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Dynamic Logic
 Handbook of Philosophical Logic
, 1984
"... ed to be true under the valuation u iff there exists an a 2 N such that the formula x = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This definition involves a metalogical operation that produces u[x=a] from u for all possibl ..."
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Cited by 823 (7 self)
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ed to be true under the valuation u iff there exists an a 2 N such that the formula x = y is true under the valuation u[x=a], where u[x=a] agrees with u everywhere except x, on which it takes the value a. This definition involves a metalogical operation that produces u[x=a] from u for all possible values a 2 N. This operation becomes explicit in DL in the form of the program x := ?, called a nondeterministic or wildcard assignment. This is a rather unconventional program, since it is not effective; however, it is quite useful as a descriptive tool. A more conventional way to obtain a square root of y, if it exists, would be the program x := 0 ; while x < y do x := x + 1: (1) In DL, such programs are firstclass objects on a par with formulas, complete with a collection of operators for forming compound programs inductively from a basis of primitive programs. To discuss the effect of the execution of a program on the truth of a formula ', DL uses a modal construct <>', which
Local Reasoning about Programs that Alter Data Structures
, 2001
"... . We describe an extension of Hoare's logic for reasoning about programs that alter data structures. We consider a lowlevel storage model based on a heap with associated lookup, update, allocation and deallocation operations, and unrestricted address arithmetic. The assertion language is based ..."
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Cited by 269 (27 self)
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. We describe an extension of Hoare's logic for reasoning about programs that alter data structures. We consider a lowlevel storage model based on a heap with associated lookup, update, allocation and deallocation operations, and unrestricted address arithmetic. The assertion language is based on a possible worlds model of the logic of bunched implications, and includes spatial conjunction and implication connectives alongside those of classical logic. Heap operations are axiomatized using what we call the \small axioms", each of which mentions only those cells accessed by a particular command. Through these and a number of examples we show that the formalism supports local reasoning: A speci cation and proof can concentrate on only those cells in memory that a program accesses. This paper builds on earlier work by Burstall, Reynolds, Ishtiaq and O'Hearn on reasoning about data structures. 1
Constructive Design of a Hierarchy of Semantics of a Transition System by Abstract Interpretation
, 2002
"... We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the bigstep semantics, termination and nontermination semantics, Plotkin’s natural, Smyth’s demoniac and Hoare’s angelic relational semantics and ..."
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Cited by 99 (18 self)
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We construct a hierarchy of semantics by successive abstract interpretations. Starting from the maximal trace semantics of a transition system, we derive the bigstep 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 EgliMilner 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
A Congruence for Gamma Programs
 in Proc. of the 5th workshop on Languages and Compilers for Parallel Computing
, 1993
"... . This paper defines a congruence relation on Gamma programs. Based on this congruence relation, laws for transforming programs are derived. We define an axiomatic semantics for Gamma based on Brookes' transition assertions. The definition of the congruence is in terms of provable satisfiability of ..."
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Cited by 43 (10 self)
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. This paper defines a congruence relation on Gamma programs. Based on this congruence relation, laws for transforming programs are derived. We define an axiomatic semantics for Gamma based on Brookes' transition assertions. The definition of the congruence is in terms of provable satisfiability of such assertions. We consider the relationship between our congruence and other orderings that have been proposed in the literature. 1 Introduction In this paper we study the problem of transforming programs expressed in the Gamma programming formalism. We aim at defining a relation between Gamma programs that captures when one program mimics another program. This relation allows us to decide when we can safely replace one program by another i.e., it gives us a means of expressing the correctness of program transformations. Gamma operates with a single data structure, the multiset, and computation proceeds by rewriting multisets of data. Gamma has proved successful in expressing a variety of...
On Hoare Logic and Kleene Algebra with Tests
"... We show that Kleene algebra with tests (KAT) subsumes propositional Hoare logic (PHL). Thus the specialized syntax and deductive apparatus of Hoare logic are inessential and can be replaced by simple equational reasoning. In addition, we show that all relationally valid inference rules are derivable ..."
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Cited by 41 (13 self)
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We show that Kleene algebra with tests (KAT) subsumes propositional Hoare logic (PHL). Thus the specialized syntax and deductive apparatus of Hoare logic are inessential and can be replaced by simple equational reasoning. In addition, we show that all relationally valid inference rules are derivable in KAT and that deciding the relational validity of such rules is PSPACEcomplete.
An observationally complete program logic for imperative higherorder functions
 In Proc. LICS’05
, 2005
"... Abstract. We propose a simple compositional program logic for an imperative extension of callbyvalue PCF, built on Hoare logic and our preceding work on program logics for pure higherorder functions. A systematic use of names and operations on them allows precise and general description of comple ..."
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Cited by 39 (11 self)
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Abstract. We propose a simple compositional program logic for an imperative extension of callbyvalue PCF, built on Hoare logic and our preceding work on program logics for pure higherorder functions. A systematic use of names and operations on them allows precise and general description of complex higherorder imperative behaviour. The proof rules of the logic exactly follow the syntax of the language and can cleanly embed, justify and extend the standard proof rules for total correctness of Hoare logic. The logic offers a foundation for general treatment of aliasing and local state on its basis, with minimal extensions. After establishing soundness, we prove that valid assertions for programs completely characterise their behaviour up to observational congruence, which is proved using a variant of finite canonical forms. The use of the logic is illustrated through reasoning examples which are hard to assert and infer using existing program logics.
Hoare Logic and Auxiliary Variables
 Formal Aspects of Computing
, 1998
"... Auxiliary variables are essential for specifying programs in Hoare Logic. They are required to relate the value of variables in different states. However, the axioms and rules of Hoare Logic turn a blind eye to the rle of auxiliary variables. We stipulate a new structural rule for adjusting auxiliar ..."
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Cited by 38 (0 self)
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Auxiliary variables are essential for specifying programs in Hoare Logic. They are required to relate the value of variables in different states. However, the axioms and rules of Hoare Logic turn a blind eye to the rle of auxiliary variables. We stipulate a new structural rule for adjusting auxiliary variables when strengthening preconditions and weakening postconditions. Courtesy of this new rule, Hoare Logic is adaptation complete, which benefits software reuse. This property is responsible for a number of improvements. Relative completeness follows uniformly from the Most General Formula property. Moreover, contrary to common belief, one can show that Hoare Logic subsumes VDM's operation decomposition rules in that every derivation in VDM can be naturally embedded in Hoare Logic. Furthermore, the new treatment leads to a significant simplification in the presentation for verification calculi dealing with more interesting features such as recursion or concurrency.
The pitfalls of verifying floatingpoint computations
 ACM Transactions on programming languages and systems
"... Current critical systems often use a lot of floatingpoint computations, and thus the testing or static analysis of programs containing floatingpoint operators has become a priority. However, correctly defining the semantics of common implementations of floatingpoint is tricky, because semantics ma ..."
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Cited by 33 (2 self)
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Current critical systems often use a lot of floatingpoint computations, and thus the testing or static analysis of programs containing floatingpoint operators has become a priority. However, correctly defining the semantics of common implementations of floatingpoint is tricky, because semantics may change according to many factors beyond sourcecode level, such as choices made by compilers. We here give concrete examples of problems that can appear and solutions for implementing in analysis software. 1
Hoare Logic and VDM: MachineChecked Soundness and Completeness Proofs
, 1998
"... Investigating soundness and completeness of verification calculi for imperative programming languages is a challenging task. Many incorrect results have been published in the past. We take advantage of the computeraided proof tool LEGO to interactively establish soundness and completeness of both H ..."
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Cited by 31 (1 self)
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Investigating soundness and completeness of verification calculi for imperative programming languages is a challenging task. Many incorrect results have been published in the past. We take advantage of the computeraided proof tool LEGO to interactively establish soundness and completeness of both Hoare Logic and the operation decomposition rules of the Vienna Development Method (VDM) with respect to operational semantics. We deal with parameterless recursive procedures and local variables in the context of total correctness. As a case study, we use LEGO to verify the correctness of Quicksort in Hoare Logic. As our main contribution, we illuminate the rle of auxiliary variables in Hoare Logic. They are required to relate the value of program variables in the final state with the value of program variables in the initial state. In our formalisation, we reflect their purpose by interpreting assertions as relations on states and a domain of auxiliary variables. Furthermore, we propose a new structural rule for adjusting auxiliary variables when strengthening preconditions and weakening postconditions. This rule is stronger than all previously suggested structural rules, including rules of adaptation. With the new treatment, we are able to show that, contrary to common belief, Hoare Logic subsumes VDM in that every derivation in VDM can be naturally embedded in Hoare Logic. Moreover, we establish completeness results uniformly as corollaries of Most General Formula theorems which remove the need to reason about arbitrary assertions.
Abstract interpretation based formal methods and future challenges, invited paper
 Informatics — 10 Years Back, 10 Years Ahead, volume 2000 of Lecture Notes in Computer Science
, 2001
"... Abstract. In order to contribute to the solution of the software reliability problem, tools have been designed to analyze statically the runtime behavior of programs. Because the correctness problem is undecidable, some form of approximation is needed. The purpose of abstract interpretation is to f ..."
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Cited by 29 (6 self)
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Abstract. In order to contribute to the solution of the software reliability problem, tools have been designed to analyze statically the runtime behavior of programs. Because the correctness problem is undecidable, some form of approximation is needed. The purpose of abstract interpretation is to formalize this idea of approximation. We illustrate informally the application of abstraction to the semantics of programming languages as well as to static program analysis. The main point is that in order to reason or compute about a complex system, some information must be lost, that is the observation of executions must be either partial or at a high level of abstraction. In the second part of the paper, we compare static program analysis with deductive methods, modelchecking and type inference. Their foundational ideas are briefly reviewed, and the shortcomings of these four methods are discussed, including when they should be combined. Alternatively, since program debugging is still the main program verification