scholarly journals Symbolic WS1S

10.29007/t28j ◽  
2018 ◽  
Author(s):  
Loris D'Antoni ◽  
Margus Veanes
Keyword(s):  
Blow Up ◽  
Finite Automata ◽  
Decision Procedures ◽  
Second Order ◽  
Order Logic ◽  
Symbolic Logic ◽  
First Order ◽  
Symbolic Automata ◽  
Second Order Logic ◽  
Monadic Second Order Logic

We extend weak monadic second-order logic of one successor (WS1S) to symbolic alphabets byallowing character predicates to range over decidable first order theories and not just finite alphabets.We call this extension symbolic WS1S (s-WS1S). We then propose two decision procedures for such alogic: 1) we use symbolic automata to extend the classic reduction from WS1S to finite automata toour symbolic logic setting; 2) we show that every s-WS1S formula can be reduced to a WS1S formulathat preserves satisfiability, at the price of an exponential blow-up.

scholarly journals First order quantifiers in monadic second order logic

2004 ◽  
Vol 69 (1) ◽  
pp. 118-136 ◽  
Author(s):  
H. Jerome Keisler ◽  
Wafik Boulos Lotfallah
Keyword(s):  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Existential Quantifier ◽  
Reachability Problem ◽  
Open Problems ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Quantifier Alternation

AbstractThis paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn (S) on properties S that says “there are n components having S”. We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.We introduce another operation alt(S) on properties which has the same relationship with the Circuit Value Problem as reach(S) (defined in [JM01]) has with the Directed Reachability Problem. We use alt(S) to show that Πn ⊈ FO(Σn), Σn ⊈ FO(∆n). and ∆n+1 ⊈ FOB(Σn), solving some open problems raised in [Mat98].

scholarly journals Where First-Order and Monadic Second-Order Logic Coincide

2016 ◽  
Vol 17 (4) ◽  
pp. 1-18 ◽  
Author(s):  
Michael Elberfeld ◽  
Martin Grohe ◽  
Till Tantau
Keyword(s):  
Second Order ◽  
Order Logic ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic

scholarly journals Using hybrid logic for querying dependency treebanks

2012 ◽  
Vol 7 ◽  
Author(s):  
Anders Søgaard ◽  
Søren Lind Kristiansen
Keyword(s):  
General Purpose ◽  
Second Order ◽  
Logic Model ◽  
Order Logic ◽  
Hybrid Logic ◽  
First Order Logic ◽  
Model Checker ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic

Existing logic-based querying tools for dependency treebanks use first order logic or monadic second order logic. We introduce a very fast model checker based on hybrid logic with operators ↓, @ and A and show that it is much faster than an existing querying tool for dependency treebanks based on first order logic, and much faster than an existing general purpose hybrid logic model checker. The querying tool is made publicly available.

scholarly journals Local Normal Forms for First-Order Logic with Applications to Games and Automata

1999 ◽  
Vol Vol. 3 no. 3 ◽  
Author(s):  
Thomas Schwentick ◽  
Klaus Barthelmann
Keyword(s):  
Normal Forms ◽  
Winning Strategy ◽  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Relational Structures ◽  
Second Order Logic ◽  
Monadic Second Order Logic

International audience Building on work of Gaifman [Gai82] it is shown that every first-order formula is logically equivalent to a formula of the form ∃ x_1,...,x_l, \forall y, φ where φ is r-local around y, i.e. quantification in φ is restricted to elements of the universe of distance at most r from y. \par From this and related normal forms, variants of the Ehrenfeucht game for first-order and existential monadic second-order logic are developed that restrict the possible strategies for the spoiler, one of the two players. This makes proofs of the existence of a winning strategy for the duplicator, the other player, easier and can thus simplify inexpressibility proofs. \par As another application, automata models are defined that have, on arbitrary classes of relational structures, exactly the expressive power of first-order logic and existential monadic second-order logic, respectively.

scholarly journals HEINRICH BEHMANN’S 1921 LECTURE ON THE DECISION PROBLEM AND THE ALGEBRA OF LOGIC

2015 ◽  
Vol 21 (2) ◽  
pp. 164-187 ◽  
Author(s):  
PAOLO MANCOSU ◽  
RICHARD ZACH
Keyword(s):  
English Translation ◽  
Decision Problem ◽  
Axiomatic Approach ◽  
Decision Procedures ◽  
Second Order ◽  
Order Logic ◽  
David Hilbert ◽  
Second Order Logic ◽  
Monadic Second Order Logic

AbstractHeinrich Behmann (1891–1970) obtained his Habilitation under David Hilbert in Göttingen in 1921 with a thesis on the decision problem. In his thesis, he solved—independently of Löwenheim and Skolem’s earlier work—the decision problem for monadic second-order logic in a framework that combined elements of the algebra of logic and the newer axiomatic approach to logic then being developed in Göttingen. In a talk given in 1921, he outlined this solution, but also presented important programmatic remarks on the significance of the decision problem and of decision procedures more generally. The text of this talk as well as a partial English translation are included.

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