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Arithmetic, First-Order Logic, and Counting Quantifiers
[article]

2002
*
arXiv
*
pre-print

This paper gives a thorough overview of what is known about

arXiv:cs/0211022v1
fatcat:erouadqpdbbtvn5gk3te66tlwm
*first*-*order**logic*with*counting**quantifiers**and*with*arithmetic*predicates. ... Applying this theorem, we obtain an easy proof of Ruhl's result that reachability (*and*similarly, connectivity) in finite graphs is not expressible in*first*-*order**logic*with unary*counting**quantifiers*... Syntax*and*Semantics*First*-*order**logic*with unary*counting**quantifiers*, FOunC, is the extension of*first*-*order**logic*obtained by adding unary*counting**quantifiers*of the form ∃ =x y. ...##
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Arithmetic, first-order logic, and counting quantifiers

2005
*
ACM Transactions on Computational Logic
*

This paper gives a thorough overview of what is known about

doi:10.1145/1071596.1071602
fatcat:xnkejnln4vdafggock5yx63yne
*first*-*order**logic*with*counting**quantifiers**and*with*arithmetic*predicates. ... For other related work that deals with*first*-*order**logic*,*counting*,*and*/or*arithmetic*from different points of view see, e.g., the articles [Lee 2003; Llima 1998; Krynicki*and*Zdanowski 2003; Mostowski ...*FIRST*-*ORDER**LOGIC*WITH*COUNTING**QUANTIFIERS*In this section we fix the syntax*and*semantics of*first*-*order**logic*with*counting**quantifiers*,*and*we summarize some important properties of this*logic*. ...##
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Page 4089 of Mathematical Reviews Vol. , Issue 98G
[page]

1998
*
Mathematical Reviews
*

Summary: “We study the expressive power in the finite setting of the

*logic*Fixed-Point +*Counting*, the extension of*first*-*order**logic*which is obtained by adding both the fixed-point constructor*and*the ... Summary: “A combinatorial criterion for two finite structures to agree on all sentences with bounded*quantifier*rank in*first*-*order**logic*with any set of unary generalized*quantifiers*is given. ...##
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First-order logic
[chapter]

1998
*
Springer Undergraduate Mathematics Series
*

This paper gives a thorough overview of what is known about

doi:10.1007/978-1-4471-0589-3_4
fatcat:n3vnowpts5cfrnlocrenwd6isy
*first*-*order**logic*with*counting**quantifiers**and*with*arithmetic*predicates. ... For other related work that deals with*first*-*order**logic*,*counting*,*and*/or*arithmetic*from different points of view see, e.g., the articles [Lee 2003; Llima 1998; Krynicki*and*Zdanowski 2003; Mostowski ...*FIRST*-*ORDER**LOGIC*WITH*COUNTING**QUANTIFIERS*In this section we fix the syntax*and*semantics of*first*-*order**logic*with*counting**quantifiers*,*and*we summarize some important properties of this*logic*. ...##
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The Nature and Purpose of Numbers

2010
*
Journal of Philosophy
*

Each is implemented through a specific device, i.e., respectively, an extra-

doi:10.5840/jphil2010107415
fatcat:7io7pj4a3vdevhfo6j7e67ltai
*logical*operator representing numerical abstraction*and*a non-standard (but still*first*-*order*) cardinality*quantifier*. ... Along the way, we investigate the*logical*status of*arithmetic*, the function of abstraction principles,*and*the respective merits of various strategies for reducing*arithmetical*notions to those of a theory ... A unary*quantifier*is*first*-*order*if*and*only if it represents a collection of subsets of D (*and*similarly, a binary*quantifier*is*first*-*order*if*and*only if it expresses a relation between subsets of ...##
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Extensions of MSO and the monadic counting hierarchy

2011
*
Information and Computation
*

We show that over unary vocabularies the

doi:10.1016/j.ic.2010.09.002
fatcat:ocgtvtbwz5gevm36kxlicumrgu
*logic*MSO(R), where MSO is monadic second-*order**logic**and*R is the*first*-*order*Rescher*quantifier*, can be characterized by Presburger*arithmetic*, whereas the*logic*... In this paper we study the expressive power of the extension of*first*-*order**logic*by the unary second-*order*majority*quantifier*Most 1 . ... Acknowledgments The authors thank Lauri Hella for valuable comments*and*suggestions. ...##
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Modulo quantifiers over functional vocabularies extending addition
[article]

2021
*
arXiv
*
pre-print

We show that

arXiv:1705.00290v6
fatcat:2jyjfjsvmze7vg6bmejumj6o44
*first**order**logic*(FO)*and**first**order**logic*extended with modulo*counting**quantifiers*(FOMOD) over purely functional vocabularies which extend addition, satisfy the Crane beach property ( ... Showing the existence of regular languages not definable in FOMOD[<, +, *] is equivalent to the separation of the circuit complexity classes ACC0*and*NC1 . ... In this paper, we identify a sufficient condition for functional vocabularies over*first**order**logic**and**first**order**logic*extended with modulo*quantifiers*, to satisfy the Crane Beach property. ...##
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Page 5204 of Mathematical Reviews Vol. , Issue 2002G
[page]

2002
*
Mathematical Reviews
*

The present paper considers

*logics*formed by extending*first*-*order**logic*with groupoidal*quantifiers*. ... The paper establishes that extending*first*-*order**logic*with*arithmetic*by all groupoidal*quantifiers*yields a*logic*that exactly characterizes LOGCFL, which is known to be equivalent to the complexity ...##
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Building a Calculus of Data Structures
[chapter]

2010
*
Lecture Notes in Computer Science
*

, multiset algebra with Presburger

doi:10.1007/978-3-642-11319-2_6
fatcat:owrtpwwto5baza2ruizloskv6i
*arithmetic*, the Bernays-Schönfinkel-Ramsey class of*first*-*order**logic*,*and*the*logic*of algebraic data types with the set content function. ... As an instance of this approach, we describe a decidable*logic*whose formulas are propositional combinations of formulas in: weak monadic second-*order**logic*of two successors, two-variable*logic*with*counting*... , multiset algebra with Presburger*arithmetic*, the Bernays-Schönfinkel-Ramsey class of*first*-*order**logic*,*and*the*logic*of algebraic data types with the set content function. ...##
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Induction and indefinite extensibility: the Gödel sentence is true, but did someone change the subject?

1998
*
Mind
*

If basic

doi:10.1093/mind/107.427.597
fatcat:7dx77ehlwndbxkzdw5wq6l5bze
*arithmetic*propositions are*logically*true-as the logicist contends-then there is tension between this conservation requirement*and*the ontological commitments of*arithmetic*. ...*and*elimination rules of the*logical*terms that occur in S. ... rules for negation, identity,*and*the*first*-*order*existential*quantifier*. ...##
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One-Variable Logic Meets Presburger Arithmetic
[article]

2019
*
arXiv
*
pre-print

We consider the one-variable fragment of

arXiv:1810.10899v3
fatcat:bnl6big7xnarbfr7ksrcppgrsq
*first*-*order**logic*extended with Presburger constraints. ... The*logic*is designed in such a way that it subsumes the previously-known fragments extended with*counting*, modulo*counting*or cardinality comparison*and*combines their expressive powers. ... The author would also like to thank the two anonymous reviewers as well as Witold Charatonik, Emanuel Kieroński*and*Antti Kuusisto for their careful proofreading*and*for pointing out numerous grammatical ...##
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Combining Theories with Shared Set Operations
[chapter]

2009
*
Lecture Notes in Computer Science
*

Algebra with Presburger

doi:10.1007/978-3-642-04222-5_23
fatcat:z43roeqdkbepbgistfmf63btsa
*Arithmetic*(with*quantifiers*over sets*and*integers), 2) weak monadic second-*order**logic*over trees (with monadic second-*order**quantifiers*), 3) two-variable*logic*with*counting*...*quantifiers*(ranging over elements), 4) the Bernays-Schönfinkel-Ramsey class of*first*-*order**logic*with equality (with ∃ * ∀ **quantifier*prefix),*and*5) the*quantifier*-free*logic*of multisets with cardinality ... Conclusion We have presented a combination result for*logics*that share operations on sets. This result yields an expressive decidable*logic*that is useful for software verification. ...##
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Expressing versus Proving: Relating Forms of Complexity in Logic

2010
*
Journal of Logic and Computation
*

of

doi:10.1093/logcom/exq008
fatcat:lpla27ghunf5fmmvdanaoasaei
*arithmetic*. ... Lastly, we will touch upon the topic of formalizing complexity theory using*logic*,*and*the meta-question of complexity of*logical*reasoning about complexity-theoretic statements. ... The results on connections between finite model theory*and*bounded*arithmetic*mentioned here come from my joint work with Stephen Cook (which resulted in my PhD thesis). ...##
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Logic and Complexity: Independence results and the complexity of propositional calculus
[chapter]

1995
*
Proceedings of the International Congress of Mathematicians
*

I want to thank Sam Buss, Jan Krajicek, Russell Irnpagliazzo,

doi:10.1007/978-3-0348-9078-6_22
fatcat:th6np34k3fehvfyffm73lbha2a
*and*Jiff Sgall for their comments on a draft of this paper. ... Bounded*arithmetic*We turn now to some*first**order*theories that are important in this field. Let us*first*recall the classical theory Peano*arithmetic*used to formalize number theory. ... Suppose we use not only the usual*quantifiers*V*and*3, but also a*counting**quantifier*, saying that the number of x 's satisfying ip is divisible by p. ...##
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Number Determiners, Numbers, and Arithmetic

2005
*
Philosophical Review
*

The latter would have two hundred

doi:10.1215/00318108-114-2-179
fatcat:dct5whply5fjjn2xafog3uev5u
*first*-*order**quantifiers*in its*logical*form. ... These*quantifiers*are part of the*first*-*order*predicate calculus,*and*this calculus, in turn, is part of*logic*,*and*thus unproblematic. ... In these cases, it is plausible that there is a plural*quantifier*involved, as in the sentence 'There are numbers between 100*and*200.' 28 Carnap's discussion can be found in his famous 1956 essay. ...
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