Agentic Logic Framework for AI Systems

• Agentic logic is a formal, algorithmic framework that specifies multi-agent AI systems using graph-based workflows and temporal assertions.
• It operationalizes a hybrid traversal algorithm that combines symbolic constraints with vector semantics to optimize multi-step task execution.
• Temporal assertions and formal verification techniques ensure safety, liveness, and compliance across complex, dynamically composed agent architectures.

Agentic logic is a rigorous formal and algorithmic framework for specifying, guiding, monitoring, and verifying the behavior of multi-agent AI systems—especially those leveraging LLMs to perform complex, multi-step tasks across dynamically composed workflows. It encompasses graph-based logical workflows, temporal expression languages for specifying behavioral invariants, and semantic modeling to support verification of safety, liveness, and functional properties across heterogeneous agentic architectures.

1. Formal Models for Agentic Logic

Recent formalizations of agentic logic integrate workflow graph structures, temporal assertion languages, and system semantic models.

Graph-based Logic:

A core abstraction is the directed graph $G=(V,E)$, where each node $n_i \in V$ encodes an instruction $t_i$, node type $$\tau_i \in \{\mathrm{Start},\mathrm{Generic},\mathrm{Transfer},\mathrm{End}\}$$, and an optional action $a_i$. Directed edges $(n_i\to n_j)\in E$ are guarded by logical conditions $C_{ij}$ over conversation or workflow state $S$. A path $p=(n_0,\dots,n_\ell)$ is valid under state $S$ iff all edge conditions $C_{n_k,n_{k+1}}(S)=\mathrm{True}$, formalized as

$$\mathrm{ValidPath}(p;S) = \bigwedge_{k=0}^{\ell-1} C_{n_k,n_{k+1}}(S) = \mathrm{True}$$

Traversal always begins at a unique Start node and terminates at an End or Transfer node (Casella et al., 9 Mar 2025).

Semantic Models for Multi-Agent Systems:

At a higher level, agentic AI systems are modeled by a pair $(\mathcal{H},\mathcal{L})$:

• $\mathcal{H}$ (“host agent model”): Top-level orchestrator that parses user intent, decomposes tasks, invokes sub-agents/tools, aggregates results, and returns responses. This model formalizes all system agents $\mathcal{A}$, external entities $\mathcal{E}$ (via A2A or MCP), tasks $\mathcal{T}$, registry $\mathcal{R}$, orchestration logic $\mathcal{O}$, a communication layer $\mathrm{CL}$, and global state $\mathbb{S}_H$.
• $\mathcal{L}$ (“task lifecycle model”): Fine-grained FSM for sub-task state management, with states such as CREATED, READY, DISPATCHING, IN_PROGRESS, COMPLETED, FAILED, and error-handling transitions, represented as $(\mathbb{S}_t, s_0, \mathcal{E}_t, \delta)$ (Allegrini et al., 15 Oct 2025).

This dual-level modeling enables precise specification and mechanized reasoning over both global and local system properties.

2. Algorithmic Realization in Conversational AI

Agentic logic is operationalized in modern LLM-driven agentic frameworks using a hybrid of symbolic constraints and vector semantics.

Hybrid Traversal Algorithm (“PAF”):

1. Node Embedding: Each node instruction $t_i$ is embedded as $e_i = \phi(t_i) \in \mathbb{R}^d$.
2. LLM-Driven Generation: At each step, the LLM generates response $r_t$, re-embedded as $x_t = \phi(r_t)$.
3. Successor Scoring: For all legal children $n_j$ of current node, compute $s_j(t) = e_j^\top x_t$.
4. Selection Rule: If the maximal score $s_{j^*}(t)\geq\theta$ (preset threshold), select $n_{j^*}$; otherwise, invoke an LLM-as-Judge to pick the next node using explicit reasoning over the workflow and dialog context.
5. Action and State Transition: Execute $a[n_{next}]$ if defined and progress dialog history. Stop on end-conditions.

By adjusting $\theta$, the system interpolates between strict symbolic path-following ($\theta\rightarrow1.0$) and aggressive vector-based “jumping” between children ($\theta\rightarrow-\infty$). Irrespective of vector confidence, transitions remain local via $E$; arbitrary hops outside $\operatorname{Children}(n_{cur})$ are prohibited (Casella et al., 9 Mar 2025).

Experiments demonstrate that this hybridized agentic logic achieves statistically significant increases in semantic alignment and throughput over purely LLM-based or symbolic baselines—e.g., in 100 simulated dialogues, “optimized PAF” achieved a 0.594 total hit rate and mean similarity 0.496, outperforming “basic PAF” (0.481/0.400) and single-shot baselines (0.391/0.387).

3. Temporal Assertions and Monitoring

Agentic logic incorporates linear-time temporal assertion languages for systematic runtime checking and regression validation.

Oroboro Temporal Language:

Predicates $\mathrm{Pred}(p)$ are asserted over agent actions (e.g., tool calls, inter-agent transfers) per cycle. The syntax supports:

• Concatenation: $A + B$
• Conditional: $A {> \!\!>} B$ (if $A$ then $B$ in next step)
• Fusion: $A/B$ (co-occurrence in the same cycle)
• Bounded repetition Formulas are interpreted against sequences $\mathcal{E}=e_0,e_1,\dots$; matches/failures trace precise locations in execution.

Correspondence to LTL: $A+B \approx A \wedge X B$, $A {> \!\!>} B \approx \neg A \vee (A \wedge X B)$, while the fusion and bounded repetition operators extend standard LTL for agentic action scenarios (Sheffler, 19 Aug 2025).

Monitoring and Regression Testing:

Temporal assertions (e.g., “xferToGreeting {> !!>} (sayHello + xferToWeather)”) are registered as “always” checks and evaluated continuously as agent executions proceed. Failures signal protocol/order violations; e.g., when agents powered by a weaker model (llama3.2) mis-sequenced transfers or failed to hand off tasks, assertion failures were immediately detected, whereas under a strong model (claude-3-7-sonnet) no failures arose across test runs.

Instrumentation leverages pre-existing hooks (e.g., ADK’s before_tool_call), bypassing brittle text matching and providing robust, model-agnostic error detection.

4. Temporal-Logic Properties for Formal Verification

Agentic logic frameworks provide an extensive set of temporal-logic properties for both global and sub-task validation.

Host-Agent Properties (select examples):

• Liveness: $AG(\mathrm{Req}_U \rightarrow AF\,\mathrm{Resp}_H)$—every user request eventually produces a response.
• Safety: $$AG( \mathrm{CL.invoke}(EE, protocol, st) \rightarrow st\in D )$$—can only invoke defined sub-tasks.
• Completeness: $$AG(\mathrm{Req}_U \rightarrow EX( \mathrm{LLM.Build\_Task\_DAG}(\cdots) \vee \mathrm{Clarify\_Intent} ))$$—every prompt triggers intent resolution.
• Fairness: All agent/tool invocations must be eventually answered.

Task Lifecycle Properties (select examples):

• Liveness: $$AG(\mathrm{state}=\mathrm{CREATED} \rightarrow AF(\mathrm{state}=\mathrm{COMPLETED} \vee \mathrm{ERROR} \vee \mathrm{CANCELED}))$$
• Safety: $$G(\mathrm{state}=\mathrm{DISPATCHING} \rightarrow \mathrm{previous\_state}=\mathrm{READY})$$
• No Starvation: $$AG(\mathrm{state}=\mathrm{AWAITING\_DEPENDENCY}\rightarrow AF(\mathrm{state}\neq \mathrm{AWAITING\_DEPENDENCY}))$$

Altogether, 31 such properties are defined: 17 for host-agents, 14 for task lifecycles, covering liveness, safety, completeness, and fairness (Allegrini et al., 15 Oct 2025).

Verification is achieved via construction of Kripke models/transition systems and use of off-the-shelf model checkers (e.g., NuSMV, SPIN, PRISM) to establish satisfaction or produce counterexamples indicating protocol/lifecycle violations.

5. Concrete Scenarios and Empirical Results

Conversational AI (PAF):

Empirical evaluation on synthetic dialogue tasks shows that hybrid agentic logic (vector+LLM) not only increases the fraction of agent responses exceeding semantic similarity thresholds but also reduces LLM call latency by ~40%, as vector methods are successful in approximately 80% of turns.

Multi-Agent Coordination:

In the three-agent “Weather/Greeting/Farewell” scenario, temporal assertions validate the order of tool invocations and transfers across runs. When using robust LLMs, all temporal properties are consistently upheld. Substituting weaker models immediately surfaces failures in agentic coordination, such as missed transfers or self-loops, as evidenced in precise, step-indexed MATCH/FAILURE traces (Sheffler, 19 Aug 2025).

Financial Planner Workflow:

In orchestrated workflows (e.g., budgeting plus investment via A2A+MCP protocols), model checking with defined properties ensures correct dependency resolution, proper invocation sequencing, successful aggregation, and robust error/fallback handling. Any deviations (e.g., DAG before discovery, unauthorized tool invocation, unhandled errors) surface as explicit property violations with counterexample traces (Allegrini et al., 15 Oct 2025).

6. Impact, Scope, and Extensions

Agentic logic, as currently formalized, provides domain-agnostic methods for:

• Provable compliance with business and safety rules
• Mechanized verification of AI workflow correctness
• Online monitoring and rapid regression detection
• Prevention of deadlocks, privilege escalation, and architectural misalignments

The framework is applicable wherever a top-level orchestrator decomposes work into asynchronous, verifiable sub-tasks managed by heterogeneous agents and tools (not limited to a particular industry or protocol stack).

A plausible implication is that agentic logic principles may serve as foundational infrastructure for rigorous certification, continuous assurance, and lifecycle governance as deployment of multi-agent LLM-based systems becomes pervasive in critical environments.

Table: Key Components of Agentic Logic Across Research

Core Element	Representative Paper	Formalism Used
Workflow graph/traversal	(Casella et al., 9 Mar 2025)	Directed labeled graph, Hybrid vector/LLM selection
Temporal monitoring	(Sheffler, 19 Aug 2025)	Oroboro temporal expr., LTL mapping
System-level semantics	(Allegrini et al., 15 Oct 2025)	Host-agent model, Task lifecycle FSM, CTL/LTL properties

Each of these elements targets a distinct but complementary aspect: operational control and efficiency, runtime monitoring, and formal specification/verification, respectively, collectively characterizing the recent state of “agentic logic.”