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Recursion Schemes and Logical Reflection

Christopher H. Broadbent, Arnaud Carayol, C.-H. Luke Ong, Olivier Serre
2010 2010 25th Annual IEEE Symposium on Logic in Computer Science  
In this paper, we prove that higher-order recursion schemes are reflective w.r.t. both modal µ-calculus and monadic second order (MSO) logic.  ...  (Order-n recursion schemes are equi-expressive with order-n CPDA for generating trees.)  ...  Let t be a Σ-labelled tree gererated by an order-n recursion scheme S and let ϕ(x) be an MSOformula. (i) There is an algorithm that transforms (S, ϕ) to an order-n CPDA A such that L(A) = ||t|| ϕ .  ... 
doi:10.1109/lics.2010.40 dblp:conf/lics/BroadbentCOS10 fatcat:qpkit35y3fb6zc6xo5z4jjw2be

Higher-order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

Christopher H. Broadbent, Arnaud Carayol, C.-H. Luke Ong, Olivier Serre
2021 ACM Transactions on Computational Logic  
This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes.  ...  We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set  ...  This logic is incomparable with MSO logic. He showed that model-checking is decidable and that both reflection and selection hold for trees generated by recursion schemes.  ... 
doi:10.1145/3452917 fatcat:e7pseftknfcxzdwunjwrk6ofkm

Higher-Order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties [article]

Christopher H. Broadbent, Arnaud Carayol, C.-H. Luke Ong, Olivier Serre
2021 arXiv   pre-print
This paper studies the logical properties of a very general class of infinite ranked trees, namely those generated by higher-order recursion schemes.  ...  We consider, for both monadic second-order logic and modal mu-calculus, three main problems: model-checking, logical reflection (aka global model-checking, that asks for a finite description of the set  ...  This logic is incomparable with MSO logic. He showed that model-checking is decidable and that both reflection and selection hold for trees generated by recursion schemes.  ... 
arXiv:2010.06366v2 fatcat:htqiqtp3rfaxxattbycq4hlhsq

Page 7161 of Mathematical Reviews Vol. , Issue 95m [page]

1995 Mathematical Reviews  
So the logic provides easy loop construction and allows one to obtain totally correct program schemes. In Nepeivoda calculi, Q actions are investigated inside a propositional logic framework.  ...  (I13-reflecting ordinals may be regarded as the recursion-theoretic analogue of weakly compact cardinals.)  ... 

Page 2857 of Mathematical Reviews Vol. , Issue 87f [page]

1987 Mathematical Reviews  
(S-UPPS) Definability and forcing in E-recursion. Math. Scand. 57 (1985), no. 1, 5-28. Set recursion is obtained by adding an enumeration scheme to the schemes for the rudimentary functions.  ...  Dag Normann (Oslo) Slaman, Theodore A. (1-CHI) 87£:03133 Reflection and the priority method in E-recursion theory.  ... 

Relational semantics of linear logic and higher-order model-checking [article]

Charles Grellois, Paul-André Melliès
2015 arXiv   pre-print
recursion schemes.  ...  Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding  ...  recursion scheme G.  ... 
arXiv:1501.04789v3 fatcat:4yd3ryhkpzgzfhpy52fctlmglq

Page 3225 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews  
., 1976; MR 57 #18202], he shows how to define (and find the least solution of) a system of recursion equations over the reducible schemes.  ...  In this very clearly written paper the author introduces new trans- formation rules for recursive program schemes.  ... 

Page 4412 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
The proofs are based on a characterization of IZ; and IM; by means of reflection principles, and make use of modal prov- ability logic, more precisely the Gédel-Léb logic GL.  ...  The author studies parameter-free induction schemes IZ; and IIl; over Kalmar elementary arithmetic. He solves a problem of R.  ... 

Page 4964 of Mathematical Reviews Vol. , Issue 86k [page]

1986 Mathematical Reviews  
E-recursion, or alternatively set-recursion, is defined by adding an enumeration scheme to the schemes for the rudimentary functions.  ...  “From the technical point of view, one can take this work 4s the theory of many-sorted, concurrent and (more significantly) second- order recursion schemes.  ... 

Page 1181 of Mathematical Reviews Vol. 44, Issue 6 [page]

1972 Mathematical Reviews  
Reflecting on this difference, the author introduces a recursion scheme for defining proper-class-valued mappings or operations by recursion on the ordinal numbers. If we define Y..  ...  Logic Colloquium ’69 (Proc. Summer School and Colloq., Manchester, 1969), pp. 67-76. North-Holland, Amster- dam, 1971.  ... 

The Stable Models of a Predicate Logic Program

V. Wiktor Marek, Anil Nerode, Jeffrey B. Remmel
1994 The Journal of Logic Programming  
Statement of problems and results In this paper we investigate and solve the problem classifying the Turing complexity of stable models of finite and recursive predicate logic programs.  ...  Here we show that, up to a recursive 1-1 coding, the set of all stable models of finite Predicate Logic Programs and the Π 0 1 classes (equivalently, the set of all infinite branches of recursive trees  ...  Acknowledgements This research was partially supported by NSF grants IRI 9012902, DMS-8902797, DMS-8702473 and ARO contract DAAG629-85-C-0018.  ... 
doi:10.1016/s0743-1066(14)80008-3 fatcat:ukrtmiky6ngvnasnonojeox4ye

Indexed linear logic and higher-order model checking

Charles Grellois, Paul-André Melliès
2015 Electronic Proceedings in Theoretical Computer Science  
and linear logic.  ...  recursion scheme G in an appropriate intersection type system associated to the automaton A.  ...  , and thus higher-order recursion schemes.  ... 
doi:10.4204/eptcs.177.4 fatcat:76shpi2nn5ckphb4myagjd54b4

Inductive Synthesis of Logic Programs by Composition of Combinatory Program Schemes [chapter]

Andreas Hamfelt, Jørgen Fischer Nilsson
1999 Lecture Notes in Computer Science  
The combinators comprise fold combinators for recursion on lists.  ...  Based on a variable-free combinatory form of definite clause logic programs we outline a methodology and supporting program environment CombInduce for inducing well-moded logic programs from examples.  ...  In [8] we prove that logic program recursion with the fold schemes suffices for expressing all partial and general recursive functions.  ... 
doi:10.1007/3-540-48958-4_8 fatcat:ndvfpzjkb5gkxgzeujadumtr34

A Framework of Variant Logic Construction for Cellular Automata [chapter]

Jeffrey Z.J., Christian H.H., Tosiyasu L.
2011 Cellular Automata - Innovative Modelling for Science and Engineering  
All block matrices have reflection symmetric distributions via horizontal and vertical reflection symmetry distributions.  ...  There are 16 blocks in a comprised matrix in which thin lines separate different recursive images as blocks and each block under a function recursively, thick lines separate four 2x2 blocks and each 2x2  ... 
doi:10.5772/15400 fatcat:pazox26tj5hzbehif6qlqbxv64

Page 6651 of Mathematical Reviews Vol. , Issue 90K [page]

1990 Mathematical Reviews  
Another model, a natural recursion-theoretic model, is constructed, in which only recursive operations from integers to integers are represented, even though the logic can be classical.  ...  The semantics is continuation-based, and so reflects the order in which expressions are evaluated. It provides a means by which lazy functional languages can be better understood.”  ... 
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