31077 Toulouse Cbdex, France
Communicated by W.M. Turski Received 8 March 1985
Keywords: Theoremproving, formal languages
Section 1
Constructive logic is acquiring an ever increasing importance in computer science [2,3,7,8,9]. The majority of the research in this direction is based on the analysis of structural properties of proofs as a whole, a n d / o r the recognizable isomorphism between algorithms and constructive proofs in certain formal systems. However, it is not yet known whether one of the most efficient (and old) proof procedures for classical logic, the resolution m e t h o d [11,12], can easily be extended to constructive logic. Here, we propose a resolution procedure for propositional intuitionistic logic, which however remains a sort of external oracle: it applies in fact to a system of modal logic treated as a formal semantics of intuitionism. This detour was originally m a d e as a heuristic approach: since intuitionistic logic has a very rich (Kripke-style) semantics e m b e d d e d in a relatively poor language, it seems natural to translate intuitionistic formulas into a richer language, to make their behaviour more transparent w.r.t, resolution. However, as it will be apparent from the definition of the system of rules proposed, the backtrack to intuitionistic language is not at all simple, for the following reasons: (i) The formulation of the rules for the modal system is not too complicated, while their reformu-
lation for a language where implication cannot be eliminated and modal operators are not explicitly present leads to an enormous loss of simplicity. (ii) A modal deduction from a set of clauses which translate intuitionistic formulas generates clauses which translate no intuitionistic formula. Obviously, we cannot conclude that intuitionistic logic is absolutely intractable by resolution methods: in fact, such problems may arise in connection with this particular resolution system only. Below, we shall define the modal system MI, the translation of intuitionistic logic LI into MI, and the resolution rules for MI together with a sketch of the completeness proof. The system of rules is inspired by Farihas Del Cerro et al. [4,5,6], although new complications arise here because of the peculiarity of MI-interpretations. The details of the proofs can be found in [1], as well as some examples and a short discussion on the difficulties connected with a reformulation of the rules in terms of the intuitionistic language. Section 2
The modal system MI is obtained by adding the following axiom to the system $4 of Lewis [10]: p ~ Ulp if p is atomic.
0020-0190/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
87
Volume 22. N umber 2
IN FORMATION PROCESSING LETTERS DV[]B[~p]
It is easily seen that MI has exactly the same type of Kripke interpretation as intuitionistic logic. If A is an LI-formula, T(A) is the M I - f o r m u l a defined inductively as follows: (i) Y ( p ) = p
Dv~(~pvB)
DVB[~p];
by
Dv~pv<~B;
(iii) replace a subformula of C of the form
(ii) T ( A & B ) = T ( A ) & T ( B ) ,
DV~(~p&B)
(iii) T ( A v B ) = T ( A ) V T ( B ) ,
by
Dv~p;
(iv) replace a subformula of C of the form '
(v) T ( A --* B ) = [ ] ( T ( A ) ~
by
(ii) replace a s u b f o r m u l a of C of the form
if p is an atom,
(iv) T(-~A) = D ~ T ( A ) ,
18 January 1986
D v ~((--~p V B)&A)
T(B)). by
It can easily be shown that A is LI-valid iff T(A) is MI-valid, i.e., i f f - ~ T ( A ) is unsatisfiable. The validation p r o b l e m for LI then reduces to the refutation p r o b l e m for MI.
D v ~p v ~(B&A). Here, B[-~p] indicates that ~ p is a subformula of B. (2) If C is a clause, then "r(C) is obtained from C by replacing one or more occurrences of [] by [-q 73. 0 ( C ; ~ p ) and "r(C) are logical consequences of C. (3) If C 1 and C 2 are clauses, the operation
Section 3
In this section we shall define the resolution rules for propositional MI.
Y"[C, ;C2]
3.1. Definition. If A is an MI-formula, then A is in Modal C o n j u n c t i v e Normal F o r m ( M C N F ) iff it has the form: C~&.--&C
n
and the property [C1;C2]
(n >~ 1),
are inductively defined as follows:
where each C,, called a clause, has the general form of a disjunction (possibly with only one disjunct) of the following form: L~ v . - . v L m v []B~ v --VO(D,&---&Dq,)V
v
(i) E O [ p ; ~ p ] =.0, [p;~p]
I--IBp
.... V O ( D , r & - - - & D q r
Here, each L~ is a literal and each Bj and D k is a clause.
is resolvable
(ii) E ° [ A l v )-
is resolvable; .-- v A , v --- v A , , ;
gl V - - . VBj v . - . V B m]
=Y°[A,;Bj]
vA, v
...
VA,_
1
V A i + I V "'- V A n 3.2. Theorem. / f A is an MI-formula, then A can be transformed into an equivalent formula A ' in MCNF. 3.3. Definitions. (1) If C is a clause which contains the literal --,p within the scope of a ~ , then 0 ( C ; ~ p ) is o b t a i n e d from C by repeatedly applying the following transformations until ~ p is no longer within the scope of ~ : (i) replace a subformula of C of the form 88
v B 1V --- VBj_ 1V Bj+ 1 v --- VB m. If [A,;Bj] is resolvable, then [A~ V - . . v A , 1 ; B ~ v --- v B,1 ] is resolvable; (iii) if A is [] or <), then
= A( =°[A; U,] & B , &
• • • &Bj_,&B,. 1 & . . . &Bin).
Volume 22, Number 2
I N F O R M A T I O N PROCESSING LETTERS
If [A ; Bj] is resolvable, then [t3A ; A(B 1& • -- &Bin) ] is resolvable. Moreover, if B does not begin with a modal operator, then 2°[~A ; B] = Z°[A ; B], and if [A ; B] is resolvable, then [~A ; B] is resolvable. The operation 2 is then defined as follows: Z[C 1; C2] is obtained from 2°[C1 ;C2] by replacing every occurrence of &O
by
~.
A (f~& B)
by
f~,
(,t~v B)
by
B,
18 January 1986
Section 4
Completeness is proved indirectly, by showing that the system including the rules (R1)-(R5) and the following rule: A[MC] (R6) A [ p ( ~ C ' ~ p ) ] is complete. The completeness theorem for the system including rules (R1)-(R5) immediately follows by the redundancy of (R6). 4.1. Definitions. (1) If C is a clause, then d(C) is defined as follows:
where A is [] or ~ . (4) If C is a clause of the form
(i) d ( p ) = 0
C [ O ( D , & - - " & D i & - ' " &D j
if p is an atom,
(ii) d ( ~ B ) = d(B),
& - ' " &Dn) ] ,
(iii) d(B&C) = d(B v C ) = d(B ~ C) = max{d(B), d(C)},
then 2'[C] is C[ O(Y.[Di ; Dj] a:D, a : ' " " & D i _ , & D i + , &--'&Dj_,&Dj+I&'"&D,)
(iv) d(E]B)
(2) If S is a set of clauses of the form {DA 1. . . . . []A,,, B, . . . . .
The resolution system for MI consists of the following rules: (R1) (R2)
C [ D V D] C[D] ' D 1 v C~ ; D 2 V C 2 D 1 V D 2 V Y~[C1; C2]
if [C 1; C2] is resolvable, CvD (R3) 2 ' [ C ] v D ' (R4)
C ,r(C) '
A[C] (R5) A [ p ( C ; ~ p ) ] if C contains ~p within the scope of a ~ .
if B has the form []B',
t d(B) + 1 otherwise.
].
If [D~; Dj] is resolvable, then C is resolvable.
f d(B)
Bm },
where the main logical symbol of each B i is not [3, then n
d * ( S ) = Y ] d ( E 3 a i ) + max (d(B,)). i=l
l~j~
(3) Let C be a clause. Then, v(C) is defined as follows: (i) if the main logical symbol of C is not v , then v(C) = 0; (ii) if C has the form C I v C2, then v(C) = 1 + v(C,) + v(C2). (4) If S is a set of clauses, then v ( S ) = E v(C). C~S
(5) Let S be a set of clauses and S* = { A [ ~ C ] " A[MC] E S}. 89
Volume 22, Number 2
INFORMATION PROCESSING LETTERS
T h e s e q u e n c e o f sets S0, S~, S 2 . . . . follows:
is d e f i n e d as
4.4. T h e o r e m . Let S be a set of clauses, 0S = (ISIA, . . . . .
(i) S0 = S w S * , (ii) Si+ 1 = S i u { o ( C ; - ~ p ) : C ~ S i a n d C c o n tains -~p in the s c o p e o f a ~ }. Then,
([]A,,...,MA
n, ~ B , . . . . .
~B re,L, .....
L k},
w h e r e e a c h Li is a literal. E(S) t h e n consists o f the f o l l o w i n g sets: S o = {AI . . . . .
A n,L1 .....
Lk)
a n d , for all j, 1 ~
An, Bj}.
4.2. T h e o r e m ( C o m p l e t e n e s s ) . L e t S be a set of
clauses. S is unsatisfiable iff S is refutable in the system of rules ( R 1 ) - ( R 6 ) . T h e p r o o f follows b y d o u b l e i n d u c t i o n o n d * ( S ) a n d v(S). T h e m a i n t h e o r e m s used f o r the case d*(S) > 0 a n d v(S) = 0 are the following. 4.3. T h e o r e m . Let S be a set of clauses such that 0S is unitary. 0S is satisfiable iff, for all i, eoery S' ~ E ( T i ( 0 S ) ) such that d*(S') < d*(S) is satisfia-
ble.
90
FIA.,, ~ ( C , & . - -
O B 1. . . . .
&Cm),
~ B p , L1 . . . . .
L k ),
and S ' = ( A 1. . . . .
0S is a finite set. (6) Let S b e a set o f clauses. {T~(S)} is the f a m i l y o f the sets w h i c h c a n be o b t a i n e d f r o m S b y a p p l y i n g ( o n c e or several times) the t r a n s f o r m a tion "r to o n e or m o r e f o r m u l a s in S. (7) Let S be a set o f u n i t a r y clauses o f the following form:
18 January 1986
A n,C~ .....
Cm}.
Then, eoery refutation of S' can be transformed into a refutation of S.
References [1] M. Cialdea, Une m&hode de r~solution pour la Iogique intuitionniste propositionnelle, Rappt. LSI No. 180, UPS Toulouse. [2] R.L. Constable, Constructive mathematics and automatic program writers, in: Information Processing '71, Proc. I FIP Congress 1971 (North-Holland, Amsterdam, 1971). [3] R.L. Constable, Programs as proofs: A synopsis, Inform. Process. Lett. 16 (3) (1983) 105-112. [4] L. Fari~aas Del Cerro, A simple deduction method for modal logic, Inform. Process. Lett. 14 (2) (1982) 49-51. [5] L. Farihas Del Cerro and A.R. Cavalli, Resolution for linear temporal logic, Rept. LITP 82-52, 1982. [6] L. Fari~as Del Cerro and E. Lauth, Linear resolution for temporal logic, Note LSI, 1983. [7] C. Goad, Computational uses of the manipulation of formal proofs, Rept. No. STAN-CS-80-819, Dept. of Computer Science, Stanford Univ., 1980. [8] C. Goad, Proofs as descriptions of computations, Proc. 5th Conf. on Automated Deduction, Lecture Notes in Computer Science (Springer, Berlin, 1980). [9] S. Goto, Program synthesis from natural deduction proofs, Internat. Joint Conf. on Artificial Intelligence, Tokyo, 1979. [10] G.E. Hughes and M.J. Cresswell, An Introduction to Modal Logic (Methuen, London, 1978). [11] J.A. Robinson, A machine oriented logic based on the resolution principle, J. ACM 12 (1962). [12] J.A. Robinson, Logic: Form and Function (Edinburgh University Press, 1979).
18 January 1986
S O M E REMARKS ON T H E POSSIBILITY OF EXTENDING R E S O L U T I O N PROOF P R O C E D U R E S T O I N T U I T I O N I S T I C LOGIC
Marta C I A L D E A Langages et Systbmes Informatiques, Laboratoire Associb au CNRS, i~quipe Cornprkhension du Raisonnement Naturel, Universitb Paul Sabatier, 118 route de Narbonne, 31077 Toulouse Cbdex, France
Communicated by W.M. Turski Received 8 March 1985
Keywords: Theoremproving, formal languages
Section 1
Constructive logic is acquiring an ever increasing importance in computer science [2,3,7,8,9]. The majority of the research in this direction is based on the analysis of structural properties of proofs as a whole, a n d / o r the recognizable isomorphism between algorithms and constructive proofs in certain formal systems. However, it is not yet known whether one of the most efficient (and old) proof procedures for classical logic, the resolution m e t h o d [11,12], can easily be extended to constructive logic. Here, we propose a resolution procedure for propositional intuitionistic logic, which however remains a sort of external oracle: it applies in fact to a system of modal logic treated as a formal semantics of intuitionism. This detour was originally m a d e as a heuristic approach: since intuitionistic logic has a very rich (Kripke-style) semantics e m b e d d e d in a relatively poor language, it seems natural to translate intuitionistic formulas into a richer language, to make their behaviour more transparent w.r.t, resolution. However, as it will be apparent from the definition of the system of rules proposed, the backtrack to intuitionistic language is not at all simple, for the following reasons: (i) The formulation of the rules for the modal system is not too complicated, while their reformu-
lation for a language where implication cannot be eliminated and modal operators are not explicitly present leads to an enormous loss of simplicity. (ii) A modal deduction from a set of clauses which translate intuitionistic formulas generates clauses which translate no intuitionistic formula. Obviously, we cannot conclude that intuitionistic logic is absolutely intractable by resolution methods: in fact, such problems may arise in connection with this particular resolution system only. Below, we shall define the modal system MI, the translation of intuitionistic logic LI into MI, and the resolution rules for MI together with a sketch of the completeness proof. The system of rules is inspired by Farihas Del Cerro et al. [4,5,6], although new complications arise here because of the peculiarity of MI-interpretations. The details of the proofs can be found in [1], as well as some examples and a short discussion on the difficulties connected with a reformulation of the rules in terms of the intuitionistic language. Section 2
The modal system MI is obtained by adding the following axiom to the system $4 of Lewis [10]: p ~ Ulp if p is atomic.
0020-0190/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
87
Volume 22. N umber 2
IN FORMATION PROCESSING LETTERS DV[]B[~p]
It is easily seen that MI has exactly the same type of Kripke interpretation as intuitionistic logic. If A is an LI-formula, T(A) is the M I - f o r m u l a defined inductively as follows: (i) Y ( p ) = p
Dv~(~pvB)
DVB[~p];
by
Dv~pv<~B;
(iii) replace a subformula of C of the form
(ii) T ( A & B ) = T ( A ) & T ( B ) ,
DV~(~p&B)
(iii) T ( A v B ) = T ( A ) V T ( B ) ,
by
Dv~p;
(iv) replace a subformula of C of the form '
(v) T ( A --* B ) = [ ] ( T ( A ) ~
by
(ii) replace a s u b f o r m u l a of C of the form
if p is an atom,
(iv) T(-~A) = D ~ T ( A ) ,
18 January 1986
D v ~((--~p V B)&A)
T(B)). by
It can easily be shown that A is LI-valid iff T(A) is MI-valid, i.e., i f f - ~ T ( A ) is unsatisfiable. The validation p r o b l e m for LI then reduces to the refutation p r o b l e m for MI.
D v ~p v ~(B&A). Here, B[-~p] indicates that ~ p is a subformula of B. (2) If C is a clause, then "r(C) is obtained from C by replacing one or more occurrences of [] by [-q 73. 0 ( C ; ~ p ) and "r(C) are logical consequences of C. (3) If C 1 and C 2 are clauses, the operation
Section 3
In this section we shall define the resolution rules for propositional MI.
Y"[C, ;C2]
3.1. Definition. If A is an MI-formula, then A is in Modal C o n j u n c t i v e Normal F o r m ( M C N F ) iff it has the form: C~&.--&C
n
and the property [C1;C2]
(n >~ 1),
are inductively defined as follows:
where each C,, called a clause, has the general form of a disjunction (possibly with only one disjunct) of the following form: L~ v . - . v L m v []B~ v --VO(D,&---&Dq,)V
v
(i) E O [ p ; ~ p ] =.0, [p;~p]
I--IBp
.... V O ( D , r & - - - & D q r
Here, each L~ is a literal and each Bj and D k is a clause.
is resolvable
(ii) E ° [ A l v )-
is resolvable; .-- v A , v --- v A , , ;
gl V - - . VBj v . - . V B m]
=Y°[A,;Bj]
vA, v
...
VA,_
1
V A i + I V "'- V A n 3.2. Theorem. / f A is an MI-formula, then A can be transformed into an equivalent formula A ' in MCNF. 3.3. Definitions. (1) If C is a clause which contains the literal --,p within the scope of a ~ , then 0 ( C ; ~ p ) is o b t a i n e d from C by repeatedly applying the following transformations until ~ p is no longer within the scope of ~ : (i) replace a subformula of C of the form 88
v B 1V --- VBj_ 1V Bj+ 1 v --- VB m. If [A,;Bj] is resolvable, then [A~ V - . . v A , 1 ; B ~ v --- v B,1 ] is resolvable; (iii) if A is [] or <), then
= A( =°[A; U,] & B , &
• • • &Bj_,&B,. 1 & . . . &Bin).
Volume 22, Number 2
I N F O R M A T I O N PROCESSING LETTERS
If [A ; Bj] is resolvable, then [t3A ; A(B 1& • -- &Bin) ] is resolvable. Moreover, if B does not begin with a modal operator, then 2°[~A ; B] = Z°[A ; B], and if [A ; B] is resolvable, then [~A ; B] is resolvable. The operation 2 is then defined as follows: Z[C 1; C2] is obtained from 2°[C1 ;C2] by replacing every occurrence of &O
by
~.
A (f~& B)
by
f~,
(,t~v B)
by
B,
18 January 1986
Section 4
Completeness is proved indirectly, by showing that the system including the rules (R1)-(R5) and the following rule: A[MC] (R6) A [ p ( ~ C ' ~ p ) ] is complete. The completeness theorem for the system including rules (R1)-(R5) immediately follows by the redundancy of (R6). 4.1. Definitions. (1) If C is a clause, then d(C) is defined as follows:
where A is [] or ~ . (4) If C is a clause of the form
(i) d ( p ) = 0
C [ O ( D , & - - " & D i & - ' " &D j
if p is an atom,
(ii) d ( ~ B ) = d(B),
& - ' " &Dn) ] ,
(iii) d(B&C) = d(B v C ) = d(B ~ C) = max{d(B), d(C)},
then 2'[C] is C[ O(Y.[Di ; Dj] a:D, a : ' " " & D i _ , & D i + , &--'&Dj_,&Dj+I&'"&D,)
(iv) d(E]B)
(2) If S is a set of clauses of the form {DA 1. . . . . []A,,, B, . . . . .
The resolution system for MI consists of the following rules: (R1) (R2)
C [ D V D] C[D] ' D 1 v C~ ; D 2 V C 2 D 1 V D 2 V Y~[C1; C2]
if [C 1; C2] is resolvable, CvD (R3) 2 ' [ C ] v D ' (R4)
C ,r(C) '
A[C] (R5) A [ p ( C ; ~ p ) ] if C contains ~p within the scope of a ~ .
if B has the form []B',
t d(B) + 1 otherwise.
].
If [D~; Dj] is resolvable, then C is resolvable.
f d(B)
Bm },
where the main logical symbol of each B i is not [3, then n
d * ( S ) = Y ] d ( E 3 a i ) + max (d(B,)). i=l
l~j~
(3) Let C be a clause. Then, v(C) is defined as follows: (i) if the main logical symbol of C is not v , then v(C) = 0; (ii) if C has the form C I v C2, then v(C) = 1 + v(C,) + v(C2). (4) If S is a set of clauses, then v ( S ) = E v(C). C~S
(5) Let S be a set of clauses and S* = { A [ ~ C ] " A[MC] E S}. 89
Volume 22, Number 2
INFORMATION PROCESSING LETTERS
T h e s e q u e n c e o f sets S0, S~, S 2 . . . . follows:
is d e f i n e d as
4.4. T h e o r e m . Let S be a set of clauses, 0S = (ISIA, . . . . .
(i) S0 = S w S * , (ii) Si+ 1 = S i u { o ( C ; - ~ p ) : C ~ S i a n d C c o n tains -~p in the s c o p e o f a ~ }. Then,
([]A,,...,MA
n, ~ B , . . . . .
~B re,L, .....
L k},
w h e r e e a c h Li is a literal. E(S) t h e n consists o f the f o l l o w i n g sets: S o = {AI . . . . .
A n,L1 .....
Lk)
a n d , for all j, 1 ~
An, Bj}.
4.2. T h e o r e m ( C o m p l e t e n e s s ) . L e t S be a set of
clauses. S is unsatisfiable iff S is refutable in the system of rules ( R 1 ) - ( R 6 ) . T h e p r o o f follows b y d o u b l e i n d u c t i o n o n d * ( S ) a n d v(S). T h e m a i n t h e o r e m s used f o r the case d*(S) > 0 a n d v(S) = 0 are the following. 4.3. T h e o r e m . Let S be a set of clauses such that 0S is unitary. 0S is satisfiable iff, for all i, eoery S' ~ E ( T i ( 0 S ) ) such that d*(S') < d*(S) is satisfia-
ble.
90
FIA.,, ~ ( C , & . - -
O B 1. . . . .
&Cm),
~ B p , L1 . . . . .
L k ),
and S ' = ( A 1. . . . .
0S is a finite set. (6) Let S b e a set o f clauses. {T~(S)} is the f a m i l y o f the sets w h i c h c a n be o b t a i n e d f r o m S b y a p p l y i n g ( o n c e or several times) the t r a n s f o r m a tion "r to o n e or m o r e f o r m u l a s in S. (7) Let S be a set o f u n i t a r y clauses o f the following form:
18 January 1986
A n,C~ .....
Cm}.
Then, eoery refutation of S' can be transformed into a refutation of S.
References [1] M. Cialdea, Une m&hode de r~solution pour la Iogique intuitionniste propositionnelle, Rappt. LSI No. 180, UPS Toulouse. [2] R.L. Constable, Constructive mathematics and automatic program writers, in: Information Processing '71, Proc. I FIP Congress 1971 (North-Holland, Amsterdam, 1971). [3] R.L. Constable, Programs as proofs: A synopsis, Inform. Process. Lett. 16 (3) (1983) 105-112. [4] L. Fari~aas Del Cerro, A simple deduction method for modal logic, Inform. Process. Lett. 14 (2) (1982) 49-51. [5] L. Farihas Del Cerro and A.R. Cavalli, Resolution for linear temporal logic, Rept. LITP 82-52, 1982. [6] L. Fari~as Del Cerro and E. Lauth, Linear resolution for temporal logic, Note LSI, 1983. [7] C. Goad, Computational uses of the manipulation of formal proofs, Rept. No. STAN-CS-80-819, Dept. of Computer Science, Stanford Univ., 1980. [8] C. Goad, Proofs as descriptions of computations, Proc. 5th Conf. on Automated Deduction, Lecture Notes in Computer Science (Springer, Berlin, 1980). [9] S. Goto, Program synthesis from natural deduction proofs, Internat. Joint Conf. on Artificial Intelligence, Tokyo, 1979. [10] G.E. Hughes and M.J. Cresswell, An Introduction to Modal Logic (Methuen, London, 1978). [11] J.A. Robinson, A machine oriented logic based on the resolution principle, J. ACM 12 (1962). [12] J.A. Robinson, Logic: Form and Function (Edinburgh University Press, 1979).