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Combining Non-stably Infinite, Non-first Order Theories

2005
*
Electronical Notes in Theoretical Computer Science
*

Notably, some

doi:10.1016/j.entcs.2004.06.066
fatcat:s3ofxia5kjbxfdqq4nsuqfyism
*non*-*stably**infinite**theories*can now be*combined*with the*theory*of arrays. ... Also, the*combination*scheme is presented here using a semantic notion of*theory*, allowing to handle*non*-*first**order**theories*. ... The second improvement in this presentation is the*combination*of*non*-*stably**infinite*I-*theories*, following [9] . ...##
###
Combination of Disjoint Theories: Beyond Decidability
[chapter]

2012
*
Lecture Notes in Computer Science
*

*First*, we show that it is possible under minor technical conditions to

*combine*a decidable (not necessarily

*stably*

*infinite*)

*theory*and a disjoint finitely axiomatized

*theory*, obtaining a refutationally ... We here study

*combinations*involving an arbitrary

*first*-

*order*

*theory*. Decidability is lost, but refutational completeness is preserved. ... Consider

*combining*Presburger arithmetic (which is decidable) with a

*first*-

*order*and finitely axiomatizable but

*non*-disjoint

*theory*defining multiplication in terms of addition. ...

##
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Combining Nonstably Infinite Theories

2005
*
Journal of automated reasoning
*

Applications of our results include the

doi:10.1007/s10817-005-5204-9
fatcat:atwl57bvxbaorercpqgkpzhz7m
*combination*of*theories*having only finite models, as well as the*combination*of*non*-*stably**infinite**theories*with the*theory*of equality, the*theories*of total ... The Nelson-Oppen*combination*method*combines*decision procedures for*first*-*order**theories*over disjoint signatures into a single decision procedure for the union*theory*. ...*Non*-*Stably**Infinite**Theories*pap.tex; 26/01/2005; 18:07; p.33 ...##
###
Combining Theories: The Ackerman and Guarded Fragments
[chapter]

2011
*
Lecture Notes in Computer Science
*

In particular, it is not required for these other

doi:10.1007/978-3-642-24364-6_4
fatcat:hndeqmbmirdo5azpeoi2mauzji
*theories*to be*stably*-*infinite*. ... In [7] we showed that any*first*-*order**theory*from the Bernays-Schönfinkel-Ramsey fragment, the two variable fragment, or the monadic fragment can be*combined*with virtually any other decidable*theory*. ... These fragments include*non**stably**infinite**theories*. ...##
###
Combinations of Theories for Decidable Fragments of First-Order Logic
[chapter]

2009
*
Lecture Notes in Computer Science
*

in a

doi:10.1007/978-3-642-04222-5_16
fatcat:wijhyhw2gbfunongjdrujkim5u
*stably**infinite**theory*has an*infinite*model. ... The design of decision procedures for*first*-*order**theories*and their*combinations*has been a very active research subject for thirty years; it has gained practical importance through the development of ... The*first*-*order*decidable classes considered in this paper contain*theories*that are not*stably**infinite*. ...##
###
Theory Combination: Beyond Equality Sharing
[chapter]

2019
*
Lecture Notes in Computer Science
*

Although it is not even semidecidable in

doi:10.1007/978-3-030-22102-7_3
fatcat:ukphko527rcpthhau3ydqyw6gy
*first*-*order*logic, it is decidable in some*first*-*order**theories*or fragments thereof (e.g., the quantifier-free fragment). ... A standard solution is the equality-sharing method by Nelson and Oppen, which requires the*theories*to be disjoint and*stably**infinite*. ... Part of this work was done when the*first*author was visiting LORIA Nancy and the Computer Science Laboratory of SRI International: the support of both institutions is greatly appreciated. ...##
###
A Rewrite Rule Based Framework for Combining Decision Procedures *
[chapter]

2002
*
Lecture Notes in Computer Science
*

½

doi:10.1007/3-540-45988-x_8
fatcat:et737pnk5jdphj3w6eltjf4ydu
*Stably**infinite*T is*stably**infinite*if for any T -satisfiable ϕ, there exists a model of T satisfying ϕ and whose domain is*infinite*. How to reduce the*non*-determinism? ... .,*Combination*Lemma Lemma: Let T 1 , T 2 be two signature-disjoint*stably*-*infinite**theories*. ...##
###
Automatic decidability and combinability

2011
*
Information and Computation
*

(x < y ⇒ x < y + y) is true for all possible interpretations of the

doi:10.1016/j.ic.2011.03.005
fatcat:43hij62ivrc3plqi2seqpzq73m
*non*-logical symbols < and +, but only for those interpretations in which < is the usual*ordering*over the integers and + is the addition ... Finally, to make the situation even more complex, most verification problems involve more than one*theory*, so that methods to*combine**theories*, such as the one pioneered by Nelson and Oppen [22], are required ...*First*, we intend to investigate further applications of Schematic Saturation. For example, in [31, 26] , new*combination*schemas for*non*-*stably**infinite**theories*are described. ...##
###
A Gentle Non-disjoint Combination of Satisfiability Procedures
[chapter]

2014
*
Lecture Notes in Computer Science
*

However, in its initial presentation, the Nelson-Oppen

doi:10.1007/978-3-319-08587-6_9
fatcat:5zdeuh6sgvfirpkd64uo6gb6su
*combination*method requires the*theories*to be signature-disjoint and*stably**infinite*(to guarantee the existence of an*infinite*model). ... In this paper, we adapt the notion of gentle*theory*to the*non*-disjoint*combination*of*theories*sharing only unary predicates (plus constants and the equality). ... This is a natural extension of previous works on*combining**non*-*stably**infinite**theories*, in the straight line of*combination*methodsà la Nelson-Oppen. ...##
###
Combining Lists with Non-stably Infinite Theories
[chapter]

2005
*
Lecture Notes in Computer Science
*

More specifically, we provide a new method that is able to

doi:10.1007/978-3-540-32275-7_4
fatcat:auatpz2ia5dkfks3nhomixufae
*combine*the*theory*of lists with any*theory*T of the elements, regardless of whether T is*stably**infinite*or not. ... This*combination*can be achieved using the Nelson-Oppen method only if T is*stably**infinite*. The goal of this paper is to relax the stable-*infiniteness*requirement. ... Acknowledgments We are grateful to Christophe Ringeissen for insightful discussions on the problem of*combining**non*-*stably**infinite**theories*. ...##
###
Combined Covers and Beth Definability
[chapter]

2020
*
Lecture Notes in Computer Science
*

A successive parallel research line inside the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as "covers") in

doi:10.1007/978-3-030-51074-9_11
fatcat:4bw5smiqljcjpkujpvru6j4qsm
*first*-*order**theories*. ... In the*non*-convex case, we show by a counterexample that cover may not exist in the*combined**theories*, even in case*combined*quantifier-free interpolants do exist. ... In the context of*first**order**theories*, we prove an even more direct connection: Theorem 1. Suppose that T is a universal*theory*. ...##
###
Combination of convex theories: Modularity, deduction completeness, and explanation

2010
*
Journal of symbolic computation
*

*Combining*

*Non*

*Stably*

*Infinite*Convex

*Theories*So far, we have only considered

*stably*

*infinite*convex

*theories*. ...

*Non*-

*stably*

*infinite*

*theories*. In [4] , the authors have shown that all convex

*theories*with no trivial models are

*stably*

*infinite*, and therefore they can be

*combined*using the NO

*combination*schema. ...

##
###
Combining abstract interpreters

2006
*
Proceedings of the 2006 ACM SIGPLAN conference on Programming language design and implementation - PLDI '06
*

*stably*

*infinite*, and disjoint. ... We also present an interesting application of logical product wherein some lattices can be reduced to

*combination*of other (unrelated) lattices with known abstract interpreters. •

*Theory*of sign. ... However, all of these five

*theories*are convex and

*stably*

*infinite*. Logical Lattices and Their

*Combination*A lattice L consists of a set DL and a partial

*order*L among elements of DL. ...

##
###
Combining abstract interpreters

2006
*
SIGPLAN notices
*

*stably*

*infinite*, and disjoint. ... We also present an interesting application of logical product wherein some lattices can be reduced to

*combination*of other (unrelated) lattices with known abstract interpreters. •

*Theory*of sign. ... However, all of these five

*theories*are convex and

*stably*

*infinite*. Logical Lattices and Their

*Combination*A lattice L consists of a set DL and a partial

*order*L among elements of DL. ...

##
###
Combined Covers and Beth Definability (Extended Version)
[article]

2020
*
arXiv
*
pre-print

However, we exhibit a cover transfer algorithm operating also in the

arXiv:1911.07774v3
fatcat:qz7fwdkqvnf57gfrrulhkxp7ai
*non*-convex case for special kinds of*theory**combinations*; these*combinations*(called 'tame*combinations*') concern multi-sorted*theories*... In the*non*-convex case, we show by a counterexample that covers may not exist in the*combined**theories*, even in case*combined*quantifier-free interpolants do exist. ... A*theory*T is*stably**infinite*iff every T -satisfiable constraint is satisfiable in an*infinite*model of T . ...
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