Papers by Mateus de Oliveira Oliveira
Synthesis and Analysis of Petri Nets from Causal Specifications
Lecture Notes in Computer Science, 2022
25th ACM International Conference on Hybrid Systems: Computation and Control
The mortality problem for a given dynamical system S consists of determining whether every trajec... more The mortality problem for a given dynamical system S consists of determining whether every trajectory of S eventually halts. In this work, we show that this problem is decidable for the class of piecewise constant derivative systems on two-dimensional manifolds, also called surfaces (PCD 2m). Two closely related open problems are point-to-point and edge-to-edge reachability for PCD 2m. Building on our technique to establish decidability of mortality for PCD 2m , we show that the edge-to-edge reachability problem for these systems is also decidable. In this way we solve the edge-toedge reachability case of an open problem due to Asarin, Mysore, Pnueli and Schneider [4]. This implies that the interval-to-interval version of the classical open problem of reachability for regular piecewise affine maps (PAMs) is also decidable. In other words, point-to-point reachability for regular PAMs can be effectively approximated with arbitrarily precision.
arXiv: Computational Complexity, Apr 22, 2014
It has been known for almost three decades that many NP-hard optimization problems can be solved ... more It has been known for almost three decades that many NP-hard optimization problems can be solved in polynomial time when restricted to structures of constant treewidth. In this work we provide the first extension of such results to the quantum setting. We show that given a quantum circuit C with n uninitialized inputs, poly(n) gates, and treewidth t, one can compute in time (n δ) exp(O(t)) a classical assignment y ∈ {0, 1} n that maximizes the acceptance probability of C up to a δ additive factor. In particular, our algorithm runs in polynomial time if t is constant and 1/poly(n) < δ < 1. For unrestricted values of t, this problem is known to be complete for the complexity class QCMA, a quantum generalization of MA. In contrast, we show that the same problem is NP-complete if t = O(log n) even when δ is constant. On the other hand, we show that given a n-input quantum circuit C of treewidth t = O(log n), and a constant δ < 1/2, it is QMA-complete to determine whether there exists a quantum state |ϕ ∈ (C d) ⊗n such that the acceptance probability of C|ϕ is greater than 1 − δ, or whether for every such state |ϕ , the acceptance probability of C|ϕ is less than δ. As a consequence, under the widely believed assumption that QMA = NP, we have that quantum witnesses are strictly more powerful than classical witnesses with respect to Merlin-Arthur protocols in which the verifier is a quantum circuit of logarithmic treewidth.
In this work we provide algorithmic solutions to five fundamental problems concerning the verific... more In this work we provide algorithmic solutions to five fundamental problems concerning the verification, synthesis and correction of concurrent systems that can be modeled by bounded p/t-nets. We express concurrency via partial orders and assume that behavioral specifications are given via monadic second order logic. A c-partial-order is a partial order whose Hasse diagram can be covered by c paths. For a finite set T of transitions, we let P(c, T, ϕ) denote the set of all T-labelled c-partial-orders satisfying ϕ. If N = (P, T) is a p/t-net we let P(N, c) denote the set of all c-partially-ordered runs of N. A (b, r)-bounded p/t-net is a b-bounded p/t-net in which each place appears repeated at most r times. We solve the following problems: 1. Verification: given an MSO formula ϕ and a bounded p/t-net N determine whether P(N, c) ⊆ P(c, T, ϕ), whether P(c, T, ϕ) ⊆ P(N, c), or whether P(N, c) ∩ P(c, T, ϕ) = ∅. 2. Synthesis from MSO Specifications: given an MSO formula ϕ, synthesize a semantically minimal (b, r)-bounded p/t-net N satisfying P(c, T, ϕ) ⊆ P(N, c). 3. Semantically Safest Subsystem: given an MSO formula ϕ defining a set of safe partial orders, and a b-bounded p/t-net N , possibly containing unsafe behaviors, synthesize the safest (b, r)-bounded p/t-net N ′ whose behavior lies in between P(N, c) ∩ P(c, T, ϕ) and P(N, c). 4. Behavioral Repair: given two MSO formulas ϕ and ψ, and a b-bounded p/t-net N , synthesize a semantically minimal (b, r)-bounded p/t net N ′ whose behavior lies in between P(N, c) ∩ P(c, T, ϕ) and P(c, T, ψ). 5. Synthesis from Contracts: given an MSO formula ϕ yes specifying a set of good behaviors and an MSO formula ϕ no specifying a set of bad behaviors, synthesize a semantically minimal (b, r)-bounded p/t-net N such that P(c, T, ϕ yes) ⊆ P(N, c) but P(c, T, ϕ no) ∩ P(N, c) = ∅.
Graph Amalgamation Under Logical Constraints
We say that a graph G is an H-amalgamation of graphs \(G_1\) and \(G_2\) if G can be obtained by ... more We say that a graph G is an H-amalgamation of graphs \(G_1\) and \(G_2\) if G can be obtained by gluing \(G_1\) and \(G_2\) along isomorphic copies of H. In the amalgamation recognition problem we are given connected graphs \(H,G_1,G_2,G\) and the goal is to determine whether G is an H-amalgamation of \(G_1\) and \(G_2\). Our main result states that amalgamation recognition can be solved in time \(2^{O(\varDelta \cdot t)}\cdot n^{O(t)}\) where \(n,t,\varDelta \) are the number of vertices, the treewidth and the maximum degree of G respectively.
ArXiv, 2021
In the Intersection Non-emptiness problem, we are given a list of finite automata A1, A2, . . . ,... more In the Intersection Non-emptiness problem, we are given a list of finite automata A1, A2, . . . , Am over a common alphabet Σ as input, and the goal is to determine whether some string w ∈ Σ∗ lies in the intersection of the languages accepted by the automata in the list. We analyze the complexity of the Intersection Non-emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-Thérien hierarchy. Automata accepting languages from the lowest levels of these hierarchies arise naturally in the context of model checking. We identify a dichotomy in the dot-depth hierarchy by showing that the problem is already NP-complete when all input automata accept languages of the levels B0 or B1/2 and already PSPACE-hard when all automata accept a language from the level B1. Conversely, we identify a tetrachotomy in the Straubing-Thérien hierarchy. More precisely, we show that the problem is in AC0 when rest...
arXiv: Quantum Physics, 2010
We prove that the family of embezzlement states defined by van Dam and Hayden [vanDamHayden2002] ... more We prove that the family of embezzlement states defined by van Dam and Hayden [vanDamHayden2002] is universal for both quantum and classical entangled two-prover non-local games with an arbitrary number of rounds. More precisely, we show that for each $\epsilon>0$ and each strategy for a k-round two-prover non-local game which uses a bipartite shared state on 2m qubits and makes the provers win with probability $\omega$, there exists a strategy for the same game which uses an embezzlement state on $2m + 2m/\epsilon$ qubits and makes the provers win with probability $\omega-\sqrt{2\epsilon}$. Since the value of a game can be defined as the limit of the value of a maximal 2m-qubit strategy as m goes to infinity, our result implies that the classes QMIP*_{c,s}[2,k] and MIP*_{c,s}[2,k] remain invariant if we allow the provers to share only embezzlement states, for any completeness value c in [0,1] and any soundness value s < c. Finally we notice that the circuits applied by each p...
In the Maximum-Duo Preservation String Mapping (Max-Duo PSM) problem, the input consists of two r... more In the Maximum-Duo Preservation String Mapping (Max-Duo PSM) problem, the input consists of two related strings A and B of length n and a nonnegative integer k. The objective is to determine whether there exists a mapping m from the set of positions of A to the set of positions of B that maps only to positions with the same character and preserves at least k duos, which are pairs of adjacent positions. We develop a randomized algorithm that solves MaxDuo PSM in time 4 · nO(1), and a deterministic algorithm that solves this problem in time 6.855 · nO(1). The previous best known (deterministic) algorithm for this problem has running time (8e)2k+o(k) ·nO(1) [Beretta et al., Theor. Comput. Sci. 2016]. We also show that Max-Duo PSM admits a problem kernel of size O(k3), improving upon the previous best known problem kernel of size O(k6). 1998 ACM Subject Classification G.2.1 Combinatorics, F.2 Analysis of Agorithms and Problem Complexity
It can be shown that each permutation group $G \sqsubseteq S_n$ can be embedded, in a well define... more It can be shown that each permutation group $G \sqsubseteq S_n$ can be embedded, in a well defined sense, in a connected graph with $O(n+|G|)$ vertices. Some groups, however, require much fewer vertices. For instance, $S_n$ itself can be embedded in the $n$-clique $K_n$, a connected graph with n vertices. In this work, we show that the minimum size of a context-free grammar generating a finite permutation group $G \sqsubseteq S_n$ can be upper bounded by three structural parameters of connected graphs embedding $G$: the number of vertices, the treewidth, and the maximum degree. More precisely, we show that any permutation group $G \sqsubseteq S_n$ that can be embedded into a connected graph with $m$ vertices, treewidth k, and maximum degree $\Delta$, can also be generated by a context-free grammar of size $2^{O(k\Delta\log\Delta)}\cdot m^{O(k)}$. By combining our upper bound with a connection between the extension complexity of a permutation group and the grammar complexity of a for...
In this work we study the validity problem in equational logic from the perspective of parameteri... more In this work we study the validity problem in equational logic from the perspective of parameterized complexity theory. We introduce a variant of equational logic in which sentences are pairs of the form \((t_1 =t_2,\omega )\), where \(t_1 =t_2\) is an equation, and \(\omega \) is an arbitrary ordering of the positions corresponding to subterms of \(t_1\) and \(t_2\). We call such pairs ordered equations. With each ordered equation, one may naturally associate a notion of width, and with each proof of validity of an ordered equation, one may naturally associate a notion of depth. We define the width of such a proof as the maximum width of an ordered equation occurring in it. Finally, we introduce a parameter b that restricts the way in which variables are substituted for terms. We say that a proof is b-bounded if all substitutions used in it satisfy such restriction.
ArXiv, 2014
In this work we provide algorithmic solutions to five fundamental problems concerning the verific... more In this work we provide algorithmic solutions to five fundamental problems concerning the verification, synthesis and correction of concurrent systems that can be modeled by bounded p/t-nets. We express concurrency via partial orders and assume that behavioral specifications are given via monadic second order logic. A c-partial-order is a partial order whose Hasse diagram can be covered by c paths. For a finite set T of transitions, we let P(c,T,\phi) denote the set of all T-labelled c-partial-orders satisfying \phi. If N=(P,T) is a p/t-net we let P(N,c) denote the set of all c-partially-ordered runs of N. A (b, r)-bounded p/t-net is a b-bounded p/t-net in which each place appears repeated at most r times. We solve the following problems: 1. Verification: given an MSO formula \phi and a bounded p/t-net N determine whether P(N,c)\subseteq P(c,T,\phi), whether P(c,T,\phi)\subseteq P(N,c), or whether P(N,c)\cap P(c,T,\phi)=\emptyset. 2. Synthesis from MSO Specifications: given an MSO f...
In the Steiner tree problem, the input consists of an edge-weighted graph G together with a set S... more In the Steiner tree problem, the input consists of an edge-weighted graph G together with a set S of terminal vertices. The goal is to find a minimum weight tree in G that spans all terminals. This fundamental NP-hard problem has direct applications in many subfields of combinatorial optimization, such as planning, scheduling, etc. In this work we introduce a new heuristic for the Steiner tree problem, based on a simple routine for improving the cost of sub-optimal Steiner trees: first, the sub-optimal tree is split into three connected components, and then these components are reconnected by using an algorithm that computes an optimal Steiner tree with 3-terminals (the roots of the three components). We have implemented our heuristic into a solver and compared it with several state-of-the-art solvers on well-known data sets. Our solver performs very well across all the data sets, and outperforms most of the other benchmarked solvers on very large graphs, which have been either obta...
In this work we study the relationship between size and treewidth of circuits computing variants ... more In this work we study the relationship between size and treewidth of circuits computing variants of the element distinctness function. First, we show that for each n, any circuit of treewidth t computing the element distinctness function delta_n:{0,1}^n -> {0,1} must have size at least Omega((n^2)/(2^{O(t)}*log(n))). This result provides a non-trivial generalization of a super-linear lower bound for the size of Boolean formulas (treewidth 1) due to Neciporuk. Subsequently, we turn our attention to read-once circuits, which are circuits where each variable labels at most one input vertex. For each n, we show that any read-once circuit of treewidth t and size s computing a variant tau_n:{0,1}^n -> {0,1} of the element distinctness function must satisfy the inequality t * log(s) >= Omega(n/log(n)). Using this inequality in conjunction with known results in structural graph theory, we show that for each fixed graph H, read-once circuits computing tau_n which exclude H as a mino...
ArXiv, 2019
In this work we introduce the notion of decisional width of a finite relational structure and the... more In this work we introduce the notion of decisional width of a finite relational structure and the notion of regular-decisional width of a regular class of finite structures. Our main result states that the first-order theory of any regular-decisional class of finite structures is decidable. Building on the proof of this decidability result, we show that the problem of counting satisfying assignments for a first-order logic formula in a structure of constant width is fixed parameter tractable when parameterized by the width parameter and can be solved in quadratic time with respect to the length of the input representation of the structure.
In its most traditional setting, the main concern of optimization theory is the search for optima... more In its most traditional setting, the main concern of optimization theory is the search for optimal solutions for instances of a given computational problem. A recent trend of research in artificial intelligence, called solution diversity, has focused on the development of notions of optimality that may be more appropriate in settings where subjectivity is essential. The idea is instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms outputting a small set of sufficiently good solutions that are sufficiently diverse from one another. This way, the user has the opportunity to choose the solution being most appropriate to the context at hand. It also displays the richness of the solution space. When combined with techniques from parameterized complexity theory, the paradigm of diversity of solutions offers a powerful algorithmic framework to address problems of practical relevance. In this work, we investigate the ...
We are studying a weighted version of a linear extension problem, given some finite partial order... more We are studying a weighted version of a linear extension problem, given some finite partial order ρ, called Completion of an Ordering. While this problem is NP-complete, we show that it lies in FPT when parameterized by the interval width of ρ. This ordering problem can be used to model several ordering problems stemming from diverse application areas, such as graph drawing, computational social choice, or computer memory management. Each application yields a special ρ. We also relate the interval width of ρ to parameterizations such as maximum range that have been introduced earlier in these applications, sometimes improving on parameterized algorithms that have been developed for these parameterizations before. This approach also gives some practical sub-exponential time algorithms for ordering problems. 2012 ACM Subject Classification Theory of computation → Fixed parameter tractability; Theory of computation → Dynamic programming; Mathematics of computing → Combinatorial optimiz...
We introduce the notion of monotone linear-programming circuits (MLP circuits), a model of comput... more We introduce the notion of monotone linear-programming circuits (MLP circuits), a model of computation for partial Boolean functions. Using this model, we prove the following results. 1. MLP circuits are superpolynomially stronger than monotone Boolean circuits. 2. MLP circuits are exponentially stronger than monotone span programs. 3. MLP circuits can be used to provide monotone feasibility interpolation theorems for Lovasz-Schrijver proof systems, and for mixed Lovasz-Schrijver proof systems. 4. The Lovasz-Schrijver proof system cannot be polynomially simulated by the cutting planes proof system. This is the first result showing a separation between these two proof systems. Finally, we discuss connections between the problem of proving lower bounds on the size of MLPs and the problem of proving lower bounds on extended formulations of polytopes.
A fundamental drawback that arises when one is faced with the task of deterministically certifyin... more A fundamental drawback that arises when one is faced with the task of deterministically certifying solutions to computational 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 precisely, we introduce a PSPACE-hard notion of symbolic constraint satisfaction problem where both instances and solutions for these instances are implicitly represented by ordered decision diagrams (i.e. read-once, oblivious, branching programs). Our main result states that given an ordered decision diagram D of length k and width w specifying a CSP instance, 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 co...
Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof t... more Let CMSO denote the counting monadic second order logic of graphs. We give a constructive proof that for some computable function f, there is an algorithm A that takes as input a CMSO sentence F, a positive integer t, and a connected graph G of maximum degree at most D, and determines, in time f(|F|,t)*2^O(D*t)*|G|^O(t), whether G has a supergraph G' of treewidth at most t such that G' satisfies F. The algorithmic metatheorem described above sheds new light on certain unresolved questions within the framework of graph completion algorithms. In particular, using this metatheorem, we provide an explicit algorithm that determines, in time f(d)*2^O(D*d)*|G|^O(d), whether a connected graph of maximum degree D has a planar supergraph of diameter at most d. Additionally, we show that for each fixed k, the problem of determining whether G has a k-outerplanar supergraph of diameter at most d is strongly uniformly fixed parameter tractable with respect to the parameter d. This result ...
Automated Deduction – CADE 27: 27th International Conference on Automated Deduction, Natal, Brazil, August 27–30, 2019, Proceedings
Automated Deduction – CADE 27, 2019
Automated Reasoning for Security Protocols
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Papers by Mateus de Oliveira Oliveira