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Combining Non-stably Infinite, Non-first Order Theories

Pascal Fontaine, Pascal Gribomont
2005 Electronical Notes in Theoretical Computer Science  
Notably, some 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] .  ... 
doi:10.1016/j.entcs.2004.06.066 fatcat:s3ofxia5kjbxfdqq4nsuqfyism

Combination of Disjoint Theories: Beyond Decidability [chapter]

Pascal Fontaine, Stephan Merz, Christoph Weidenbach
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.  ... 
doi:10.1007/978-3-642-31365-3_21 fatcat:4hrpnrpfmjg4jfqhytlhph2bmu

Combining Nonstably Infinite Theories

Cesare Tinelli, Calogero G. Zarba
2005 Journal of automated reasoning  
Applications of our results include the 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  ... 
doi:10.1007/s10817-005-5204-9 fatcat:atwl57bvxbaorercpqgkpzhz7m

Combining Theories: The Ackerman and Guarded Fragments [chapter]

Carlos Areces, Pascal Fontaine
2011 Lecture Notes in Computer Science  
In particular, it is not required for these other 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.  ... 
doi:10.1007/978-3-642-24364-6_4 fatcat:hndeqmbmirdo5azpeoi2mauzji

Combinations of Theories for Decidable Fragments of First-Order Logic [chapter]

Pascal Fontaine
2009 Lecture Notes in Computer Science  
in a 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.  ... 
doi:10.1007/978-3-642-04222-5_16 fatcat:wijhyhw2gbfunongjdrujkim5u

Theory Combination: Beyond Equality Sharing [chapter]

Maria Paola Bonacina, Pascal Fontaine, Christophe Ringeissen, Cesare Tinelli
2019 Lecture Notes in Computer Science  
Although it is not even semidecidable in 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.  ... 
doi:10.1007/978-3-030-22102-7_3 fatcat:ukphko527rcpthhau3ydqyw6gy

A Rewrite Rule Based Framework for Combining Decision Procedures * [chapter]

Deepak Kapur
2002 Lecture Notes in Computer Science  
½ 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.  ... 
doi:10.1007/3-540-45988-x_8 fatcat:et737pnk5jdphj3w6eltjf4ydu

Automatic decidability and combinability

Christopher Lynch, Silvio Ranise, Christophe Ringeissen, Duc-Khanh Tran
2011 Information and Computation  
(x < y ⇒ x < y + y) is true for all possible interpretations of the 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.  ... 
doi:10.1016/j.ic.2011.03.005 fatcat:43hij62ivrc3plqi2seqpzq73m

A Gentle Non-disjoint Combination of Satisfiability Procedures [chapter]

Paula Chocron, Pascal Fontaine, Christophe Ringeissen
2014 Lecture Notes in Computer Science  
However, in its initial presentation, the Nelson-Oppen 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.  ... 
doi:10.1007/978-3-319-08587-6_9 fatcat:5zdeuh6sgvfirpkd64uo6gb6su

Combining Lists with Non-stably Infinite Theories [chapter]

Pascal Fontaine, Silvio Ranise, Calogero G. Zarba
2005 Lecture Notes in Computer Science  
More specifically, we provide a new method that is able to 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.  ... 
doi:10.1007/978-3-540-32275-7_4 fatcat:auatpz2ia5dkfks3nhomixufae

Combined Covers and Beth Definability [chapter]

Diego Calvanese, Silvio Ghilardi, Alessandro Gianola, Marco Montali, Andrey Rivkin
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 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.  ... 
doi:10.1007/978-3-030-51074-9_11 fatcat:4bw5smiqljcjpkujpvru6j4qsm

Combination of convex theories: Modularity, deduction completeness, and explanation

Duc-Khanh Tran, Christophe Ringeissen, Silvio Ranise, Hélène Kirchner
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.  ... 
doi:10.1016/j.jsc.2008.10.006 fatcat:mej5vvsqe5ehhmmf4hfhnwgwta

Combining abstract interpreters

Sumit Gulwani, Ashish Tiwari
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.  ... 
doi:10.1145/1133981.1134026 dblp:conf/pldi/GulwaniT06 fatcat:y5zkvl77dfasxosmfz4hucjjva

Combining abstract interpreters

Sumit Gulwani, Ashish Tiwari
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.  ... 
doi:10.1145/1133255.1134026 fatcat:grdxypzkbncwtk2djze4feo23a

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

Diego Calvanese and Silvio Ghilardi and Alessandro Gianola and Marco Montali and Andrey Rivkin
2020 arXiv   pre-print
However, we exhibit a cover transfer algorithm operating also in the 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 .  ... 
arXiv:1911.07774v3 fatcat:qz7fwdkqvnf57gfrrulhkxp7ai
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