Symbolic Solutions for Symbolic Constraint Satisfaction Problems Alexsander Andrade de Melo 1 , Mateus de Oliveira Oliveira 2 1 Federal University of Rio de Janeiro, Rio de Janeiro, Brazil 2 University of Bergen, Bergen, Norway aamelo@cos.ufrj.br, mateus.oliveira@uib.no Abstract A fundamental drawback that arises when one is faced with the task of deterministically certifying solutions to compu- tational problems in PSPACE is the fact that witnesses may have superpolynomial size, assuming that NP is not equal to PSPACE. Therefore, the complexity of such a deterministic verifier may already be super-polynomially lower-bounded by the size of a witness. In this work, we introduce a new symbolic framework to address this drawback. More pre- cisely, we introduce a PSPACE-hard notion of symbolic con- straint satisfaction problem where both instances and solu- tions for these instances are implicitly represented by ordered decision diagrams (i.e. read-once, oblivious, branching pro- grams). Our main result states that given an ordered decision diagram D of length k and width w specifying a CSP in- stance, one can determine in time f (w, w ′ ) · k whether there is an ODD of width at most w ′ encoding a solution for this instance. Intuitively, while the parameter w quantifies the complexity of the instance, the parameter w ′ quantifies the complexity of a prospective solution. We show that CSPs of constant width can be used to formalize natural PSPACE hard problems, such as reachability of configurations for Tur- ing machines working in nondeterministic linear space. For such problems, our main result immediately yields an algo- rithm that determines the existence of solutions of width w in time g(w) · n, where g : N → N is a suitable computable function, and n is the size of the input. 1 Introduction One of the main drawbacks when dealing with compu- tational problems which lie in complexity classes beyond NP is the fact that witnesses for YES instances of these prob- lems may not have polynomial size. Indeed, in the case of PSPACE-hard problems, YES instances cannot be certified in polynomial time, unless NP=PSPACE. One approach to- wards solving PSPACE complete problems is to use gen- eralizations of propositional proof systems to the context of quantified formulas (Balabanov, Widl, and Jiang 2014; Narizzano et al. 2009). In this case, proofs of satisfiabil- ity/unsatisfiability play the role of certificates for YES/NO instances respectively. The drawback however, is that proofs may still have exponential size, and that the search for such proofs is non-deterministic. In this work, we introduce a new symbolic framework for certifying YES instances of PSPACE-complete prob- lems. More precisely, we introduce a new symbolic no- tion of constraint satisfaction problem (CSP) where both in- stances and assignments are encoded using the notion of or- dered decision diagrams (ODDs), a straightforward gener- alization of the notion of ordered binary decision diagrams (OBDDs) to non-binary alphabets. Our main result (Theo- rem 1) states that for each w, w ′ ∈ N, the process of de- termining whether a given CSP of width w has a solution of width w ′ can be solved in time f (w, w ′ ) · k, where f is a suitable function depending only on w and on w ′ and k is the length of the input ODD. In the terminology of pa- rameterized complexity theory (Downey and Fellows 1999; Cygan et al. 2015), our algorithm is fixed parameter tractable with respect to the parameters w and w ′ . Our second main result (Theorem 11) states that the task of determining whether a CSP of constant width has a so- lution is already PSPACE-hard. The family of CSPs that we use to prove this hardness result encode natural grid-like CSPs with uniform horizontal and vertical nearest-neighbor constraints. Satisfiability for these CSPs generalize natural PSPACE-hard reachability problems, such as configuration reachability for cellular automata and for non-deterministic linear space Turing machines. Interestingly, for such CSPs of constant width, the complexity of satisfiability is directly correlated with the complexity of a prospective satisfying assignment. In particular, our main result immediately gives rise to an algorithm running in time g(w) · k that deter- mines whether a CSP instance encoded by an ODD of length k has a satisfying assignment of width at most w, where g : N → N is a suitable function depending only on w. It is worth highlighting some fundamental differences in the way in which ODDs are used in our work and in the way in which other notions of ordered decision di- agrams have been used in previous works (Bollig 2012; Bollig 2014; Hachtel and Somenzi 1993; Woelfel 2006; Sawitzki 2004). The main difference is that in previous works each path in an ODD (or analogue formalisms) en- codes a whole assignment for the variables of the input in- stance. Therefore, in these contexts, an ODD encodes a set of solutions. On the other hand, in our approach, a path in an ODD encodes the assignment of a single variable, and the whole ODD is used to encode a single assignment. As a consequence, in our approach we can encode assignments for CSP instances containing exponentially many variables. Proceedings of the 17th International Conference on Principles of Knowledge Representation and Reasoning (KR 2020) Main Track 49