This paper investigates the impact of perturbations on the best-response-based algorithms approximating Nash equilibria in zero-sum games, namely Double Oracle and Fictitious Play. More precisely, we assume that the oracle computing the best responses perturbs the utilities before selecting the best response. We show that using such an oracle reduces the number of iterations for both algorithms. For some cases, suitable perturbations ensure the expected number of iterations is logarithmic. Although the utility perturbation is computationally demanding as it requires iterating through all pure strategies, we demonstrate that one can efficiently perturb the utilities in games where pure strategies have further inner structure.

The application of graph neural networks (GNNs) to learn heuristic functions in classical planning is gaining traction. Despite the variety of methods proposed in the literature to encode classical planning tasks for GNNs, a comparative study evaluating their relative performances has been lacking. Moreover, some encodings have been assessed solely for their expressiveness rather than practical effectiveness in planning. This paper provides an extensive comparative analysis of existing encodings. Our results indicate that the smallest encoding based on Gaifman graphs, not yet applied in planning, outperforms the rest due to its fast evaluation times and the informativeness of the resulting heuristic. The overall coverage measured on the IPC almost reaches that of the state-of-the-art planner LAMA while exhibiting rather complementary strengths across different domains.

This paper proposes a novel method to predict uncertain action costs in classical planning, considering the resulting plan’s quality rather than the prediction’s quality. Unlike the solution offered by decision-focused learning (DFL), our method does not compute a gradient of the regret loss function. Instead, it starts with any trained model, e.g. by the usual mean square error (MSE), to obtain a tuple of the approximate model’s parameters. Next, it randomly samples a collection of tuples of parameters in the neighborhood of the approximate parameters. Finally, it employs the Multiplicative Weights Update algorithm to compute a probability distribution over this collection reflecting the quality of the sampled tuples of parameters w.r.t. the regret loss function. A weighted average of the predictions w.r.t. the sampled tuples of parameters gives us the resulting action costs prediction that considerably reduces the average regret compared to the MSE-trained predictor.

Graph Neural Networks (GNNs) have become the standard method of choice for learning with structured data, demonstrating particular promise in classical planning. Their inherent invariance under symmetries of the input graphs endows them with superior generalization capabilities, compared to their symmetry-oblivious counterparts. However, this comes at the cost of limited expressive power. Particularly, GNNs cannot distinguish between graphs that satisfy identical sentences of C2 logic. To leverage GNNs for learning policies in PDDL domains, one needs to encode the contextual representation of the planning states as graphs. The expressiveness of this encoding, coupled with a specific GNN architecture, then hinges on the absence of indistinguishable states necessitating distinct actions. This paper provides a comprehensive theoretical and statistical exploration of such situations in PDDL domains across diverse natural encoding schemes and GNN models.

We introduce the metric induced by Gaifman graphs into lifted planning. We analyze what kind of information this metric carries and how it can be utilized for constructing lifted delete-free relaxation heuristics. In particular, we prove how the action dynamics influence the distances between objects. As a corollary, we derive a lower bound on the length of any plan. Finally, we apply our theoretical findings on the Gaifman graphs to improve the delete-free relaxation heuristics induced by PDDL homomorphisms.

Classical planning tasks are modelled in PDDL which is a schematic language based on first-order logic. Most of the current planners turn this lifted representation into a propositional one via a grounding process. However, grounding may cause an exponential blowup. Therefore it is important to investigate methods for searching for plans on the lifted level. To build a lifted state-based planner, it is necessary to invent lifted heuristics. We introduce maps between PDDL tasks preserving plans allowing to transform a PDDL task into a smaller one. We propose a novel method for computing lifted (admissible) delete-free relaxed heuristics via grounding of the smaller task and computing the (admissible) delete-free relaxed heuristics there. This allows us to transfer the knowledge about relaxed heuristics from the grounded level to the lifted level.

Effective planning while competing for limited resources is crucial in many real-world applications such as on-demand transport companies competing for passengers. Planning techniques therefore have to take into account possible actions of an adversarial agent. Such a challenge that can be tackled by leveraging game-theoretical methods such as Double Oracle. This paper aims at the scalability issues arising from combining planning techniques with Double Oracle. In particular, we propose an abstraction-based heuristic for deciding how resources will be collected (e.g. which car goes for which passenger and in which order) and we propose a method for decomposing planning tasks into smaller ones (e.g. generate plans for each car separately). Our empirical evaluation shows that our proposed approach considerably improves scalability compared to the state-of-the-art techniques.

Many efficient algorithms have been designed to recover Nash equilibria of various classes of finite games. Special classes of continuous games with infinite strategy spaces, such as polynomial games, can be solved by semidefinite programming. In general, however, continuous games are not directly amenable to computational procedures. In this contribution, we develop an iterative strategy generation technique for finding a Nash equilibrium in a whole class of continuous two-person zero-sum games with compact strategy sets. The procedure, which is called the double oracle algorithm, has been successfully applied to large finite games in the past. We prove the convergence of the double oracle algorithm to a Nash equilibrium. Moreover, the algorithm is guaranteed to recover an approximate equilibrium in finitely-many steps. Our numerical experiments show that it outperforms fictitious play on several examples of games appearing in the literature. In particular, we provide a detailed analysis of experiments with a version of the continuous Colonel Blotto game.

Detection of redundant operators that can be safely removed from the planning task is an essential technique allowing to greatly improve performance of planners. In this paper, we employ structure-preserving maps on labeled transition systems (LTSs), namely endomorphisms well known from model theory, in order to detect redundancy. Computing endomorphisms of an LTS induced by a planning task is typically infeasible, so we show how to compute some of them on concise representations of planning tasks such as finite domain representations and factored LTSs. We formulate the computation of endomorphisms as a constraint satisfaction problem (CSP) that can be solved by an off-the-shelf CSP solver. Finally, we experimentally verify that the proposed method can find a sizeable number of redundant operators on the standard benchmark set.

Effective plan generation in adversarial environments has to take into account possible actions of adversary agents, i.e., the agent should know what the competitor will likely do. In this paper we propose a novel approach for estimating strategies of the adversary, sampling actions that interfere with the agent's ones. The estimated competitor strategies are used in plan generation by considering that agent's actions have to be applied prior to the ones of the competitor, whose estimated times dictate the agent's deadlines. Missing these deadlines entails additional plan cost.

Effective and efficient reasoning in adversarial environments is important for many real-world applications ranging from cybersecurity to military operations. Deliberative reasoning techniques, such as Automated Planning, often restrict to static environments where only an agent can make changes by its actions. On the other hand, such techniques are effective and can generate non-trivial solutions. To explicitly reason in environments with an active adversary such as zero-sum games, the game-theoretic framework such as the Double Oracle algorithm can be leveraged. In this paper, we leverage the notions of critical and adversary actions, where critical actions should be applied before the adversary ones. We propose heuristics that provide a guidance for planners about what (critical) actions and in which order have to be applied in a good plan. We empirically evaluate our approach in terms of quality of generated strategies (by leveraging Double Oracle) and CPU time required to generated such strategies.

A constraint satisfaction problem (CSP) is a computational problem where the input consists of a finite set of variables and a finite set of constraints, and where the task is to decide whether there exists a satisfying assignment of values to the variables. Depending on the type of constraints that we allow in the input, a CSP might be tractable, or computationally hard. In recent years, general criteria have been discovered that imply that a CSP is polynomial-time tractable, or that it is NP-hard. Finite-domain CSPs have become a major common research focus of graph theory, artificial intelligence, and finite model theory. It turned out that the key questions for complexity classification of CSPs are closely linked to central questions in universal algebra. This thesis studies CSPs where the variables can take values from an infinite domain. This generalization enhances dramatically the range of computational problems that can be modeled as a CSP. Many problems from areas that have so far seen no interaction with constraint satisfaction theory can be formulated using infinite domains, e.g. problems from temporal and spatial reasoning, phylogenetic reconstruction, and operations research. It turns out that the universal-algebraic approach can also be applied to study large classes of infinite-domain CSPs, yielding elegant complexity classification results. A new tool in this thesis that becomes relevant particularly for infinite domains is Ramsey theory. We demonstrate the feasibility of our approach with two complete complexity classification results: one on CSPs in temporal reasoning, the other on a generalization of Schaefer's theorem for propositional logic to logic over graphs. We also study the limits of complexity classification, and present classes of computational problems provably do not exhibit a complexity dichotomy into hard and easy problems.

Abstract We prove that the set of formulae provable in the full Lambek calculus with the structural rule of contraction is undecidable. In fact, we show that the positive fragment of this logic is undecidable.

Among the class of finite integral commutative residuated chains (ICRCs), we identify those algebras which can be obtained as a nuclear retraction of a conuclear contraction of a totally ordered Abelian ℓ-group. We call the ICRCs satisfying this condition regular. Then we discuss the structure of finite regular ICRCs. Finally, we prove that the class of regular members generate a strictly smaller variety than the variety generated by ICRCs.

In this article, solutions to three open problems on ordered commutative monoids posed in Evans et al. (2001, Semigroup forum, 62, 249-278) are presented. By an ordered monoid, we always mean a totally ordered monoid. All the problems are related to the class of ordered commutative monoids which are homomorphic images of ordered free commutative monoids.

Starting from a decomposition result of monoidal t-norm-based logic (MTL)-chains as ordinal sums, we focus our attention on a particular kind of indecomposable semihoops, namely weakly cancellative semihoops. The weak cancellation property is proved to be the difference between cancellation and pseudocomplementation, so it gives a new axiomatization of product logic and ΠMTL. By adding this property, some new fuzzy logics (propositional and first-order) are defined and studied obtaining some results about their (finite) strong standard completeness and other logical and algebraic properties.