Abstract

We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the big-step semantics, termination and nontermination semantics, Plotkin's natural, Smyth's demoniac and Hoare's angelic relational semantics and equivalent nondeterministic denotational semantics (with alternative powerdomains to the Egli-Milner and Smyth constructions), D. Scott's deterministic denotational semantics, the generalized and Dijkstra's conservative/liberal predicate transformer semantics, the generalized/total and Hoare's partial correctness axiomatic semantics and the corresponding proof methods. All the semantics are presented in a uniform fixpoint form and the correspondences between these semantics are established through composable Galois connections, each semantics being formally calculated by abstract interpretation of a more concrete one using Kleene and/or Tarski fixpoint approximation transfer theorems.

Citations

1635	Abstract interpretation: A unified lattice model for static analysis of programs by construction or approximation of fixpoints - Cousot, Cousot - 1977
1219	An axiomatic basis for computer programming - Hoare - 1969
1182	A structural approach to operational semantics - Plotkin - 1981
544	Systematic design of program analysis frameworks - Cousot, Cousot - 1979
504	Assigning meanings to programs - FLOYD - 1967
402	Domain Theory, in - Abramsky, Jung - 1994
363	Guarded commands, nondeterminacy and formal derivation of programs - Dijkstra - 1975
280	Natural semantics - Kahn - 1988
215	Predicate Calculus and Program Semantics - Dijkstra, Scholten - 1990
178	Outline of a mathematical theory of computation - Scott - 1970
169	Temporal abstract interpretation - Cousot, Cousot - 2000
136	Semantic Domains - Gunter, Scott - 1990
118	Operational and algebraic semantics of concurrent processes - Milner - 1990
116	A generalization of Dijkstra’s calculus - Nelson - 1989
91	Inductive definitions, semantics and abstract interpretation - Cousot, Cousot - 1992
84	Semantic foundations of program analysis - Cousot - 1981
78	Methodes iteratives de construction et d'approximation de points xes d'operateurs monotones sur un treillis, analyse semantique des programmes - Cousot - 1978
74	Co-induction in relational semantics - Milner, Tofte - 1991
73	Countable nondeterminism and random assignment - Apt, Plotkin - 1986
67	Methods and logics for proving programs - Cousot - 1990
60	Higherorder abstract interpretation (and application to comportment analysis generalizing strictness, termination, projection and PER analysis of functional languages), invited paper - Cousot, Cousot - 1994
57	On the semantics of fair parallelism - Park - 1980
44	Game semantics - Abramsky, McCusker - 1999
40	Constructive versions of Tarski’s fixed point theorems - Cousot, Cousot - 1979
36	Proof of algorithms by general snapshots - Naur - 1966
30	Operational semantics and program equivalence - Pitts - 2002
30	The closure operators of a lattice - Ward - 1942
24	The lattice of flow diagrams - Scott
23	Compositional and inductive semantic definitions in fixpoint, equational, constraint, closure-condition, rule-based and game-theoretic form, invited paper - Cousot, Cousot - 1995
18	The linear time-branching time spectrum (Extended abstract - Glabbeek
16	The structure of Galois connections - Shmuely - 1974
13	Abstract interpretation. Symposium on Models of Programming Languages - Cousot - 1996
10	A continuous semantics for unbounded nondeterminism - Back - 1983
10	Metric semantics from partial order semantics - Baier, Majster-Cederbaum - 1997
8	Semantics of nondeterministic and noncontinuous constructs - Broy, Gnatz, et al. - 1979
8	On infinite computations in denotational semantics - Bakker, Meyer, et al. - 1983
7	Formal computations of non deterministic recursive program schemes - Arnold, Nivat - 1980
6	Induction principles for proving invariance properties of programs - Cousot, Cousot
6	A simple fixpoint argument without the restriction to continuity - Dijkstra - 1986
6	Axiomatic approach to total correctness - Manna, Pnueli - 1974
3	General correctness: A unification of partial and total correctness - Jacobs, Gries - 1985
3	A simple relation between relational and predicate transformer semantics for nondeterministic programs - Majster-Cederbaum - 1980
2	Guarded commands, nondeterminacy and formal derivation of programs - Dijkstra - 1975
1	s 2 \Sigma g thus allowing the program slice with respect to variable X to be X:=1 with semantics fs s [X / 1] j s 2 \Sigma g. Slicing would not be consistent when considering the trace fs ! j s 2 \Sigma g or denotational semantics s \Delta ? of the progr - s
1	Can fair choice be added to Dijkstra’s calculus - Broy, Nelson - 1994
1	On infinite computations in denotational semantics - Zucker - 1983
1	induction and proofs of program properties - Fixpoint - 1969
1	Can fair choice be added to Dijkstra’s calculus - Broy, Nelson - 1994
1	A simple fixpoint argument without the restriction to continuity - Dijkstra, Gasteren - 1986
1	A generalization of Dijkstra’s calculus - Nelson - 1989