1. Introduction, Definitions, Conventions
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1 Bulletin of the Section of Logic Volume 29/4 (2000), pp V. V. Rybakov B. Fedorishin FACES OF MONOTONICITY AND WISDOM FORMULAS PROBLEM Abstract In the paper 1 we consider several aspects of monotonicity in propositional logics and the related property of wisdom of formulas connected with the problem of full logical omniscience. We present theorems providing necessary and sufficient conditions for formulas to be monotone and to be wisdom in certain transitive modal logics. A large class of transitive modal logics with the property of inward monotonicy of formulas decidable is found. KEY WORDS: non-standard logic, modal logic, knowledge representation, logical monotonicity, inward monotonicity 1. Introduction, Definitions, Conventions A common and very well known observation concerning human reasoning is that, in general, it has not to be monotone, though mathematical proof and truth of mathematical results assume a kind of monotonicity. Also, a very well known observation coming from Hintikka is that in knowledge as a side effect we very often have logical omniscience, which, in particular, appears if the following holds: for any provable formula A any agent i knows A (which is not very compatible with our intuition). Much research 1 Supported by Russian Foundation of Fundamental Studies (RFFI) and Istanbul Bilgi University in 2000/20001 and by Bern University, Switzerland in 1999
2 182 V. V. Rybakov and B. Fedorishin has been devoted to the question how to avoid omniscience, as well as to the genral question of description of knowledge in logial terms (cf. [1] [9], for instance). And primary, for these descriptions, authors use propositional modal-like logics. Therefore, we attempt to analyze these effects presented in the language of propositional logic. Consider a propositional logic L containing as a logical operation. Having a look on axiomatizations we can imagine L to be determined by a set of propositional formulas as axioms and by a set of inference rules in form r := A 1,..., A n /B, like modus ponens, and the logic L is the set of all formulas which are derivable in this system, in particular, it means that L is closed w.r.t. substitutions. Thus L A means A L, and vise versa. In what follows A(x) denotes a formula containing a variable x. For any formula A(x), this formula is saying something about x. In particular, there is some knowledge about x expressed by this formula. To comprehend formally that this knowledge is monotone, we offer the following Definition 1.1. Let L be a logic. We say a formula A(x) is x-monotone in L iff, for any formulas B and C, if L B C then L A(B) A(C) i.e. if the rule is admissible in L. x y A(x) A(y) In knowledge logics ([3]) the operation knows is often expressed as logical operation, then the meaning α is: the agent knows α. If we consider logical omniscience property for the case of empty pre-assumptions, this problem can be posed as follows: if α is provable then the agent knows α, which under the above description can be understood as α is provable. Instead of α many formulas A(α) can be offered (taking in account some specific properties of the agent knowledge in a given environment) as formulas expressing the notion the agent knows α. And if they have the same properties as described above (if α is provable then A(α) is provable) we can say that formulas A(x) are x-wisdom formulas - they know what is provable. This allows us to introduce the following
3 Faces of Monotonicity and Wisdom Formulas Problem 183 Definition 1.2. A formula A(x) is x-wisdom in a logic L iff, for any formula B, if L B then L A(B) i.e. if the rule is admissible in L. x A i (x) Wisdom formulas problem is how to determine whether a formula is wisdom. An evident proposition connecting these two notions for propositional logics L obeying at least all laws of intuitionistic propositional logic is Lemma 1.3. If the formula A(x) is x-monotone for L, the rules x/ x and modus ponens are admissible for L, and L A( )), then A(x) is an x-wisdom formula. Indeed, L B implies L B and then, since A(x) is x-monotone, we get L A( ) A(B), and applying L A i ( ) and modus ponens we conclude L A(B). Thus these notions are close but examples below will show they are in fact distinct. And we begin below to expound our speculations on monotonicity and wisdom formulas. 2. Simple Facts, Examples A simple sufficient condition for A(x) to be x-monotone is Lemma 2.1. If L is based on classical propositional calculus and all additional logical operations of L are monotone in L and A(x) is positive w.r.t. x then A(x) is x-monotone in L. The proof follows by very common and well known argument, by induction on the length of A(x): (i) if A(x) is positive w.r.t x then x y L A(x) A(y), (ii) if A(x) is negative w.r.t x then x y L A(y) A(x), which is as usual for boolean operations, and, for all additional logical operations employed, is the case because these operations are assumed to be monotone in L.
4 184 V. V. Rybakov and B. Fedorishin In examples which follow we will use Kripke semantics for modal logics. Recall, a frame is a pair F := F, R with a non-empty set F of elements and a binary accessibility relation R between them, though a F, as accepted, is an abbreviation for a is an element of F. And in what follows some knowledge of background facts concerning modal logic is assumed. Example 2.2. The formula (p q) is p-monotone in any normal modal logic but is not an x-wisdom formula for all such Kripke complete and consistent logics L. Indeed, since this formula is positive w.r.t p by Lemma 2.1 this formula is monotone by p in L, but it is not p-wisdom formula, because / ( q) is not admissible in L: take q :=. Consider any Kripke complete normal modal logic L such that there is a L-frame F with a certain element from which at least two distinct elements are R-accessible. Lemma 2.3. The formula ϕ(x) := ( x x) is not x-monotone in L but is x-wisdom in L. Proof. To verify this is x-wisdom formula assume L B. Suppose L ϕ(b). Using Kripke completeness we conclude, for a certain Kripke model M based on an L-frame, M ϕ(b). Then for a certain element a F, the following hold: a B and a B, in particular, b F such that arb and b B which contradicts the fact that F L. To show that ϕ(x) is not x-monotone, take x to be and y to be itself, then L y, but L ϕ( ) ϕ(y) does not hold. Indeed, take the L-frame F which has a F, where arb and arc but c b. Then taking y to be valid under V in c only, we get a V y and a V y, but a V ( ), therefore L ϕ( ) ϕ(y). Thus the formula is not x-monotone in L. Consider any normal modal logic L which is Kripke complete and which posses an L-frame F with a certain element from which at least one element is R-accessible
5 Faces of Monotonicity and Wisdom Formulas Problem 185 Lemma 2.4. The formula ϕ(x) := x x is not x-wisdom in L but is x-monotone formula for L, though this formula is not positive w.r.t. x. Proof. Evidently, L and is false at L-frames F which have R-accessible elements, and therefore the last formula is not a theorem of L. Thus the formula x x is not x-wisdom in L. As to monotonicity, we claim that even L ϕ(b) ϕ(c), for any B and C. Indeed, to disprove the last formula using Kripke completeness we would have, for a certain L-frame F and an element a F, (F, a) V ϕ(b) ϕ(c). But a V ϕ(b) would imply a is R-maximal and R-irreflexive element. But then a but it is not positive w.r.t. x. V ϕ(c). Thus ϕ(x) is x-monotone If we take A(x) := x then it is evident that this formula is x- monotone but not x-wisdom in L provided the logic L has L-frames with R-maximal elements which are R-irreflexive. Again, the formula (x q) is x-monotone but not x-wisdom for logics L admitting frames with R accessible elements. Now we vary a little the notion of x-monotonic in L formula: Definition 2.5. A formula A(x) is inwardly x-monotone in L iff, for any formulas B and C, if L A(B C) then L A(B) A(C) i.e. if the rule is admissible in L. A(x y) A(x) A(y) The motivation to consider this notion is as follows: if A(x) has meaning the agent knows x than our definition concerning monotonicity could be refined: only provided provability A(B C) (which means that not B C is provable but that the agent knows that B C) and probability A(B), which means the agent knows B, we can conclude that the agent knows C, i.e. A(C). To connect these two distinct monotonicity definitions we immediately derive Lemma 2.6. If A(x) is inwardly x-monotone and is x-wisdom formula in L then A(x) is x-monotone for L.
6 186 V. V. Rybakov and B. Fedorishin Proof. It is evident, if A(x) is x-wisdom formula and L B C we get L A(B C) and applying the rule A(x y)/a(x) A(y) it follows L A(B) A(C). Lemma 2.7. If A(x) is an x-monotone formula and A(x)/x is admissible for L, then A(x) is inwardly x-monotone in L. Proof. If L A(B C) then L B C and using usual x-monotonicity we conclude L A(B) A(C). But we will show that inwardly x-monotone and x-monotone formulas do not coincide. Consider any normal modal logic L which is Kripke complete w.r.t. a family K of Kripke frames F, where for each a in F there is single-element R-maximal cluster C := {b} such that arb, and there is a frame F K which has an element a, where arb and b is not R-maximal. Lemma 2.9. The formula x is x-monotone but not inwardly x-monotone in any such logic L. Proof. We know already that this formula is x-monotone because it is x-positive. To show it is not inwardly x-monotone we have to disprove admissibility of the rule (x y) x y. Take y := x and x as it is. Then L (x x) because K characterizes L. But L x x. Indeed, take a frame F from K with a not R- maximal element b which is R-accessible from another element a and the valuation V (x) := {b}. Then a V x x. Therefore we derive that x x is not provable in L. Now consider any normal modal logic L which is Kripke complete w.r.t. a family K of Kripke frames F, where there is a frame in K with a single-element R-maximal cluster C := {b}, and there is a frame F K which has an element a, where arb and arc but c b, i.e. a frame with two distinct R-accessible elements. Lemma 2.9. The formula x x is inwardly x-monotone formula but not x-monotone and not x-wisdom formula in L.
7 Faces of Monotonicity and Wisdom Formulas Problem 187 Proof. This formula is not x-wisdom for L because K has frames with single-element R-maximal R-clusters. This formula is inwardly x-monotone again by the same reason - the premise of the rule (x y) (x y) x x y y can not be unified in L. It remains only to show that this formula is not x-monotone. Take x := x and y :=. Then L x. But L x x. Indeed last formula can be disproved by any frame from K with two distinct R-accessible elements. 3. Recognizing of Monotone or Wisdom Formulas The previous results of this paper were simple observations about properties of monotonicity and wisdom. Now we turn to the question how to recognize whether a given formula A(x) is x-monotone, inwardly x-monotone or x- wisdom which is much more solid subject. This part requires already more robust mathematical analyses and requires knowledge from [10] about n- characterizing models and recognizing admissibility rules by means of them. Theorem 3.1. Let L be any normal modal logic extending K4 and having fmp. For any formula A(x) in the language of L, the rule x/a(x) is admissible for L iff x x A(x) L. That is any admissible in L rule in the form x/a(x) is derivable in L, i.e. L is structurally complete w.r.t such rules. Proof. Indeed if we have x x A(x) L then the rule x/a(x) is evidently admissible in L. Conversely, assume the formula x x A(x) is not derivable in L. Then there is a finite L-frame F disproving this formula. Therefore we may assume that the frame F is rooted by a cluster C and, for some valuation V in F of the formula x x A(x) the following holds
8 188 V. V. Rybakov and B. Fedorishin F V x and C V A(x). (1) Since the logic L has fmp and extends K4, this frame F is an open subframe of a certain m-characterizing model Ch L (m) for L (see the construction of Ch L (m) and its properties in [10]). We extend the valuation V from F to whole Ch L (m) as follows: y, (y x) V 1 (y) := V (y), V 1 (x) := V (x) {a a [Ch L (m) F]}. Since each element of the model Ch L (m) is definable by a formula (cf. [10]) and since F is finite, the valuation V 1 is definable by formulas too, i.e. for any z from the domain of V 1 there is a modal formula α z such that V 1 (z) = V 2 (α z ), where V 2 is the original valuation of Ch L (m). Then by (3.1) Ch L (m) V 1 x and Ch L (m) V 1 A(x). By Theorem from [10] any rule is admissible in L iff this rule is valid in all models Ch L (m) for all possible definable valuations. Therefore the rule x/a(x) is not admissible in L, i.e. for some substitution ε of formulas in place of x and other letters from A(x), ε(x) L but ε(a(x)) L 1. From this theorem we directly extract our main result concerning recognizing wisdom-property of formulas: Corollary 3.2. Let L be a decidable normal modal logic extending K4 and having fmp. Then there is an algorithm which recognizes x-wisdom for L formulas. Incidentally, let us briefly discuss how to avoid fully logical omniscience for agents, if we conventionally assume A(x) is formula coding the agent knows x. In the light of our previous theorem only the way to avoid this effect is to take formulas A(x) for coding the agent knows x for which L x x A(x). So we got a necessary and sufficient condition for avoiding fully logical omniscience for the case of logical consequence with no pre-assumptions for the case derivable formulas. So, as an example knows x formula we could take ( x x) which is dropping effect of
9 Faces of Monotonicity and Wisdom Formulas Problem 189 fully logical omniscience. Though in this approach A(x) means rather knows some properties of x than knows x comprehensively. Note also that for classical first order theories, in particular for the classical predicate calculus itself the question of recognizing wisdom formulas is undecidable (cf., for instance, [11]). Turning to recognizing monotonicity we will use a scheme similar to what we have employed for the case of wisdom formulas. Theorem 3.3. Let L be a normal modal logic extending K4 and having fmp. For any formula A(x), the rule x y/a(x) A(y) is admissible for L iff (x y) (x y) (A(x) A(y)) L. Thus all such L are structurally complete w.r.t. rules in this form. Proof. In fact, if (x y) (x y) (A(x) A(y)) L, then the rule x y/a(x) A(y) is trivially admissible in L. Conversely, assume (x y) (x y) (A(x) A(y)) L. Then, by fmp of L, there is a finite L-frame F disproving this formula. Therefore we may assume that the frame F is rooted by a cluster C and, for some valuation V in F of our formula the following holds F V x y and C V A(x) A(y). (2) Since the logic L is a transitive modal logic, this frame F is an open subframe of a certain m-characterizing model Ch L (m) for L (see the construction of Ch L (m) and its properties in [10]). We extend the valuation V from F to whole Ch L1 (m) as follows: z, (z y, x) V 1 (z) := V (z), V 1 (y) := V (y) {a a [Ch L (m) F]}, V 1 (x) := V (x). Since each element of Ch L (m) is definable by a formula (cf. [10]) and F is finite, the valuation V is definable by formulas also, i.e. for any z from the domain of V 1 there is a modal formula α z such that V 1 (z) = V 2 (α z ), where V 2 is the original valuation of Ch L1 (m). Then by (3.2)
10 190 V. V. Rybakov and B. Fedorishin Ch L (m) V 1 x y and Ch L (m) V 1 A(x) A(y). So the rule x y/a(x) A(y) is disproved in Ch L (m) by a definable valuation. Then by Theorem from [10] the rule x y/a(x) A(y) is not admissible in L. Corollary 3.4. Let L be a decidable normal modal logic extending K4 and having fmp. Then there is an algorithm which recognizes formulas which are x-monotone in L. Again, if we are to avoid monotonicity as an unpleasant side effect while choosing formulas to encode knows we can use our previous theorem describing monotone formulas. Now we turn to the question how to recognize inwardly monotone formulas. The approach used above for monotone formulas does not work for inward monotonicity, and we will employ some advanced technique using m-characterizing models Ch L (m) for normal modal logics L extending K4 (cf. [10]) - i.e. for transitive normal modal logics. By its definition inward monotonicity of formulas is expressed in terms of admissibility of special inference rules. Therefore, if the admissibility problem is decidable for a logic L then we can effectively recognize inwardly monotone formulas, so it remains only to borrow results from [10] for description logics which are decidable w.r.t. admissibility. Directly applying Theorem from [10] we derive Theorem 3.5. If a modal transitive logic L has (i) the property of branching below m, (ii) the effective m-drop points property, and (iii) is finitely axiomatizable then there is an algorithm recognizing admissibility of inference rules in L, in particular, this algorithm can effectively recognize inwardly monotone formulas. Since total majority of finitely axiomatizable transitive modal logics studied in literature obey this theorem (for instance K4, S4, S4.1, S4.2, S4.3, S5), this criterion for recognizing inwardly monotone formulas works sufficiently well. Though we did not show that logics within this theorem enjoy with structural completeness for rules describing inward monotonicity as it was proved for the case of monotonicity. And the question whether inward
11 Faces of Monotonicity and Wisdom Formulas Problem 191 monotonicity of formulas can be described through only decidability of modal logics themselves is open. Acknowledgments. I am grateful to Professor Gerhard Jaeger for good conditions at Bern University to begin the research presented in this paper, for his kind attention, support, discussion and advice. References [1] J. Barwise, Three Views of Common Knowledge, [in:] Yardi (ed.), Proc. Second Conference on Theoretical Aspects of Reasoning about Knowledge (1988), San Francisco. Calif.: Morgan Kaufmann, pp [2] C. Dwork and Y. Moses, Knowledge and Common Knowledge in a Byzantine Environment: Crash Failures, Information and Computation, Vol. 88 (1990), No. 2, pp [3] R. Fagin and J. Halpern, Reasoning about Knowledge and Probability, Journal of the ACM 41 (1994), no. 2, pp [4] R. Fagin, J. Halpern, Y. Moses, M. Vardi, A nonstandard Approach to the Logical Omniscience Problem, Artificial Intelligence, Vol. 79 (1995), no. 2, pp [5] R. Fagin, J. Halpern, Y. Moses, M. Vardi, Reasoning About Knowledge, The MIT Press, Cambridge, Massachusetts, London, England, 1995, 477 pp. [6] M. Kifer, L. Lozinski, A Logic for Reasoning with Inconsistency, J. Automated Deduction, Vol 9 (1992), pp [7] S. Kraus, D. L. Lehmann, Knowledge, Belief, and Time, Theoretical Computer Science, Vol. 58 (1988), pp [8] Y. Moses, Y. Shoham, Belief and Defeasible Knowledge, Artificial Intelligence, Vol. 64 (1993), No. 2, pp [9] G. Neiger, M. R. Tuttle, Common knowledge and consistent simultaneous coordination, Distributed Computing, Vol 6 (1993), No. 3, pp [10] V. V. Rybakov, Admissible Logical Inference Rules, Printed in: Studies in Logic and the Foundations of Mathematics, Vol. 136, Elsevier Sci. Publ., North-Holland, New-York-Amsterdam, 1997, 617 pp.
12 192 V. V. Rybakov and B. Fedorishin [11] V. V. Rybakov, Logics of Schemes and Admissible Rules for First-Order Theories, Stud. Fuzziness. Soft. Comp., Vol. 24 (1999), pp Department of Mathematics, Science and Letters Faculty Istanbul Biligi University, Istanbul, Turkey and Mathematical Department, Krasnoyarsk University pr. Svobodnyi 79, , Krasnoyarsk, Russia Mathematical Department, Krasnoyarsk University pr. Svobodnyi 79, , Krasnoyarsk, Russia
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