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

2005
*
Electronical Notes in Theoretical Computer Science
*

This requirement is essential in correctness

doi:10.1016/j.entcs.2005.02.011
fatcat:ukosrgfnefc43ksz4pziwbdxoq
*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. ...##
###
Special Issue: Fixpoints in Computer Science 2000

2002
*
Journal of Logic and Computation
*

As

doi:10.1093/logcom/12.1.211
fatcat:yowegpu725gxpbpi7eabhc6caa
*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. ...##
###
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]

2011
*
Lecture Notes in Computer Science
*

as

doi:10.1007/978-3-642-22944-2_4
fatcat:v2klxrhr6vfpfagwncfzpg56v4
*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). ...##
###
A Note on Negative Tagging for Least Fixed-Point Formulae

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*. ...

##
###
A Weighted μ-Calculus on Words
[chapter]

2009
*
Lecture Notes in Computer Science
*

We define

doi:10.1007/978-3-642-02737-6_31
fatcat:y3ns2jgygnbs5kbyclxjvonxpm
*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. ...##
###
UTP by Example: Designs
[chapter]

2017
*
Lecture Notes in Computer Science
*

First, we give

doi:10.1007/978-3-319-56841-6_2
fatcat:ca465gaivzcxfj2xre6ymynkii
*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. ...##
###
A Calculus of Circular Proofs and its Categorical Semantics

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. ...

##
###
Preface

2002
*
Theoretical Computer Science
*

Least and

doi:10.1016/s0304-3975(00)00345-5
fatcat:bc3qw2ifpncfvdxt3qhyvcoue4
*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 ...##
###
A Calculus of Circular Proofs and Its Categorical Semantics
[chapter]

2002
*
Lecture Notes in Computer Science
*

The main challenge in developing

doi:10.1007/3-540-45931-6_25
fatcat:3ifiiwwjmzgujneci4s2vg7alm
*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. ...##
###
Data flow analysis is model checking of abstract interpretations

1998
*
Proceedings of the 25th ACM SIGPLAN-SIGACT symposium on Principles of programming languages - POPL '98
*

In particular, the classic %ow equations

doi:10.1145/268946.268950
dblp:conf/popl/Schmidt98
fatcat:2muvgfeo6bhvva6gnqkukobs7e
*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 " ...##
###
Inf-datalog, Modal Logic and Complexities

2007
*
RAIRO - Theoretical Informatics and Applications
*

Inf-Datalog extends the usual least fixpoint

doi:10.1051/ita:2007043
fatcat:4lvx2m6mafbfbke2nf5hwq3pdm
*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*). ...##
###
Induction and recursion on datatypes
[chapter]

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. ...

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