Verifying Process Algebra Proofs in Type Theory

Abstract

In this paper we study automatic veri cation of proofs in process algebra. Formulas of process algebra are represented by t ypes in typed-calculus. Inhabitants (terms) of these types represent proofs. The speci c typed-calculus we use is the Calculus of Inductive Constructions as implemented in the interactive p r o o f construction program COQ. Axiom A1. Assumes <proc>(alt x y)=(alt y x). Axiom A2. Assumes <proc>(alt x (alt y z))=(alt (alt x y) z). Axiom A3. Assumes <proc>x=(alt x x). Axiom A4. Assumes <proc>(alt (seq x z) (seq y z))=(seq (alt x y) z). Axiom A5. Assumes <proc>(seq x (seq y z))=(seq (seq x y) z). Axiom A6. Assumes <proc>x=(alt x Delta). Axiom A7. Assumes <proc>Delta=(seq Delta x). Axiom CM1. Assumes <proc>(alt (alt (Lmer x y) (Lmer y x)) (comm x y))= (mer x y). Axiom CM2. Assumes <proc>(seq (ia D a d) x)=(Lmer (ia D a d) x). Axiom CM3. Assumes <proc>(seq (ia D a d) (mer x y))= (Lmer (seq (ia D a d) x) y). Axiom CM4. Assumes <proc>(alt (Lmer x z) (Lmer y z))=(Lmer (alt x y) z). Axiom CM5. Assumes <proc>(seq(comm (ia D a d) (ia E b e)) x)= (comm (seq (ia D a d) x) (ia E b e)). Axiom CM6. Assumes <proc>(seq (comm (ia D a d) (ia E b e)) x)= (comm (ia D a d) (seq (ia E b e) x)). Axiom CM7. Assumes <proc>(seq (comm (ia D a d) (ia E b e)) (mer x y))= (comm (seq (ia D a d) x) (seq (ia E b e) y)). Axiom CM8. Assumes <proc>(alt (comm x z) (comm y z))=(comm (alt x y) z). Axiom CM9. Assumes <proc>(alt (comm x y) (comm x z))=(comm x (alt y z)). Axiom D1. Assumes ((In ehlist a L))-><proc>(ia D a d)= (enc L (ia D a d)). Axiom D2. Assumes (In ehlist a L)-><proc>Delta=(enc L (ia D a d)). Axiom D3. Assumes <proc>(alt (enc L x) (enc L y))=(enc L (alt x y)). Axiom D4. Assumes <proc>(seq (enc L x) (enc L y))=(enc L (seq x y)). End ACP.

Definitions by rewriting in the Calculus of Constructions

“…For example, this includes the type ord of Brouwer's ordinals whose constructors are 0 : ord, s : ord ⇒ ord and lim : (nat ⇒ ord) ⇒ ord, the process algebra µCRL which can be formalised as a type proc with a choice operator Σ : (data ⇒ proc) ⇒ proc (Sellink 1993), or the type form of the formulas of first-order predicate calculus whose constructors are ¬ : form ⇒ form, ∨ : form ⇒ form ⇒ form and ∀ : (term ⇒ form) ⇒ form. For example, this includes the type ord of Brouwer's ordinals whose constructors are 0 : ord, s : ord ⇒ ord and lim : (nat ⇒ ord) ⇒ ord, the process algebra µCRL which can be formalised as a type proc with a choice operator Σ : (data ⇒ proc) ⇒ proc (Sellink 1993), or the type form of the formulas of first-order predicate calculus whose constructors are ¬ : form ⇒ form, ∨ : form ⇒ form ⇒ form and ∀ : (term ⇒ form) ⇒ form.…”

Example Verifications Using Alphabet Axioms

“…does not rely on auxiliary machinery (meta-theory) such as implicit set theory and α/β-calculus [BBK87]. This means that the new proof can be checked by a verification tool (like Coq) in a straightforward way (see [CoH88,Sel93]). Coq is a proof tool, based on type theory, which has been used for checking a number of µCRL proofs (see [BBG95,Gvd93,KoS94]).…”

“…sum operator) and implicit set theory. Second, the our proof given below is ready to be proof-checked conform standard methods (see [CoH88,Sel93]). For a computer-checked verification involving a part of the alphabet axioms given in this paper, one is referred to [KoS93].…”

“…See [Sel93] or [Sel96] for more information on using type theory for protocol verification. Examples of using Coq for protocol verification can also be found in [BBG95,Gvd93,KoS93,KoS94].…”

Size-based termination of higher-order rewriting

“…However, when considering type constructors taking functions as arguments (e.g. Sellink's model of µCRL (Sellink, 1993), Howard's constructive ordinals in Example 5), the size of a term is generally not a finite natural number but a transfinite ordinal number. However, abstract size expressions can also handle transfinite sizes.…”