AI planning is the task of finding a sequence of actions for a declaratively described system to reach some goal, while (optionally) optimizing some measures. In its classical version, the initial state is fully known and actions have deterministic, known effects. Besides its conceptual simplicity, classical planning is PSPACE-complete, both for satisficing and optimal planning and, consequently, a plethora of heuristic planners do exist, coping with the inherent complexity of the problem in different ways. Nevertheless, SAT based exact methods are competitive with heuristic methods for classical planning in many cases. Several declarative languages have been proposed for classical planning, such as STRIPS or PDDL, with PDDL being the de facto standard that most planners support, at least to some extent. PDDL formulations contain first-order formulas that are usually grounded in order to solve the planning problem, either with heuristic or exact methods, sometimes leading to a blow-up that makes the resulting problem instance intractable. With this in mind, in this paper we present a study on the translation of PDDL formulations to lifted representations in SMT, with the aim of obtaining more compact and, at the same time, efficient formulations for classical planning. We show that SMT formulations often outperform SAT based ones on hard-to-ground instances.

The B2B Meeting Scheduling Optimization Problem (B2BSP) consists of scheduling a set of meetings between given pairs of participants to an event, minimizing idle time periods in participants' schedules, while taking into account participants’ availability and accommodation capacity. Therefore, it constitutes a challenging combinatorial problem in many real-world B2B events. This work presents a comparative study of several approaches to solve this problem. They are based on Constraint Programming (CP), Mixed Integer Programming (MIP) and Maximum Satisfiability (MaxSAT). The CP approach relies on using global constraints and has been implemented in MiniZinc to be able to compare CP, Lazy Clause Generation and MIP as solving technologies in this setting. A pure MIP encoding is also presented. Finally, an alternative viewpoint is considered under MaxSAT, showing the best performance when considering some implied constraints. Experimental results on real world B2B instances, as well as on crafted ones, show that the MaxSAT approach is the one with the best performance for this problem, exhibiting better solving times, sometimes even orders of magnitude smaller than CP and MIP.

When solving a combinatorial problem using propositional satisfiability (SAT), the encoding of the constraints is of vital importance. Pseudo-Boolean (PB) constraints appear frequently in a wide variety of problems. When PB constraints occur together with at-most-one (AMO) constraints over the same variables, they can be combined into PB(AMO) constraints. In this paper we present new encodings for PB(AMO) constraints. Our experiments show that these encodings can be substantially smaller than those of PB constraints and allow many more instances to be solved within a time limit. We also observed that there is no single overall winner among the considered encodings, but efficiency of each encoding may depend on PB(AMO) characteristics such as the magnitude of coefficient values.

The Business-to-Business Meeting Scheduling problem consists of scheduling a set of meetings between given pairs of participants to an event, while taking into account participants’ availability and accommodation capacity. A crucial aspect of this problem is that breaks in participants’ schedules should be avoided as much as possible. It constitutes a challenging combinatorial problem that needs to be solved for many real world brokerage events. In this paper we present a comparative study of Constraint Programming (CP), MixedInteger Programming (MIP) and Maximum Satisfiability (MaxSAT) approaches to this problem. The CP approach relies on using global constraints and has been implemented in MiniZinc to be able to compare CP, Lazy Clause Generation and MIP as solving technologies in this setting. We also present a pure MIP encoding. Finally, an alternative viewpoint is considered under MaxSAT, showing best performance when considering some implied constraints. Experiments conducted on real world instances, as well as on crafted ones, show that the MaxSAT approach is the one with the best performance for this problem, exhibiting better solving times, sometimes even orders of magnitude smaller than CP and MIP.

One aspect that has been poorly studied in multiple-valued logics, and in particular in Łukasiewicz logic, is the generation of instances of varying difficulty for evaluating, comparing and improving satisfiability solvers. With the ultimate goal of finding challenging benchmarks for Łukasiewicz satisfiability solvers, we start by defining a natural and intuitive class of clausal forms (simple Ł-clausal forms) and studying their complexity. Since we prove that the satisfiability problem of simple Ł-clausal forms can be solved in linear time, we then define two new classes of clausal forms (Ł-clausal forms and restricted Ł-clausal forms) that truly exploit the non-lattice operations of Łukasiewicz logic and whose satisfiability problems are NP-complete when clauses have at least three literals, and admit linear-time algorithms when clauses have at most two literals. We also define an efficient satisfiability preserving translation of Łukasiewicz logic formulas into Ł-clausal forms. Finally, we describe a random generator of Ł-clausal forms and report on an empirical investigation in which we identify an easy-hard-easy pattern and a phase transition phenomenon for Ł-clausal forms.

Abstract When programs feature a complex control flow, existing techniques for resource analysis produce cost relation systems (CRS) whose cost functions retain the complex flow of the program and, consequently, might not be solvable into closed-form upper bounds. This paper presents a novel approach to resource analysis that is driven by the result of a termination analysis. The fundamental idea is that the termination proof encapsulates the flows of the program which are relevant for the cost computation so that, by driving the generation of the CRS using the termination proof, we produce a linearly-bounded CRS (LB-CRS). A LB-CRS is composed of cost functions that are guaranteed to be locally bounded by linear ranking functions and thus greatly simplify the process of CRS solving. We have built a new resource analysis tool, named MaxCore, that is guided by the VeryMax termination analyzer and uses CoFloCo and PUBS as CRS solvers. Our experimental results on the set of benchmarks from the Complexity and Termination Competition 2019 for C Integer programs show that MaxCore outperforms all other resource analysis tools.

Pseudo-Boolean (PB) constraints are usually encoded into Boolean clauses using compact Binary Decision Diagram (BDD) representations. Although these constraints appear in many problems, they are particularly useful for representing resource constraints in scheduling problems. Sometimes, the Boolean variables in the PB constraints have implicit at-most-one relations. In this work we introduce a way to take advantage of these implicit relations to obtain a compact Multi-Decision Diagram (MDD) representation for those PB constraints. We provide empirical evidence of the usefulness of this technique for some Resource-Constrained Project Scheduling Problem (RCPSP) variants, namely the Multi-Mode RCPSP (MRCPSP) and the RCPSP with Time-Dependent Resource Capacities and Requests (RCPSP/t). The size reduction of the representation of the PB constraints lets us decrease the number of Boolean variables in the encodings by one order of magnitude. We close/certify the optimum of many instances of these problems.

Planning Modulo Theories (PMT), inspired by Satisfiability Modulo Theories (SMT), allows the integration of arbitrary first order theories, such as linear arithmetic, with propositional planning. Under this setting, planning as SAT is generalized to planning as SMT. In this paper we introduce a new encoding for planning as SMT, which adheres to the relaxed relaxed ∃-step (R 2 ∃-step) semantics for parallel plans. We show the benefits of relaxing the requirements on the set of actions eligible to be executed at the same time, even though many redundant actions can be introduced. We also show how, by a MaxSMT based post-processing step, redundant actions can be efficiently removed, and provide experimental results showing the benefits of this approach.

The Multi-Mode Resource-Constrained Project Scheduling Problem (MRCPSP) is a generalization of the well known Resource-Constrained Project Scheduling Problem (RCPSP). The most common exact approaches for solving this problem are based on branch-and-bound algorithms, mixed integer linear programming and Boolean satisfiability (SAT). In this paper, we present a new exact approach for solving this problem, using Satisfiability Modulo Theories (SMT). We provide two encodings into SMT and several reformulation and preprocessing techniques. The optimization algorithm that we propose uses an SMT solver as an oracle, and depending on its answer is able to update the encoding for the next optimization step. We report extensive performance experiments showing the utility of the proposed techniques and the good performance of our approach that allows us to close several open instances.

In most termination tools two ingredients, namely recursive path orderings (RPOs) and polynomial interpretation orderings (POLOs), are used in a consecutive disjoint way to solve the final constraints generated from the termination problem. In this article we present a simple ordering that combines both RPO and POLO and defines a family of orderings that includes both, and extend them with the possibility of having, at the same time, an RPO-like treatment for some symbols and a POLO-like treatment for the others. The ordering is extended to higher-order terms, providing a new fully automatable use of polynomial interpretations in combination with beta-reduction.

The presence of uncertainty in the real world makes robustness to be a desired property of solutions to constraint satisfaction problems. Roughly speaking, a solution is robust if it can be easily repaired when unexpected events happen. This issue has already been addressed in the frameworks of Boolean satisfiability (SAT) and Constraint Programming (CP). Most works on robustness implement search algorithms to look for such solutions instead of taking the declarative approach of reformulation, since reformulation tends to generate prohibitively large formulas, especially in the CP setting.

Abstract For many years, all existing completeness results for Knuth-Bendix completion and ordered paramodulation required the term order-ing ≻ to be well-founded, monotonic and total(izable) on ground terms. Then, it was shown that well-foundedness and the subterm property were enough for ensuring completeness of ordered paramodulation. Here we show that the subterm property is not necessary either. By us-ing a new restricted form of rewriting, we obtain a completeness proof of ordered paramodulation for Horn clauses with equality, where well-foundedness of the ordering suffices. Apart from the theoretical signifi-cance of this result, some potential applications motivating the interest of dropping the subterm property are given. The proof of the results included in this paper, being still technical in some parts, is pretty much shorter and easier to read than the one we have in the preliminary version of this work presented at the CADE 2002 conference [8]. Key words: term rewriting, equational reasoning, theorem proving, paramodulation, Knuth-Bendix completion 1

The extensional theory of arrays is one of the most important ones for applications of SAT modulo theories (SMT) to hardware and software verification. Here we present a new T-solver for arrays in the context of the DPLL(T) approach to SMT. The main characteristics of our solver are: (i) no translation of writes into reads is needed, (ii) there is no axiom instantiation, and (iii) the T-solver interacts with the Boolean engine by asking to split on equality literals between indices. Unlike most state-of-the-art array solvers, it is not based on a lazy instantiation of the array axioms. This novelty might make it more convenient to apply this solver in some particular environments. Moreover, it is very competitive in practice, specially on problems that require heavy reasoning on array literals.

All current completeness results for ordered paramodulation require the term ordering > to be well-founded, monotonic and total(izable) on ground terms. Here we introduce a new proof technique where the only properties required for > are well foundedness and the subterm property: The technique is a relatively simple and elegant application of some fundamental results on the termination and confluence of ground term rewrite systems (TRS). By a careful further analysis of our technique, we obtain the first Knuth-Bendix completion procedure that finds a convergent TRS for a given set of equations E and a (possibly non-totalizable) reduction ordering p whenever it exists. Note that being a reduction ordering is the minimal possible requirement on >, since a TRS terminates if, and only if, it is contained in a reduction ordering.