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1
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Choice functions and well-orderings over the infinite binary tree

100%
Carayol A., Löding C., Niwinski D., Walukiewicz I.
Open Mathematics
|
2010
|
tom 8
|
nr 4
662-682
EN
We give a new proof showing that it is not possible to define in monadic second-order logic (MSO) a choice function on the infinite binary tree. This result was first obtained by Gurevich and Shelah using set theoretical arguments. Our proof is much simpler and only uses basic tools from automata theory. We show how the result can be used to prove the inherent ambiguity of languages of infinite trees. In a second part we strengthen the result of the non-existence of an MSO-definable well-founded order on the infinite binary tree by showing that every infinite binary tree with a well-founded order has an undecidable MSO-theory.
2
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Propositional Linear Temporal Logic with Initial Validity Semantics1

81%
Giero M.
Formalized Mathematics
|
2015
|
tom 23
|
nr 4
379-386
EN
In the article [10] a formal system for Propositional Linear Temporal Logic (in short LTLB) with normal semantics is introduced. The language of this logic consists of “until” operator in a very strict version. The very strict “until” operator enables to express all other temporal operators. In this article we construct a formal system for LTLB with the initial semantics [12]. Initial semantics means that we define the validity of the formula in a model as satisfaction in the initial state of model while normal semantics means that we define the validity as satisfaction in all states of model. We prove the Deduction Theorem, and the soundness and completeness of the introduced formal system. We also prove some theorems to compare both formal systems, i.e., the one introduced in the article [10] and the one introduced in this article. Formal systems for temporal logics are applied in the verification of computer programs. In order to carry out the verification one has to derive an appropriate formula within a selected formal system. The formal systems introduced in [10] and in this article can be used to carry out such verifications in Mizar [4].
3
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Simple-Named Complex-Valued Nominative Data – Definition and Basic Operations

61%
Ivanov I., Nikitchenko M., Kryvolap A., Korniłowicz A.
Formalized Mathematics
|
2017
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tom 25
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nr 3
205-216
EN
In this paper we give a formal definition of the notion of nominative data with simple names and complex values [15, 16, 19] and formal definitions of the basic operations on such data, including naming, denaming and overlapping, following the work [19]. The notion of nominative data plays an important role in the composition-nominative approach to program formalization [15, 16] which is a development of composition programming [18]. Both approaches are compared in [14]. The composition-nominative approach considers mathematical models of computer software and data on various levels of abstraction and generality and provides mathematical tools for reasoning about their properties. In particular, nominative data are mathematical models of data which are stored and processed in computer systems. The composition-nominative approach considers different types [14, 19] of nominative data, but all of them are based on the name-value relation. One powerful type of nominative data, which is suitable for representing many kinds of data commonly used in programming like lists, multidimensional arrays, trees, tables, etc. is the type of nominative data with simple (abstract) names and complex (structured) values. The set of nominative data of given type together with a number of basic operations on them like naming, denaming and overlapping [19] form an algebra which is called data algebra. In the composition-nominative approach computer programs which process data are modeled as partial functions which map nominative data from the carrier of a given data algebra (input data) to nominative data (output data). Such functions are also called binominative functions. Programs which evaluate conditions are modeled as partial predicates on nominative data (nominative predicates). Programming language constructs like sequential execution, branching, cycle, etc. which construct programs from the existing programs are modeled as operations which take binominative functions and predicates and produce binominative functions. Such operations are called compositions. A set of binominative functions and a set of predicates together with appropriate compositions form an algebra which is called program algebra. This algebra serves as a semantic model of a programming language. For functions over nominative data a special computability called abstract computability is introduces and complete classes of computable functions are specified [16]. For reasoning about properties of programs modeled as binominative functions a Floyd-Hoare style logic [1, 2] is introduced and applied [12, 13, 8, 11, 9, 10]. One advantage of this approach to reasoning about programs is that it naturally handles programs which process complex data structures (which can be quite straightforwardly represented as nominative data). Also, unlike classical Floyd-Hoare logic, the mentioned logic allows reasoning about assertions which include partial pre- and post-conditions [11]. Besides modeling data processed by programs, nominative data can be also applied to modeling data processed by signal processing systems in the context of the mathematical systems theory [4, 6, 7, 5, 3].
4
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The isomorphism relation between tree-automatic Structures

52%
Finkel O., Todorčević S.
Open Mathematics
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2010
|
tom 8
|
nr 2
299-313
EN
An ω-tree-automatic structure is a relational structure whose domain and relations are accepted by Muller or Rabin tree automata. We investigate in this paper the isomorphism problem for ω-tree-automatic structures. We prove first that the isomorphism relation for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is not determined by the axiomatic system ZFC. Then we prove that the isomorphism problem for ω-tree-automatic boolean algebras (respectively, partial orders, rings, commutative rings, non commutative rings, non commutative groups, nilpotent groups of class n ≥ 2) is neither a Σ21-set nor a Π21-set.
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