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

Etienne Grandjean
1983 Information and Control  
A 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.  ... 
doi:10.1016/s0019-9958(83)80043-6 fatcat:bjodas7nw5hrnhccna2h6lvhka

Reflections on Finite Model Theory

Phokion G. Kolaitis
2007 22nd Annual IEEE Symposium on Logic in Computer Science (LICS 2007)  
Main Finding: 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.  ... 
doi:10.1109/lics.2007.39 dblp:conf/lics/Kolaitis07 fatcat:3eulyaqcgbfdzihya6kzjxm2ru

Finite-model theory - a personal perspective

Ronald Fagin
1993 Theoretical Computer Science  
There is a remarkable phenomenon which says that certain properties (such as those expressible in 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.  ... 
doi:10.1016/0304-3975(93)90218-i fatcat:hmpnxjt5bvg37nwy6agiccmlm4

Finite-model theory—a personal perspective [chapter]

Ronald Fagin
1990 Lecture Notes in Computer Science  
There is a remarkable phenomenon which says that certain properties (such as those expressible in 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.  ... 
doi:10.1007/3-540-53507-1_67 fatcat:bo7ne5kfbbfzdn2gkaw4fy2vbu

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.  ... 
doi:10.1515/9783110719710-201 fatcat:fswuu6z2bndd7arwiqgbgol7re

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

Kevin J. Compton
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.  ... 
doi:10.1016/0890-5401(88)90032-6 fatcat:nd272a33kvg3th4jygy4k3y5ku

Finite and infinite model theory - a historical perspective

J Baldwin
2000 Logic Journal of the IGPL  
We describe 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.  ... 
doi:10.1093/jigpal/8.5.605 fatcat:yfgciztydzap3cdtw3pdozrf3y

A zero-one law for logic with a fixed-point operator

Andreas Blass, Yuri Gurevich, Dexter Kozen
1985 Information and Control  
Contrary to what one might expect, our equivalence result does not allow us to transfer PSPACE completeness 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  ... 
doi:10.1016/s0019-9958(85)80027-9 fatcat:pvnvemhlybfx5bq43zxugi4s2y

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: “Theordered 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

R. Foote
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."  ... 
doi:10.1126/science.1141754 pmid:17947574 fatcat:kkkhppiv5janjlofcroyivwr3y

Logical Definability of Counting Functions

Kevin J. Compton, Erich Grädel
1996 Journal of computer and system sciences (Print)  
For a logic L, *L is 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.  ... 
doi:10.1006/jcss.1996.0069 fatcat:fksc6h5r2jdg5nohcko3pxeawq
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