Modular Task Queries (MTQs)

• Modular Task Queries (MTQs) are a framework that decomposes complex tasks into modular sub-tasks with clear interfaces and heterogeneous solver integration.
• MTQs employ an iterative, lazy solving algorithm that propagates constraints between modules to efficiently prune the search space.
• The approach unifies methods like SMT, ILP, and ASP, enhancing scalability through language independence and early conflict resolution.

Modular Task Queries (MTQs) are a principled approach to decomposing, specifying, and solving complex computational tasks by representing them as interacting modules or sub-tasks, each with explicit interfaces and well-defined specification languages. The MTQ paradigm provides a framework to design and execute systems in which separate modules, possibly implemented using different languages or solvers, work together to solve a global query, propagating constraints and information among them until a joint solution is found. This model is foundational across AI, formal verification, combinatorial search, and heterogeneous problem solving, informed by the general theory and algorithms of modular model expansion tasks.

1. Formal Foundations of Modular Task Queries

MTQs extend the classic model expansion (MX) task where, given an interpretation over an instance vocabulary $\sigma$ and a logical specification $\varphi$ (written over $\sigma \cup \epsilon$), the goal is to compute an expansion (by interpreting symbols in $\epsilon$) so that the full structure satisfies $\varphi$. In the modular setting, the global MX task is decomposed into several modules; each is itself an MX problem and communicates with others via structured interfaces.

Formally, modular model expansion constructs the overall problem as a modular system, composed of primitive MX modules via operations such as projection, composition, feedback (loops), and intersection. Each module operates over its native vocabulary and can be combined with others, capturing rich task interdependencies and communication flows.

This modular viewpoint directly supports MTQs: each query is expressed in terms of submodules that solve subqueries, potentially using different representation languages or specialized solvers. The decomposition enables transparent specification, stepwise refinement, and scalable execution of complex tasks.

2. Iterative Lazy Solving Algorithm and Propagation

The core computational mechanism for MTQs, as abstracted from modular model expansion, is an iterative, “lazy” solving algorithm. The process operates as follows:

1. The system initializes with an empty expansion (all expansion symbols are unspecified).
2. An online solver maintains a working set of partial structures (potential solutions).
3. For each iteration:
   • The solver proposes a partial expansion $R$.
   • Each module $M_k$ with its oracle $O_k$ checks if the projection of $R$ onto its vocabulary is acceptable.
   • If any module rejects $R$, the oracle supplies a “reason”—a constraint (typically a formula of the form $\mathrm{Pre} \implies \mathrm{Post}$)—added to prune the search.
   • If all modules accept and $R$ is total, $R$ is returned as a solution.
   • Otherwise, the solver integrates “advices” (propagation hints) and constraints, refines its working set, and repeats.

Pseudocode (from Algorithm 1):

Initialize solver S with empty expansion. while true: if S is UNSAT: return "Unsatisfiable" else if S proposes R: Add advices from all oracles to S if some M_k rejects R: Get reason ψ from O_k; add ψ to S else if R is total: return R

To accelerate convergence, the algorithm augments search with propagation/approximation techniques based on monotonicity (positive/negative feedback in modules). For instance, an iterative propagation sequence for positive feedback is:

$$L_0 := S^+, \qquad L_{i+1} := R^{M(|_{\tau} \,\|\, S = L_i)}$$

The limit $L^*$ (positive feedback) or a pair $(L^*, U^*)$ (negative feedback) is then propagated to tightly restrict the search space.

3. Applications: Paradigms Modeled as Modular Task Queries

The MTQ and modular MX framework subsume key paradigms in search and reasoning:

• Satisfiability Modulo Theories (SMT):

A prototypical modular system composes a SAT solver with theory solvers (e.g., ILP, EUF), exchanging information in a feedback loop (DPLL($\mathcal{T}$) architecture). For example:

$$\mathrm{SMT} := \pi_{\{F, M, L\}}((\mathrm{ILP} \circ \mathrm{SAT})[L = L'])$$

Here, SAT manages the Boolean structure, ILP the theory, and their oracles propagate reasons/advices for conflict learning and propagation.

• Integer Linear Programming (ILP):

ILP solvers that use branch-and-cut are represented by modular partitioning of “cutting plane” (ILP$_c$), branch modules, and an LP solver. Their composition leverages module-level propagation to drive convergence to integrality.

• Answer Set Programming (ASP) and Hybrids (ASP-CP):

Logic programs (ASP) and constraints (CP) are handled by respective modules, integrating conflict learning, propagation, and backjumping.

The MTQ abstraction thus captures both the theory and practice of combined solving, enabling new modular systems to be built as compositions of reusable, independently specified, and tightly communicating modules.

4. Comparison with Monolithic and Specialized Methods

The modular approach to MTQs confers several distinctive advantages over traditional, monolithic, or highly specialized solvers:

• Language and Architecture Independence:

Each module can be specified in its own logic/language (e.g., SAT, ILP, CP) and solved using domain-optimized methods, while modular combination ensures overall coherence.

• Incremental “Lazy” Search and Early Pruning:

The iterative process, coupled with early constraint addition (via “reasons”), efficiently prunes large inconsistent or infeasible subspaces before expensive full expansion.

• Approximation and Propagation Integration:

The use of propagation techniques at the module and system level enables early elimination of inconsistent partial structures, improving scaling.

• Unification and Systematization:

The framework places disparate “combined” solvers (e.g., DPLL($\mathcal{T}$), branch-and-cut, CDCL-based methods) under a single algorithmic and representational paradigm, facilitating both theoretical understanding and practical extensibility.

• Scalability and Use of Heterogeneous Solvers:

By requiring modules to only weakly accept partial structures (never falsely rejecting viable paths), each module maximally leverages the efficiency of its native solver while benefitting from global feedback.

5. Implications for MTQs and Complex Query Systems

The theoretical and algorithmic results have direct implications for future MTQ systems:

• Expressive Modular Specification:

By decomposing tasks into modules with explicit interfaces (projections, feedback), system designers can readily express and reason about complicated search problems.

• Inter-module Communication Protocols:

The “advices and reasons” mechanism provides a principled protocol for information exchange: when a module cannot decide due to insufficient data, it provides hints; when it detects inconsistency, it returns effective, formalizable counterexamples.

• Integration with Heterogeneous Toolchains:

Provided the modules expose a CCAV oracle interface, arbitrarily heterogeneous tools (BDD engines, constraint databases, ASP solvers, etc.) can be composed into a unified MTQ system.

• Enhanced Efficiency via Propagation:

The explicit propagation sequences (positive/negative feedback) enable rapid exclusion of large swathes of the search space even without complete instantiation, streamlining solution of complex tasks.

• Declarative Execution Semantics:

The system admits a clear, logic-based semantics for the composition and solution of modular queries, enabling formal analysis and correctness proofs.

6. Conclusion and Outlook

The modular model expansion framework provides a general, rigorous foundation for the design and execution of Modular Task Queries, leveraging iterative lazy search, module-level propagation, and “reason/advice” mechanisms for coordination. It subsumes and abstracts important combined solvers in SMT, ILP, ASP, and CP, while affording scalability, flexibility, and deployment of heterogeneous and domain-optimized module solvers. The analogy and shared structure with MTQs highlight a path forward for declarative, modular, and efficient specification and solution of complex queries in AI, formal methods, computational biology, and beyond.