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Complexity of the first-order theory of almost all finite structures

1983
*
Information and Control
*

A

doi:10.1016/s0019-9958(83)80043-6
fatcat:bjodas7nw5hrnhccna2h6lvhka
*first*-*order*sentence*of*a relational type Y is true*almost*everywhere if*the*proportion*of*its models among*the**structures**of*type Y and cardinality m tends to 1 when m tends to 0o. ... Fagin (1976) proved that*the**theory*, called Th(Y),*of**the**first*-*order*sentences (*of*type Y) true a.e. coincides with*the**theory**the*axioms*of*which say roughly: "any*finite*substructure (possibly empty ... Mac Aloon for making me aware*of*Fagin's work (1976). Thanks to Peter Ctote and to*the*working group on*complexity**of*University Paris 7. ...##
###
Reflections on Finite Model Theory

2007
*
22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)
*

Main Finding:

doi:10.1109/lics.2007.39
dblp:conf/lics/Kolaitis07
fatcat:3eulyaqcgbfdzihya6kzjxm2ru
*All*major computational*complexity*classes, including P, NP, and PSPACE, can be characterized in terms*of*definability in various logics on classes*of**finite**structures*. ... Reinforces*the*unity*of*computation and logic. Yields machine-independent characterizations*of*computational*complexity*classes. ...*Almost*Sure*First*-*Order*Truth Testing if a FO-sentence is*almost*surely true on*all**finite*graphs is a decidable problem; in fact, it is PSPACEcomplete. ...##
###
Finite-model theory - a personal perspective

1993
*
Theoretical Computer Science
*

There is a remarkable phenomenon which says that certain properties (such as those expressible in

doi:10.1016/0304-3975(93)90218-i
fatcat:hmpnxjt5bvg37nwy6agiccmlm4
*first*-*order*logic) are either*almost*surely true or*almost*surely false. (4) Descriptice*complexity**theory*... .,*Finite*-mode1*theory*-a personal perspective, Theoretical Computer Science 116 (1993) 3-31.*Finite*-model*theory*is a study*of**the*logical properties*of**finite*mathematical*structures*. ... I thank them, along with Miki Ajtai, Kevin Compton, Joe Halpern, Neil Immerman, Russell Impagliazzo, Jim Lynch, and Bob Vaught, for reading a draft*of**the*paper and giving me comments. ...##
###
Finite-model theory—a personal perspective
[chapter]

1990
*
Lecture Notes in Computer Science
*

There is a remarkable phenomenon which says that certain properties (such as those expressible in

doi:10.1007/3-540-53507-1_67
fatcat:bo7ne5kfbbfzdn2gkaw4fy2vbu
*first*-*order*logic) are either*almost*surely true or*almost*surely false. (4) Descriptice*complexity**theory*... .,*Finite*-mode1*theory*-a personal perspective, Theoretical Computer Science 116 (1993) 3-31.*Finite*-model*theory*is a study*of**the*logical properties*of**finite*mathematical*structures*. ... I thank them, along with Miki Ajtai, Kevin Compton, Joe Halpern, Neil Immerman, Russell Impagliazzo, Jim Lynch, and Bob Vaught, for reading a draft*of**the*paper and giving me comments. ...##
###
Introduction
[chapter]

2021
*
Groups and Model Theory
*

*The*objective is to investigate what kind

*of*groups are definable in

*the*given

*first*-

*order*context or ambient

*structure*. ...

*The*

*first*book, "

*Complexity*and Randomness in Group

*Theory*. GAGTA book 1," was published by De Gruyter in 2020. ...

*All*

*the*rich groups mentioned above are

*first*-

*order*rigid. ...

##
###
Page 2294 of Mathematical Reviews Vol. , Issue 86f
[page]

1986
*
Mathematical Reviews
*

Unfortunately no extension

*of**the**first*-*order**theory**of**finite**structures*is recur- sively axiomatisable. ... One would like to enrich*first*-*order*logic so that*the*enriched logic fits better*the*case*of**finite**structures*. ...##
###
The computational complexity of asymptotic problems I: Partial orders

1988
*
Information and Computation
*

*The*computational

*complexity*

*of*

*the*set

*of*sentences with asymptotic probability 1 is determined. For

*first*-

*order*logic, it is PSPACE-complete. For inductive fixed-point logic, it is EXPTIME-complete. ...

*The*class

*of*partial

*orders*is shown to have Ol laws for

*first*-

*order*logic and for inductive fixed-point logic, a logic which properly contains

*first*-

*order*logic. ...

*COMPLEXITY*

*OF*

*THE*

*ALMOST*SURE

*THEORIES*

*OF*PARTIAL

*ORDERS*We now determine

*the*computational

*complexities*

*of*

*the*

*first*-

*order*and inductive fixed-point

*theories*

*of*

*almost*

*all*partial

*orders*. ...

##
###
Finite and infinite model theory - a historical perspective

2000
*
Logic Journal of the IGPL
*

We describe

doi:10.1093/jigpal/8.5.605
fatcat:yfgciztydzap3cdtw3pdozrf3y
*the*progress*of*model*theory*in*the*last half century from*the*standpoint*of*how*finite*model*theory*might develop. 1 ... Theorem 4.35 For appropriate functions f determining*the*interpretation*of**the*Ramsey quantifier*the*logic L ω,ω (Q ram,f ) satisfies*the*0-1 law on graphs with respect to edge probability n −α for irrational ... Abstact and*Finite*Model*Theory**The*'weakness'*of**first**order*logic on*finite**structures*has led to*the*study*of*various extensions*of**first**order*logic. ...##
###
A zero-one law for logic with a fixed-point operator

1985
*
Information and Control
*

Contrary to what one might expect, our equivalence result does not allow us to transfer PSPACE completeness

doi:10.1016/s0019-9958(85)80027-9
fatcat:pvnvemhlybfx5bq43zxugi4s2y
*of**the**theory**of*random*structures*from*first*-*order*logic to*the*fixed-point operators. ... For any sentence q~*of**the*extended logic,*the*proportion*of*models*of*q~ among*all**structures*with universe {1, 2,..., n} approaches 0 or 1 as n tends to infinity. ... A zero-one law for*the**first*-*order**theory**of**almost**all**structures*in a*first*-*order*definable class can be transferred to*the*FO + IFP*theory*by our methods provided*the**almost*surely true*first*-*order*...##
###
Page 3011 of Mathematical Reviews Vol. , Issue 2002E
[page]

2002
*
Mathematical Reviews
*

While work

*of*Ax and Kochen and*of*Ershov has produced a good understanding*of**first*-*order**theories**of*Henselian valued fields*of*characteristic 0 in general, and p-adic fields in particular,*all**the*analogous ...*The*proof method adopted here is based on an analysis*of**the**structure**of*Skolem functions arising from certain kinds*of**first*-*order*prefixes, and hence bounding*the*number*of*witnesses*of**the*existential ...##
###
Page 4358 of Mathematical Reviews Vol. , Issue 87h
[page]

1987
*
Mathematical Reviews
*

Another immediate corollary is that if X is as in

*the*main theorem, then for*almost**all*primes p*the*exponent*of*X at p is*finite*;*the*exponent*of*X is*the*least upper bound*of**the**orders**of*p-primary ... By work*of*D. J. Anick [Pacific J. Math. 123 (1986), no. 2, 257-262], there are*finite*1-connected*complexes*K such that H,(.K; Z) has p-torsion*of*every possible*order*. ...##
###
Page 1374 of Mathematical Reviews Vol. , Issue 98C
[page]

1998
*
Mathematical Reviews
*

Summary: “

*The*‘*ordered*conjecture’*of*Ph. G. Kolaitis and M. Y. Vardi asks whether fixed-point logic differs from*first*-*order*logic on every infinite class*of**finite**ordered**structures*. ...*the*existential problem in*the**first*-*order**theory**of*real closed fields, are obtained.” ...##
###
Page 2431 of Mathematical Reviews Vol. , Issue 2000d
[page]

2000
*
Mathematical Reviews
*

*The*authors work with recursive

*structures*X (for a

*finite*

*first*-

*order*relational vocabulary) that are homogeneous (meaning that

*all*isomorphisms between

*finite*substructures extend to automor- phisms ... ; Beijing) ; Ning, Shucheng (PRC-ASBJ; Beijing) Practically solving some problems expressed in

*the*

*first*

*order*

*theory*

*of*real closed fields. ...

##
###
Mathematics and Complex Systems

2007
*
Science
*

*The*Classification

*of*

*Finite*Simple Groups In

*the*1880s

*the*quest began to classify

*finite*simple groups, in

*the*sense

*of*listing them

*all*in "families" that enjoy common

*structural*properties. ... In hindsight, Gagnnon [(12), p. 27] observed, "

*Almost*every facet

*of*Moonshine finds a natural formulation in conformal field

*theory*, where it often was discovered

*first*." ...

##
###
Logical Definability of Counting Functions

1996
*
Journal of computer and system sciences (Print)
*

For a logic L, *L is

doi:10.1006/jcss.1996.0069
fatcat:fksc6h5r2jdg5nohcko3pxeawq
*the*class*of*functions on*finite**structures*counting*the*tuples (T , cÄ ) satisfying a given formula (T , cÄ ) in L. ... Saluja, Subrahmanyam, and Thakur showed that on classes*of**ordered**structures**FO=*P (where FO denotes*first*-*order*logic) and that every function in * 1 has a fully polynomial randomized approximation ...*The*other logical characterizations*of**complexity*classes cited in*the**first*paragraph apply only to classes*of**ordered**structures*. ...
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