, Let a in H S . Let V be the valuation on S Let V ? be the valuation on S defined by a |= ?(p) iff (S, V ? ), a |= ?(p) Since (S, V ), a |= ?(p), (S, V ? ), a |= ?(p) Therefore, by ( * ) Suppose [C] is definable with [N ] and [P ] in the class of all normal frames. Hence, there exists a formula ?(p) in [N ] and [P ] such that ( * ) for all normal frames Let a in H S . Let V be the valuation on S defined by: a contradiction. (3) Suppose [P ] is definable with, Moreover, for all formulas ?(p) in the class of all normal frames. Hence, there exists a formula ?(p) in [N ] and [C] such that ( * ) for all standard frames Let a in H S . By Sym(C), Sym(P ) and Inf 1 (C, P ), there exists b in H S such that C S (a, b) and not P S (a, b). Let V be the valuation on S defined by V (p) = [a] C S \{b}. Obviously, (S, V ), a |= [P ]p. Thus, by ( * ), (S, V ), a |= ?(p). By Sym(P ) and Inf 1 (P, ?), there exists c in H S such that P SS, V ? ), a |= [P ]p. Moreover, for all formulas ?(p) in [N ] and [C], (S, V ), a |= ?(p) iff (S, V ? ), a |= ?(p)
, Therefore Variants Other primitives may be defined as well. In this section, we consider the predicate symbol < of precedence between positive hyperreals and the function symbol + of addition between positive hyperreals
, Obviously, the second player wins all Ehrenfeucht games over (S P H , [a, b]) and (S P H , [b, a]) with respect to N , C, P and ?. Thus, by [12, theorem 2.2.8], for all formulas ?(x, y) b: a contradiction. ? What about the axiomatization/completeness or the decidability/complexity of the first-order theory based on the predicates N , C, P and <? As for the modal logic option, the obvious road consists in adding a connective [<] interpreted in S P H in the following way Adding addition Let us add a function symbol + of arity 2 to our first-order language, What about the axiomatization/completeness or the decidability/complexity of the modal logic based on the connectives The formulas are now given by the rule: ? ? ::= N (s, t) | C(s, t) | P (s, t) | s ? t | ? | ¬? | (? ? ?) | ?x.? where s and t range over the set of terms defined by the rule ? s ::= x |
, it appears that if we restrict the language to the predicate N or if we restrict the language to the predicates N and C then the function symbol + can be eliminated To see this, it suffices to observe that the following sentences hold in S P H : ? ?x, ? ?x.?y.?z.(C(x, y + z) ? (C(x, y) ? ¯ N (x, z)) ? (C(x, z) ? ¯ N (x, y)))
, Since Obviously, the second player wins all Ehrenfeucht games over(x, y, z) [a ?1 , b ?1 , c ?1 ] Since ( * ), C S P H (a ?1 + S P H b ?1 , c ?1 ): a contradiction Suppose there exists a formula ?(x, y, z) in N , C, P and ? such that ( * ) for all a, b, c in H Since ( * ), S |= ?(x, y, z) [a, b, c]. Obviously, the second player wins all Ehrenfeucht games over ): a contradiction. (3) Suppose there exists a formula ?(x, y, z) in N , C, P and ? such that ( * ) for all a, b, Thus, by [12, theorem 2.2.8], for all formulas ? H S P H be such that a + S P H b = c and a 2 + S P H b 2 = c 2 . Since ( * ), S |= ?(x Obviously, the second player wins all Ehrenfeucht games over (S P H Since ( * ), a 2 + S P H b 2 = c 2 : a contradiction. ? What about the axiomatization/completeness or the decidability/complexity of the first-order theory based on the predicates N , C and P and the function +? As for the modal logic option, the obvious road consists in adding a connective ? interpreted in S P H in the following way: ? (S P H , V ), a |= ? ? ? iff there exist b
, A Qualitative Model for Time Granularity, Computational Intelligence, vol.16, issue.2, pp.137-168, 2000.
DOI : 10.1111/0824-7935.00110
, Order of Magnitude Qualitative Reasoning with Bidirectional Negligibility, Current Topics in Artificial Intelligence, pp.370-378, 2006.
DOI : 10.1007/11881216_39
, Relational Approach to Order-of-Magnitude Reasoning, pp.105-124, 2006.
DOI : 10.1007/11964810_6
, A Nonstandard Approach to Option Pricing, Mathematical Finance, vol.9, issue.4, pp.1-38, 1991.
DOI : 10.1016/0304-4149(89)90079-3
, Numeric reasoning with relative orders of magnitude, Proceedings of the Eleventh National Conference on Artificial Intelligence, AAAI, pp.541-547, 1993.
, Order of magnitude comparisons of distance, J. Artif. Intell. Res, vol.10, pp.1-38, 1999.
, A Decision Procedure for the First Order Theory of Real Addition with Order, SIAM Journal on Computing, vol.4, issue.1, pp.69-76, 1975.
DOI : 10.1137/0204006
, A Non-standard Temporal Deductive Database System, Journal of Symbolic Computation, vol.22, issue.5-6, pp.649-664, 1996.
DOI : 10.1006/jsco.1996.0070
, Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, 1998.
DOI : 10.1007/978-1-4612-0615-6
, Relational approach for a logic for order of magnitude qualitative reasoning with negligibility, non-closeness and distance, Logic Journal of IGPL, vol.6, issue.4, pp.375-394, 2009.
DOI : 10.1093/jigpal/jzp016
, Reasoning with orders of magnitude and approximate relations, Proceedings of the Sixth National Conference on Artificial Intelligence, AAAI, pp.626-630, 1987.
, Order of magnitude reasoning, Proceedings of the Fifth National Conference on Artificial Intelligence, AAAI, pp.100-104, 1986.
, Mathematical foundations of qualitative reasoning, AI Mag, pp.91-106, 2003.
, Relative and absolute order-of-magnitude models unified, Annals of Mathematics and Artificial Intelligence, vol.24, issue.4, pp.45-323, 2005.
DOI : 10.1007/s10472-005-9002-1
, Qualitative reasoning about physical systems: A return to roots, Artificial Intelligence, vol.51, issue.1-3, pp.1-9, 1991.
DOI : 10.1016/0004-3702(91)90106-T
, Nonstandard versions of conventionally infinite networks, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol.48, issue.10, pp.1261-1265, 2001.
DOI : 10.1109/81.956025