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A Proof Calculus for Natural Semantics Based on Greatest Fixed Point Semantics

Sabine Glesner
2005 Electronical Notes in Theoretical Computer Science  
This requirement is essential in correctness proofs for compilers. We show that a greatest fixed point interpretation of natural semantics is able to model both aspects equally well.  ...  Furthermore, we develop a proof calculus based on it and demonstrate its application for two typical problems.  ...  Acknowledgement The author would like to thank the anonymous reviewers for their helpful comments.  ... 
doi:10.1016/j.entcs.2005.02.011 fatcat:ukosrgfnefc43ksz4pziwbdxoq

Special Issue: Fixpoints in Computer Science 2000

2002 Journal of Logic and Computation  
As a consequence of our proof, we obtain that there is no finite base temporal logic which is expressively complete for the -calculus. -Calculus with Explicit Points and Approximations M. Dam and D.  ...  Gurov We present a Gentzen-style sequent calculus for program verification which accommodates both model checkinglike verification based on global state space exploration, and compositional reasoning.  ...  As a consequence of our proof, we obtain that there is no finite base temporal logic which is expressively complete for the -calculus. -Calculus with Explicit Points and Approximations M.  ... 
doi:10.1093/logcom/12.1.211 fatcat:yowegpu725gxpbpi7eabhc6caa

Page 3496 of Mathematical Reviews Vol. , Issue 94f [page]

1994 Mathematical Reviews  
A soundness and adequacy theorem is proved, showing the equivalence of the fixed point semantics to a transition-based one.  ...  points (vs. greatest, or others) is given: the least fixed point operator is the only uniform fixed point operator.  ... 

Page 1818 of Mathematical Reviews Vol. 56, Issue 5 [page]

1978 Mathematical Reviews  
Since there are programs for which there exists no greatest fixed point or there exist more than one maximal fixed point, the optimal fixed point is suggested as a good choice.  ...  A natural correspondence is established between the semantics of programming languages and first-order predicate calculus, whereby so-called operational semantics is viewed as syntactic, proof-theoretic  ... 

Page 3443 of Mathematical Reviews Vol. , Issue 99e [page]

1999 Mathematical Reviews  
We propose a semantics for Prolog programs based on a four-valued logic.  ...  The semantics of BUMHFs is characterized in terms of fixed points and canonical models, and an efficient proof method is presented as its operational semantics based on SLD-resolution with constraints.  ... 

Infinite Computation, Co-induction and Computational Logic [chapter]

Gopal Gupta, Neda Saeedloei, Brian DeVries, Richard Min, Kyle Marple, Feliks Kluźniak
2011 Lecture Notes in Computer Science  
as a solver for the full µ-calculus, as it permits unrestricted nesting of least and greatest fixed point operators.  ...  An operational semantics-similar to SLD resolutionwas given for computing those answers to a query that are in the greatest fixed point of a logic program (the semantics is discussed below).  ... 
doi:10.1007/978-3-642-22944-2_4 fatcat:v2klxrhr6vfpfagwncfzpg56v4

A Note on Negative Tagging for Least Fixed-Point Formulae

Dilian Gurov, Bruce Kapron
1999 RAIRO - Theoretical Informatics and Applications  
Proof systems with sequents of the form U Φ for proving validity of a propositional modal µ-calculus formula Φ over a set U of states in a given model usually handle fixed-point formulae through unfolding  ...  Tagging is a technique originated by Winskel for annotating fixed-point formulae with information about the proof states at which these are unfolded.  ...  One therefore needs conditions for terminating the proof search process based on identifying certain "loops" in a proof.  ... 
doi:10.1051/ita:1999124 fatcat:ysa35jb42vbu5dwwramabdkomm

A Weighted μ-Calculus on Words [chapter]

Ingmar Meinecke
2009 Lecture Notes in Computer Science  
We define a weighted μ-calculus on finite and infinite words.  ...  For important semirings like distributive complete lattices, the tropical and the probabilistic semiring, we show that the formulas of the conjunction-free weighted μ-calculus define exactly the class  ...  The author would like to thank Paul Gastin and Manfred Droste for some thought-provoking impulses.  ... 
doi:10.1007/978-3-642-02737-6_31 fatcat:y3ns2jgygnbs5kbyclxjvonxpm

UTP by Example: Designs [chapter]

Jim Woodcock, Simon Foster
2017 Lecture Notes in Computer Science  
First, we give a simple relational semantics that accounts for a theory of partial correctness.  ...  Second, we give a semantics based on the theory of precondition-postcondition pairs, known in UTP as designs. This paper should be read in conjunction with the UTP book by Hoare & He.  ...  The weakest precondition calculus is based on this idea: it fixes the program Q and a postcondition r and provides the weakest solution for p.  ... 
doi:10.1007/978-3-319-56841-6_2 fatcat:ca465gaivzcxfj2xre6ymynkii

A Calculus of Circular Proofs and its Categorical Semantics

Luigi Santocanale
2001 BRICS Report Series  
Fixed point calculi, mu-calculi.<br />Bicompletion of categories. Models of interactive computation.</p>  ...  To each proof of the calculus we associate<br />a system of equations which has a meaning in every mu-bicomplete<br />category.  ...  Introduction In this paper we introduce a calculus of proofs for a simple fixed point logic.  ... 
doi:10.7146/brics.v8i15.20472 fatcat:p74h26e4j5hqbktjhxfhx6a5cm

Preface

Mariangiola Dezani-Ciancaglini, Mitsu Okada, Masako Takahashi
2002 Theoretical Computer Science  
Least and Greatest Fixed-Points in Intuitionistic Natural Deduction by Tarmo Uustalu and Varmo Vene is a comparative study of a number of (intensionalsemantically distinct) least and greatest ÿxed-point  ...  Preface This special volume of Theoretical Computer Science is based on a selection of papers presented at, or inspired by the two week MSJ regional Workshop on Theories of Types and Proofs held in Tokyo  ... 
doi:10.1016/s0304-3975(00)00345-5 fatcat:bc3qw2ifpncfvdxt3qhyvcoue4

A Calculus of Circular Proofs and Its Categorical Semantics [chapter]

Luigi Santocanale
2002 Lecture Notes in Computer Science  
The main challenge in developing a theory for the calculus is to define the semantics of proofs, since the usual method by induction on the structure is not available.  ...  We present a calculus of "circular proofs": the graph underlying a proof is not a finite tree but instead it is allowed to contain a certain amount of cycles.  ...  Introduction In this paper we introduce a calculus of proofs for a simple fixed point logic.  ... 
doi:10.1007/3-540-45931-6_25 fatcat:3ifiiwwjmzgujneci4s2vg7alm

Data flow analysis is model checking of abstract interpretations

David A. Schmidt
1998 Proceedings of the 25th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '98  
In particular, the classic %ow equations for bit-vector-based d-Jo. s reformat trivially into modal mu-Cal&us formulas., A surprising consequence is that two of the classical d&a. s are exposed as unsound  ...  Kozen's modal mu-calculus to express trace properties, we express in simplest possible terms that a d&a. is a model check of a program's a.i. trace.  ...  with greatest-fixed point operators, because the initial approximations for least-fixed point equations are "false" (empty sets) and the initial approximations for greatest-fixed point equations are "  ... 
doi:10.1145/268946.268950 dblp:conf/popl/Schmidt98 fatcat:2muvgfeo6bhvva6gnqkukobs7e

Inf-datalog, Modal Logic and Complexities

Eugénie Foustoucos, Irène Guessarian
2007 RAIRO - Theoretical Informatics and Applications  
Inf-Datalog extends the usual least fixpoint semantics of Datalog with greatest fixpoint semantics: we defined inf-Datalog and characterized the expressive power of various fragments of inf-Datalog in  ...  We deduce a unified and elementary proof that global model-checking (computing all nodes satisfying a formula in a given structure) has 1. quadratic data complexity in time and linear program complexity  ...  Same proof if all IDBs are tagged (computed as greatest fixed points).  ... 
doi:10.1051/ita:2007043 fatcat:4lvx2m6mafbfbke2nf5hwq3pdm

Induction and recursion on datatypes [chapter]

Henk Doornbos, Roland Backhouse
1995 Lecture Notes in Computer Science  
greatest fixed point of d B · (e S + r S ;(-)).  ...  or T (x) = wlp S (x) · wlp T (x) and wp S or T (x) = wp S (x) · wp T (x), (h) wlp while B do S (x) is the greatest fixed point of d B ·x+d B ·wlp S (-) and wp while B do S (x) is the least fixed point  ...  The use of this algebra is illustrated by derivations of a shortest path, a hamiltonian circuits, and a sorting algorithm. All derivations are formal, understandable and concise.  ... 
doi:10.1007/3-540-60117-1_14 fatcat:namzjvik6vapne363qzzvaqdlu
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