Closed and Logical Relations for Over- and Under-Approximation of Powersets

Abstract

Abstract. We redevelop and extend Dams's results on over-and underapproximation with higher-order Galois connections: (1) We show how Galois connections are generated from U-GLB-L-LUBclosed binary relations, and we apply them to lower and upper powerset constructions, which are weaker forms of powerdomains appropriate for abstraction studies.(2) We use the powerset types within a family of logical relations, show when the logical relations preserve U-GLB-L-LUB-closure, and show that simulation is a logical relation. We use the logical relations to rebuild Dams's most-precise simulations, revealing the inner structure of overand under-approximation. (3) We extract validation and refutation logics from the logical relations, state their resemblance to Hennessey-Milner logic and description logic, and obtain easy proofs of soundness and best precision.Almost all Galois-connection-based static analyses are over-approximating: For Galois connection, (P(C), ⊆) α o , γ (A, A ), an abstract value a ∈ A proclaims a property of all the outputs of a program. For example, even ∈ Parity (see Figure 2 for the abstract domain Parity) asserts, "∀even" -all the program's outputs are even numbers, that is, the output is a set from {S ∈ P(Nat) | S ⊆ γ(even)}.An under-approximating Galois connection, (P(C), ⊇) α u , γ A op , where A op = (A, A ), is the dual. Here, even ∈ Parity op asserts that all even numbers are included in the program's outputs -a strong assertion. Also, we may reuse γ : A → P(C) as the upper adjoint from A op to P(C) op iff γ preserves joins in (A, A ) -another strong demand.Fortunately, there is an alternative view of under-approximation: a ∈ A op asserts an existential property -there exists an output with property a. For example, even ∈ Parity op asserts "∃even" -there is an even number in the program's outputs, which is a set from {S ∈ P(Nat) | S ∩ γ(even) = ∅}. Now, we can generalize both over-and under-approximation to multiple properties, e.g., ∀{even, odd } ≡ ∀(even ∨ odd ) -all outputs are even-or odd-valued; and ∃{even, odd } ≡ ∃even ∧ ∃odd -the output set includes an even value and an odd value. These examples "lift" A and A op into the powerset lattices, P L (A) and P U (A), respectively, and set the stage for the problem studied in this paper.schmidt@cis.ksu.edu.

Systematic Construction of Abstractions for Model-Checking

“…Among the attempts of using AI to systematically derive best abstractions, the work of Loiseaux et al [22] and Schmidt [23] are the closest to ours. [22] showed how to derive a simulation-based sound abstract transition system from Galois connections within the AI framework, but their results apply only to the universal fragment of L µ .…”

“…[22] showed how to derive a simulation-based sound abstract transition system from Galois connections within the AI framework, but their results apply only to the universal fragment of L µ . Motivated by the study of MixTSs, [23] showed how to capture over-and underapproximations between transition systems using AI and systematically derived Dams's most precise results. However, the starting goal of this work was formalizing the overand the under-approximations, restricting the result to the specific L µ models, namely, transition systems.…”

“…The approaches of [8,23] develop best over-and under-approximations separately, whereas our combination of AI with bilattices provides a uniform way for abstraction of both satisfaction and refutation of L µ . Multi-valued logic has been previously combined with abstraction in the form of 3-valued transition systems (e.g.…”

Behavioural model fusion

“…K 1 ), and then lifted to the merge. Thus, in the merged model, we may have must transitions that are distinct from may transitions because the successors of must transitions are less refined than those of the may transitions (see (Schmidt, 2004), , (Wei et al, 2008)). …”

3-Valued Abstraction: More Precision at Less Cost

“…Precision of modal (or mixed) transition systems, with ordinary may and must transitions, is studied in [4,7,21]. They suggest constructions of such abstract models which are most precise among all models from this specific class.…”