Modular Interpreted Systems: A Formal Approach

• Modular Interpreted System (MIS) is a formal framework modeling synchronous multi-agent interactions through modular agent modules and explicit interference interfaces.
• The revised MIS employs multiset-based, nondeterministic interference to support flexible communication (broadcast, multicast, point-to-point) and enhance system openness.
• Quantitative metrics like Interaction Complexity and Global Complexity enable rigorous evaluation of scalability and modularity, facilitating compositional multi-agent system design.

A Modular Interpreted System (MIS) is a formal model for synchronous multi-agent systems, emphasizing modularity and openness by organizing systems as synchronous compositions of agent modules. Each module consists of a local state machine, local actions, and a transition mechanism mediated exclusively through an explicit interference interface, while the global semantics unfolds as a product structure over the modules' local states and a concurrent game-based operational model. The revised MIS formalism explicitly addresses several limitations of prior approaches, enabling compositionality, broadcast/multicast communication, and easy incremental extension (Jamroga et al., 2013).

1. Formal Structure of Modular Interpreted Systems

A Modular Interpreted System is formally defined as a tuple

$$\mathbf{MIS} = \bigl(\mathit{Agtnames},\,\mathit{Act},\,\Lambda,\;\{a_1,\dots,a_k\}\bigr)$$

where:

• $\mathit{Agtnames}$: finite set of agent identifiers,
• $\mathit{Act}$: finite set of local action names,
• $\Lambda$: finite alphabet of interaction symbols (tokens),
• $\{a_1, \dots, a_k\}$: agent modules, each $$a_j = (\mathsf{id}(a_j), \{m_{j,1}, \dots, m_{j,n_j}\})$$ with modules $m$ defined as

$m = (St, Init, d, out, in, o, \Pi, \pi)$

with:

• $St$: finite set of local states,
• $Init \subseteq St$: set of initial states,
• $d: St \to \wp(\mathit{Act})$: enabled local actions per state,
• $out: D \to \wp(\wp(T))$, with $$D = \{(s,\alpha) \mid s \in St,\, \alpha \in d(s)\}$$ and $$T = \Lambda \times (\mathit{Agtnames} \cup \{\epsilon\})$$: possible multisets of emitted tokens,
• $$in: St \times \mathcal{B}(\Lambda) \to \wp(\Lambda)$$: nondeterministically maps state and bag of incoming symbols to impressions,
• $o: D \times \Lambda \to \wp(St)$: (possibly nondeterministic) local transition function,
• $\Pi$: private atomic proposition vocabulary,
• $\pi: \Pi \to \wp(St)$: state labeling.

Inter-module communication is realized solely via $out$ (emission) and $in$ (reception/impression), providing a clear interface between internal computation and external interaction. Global behavior is given by an unfolding into a concurrent epistemic game structure $NCEGS(\mathbf{MIS})$, whose state space is the product of all module states, with joint moves synchronized and interference propagated via the defined interface.

2. Original MIS Formalism and Its Deficiencies

The original MIS framework (Jamroga & Tabatabai 2007) specified deterministic interference structures, requiring each agent module to communicate with every other via fixed-length tuples: $$out_j: D_j \to T^k \quad\text{and}\quad in_j: St_j \times T^k \to \Lambda,$$ where $k$ is the system size. This approach imposed several constraints:

• The number of modules (and agents) was hard-coded, making system extension or reduction cumbersome.
• Broadcast/multicast semantics required redundant encoding.
• No support for nondeterministic or randomized protocols.
• Openness and modularity were compromised, since the addition or removal of agents necessitated rewrites in every module’s interface code. These limitations hindered the generality and scalability of the framework for modeling true multi-agent openness and compositionality (Jamroga et al., 2013).

3. Enhanced Interference Mechanisms

The key improvement involves adopting a multiset-based and nondeterministic definition of interference. Specifically:

• $out_j: D_j \to \wp(\wp(T))$ enables each agent to emit arbitrary multisets of tokens, supporting any mix of point-to-point, broadcast (by using $\epsilon$ as recipient), or multicast communications without direct reference to the system size.
• $$in_j: St_j \times \mathcal{B}(\Lambda) \to \wp(\Lambda)$$ allows agents to pattern-match on the received bag (multiset) of symbols, interpreting incoming interference non-deterministically.

This flexibility eliminates dependency on the total number of modules in both emission and reception, and permits representing protocols with randomized behavior, lossy communication, or dynamic topology. Notably, system designers can write and replicate agent code without modification when the overall system scale changes, except possibly for specialized roles.

4. Benchmark Examples: Trains–Tunnel–Controller and Dining Cryptographers

Two canonical benchmarks illustrate the expressiveness and modularity of the revised MIS:

A. Trains–Tunnel–Controller (TTC$_n$): $n$ identical trains interact with a central controller to enforce mutual exclusion on a shared tunnel. Trains send requests via tokens to the controller, which in turn grants permission. Each train’s module defines states (e.g., $out, \tunneeded, granted, in$), local actions, emission logic (e.g., $(try_i, ctrl)$), and a simple impression function to process controller responses. TTC$_n$ instantiates a mutual-exclusion game with $n+1$ players, with interaction complexity $IC(TTC_n) = O(n)$ while the global complexity $GC(TTC_n)$ grows exponentially in $n$; hence, the system is properly multi-agent.

B. Dining Cryptographers (DC): The broadcast version DC1 employs $n$ cryptographer agents plus two special modules. Each cryptographer shares a coin with neighbors, announces parities, and the $Counter$ XORs the results. The system is $O(1)$-open, as addition of a new cryptographer only requires constant adjustments to the special modules.

Comparison with alternative variants demonstrates that point-to-point versions (DC2) are $O(n)$-open, and fully anonymous versions (DC0) can achieve $O(0)$-open design, evidencing the spectrum of openness depending on communication structure.

5. Quantitative Measures: Interaction Complexity and Openness

Rigorous metrics provide a basis for evaluating modularity and multi-agency:

• Interaction Complexity (IC): For a module $i$, define

$$\#out_i(s) = \max\{|W| \mid W \in out_i(s,\alpha)\ \text{for some}\ \alpha\}, \quad \#in_i(s) = \max\{|H| \mid H \in \mathrm{dom}(in_i)\}$$

with

$$IC(i) = \sum_{s\in St_i}(\#out_i(s) + \#in_i(s)), \quad IC(M) = \sum_{i} IC(i)$$

• Global Complexity (GC): $GC(M)$ is the number of transitions in $NCEGS(M)$.

A class $\mathcal{M}$ has $\mathcal{C}$-sparse interaction if there is $f \in \mathcal{C}$ with $IC(M) \leq f(GC(M))$. Proper multi-agent design is associated with $O(\log)$-sparse interaction for $|Agt(M)| \geq 2$.

• Openness: The openness measure quantifies the minimal syntactic cost (in terms of rewriting steps) to assimilate an added agent and restore the system class, resulting in categories like $O(1)$-open, $O(n)$-open, or $O(0)$-open. The broadcast DC1$_n$ system is $O(1)$-open; point-to-point DC2$_n$ is $O(n)$-open.

These metrics allow for principled assessment of compositionality and scalability.

6. Modeling Advantages and Guidelines

The adoption of multiset-based, nondeterministic interference yields several advantages:

• Modules remain oblivious to the total population, supporting protocol replication and rule-based composition.
• Broadcast, point-to-point, and arbitrary group communication are encoded uniformly.
• System size can be increased or decreased with minimal to zero adjustment, enhancing openness.
• Local verification suffices for a wide range of correctness and information-flow properties due to the explicit interference interface.
• Protocols can be structured so that addition or removal of an agent is a constant or zero-step operation, maximizing openness.

Recommended best practices include modeling logical agents as one or more modules, ensuring all communication passes through the $out$/$in$ interface, exploiting multisets and broadcast where possible, supporting nondeterminism for randomized behavior, using decision lists or guarded commands for infinite $in$-domains, and validating the multi-agent structure via $IC$ versus $GC$ analysis (Jamroga et al., 2013).

7. Significance for Multi-Agent Systems Research

The revised MIS formalism directly supports modular reasoning, incremental design, and rigorous formal analysis (e.g., in ATL, CTL*, or epistemic logics) of multi-agent systems. By ensuring that communication, coordination, and extension are handled at the interference layer, designers can develop scalable, analyzable systems with minimal code coupling. The benchmark examples demonstrate how seemingly minor changes in interference modeling eliminate otherwise non-scalable engineering burdens. Openness and interaction complexity emerge as formal, quantifiable metrics for system design, setting a methodological foundation for principled multi-agent model construction and analysis (Jamroga et al., 2013).