Solving Satisfiability Modulo Counting for Symbolic and Statistical AI Integration
With Provable Guarantees

Symbolic and statistical approaches are two fundamental driving forces of Artificial Intelligence (AI). Symbolic AI, exemplified by SATisfiability (SAT) and constraint programming, finds solutions satisfying constraints but requires rigid formulations and is difficult to include probabilities. Statistical AI captures uncertainty but often lacks constraint satisfaction. Integrating symbolic and statistical AI remains an open field and has gained research attention recently (cpm 2023; nes 2023; Munawar et al. 2023).

Satisfiability Modulo Counting (SMC) is an umbrella problem at the intersection of symbolic and statistical AI. It encompasses problems that carry out symbolic decision-making (satisfiability) mixed with statistical reasoning (model counting). SMC searches for policy interventions to control probabilistic outcomes. Formally, SMC is an SAT problem involving predicates on model counts. Model counting computes the number of models (i.e., solutions) to an SAT formula. Its weighted form subsumes probabilistic inference on Machine Learning (ML) models.

As a motivating SMC application, stochastic connectivity optimization searches for the optimal plan to reinforce the network structure so its connectivity is preserved under stochastic events – a central problem for a city planner who works on securing her residents multiple paths to emergency shelters in case of natural disasters. This problem is useful for disaster preparation (Wu, Sheldon, and Zilberstein 2015), bio-diversity protection (Dilkina and Gomes 2010), internet resilience (Israeli and Wood 2002), social influence maximization (Kempe, Kleinberg, and Tardos 2005), energy security (Almeida et al. 2019), etc. It requires symbolic reasoning (satisfiability) to decide which roads to reinforce and where to place emergency shelters, and statistical inference (model counting) to reason about the number of paths to shelters and the probabilities of natural disasters. Despite successes in many use cases, previous approaches (Williams and Snyder 2005; Conrad et al. 2012; Sheldon et al. 2010; Wu, Sheldon, and Zilberstein 2014) found solutions lack of certifiable guarantees, which are unfortunately in need for policy adoption in this safety-related application. Besides, their surrogate approximations of connectivity may overlook important probabilistic scenarios. This results in suboptimal quality of the generated plans. As application domains for SMC solvers, this paper considers emergency shelter placement and supply chain network management – two important stochastic connectivity optimization problems.

It is challenging to solve SMC because of their highly intractable nature ($\text{NP}^{\text{PP}}$-complete) (Park and Darwiche 2004) – still intractable even with good satisfiability solvers (Biere et al. 2009; Rossi, van Beek, and Walsh 2006; Braunstein, Mézard, and Zecchina 2005) and model counters (Gomes, Sabharwal, and Selman 2006a; Ermon et al. 2013b; Achlioptas and Theodoropoulos 2017; Chakraborty, Meel, and Vardi 2013; Kisa et al. 2014; Cheng et al. 2012; Gogate and Dechter 2012). Previous research on SMC solves either a special case or domain-specific applications (Belanger and McCallum 2016; Welling and Teh 2003; Yedidia, Freeman, and Weiss 2000; Wainwright and Jordan 2008; Fang, Stone, and Tambe 2015; Conitzer and Sandholm 2006; Sheldon et al. 2010). The special case is called the Marginal Maximum-A-Posterior (MMAP) problem, whose decision version can be formulated as a special case of SMC (Marinescu, Dechter, and Ihler 2014; Liu and Ihler 2013; Mauá and de Campos 2012; Jiang, Rai, and III 2011; Lee et al. 2016; Ping, Liu, and Ihler 2015). Both cases are solved by optimizing the surrogate representations of the intractable model counting in variational forms (Liu and Ihler 2012; Kiselev and Poupart 2014), or via knowledge compilation (Choi, Friedman, and den Broeck 2022; Ping, Liu, and Ihler 2015; Mei, Jiang, and Tu 2018) or via sample average approximation (Kleywegt, Shapiro, and Homem-de-Mello 2002; Shapiro 2003; Swamy and Shmoys 2006; Sheldon et al. 2010; Dyer and Stougie 2006; Wu et al. 2017; Xue, Fern, and Sheldon 2015; Verweij et al. 2003).

Nevertheless, previous approaches either cannot quantify the quality of their solutions, or offer one-sided guarantees, or offer guarantees which can be arbitrarily loose. The lack of tight guarantees results in delayed policy adoption in safety-related applications such as the stochastic connectivity optimization considered in this paper. Second, optimizing surrogate objectives without quantifying the quality of approximation leads to sub-optimal behavior empirically. For example, previous stochastic connectivity optimization solvers occasionally produce suboptimal plans because their surrogate approximations overlook cases of significant probability. This problem is amplified when combinatorial constraints are present.

We propose $\mathtt{XOR}$-$\mathtt{SMC}$, a polynomial algorithm accessing NP-oracles, to solve highly intractable SMC problems with constant approximation guarantees. These guarantees hold with high (e.g. $>99\%$) probability. The strong guarantees enable policy adoption in safety-related domains and improve the empirical performance of SMC solving (e.g., eliminating sub-optimal behavior and providing constraint satisfaction guarantees). The constant approximation means that the solver can correctly decide the truth of an SMC formula if tightening or relaxing the bounds on the model count by a multiplicative constant do not change its truth value. The embedding algorithms allow us to find approximate solutions to beyond-NP SMC problems via querying NP oracles. It expands the applicability of the state-of-the-art SAT solvers to highly intractable problems.

The high-level idea behind $\mathtt{XOR}$-$\mathtt{SMC}$ is as follows. Imagine a magic that randomly filters out half of the models (solutions) to an SAT formula. Model counting can be approximated using this magic and an SAT solver – we confirm the SAT formula has more than $2^{k}$ models if it is satisfiable after applying this magic $k$ times. This magic can be implemented by introducing randomized constraints. The idea is developed by researchers (Valiant and Vazirani 1986; Jerrum, Valiant, and Vazirani 1986; Gomes, Sabharwal, and Selman 2006b, a; Ermon et al. 2013b, a; Kuck et al. 2019; Achlioptas and Theodoropoulos 2017; Chakraborty, Meel, and Vardi 2013; Chakraborty et al. 2014). In these works, model counting is approximated with guarantees using polynomial algorithms accessing NP oracles. $\mathtt{XOR}$-$\mathtt{SMC}$ notices such polynomial algorithms can be encoded as SAT formulae. Hence, SAT-Modulo-Counting can be written as SAT-Modulo-SAT (or equivalently SAT), when we embed the SAT formula compiled from algorithms to solve model counting into SMC. The constant approximation guarantee also carries.

We evaluate the performance of $\mathtt{XOR}$-$\mathtt{SMC}$ on real-world stochastic connectivity optimization problems. In particular, we consider applied problems of emergency shelter placement and supply chain management. For the shelter placement problem, our $\mathtt{XOR}$-$\mathtt{SMC}$ finds high-quality shelter assignments with less computation time and better quality than competing baselines. For wheat supply chain management, the solutions found by our $\mathtt{XOR}$-$\mathtt{SMC}$ are better than those found by baselines. $\mathtt{XOR}$-$\mathtt{SMC}$ also runs faster than baselines¹¹1The code is available at: https://github.com/jil016/xor-smc. Please refer to https://arxiv.org/abs/2309.08883 for the Appendix..

Satisfiability Modulo Counting (SMC) is Satisfiability Modulo Theory (SMT) (Barrett et al. 2009) with model counting as the background theory. A canonical definition of the SMC problem is to determine if there exists $\mathbf{x}=(x_{1},\dots,x_{n})\in\{0,1\}^{n}$ and $\mathbf{b}=(b_{1},\dots,b_{k})\in\{0,1\}^{k}$ that satisfies the formula:

Here each $b_{i}$ is a Boolean predicate that is true if and only if the corresponding model count exceeds a threshold. Bold symbols (i.e., $\mathbf{x}$, $\mathbf{y}_{i}$ and $\mathbf{b}$) are vectors of Boolean variables. $\phi,f_{1},\dots,f_{k}$ are Boolean functions (i.e., their input is Boolean vectors, and their outputs are also Boolean). We use $0$ to represent false and $1$ to represent true. Hence $\sum f_{i}$ computes the number of satisfying assignments (model counts) of $f_{i}$. The directions of the inequalities do not matter much because one can always negate each $f_{i}$. For instance, let $f(\mathbf{x},\mathbf{y})$ be a Boolean function (output is 0 or 1). $\sum_{\mathbf{y}\in\mathcal{Y}}f(\mathbf{x},\mathbf{y})\leq 2^{q}$ can be converted by negating $f$ and modifying the threshold to $|\mathcal{Y}|-2^{q}$, resulting in an equivalent predicate $\sum_{\mathbf{y}\in\mathcal{Y}}(\neg f(\mathbf{x},\mathbf{y}))\geq|\mathcal{Y}|-2^{q}$.

Our $\mathtt{XOR}$-$\mathtt{SMC}$ algorithm obtains the constant approximation guarantee to the following slightly relaxed SMC problems. The problem $\mathtt{SMC}(\phi,f_{1},\dots,f_{k},$ $q_{1},\dots,q_{k})$ finds a satisfying assignment $(\mathbf{x},\mathbf{b})$ for:

The only difference compared to the full-scale problem in Eq. (2)) is the replacement of $\Leftrightarrow$ with $\Rightarrow$. This change allows us to derive a concise constant approximation bound. We also mention that all the applied SMC problems considered in this paper can be formulated in this relaxed form. We thank the reviewers for pointing out the work of (Fredrikson and Jha 2014), who came up with a slightly different SMC formulation with focused applications in privacy and an exact solver. Their formulation was a little more general than ours, since theirs allows for predicates like $\sum f\geq\sum g$, while ours only allows for $\sum f\geq$ constant. However, our formulation can handle $\sum f\geq\sum g$ by formulating it with $(\sum f\geq\alpha)\wedge(\sum g\leq\alpha)$ and binary searching on $\alpha$.

The key motivation behind our proposed $\mathtt{XOR}$-$\mathtt{SMC}$ algorithm is to notice that Algorithm 1 itself can be written as a Boolean formula due to the Cook-Levin reduction. When we embed this Boolean formula into Eq. (3), the Satisfiability-Modulo-Counting problem translates into a Satisfiability-Modulo-SAT problem, or equivalently, an SAT problem. This embedding also ensures a constant approximation guarantee (see Theorem 2).

To illustrate the high-level idea, let us consider replacing each $\sum_{\mathbf{y}_{i}\in\mathcal{Y}_{i}}f_{i}(\mathbf{x},\mathbf{y}_{i})\geq 2^{q_{i}}$ in Eq. (3) with formula

We denote the previous equation (4) as $\gamma(f_{i},\mathbf{x},q_{i},\mathbf{y}_{i})$. This replacement results in the Boolean formula:

We argue that the satisfiability of formula (5) should be closely related to that of formula (3) due to the connection between model counting and satisfiability testing subject to randomized constraints (discussed in Section 2). To see this, Eq. (5) is satisfiable if and only if there exists $(\mathbf{x},\mathbf{b},\mathbf{y}_{1},\ldots,\mathbf{y}_{k})$ that render Eq. (5) true (notice $\mathbf{y}_{1},\ldots,\mathbf{y}_{k}$ are also its variables). Suppose $\texttt{SMC}(\phi,f_{1},\dots,f_{k},q_{1}+c,\dots,q_{k}+c)$ is satisfiable (a.k.a., Eq. (3) is satisfiable when $q_{i}$ is replaced with $q_{i}+c$). Let $(\mathbf{x},\mathbf{b})$ be a satisfying assignment. For any $b_{i}=1$ (true) in $\mathbf{b}$, we must have $\sum_{\mathbf{y_{i}}\in\mathcal{Y}_{i}}f_{i}(\mathbf{x},\mathbf{y}_{{i}})\geq 2^{q_{i}+c}$. This implies with a good chance, there exists a $\mathbf{y}_{i}$ that renders $\gamma(f_{i},\mathbf{x},q_{i},\mathbf{y}_{i})$ true. This is due to the discussed connection between model counting and SAT solving subject to randomized constraints. Hence $b_{i}\Rightarrow\gamma(f_{i},\mathbf{x},q_{i},\mathbf{y}_{i})$ is true. For any $b_{i}=0$ (false), the previous equation is true by default. Combining these two facts and $\phi(\mathbf{x},\mathbf{b})$ is true, we see Eq. (5) is true.

Conversely, suppose $\texttt{SMC}(\phi,f_{1},\dots,f_{k},q_{1}-c,\dots,q_{k}-c)$ is not satisfiable. This implies for every $(\mathbf{x},\mathbf{b})$, either $\phi(\mathbf{x},\mathbf{b})$ is false, or there exists at least one $j$ such that $b_{j}$ is true, but $\sum_{\mathbf{y}_{{j}}\in\mathcal{Y}_{{j}}}f_{j}(\mathbf{x},\mathbf{y}_{{j}})<2^{q_{j}-c}$. The first case implies Eq. (5) is false under the assignment. For the second case, $\sum_{\mathbf{y}_{{j}}\in\mathcal{Y}_{{j}}}f_{j}(\mathbf{x},\mathbf{y}_{{j}})<2^{q_{j}-c}$ implies with a good chance there is no $\mathbf{y}_{j}$ to make $\gamma(f_{j},\mathbf{x},q_{j},\mathbf{y}_{j})$ true. Combining these two facts, with a good chance Eq. (5) is not satisfiable.

In practice, to reduce the error probability the determination of the model count needs to rely on the majority satisfiability status of a series of equations (4) (instead of a single one). Hence we develop Algorithm 2, which is a little bit more complex than the high-level idea discussed above. The idea is still to transform the highly intractable SMC problem into solving an SAT problem of its polynomial size, while ensuring a constant approximation guarantee. Fig. 1 displays the encoding of Algorithm 2. We can see the core is still to replace the intractable model counting with satisfiability problems subject to randomized constraints. We prove $\mathtt{XOR}$-$\mathtt{SMC}$ has a constant approximation guarantee in Theorem 2. We leave the implementation of $\mathtt{XOR}$-$\mathtt{SMC}$ in the Appendix B.

Given a map $G=(V,E)$ where nodes in $V=\{v_{1},\dots,v_{N}\}$ represent $N$ areas and an edge $e=(v_{i},v_{j})\in E$ indicates a road from $v_{i}$ to $v_{j}$, $N$ and $M$ denote the number of nodes and edges, respectively. Given a subset of nodes $R=\{v_{r_{1}},\dots,v_{r_{k}}\}\subseteq V$ indicates the residential areas, the task is to choose at most $m$ nodes as shelters from the rest of the nodes, such that the number of routes that can reach a shelter from each residential area is maximized. Fig. 2 gives an example with $m=4$ shelters and there are sufficiently many roads connecting the resident area to those shelters.

Current methods (Bayram and Yaman 2018; Amideo, Scaparra, and Kotiadis 2019) considered finding shelter locations that have at least one single path from a residential area. However, those proposed methods cannot be generalized to solve the problem that requires sufficient alternative routes from residential area to shelters, primarily because counting the number of paths is intractable. This complexity makes it difficult to solve large-scale problems of this type.

Given a supply chain of $N$ suppliers, they form a supply-demand network $(V,E)$ where each node $v\in V$ represents a supplier and edges $e\in E$ represent supply-to-demand trades. Each supplier $v$ acts as a vendor to downstream suppliers and also as a buyer from upstream suppliers. We assume a conservation of production conditions for each node, i.e., the input-output ratio is 1. To guarantee substantial production, supplier $v$ should order necessary raw materials from upstream suppliers in advance. Denote the cost of trade between vendor $u$, and buyer $v$ as $c(u,v)$. Let the amount of goods of the trade between $u$ and $v$ be $f(u,v)$ and $B(v)$ be the total budget to get all his materials ready. Due to unpredictable natural disasters and equipment failure, the trade between $u$ and $v$ may fail. Denote $\boldsymbol{\theta}=(\theta_{1},\ldots,\theta_{L})\in\{0,1\}^{L}$ as the state of $L$ different stochastic events, where $\theta_{l}=1$ indicates event $l$ occurs. The objective is to design a global trading plan maximizing the expected total production output, accounting for stochastic influences. This is measured by the output of final-tier suppliers, who produce end goods without supplying others.

Existing works in supply chain optimization often gravitate towards mathematical programming approaches. However, when delving into more complex scenarios involving stochastic events—such as uncertain demand or supply disruptions—the task becomes considerably more intricate. Specifically, formulating counting constraints, like guaranteeing a certain amount of supplies across multiple vendors under stochastic events, is intractable. These complexities necessitate innovative approaches that can capture the randomness and dynamism inherent in real-world supply chain systems without sacrificing optimality.

We presented $\mathtt{XOR}$-$\mathtt{SMC}$, an algorithm with polynomial approximation guarantees to solve the highly intractable Satisfiability Modulo Counting (SMC) problems. Solving SMC problems presents unique challenges due to their intricate nature, integrating statistical inference and symbolic reasoning. Prior work on SMC solving offers no or loose guarantees and may find suboptimal solutions. $\mathtt{XOR}$-$\mathtt{SMC}$ transforms the intractable SMC problem into satisfiability problems by replacing intricate model counting with SAT formulae subject to randomized XOR constraints. $\mathtt{XOR}$-$\mathtt{SMC}$ also obtains constant approximation guarantees on the solutions obtained. SMC solvers offer useful tools for many real-world problems at the nexus of symbolic and statistical AI. Extensive experiments on two real-world applications in AI for social good demonstrate that $\mathtt{XOR}$-$\mathtt{SMC}$ outperforms other approaches in solution quality and running time.

We thank all the reviewers for their constructive comments. This research was supported by NSF grant CCF-1918327.