The succinctness of first-order logic on linear orders

Succinctness is a natural measure for comparing the strength of different logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all properties that can be expressed in L_2 can be expressed in L_1 by formulas of (approximately) the same size, but some properties can be expressed in L_1 by (significantly) smaller formulas. We study the succinctness of logics on linear orders. Our first theorem is concerned with the finite variable fragments of first-order logic. We prove that: (i) Up to a polynomial factor, the 2- and the 3-variable fragments of first-order logic on linear orders have the same succinctness. (ii) The 4-variable fragment is exponentially more succinct than the 3-variable fragment. Our second main result compares the succinctness of first-order logic on linear orders with that of monadic second-order logic. We prove that the fragment of monadic second-order logic that has the same expressiveness as first-order logic on linear orders is non-elementarily more succinct than first-order logic

Equality elimination for the inverse method and extension procedures

We demonstrate how to handle equality in the inverse method using equality elimination. In the equality elimination method, proofs consist of two parts. In the first part we try to solve equations obtaining so called solution clauses. Solution clauses are obtained by a very refined strategy — basic superposition with selection function. In the second part, we perform the usual sequent proof search by the inverse method. Our approach is called equality elimination because we eliminate all occurrences of equality in the first part of the proof. Unlike the previous approach proposed by Maslov, our method uses most general substitutions, orderin

Model Checking Probabilistic Pushdown Automata

We consider the model checking problem for probabilistic pushdown automata (pPDA) and properties expressible in various probabilistic logics. We start with properties that can be formulated as instances of a generalized random walk problem. We prove that both qualitative and quantitative model checking for this class of properties and pPDA is decidable. Then we show that model checking for the qualitative fragment of the logic PCTL and pPDA is also decidable. Moreover, we develop an error-tolerant model checking algorithm for PCTL and the subclass of stateless pPDA. Finally, we consider the class of omega-regular properties and show that both qualitative and quantitative model checking for pPDA is decidable

10161 Abstracts Collection -- Decision Procedures in Software, Hardware and Bioware

From April 19th, 2010 to April 23rd, 2010, the Dagstuhl Seminar 10161 "Decision Procedures in Soft, Hard and Bio-ware" was held in Schloss Dagstuhl Leibniz Center for Informatics. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as links to slides and links to papers behind the presentations and papers produced as a result of the seminar are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

05431 Abstracts Collection -- Deduction and Applications

From 23.10.05 to 28.10.05, the Dagstuhl Seminar 05431 ``Deduction and Applications\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available

Conflict resolution

This thesis proposes a new method for solving systems of linear constraints over the rational and real numbers (or, equivalently, linear programming) - the conflict resolution method. The method is a new approach to a classic problem in mathematics and computer science, that has been known since the 19th century. The problem has a wide range of real-life applications of increasing importance in both academic and industrial areas. Although, the problem has been a subject of intensive research for the past two centuries only a handful of methods had been developed for solving it. Consequently, new results in this field may be of particular value, not mentioning the development of new approaches. The motivation of our research did not arise solely from the field of linear programming, but rather was instantiated from problems of Satisfiability Modulo Theories (or shortly SMT). SMT is a new and rapidly developing branch of automated reasoning dedicated to reasoning in first-order logic with (combination) of various theories, such as, linear real and integer arithmetic, theory of arrays, equality and uninterpreted functions, and others. The role of linear arithmetic in solving SMT problems is very significant, since a considerable part of SMT problems arising from real-life applications involve theories of linear real and integer arithmetic. Reasoning on such instances incorporates reasoning in linear arithmetic. Our research spanned the fields of SMT and linear programming. We propose a method, that is not only used for solving linear programming problems, but also is well-suited to SMT framework. Namely, there are certain requirements imposed on theory reasoners when they are integrated in SMT solving. Our conflict resolution method possesses all the attributes necessary for integration into SMT. As the experimental evaluation of the method has shown, the method is very promising and competitive to the existing ones.EThOS - Electronic Theses Online ServiceGBUnited Kingdo

Maser action in methanol transitions

We report the detection with the ATCA of 6.7 GHz methanol emission towards OMC-1. The source has a size between 40'' and 90'', is located to the south-east of Ori-KL and may coincide in position with the 25 GHz masers. The source may be an example of an interesting case recently predicted in theory where the transitions of traditionally different methanol maser classes show maser activity simultaneously. In addition, results of recent search for methanol masers from the 25 and 104.3 GHz transitions are reported.Comment: To appear in the Proceedings of the 2004 European Workshop: "Dense Molecular Gas around Protostars and in Galactic Nuclei", Eds. Y.Hagiwara, W.A.Baan, H.J. van Langevelde, 2004, a special issue of ApSS, Kluwer; author list has been corrected, text is unchange

Theories for TC0 and Other Small Complexity Classes

We present a general method for introducing finitely axiomatizable "minimal" two-sorted theories for various subclasses of P (problems solvable in polynomial time). The two sorts are natural numbers and finite sets of natural numbers. The latter are essentially the finite binary strings, which provide a natural domain for defining the functions and sets in small complexity classes. We concentrate on the complexity class TC^0, whose problems are defined by uniform polynomial-size families of bounded-depth Boolean circuits with majority gates. We present an elegant theory VTC^0 in which the provably-total functions are those associated with TC^0, and then prove that VTC^0 is "isomorphic" to a different-looking single-sorted theory introduced by Johannsen and Pollet. The most technical part of the isomorphism proof is defining binary number multiplication in terms a bit-counting function, and showing how to formalize the proofs of its algebraic properties.Comment: 40 pages, Logical Methods in Computer Scienc