Weak-Exact Equivalence in PEPA & CTMC Analysis

• Weak-exact equivalence is a behavioral equivalence in PEPA that relaxes the matching of internal τ actions while exactly preserving quantitative constraints for visible actions.
• It ensures that the underlying CTMC exhibits exact rate equivalence, making equivalent states equiprobable in the stationary distribution.
• This approach offers a stricter performance guarantee for stochastic non-interference compared to lumpable bisimulation, aiding security analysis.

Weak-exact equivalence is a behavioral equivalence for PEPA components introduced as part of a performance-oriented reformulation of Persistent Stochastic Non-Interference. In that setting, it extends exact equivalence with a relaxed treatment of internal actions $\tau$, while preserving exact quantitative constraints on visible behavior and inducing exact equivalence on the underlying continuous-time Markov chain (CTMC). Its purpose is to retain the incoming-rate-oriented precision of exact equivalence without letting unobservable internal structure block non-interference arguments (Piazza et al., 26 Aug 2025).

1. Formal setting and motivation

Weak-exact equivalence is defined in the setting of PEPA, where an activity is a pair $(\alpha,r)$, with $\alpha\in\mathcal A$ an action type and $r$ an exponential rate. The special action $\tau$ is unobservable. A finite complete PEPA model induces a CTMC with transition rates

$q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$

and conditional transition rates

$$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$

The paper also uses aggregated rates

$$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$

together with the total outgoing rate for action $\alpha$, written $q[P,\alpha]$ in the exact-equivalence definitions (Piazza et al., 26 Aug 2025).

The motivating background is exact equivalence at CTMC level. Exact equivalence is described as an incoming-rate-oriented CTMC equivalence: unlike lumpable or strong views, which compare outgoing rates to classes, exact equivalence constrains rates entering equivalent states. This makes it quantitatively tighter, but also too rigid for security settings in which internal $(\alpha,r)$0-actions are intended to be unobservable. The paper states that “the presence of internal ($(\alpha,r)$1) actions—which are intended to be unobservable—clashes with the strict matching requirements of exact equivalence.” Weak-exact equivalence is therefore designed to preserve the exact quantitative discipline on visible behavior while weakening only the treatment of internal actions (Piazza et al., 26 Aug 2025).

2. Definition and semantic content

The starting point is exact equivalence on PEPA components. An equivalence relation $(\alpha,r)$2 is an exact equivalence if, whenever $(\alpha,r)$3, then for all $(\alpha,r)$4 and all $(\alpha,r)$5,

$(\alpha,r)$6

and

$(\alpha,r)$7

Thus exact equivalence requires equality of total outgoing rates for each action type and equality of total incoming rates from each equivalence class (Piazza et al., 26 Aug 2025).

Weak-exact equivalence keeps those requirements unchanged for visible actions and alters only the $(\alpha,r)$8-case. An equivalence relation $(\alpha,r)$9 is a weak-exact equivalence if, whenever $\alpha\in\mathcal A$0, then for all $\alpha\in\mathcal A$1,

• if $\alpha\in\mathcal A$2, then

$\alpha\in\mathcal A$3

and for all $\alpha\in\mathcal A$4,

$\alpha\in\mathcal A$5

• if $\alpha\in\mathcal A$6, then for every class $\alpha\in\mathcal A$7 with $\alpha\in\mathcal A$8,

$\alpha\in\mathcal A$9

while for the class $r$0 containing $r$1,

$r$2

Two PEPA components $r$3 and $r$4 are weak-exact equivalent, written $r$5, if $r$6 for some weak-exact equivalence $r$7 (Piazza et al., 26 Aug 2025).

This definition is weak because $r$8-behavior is treated specially: incoming $r$9-rates from classes other than the current class must still match exactly, but within the current class self/internal $\tau$0-contributions are discounted through the difference form

$\tau$1

It remains exact because for every visible action $\tau$2, the exact-equivalence constraints are preserved unchanged, and even for $\tau$3 the comparison is still phrased as precise CTMC rate equalities. The paper is explicit that this is not “weak” in the Milner-style path-compression sense; it is a rate-exact equivalence with weakened local accounting for internal actions (Piazza et al., 26 Aug 2025).

3. Relation to exact equivalence and lumpable bisimulation

Weak-exact equivalence is best understood against two neighboring relations: exact equivalence and lumpable bisimulation. Exact equivalence is finer, because weak-exact equivalence differs from it only by relaxing the $\tau$4-case. Lumpable bisimulation, by contrast, is outgoing-class-oriented. An equivalence relation $\tau$5 is a lumpable bisimulation if, whenever $\tau$6, then for all $\tau$7 and all $\tau$8 such that either $\tau$9, or $q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$0 and $q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$1, it holds that

$q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$2

The paper emphasizes the contrast succinctly: “strong equivalence compares the rates leaving a state, whereas exact equivalence focuses on the rates entering a state” (Piazza et al., 26 Aug 2025).

The following comparison captures the distinction.

Relation	Quantitative focus	Treatment of $q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$3
Exact equivalence	Outgoing rate per action and incoming rate from each class	No relaxation
Lumpable bisimulation	Outgoing rates to classes	No matching of $q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$4-moves into the current class
Weak-exact equivalence	Outgoing rate per action and incoming rate from each class	Current-class $q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$5-accounting weakened by subtracting $q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$6

Weak-exact equivalence is therefore not a variant of lumpable bisimilarity. The two relations preserve different observables, and the paper does not prove a general inclusion either way between $q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$7 and lumpable bisimilarity. A common misconception is to read weak-exact equivalence as merely “exact equivalence with hidden steps ignored.” The definition is narrower and more quantitative: visible actions retain exact-equivalence constraints, while only local accounting for internal actions is relaxed (Piazza et al., 26 Aug 2025).

4. Role in stochastic non-interference

The principal application is to non-interference in stochastic process algebra. The action set is partitioned as

$q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$8

where $q(P_i,P_j)=\sum_{a\in \mathcal A(P_i\mid P_j)} r_a$9 and $$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$0 are disjoint high and low sets. A high-level component $$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$1 can perform only actions in $$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$2. Original Persistent Stochastic Non-Interference (PSNI) was defined through lumpable bisimilarity. The new framework replaces that relation with weak-exact equivalence and introduces Exact SNI (ESNI) and Exact Persistent Stochastic Non-Interference (EPSNI) (Piazza et al., 26 Aug 2025).

The definitions are

$$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$3

and

$$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$4

Using the restriction notation $$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$5 for $$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$6, this becomes

$$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$7

The significance of this replacement is quantitative. The paper states that weak-exact equivalence induces an exact equivalence on the underlying Markov chain of the system. Therefore states in the same $$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$8-class are equiprobable in the stationary distribution, and a low-level observer cannot distinguish whether high interaction is present by looking at those quantitative observables. This suggests a stricter performance-oriented guarantee than the original lumpability-based PSNI formulation, while still abstracting from hidden or high activity after hiding (Piazza et al., 26 Aug 2025).

5. Characterizations and structural results

The paper proves that EPSNI admits the same style of characterizations as PSNI. First, EPSNI is invariant under weak-exact equivalence: $$q(P_i,P_j,\alpha)=\sum_{P_i \xrightarrow{(\alpha,r_\alpha)} P_j} r_\alpha.$$9 Second, it gives a contextual characterization by introducing weak-exact equivalence on high contexts. For a high context $$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$0, the rates

$$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$1

are used to define $$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$2, and the main theorem states

$$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$3

This absorbs persistent quantification over all reachable states into a bisimulation-style contextual condition (Piazza et al., 26 Aug 2025).

A further local characterization treats high actions like $$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$4. Weak-exact equivalence up to $$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$5, written $$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$6, uses the exact-equivalence clauses for $$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$7 and the weakened current-class accounting for $$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$8. The paper proves

$$q[P,S,\alpha]=\sum_{P'\in S} q(P,P',\alpha), \qquad q[S,P,\alpha]=\sum_{P'\in S} q(P',P,\alpha),$$9

and hence

$\alpha$0

The intended unwinding-style consequence is also stated: whenever a reachable state $\alpha$1 may execute a high-level activity leading to $\alpha$2, then $\alpha$3 and $\alpha$4 are indistinguishable in the sense that both

$\alpha$5

hold (Piazza et al., 26 Aug 2025).

The paper also establishes compositionality properties. For ordinary weak-exact equivalence,

$\alpha$6

For the high-insensitive variant, if $\alpha$7, then

$\alpha$8

For EPSNI itself, if $\alpha$9, then

$q[P,\alpha]$0

The principal limitation is equally explicit: unlike the original PSNI setting, weak-exact equivalence is not compositional for prefix (Piazza et al., 26 Aug 2025).

6. Quantitative interpretation, limits, and related terminology

Weak-exact equivalence is an aggregation condition over equivalence classes. It preserves visible action rates exactly, preserves incoming rates from each class exactly, and allows $q[P,\alpha]$1-rates to vary only inside the current class in a controlled difference form. Because PEPA resolves nondeterministic choice through race policy, the semantic object is a CTMC, not an MDP, and no scheduler machinery is involved. For finite irreducible models, the stationary distribution $q[P,\alpha]$2 satisfies

$q[P,\alpha]$3

The paper’s key quantitative claim is that weak-exact equivalence induces exact CTMC equivalence, hence equivalent states are equiprobable in steady state (Piazza et al., 26 Aug 2025).

At the same time, the scope of the notion is specific. It is a PEPA-level equivalence introduced for stochastic security, not a general-purpose weak behavioral equivalence across process calculi. It does not collapse to lumpable bisimulation, and it is not presented as a generic weak-step semantics. A plausible implication is that its natural domain is analysis in which incoming-rate exactness and stationary quantitative observables matter more than path-compression intuitions.

Related but distinct terminology appears in neighboring areas. In exact categories with weak equivalences, “quasi-weak equivalences” are introduced as the chain-complex weak equivalences whose cones are $q[P,\alpha]$4-acyclic; the paper describes them as a “weak-exact” analogue of quasi-isomorphism (Hiranouchi et al., 2010). In integrated-time Markovian process algebra, weak Markovian bisimulation is developed to obtain exact CTMC-level aggregation at steady state, but there the relevant term is not weak-exact equivalence and the preservation target is steady-state exactness under abstraction of exponentially timed internal actions (Bernardo, 2012). This suggests that across fields, closely related phrases tend to name constructions that relax a strong equivalence while preserving a specifically chosen exact invariant.