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--- ws1s_expressive_power_and_quantifier_elimination [2007/06/29 16:39]
vaibhav.rajan
+++ ws1s_expressive_power_and_quantifier_elimination [2007/06/29 17:12]
vaibhav.rajan
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-We know that Weak Monadic Second-order theory of 1 Successor (WS1S) is decideable through automata, that is, the set of satisfying interpretations of a subformula is represented by a finite-state automaton [Buc60b, Elg61]. The decision procedures (using the automata technique) have been implemented efficiently (for WS1S with certain restrictions) in MONA.
+Introduction:
-The main problem with this approach is exponential blowup in the states of the automaton due to nested quantifiers [Nil01].
+We know that Weak Monadic Second-order theory of 1 Successor (WS1S) is decidable through automata, that is, the set of satisfying interpretations of a subformula is represented by a finite-state automaton [Buc60b, Elg61]. The decision procedures (using the automata technique) have been implemented efficiently (for WS1S with certain restrictions) in MONA.
-We were interested in finding another way of deciding WS1S without using the automata. We began by investigating whether we can characterize relations in WS1S in a way that helps us eliminate quantifiers. Since we couldn't get any substantial results, we began looking at a subset of WS1S. This paper describes the key results that we obtained.
+The main problem with this approach is exponential blow-up in the states of the automaton due to nested quantifiers [Kla01].
+We were interested in finding another way of deciding WS1S without using the automata. We began by investigating whether we can characterize relations in WS1S in a way that helps us eliminate quantifiers. Since we couldn't get any substantial results, we began looking at a subset of WS1S. This paper describes the key results that we obtained.
+Conclusions and Future Directions:
+In the course of our mini-project we proved an interesting result about the lengths of words of a regular language. We used this result to characterize relations in the language $L^R$. We hope that this is a step in the direction of our original aim: characterizing relations in WS1S.
-* why the problem is interesting
-deciding WS1S - done through automata construction (mona)\\
-main problem: quantifier elimination\\
-is there another way of QE? - can we characterize relations in WS1S in some other way?\\
-WS1S too big: look at a smaller subset of the language\\
-our L: characterize unary relations, binary relations,...n-ary\\
-* what was done before
-p-recognizability
-* what the main idea is
-* what you did (precise description)
-* the most surprising/interesting things you found
-* limitations
-* future directions
-look at other subsets of WS1S, whole of WS1S
 **START**
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 The above reasoning can be extended to the n-ary case. The input string has n non-zero elements and n+1 expressions of the form $(\{(0...0)^{-1}\})^k$. A tuple $(x_1, x_2,...x_n)$ belongs to an n-ary relation iff the possible values of each exponent $k$  forms an ultimately periodic set.
+References:\\
+[Buc60a] Julius Richard Buchi. On a decision method in restricted second-order arithmetic. In Proc. Internat. Cong. on Logic, Methodol., and Philos. of Sci. Stanford University Press, 1960.\\
+[Elg61] Calvin C. Elgot. Decision problems of Finite automata design and related arithmetics. Transactions of the American Mathematical Society, 98:21{52, 1961.\\
+[Kla01] Nils Klarlund, Anders Moller. MONA Version 1.4 User Manual. BRICS, Department of Computer Science, University of Aarhus, 2001.\\