Equational reasoning is one of the most intuitive and widely used types of symbolic reasoning. In this setting, the goal is to determine whether a given ground equation t=t' follows as a consequence of a set of equational axioms E using the process of replacing equals with equals. An equation t=t' is variable-preserving if each variable occurring in t occurs in t' and vice versa. Such equations capture essential algebraic properties, such as commutativity, associativity and distributivity, which arise in a wide variety of contexts. In this work, we show that for each fixed set E of variable-preserving equations, the set of ground equations derivable from E in depth at most d is soundly over-approximable in fixed-parameter tractable time. More specifically, we devise an algorithm that takes as input a set E of variable-preserving equations and a target ground equation t=t', and always halts with a YES or NO answer. 1) If equation t=t' can be derived from E in depth at most d, the algorithm always halts with a YES. 2) If equation t=t' does not belong to the equational closure of E, then the algorithm always halts with a NO. In other words, the set of YES instances contains the set of ground equations that can be deduced from E in depth at most d, and possibly other equations that require derivations of higher depth. However, this set contains no ground equation that is not in the equational closure of E. For this reason, the algorithm is sound. Our algorithm works in time f(d) * |t| * |t'|, where |t| and |t'| are the number of symbols in t and t' respectively, d is the depth parameter, and f(d) is a function whose growth depends only on d and on parameters associated with the equations in E.

Functional decomposition is the process of breaking down a function f into a composition f=g(f_1,...,f_k) of simpler functions f_1,...,f_k belonging to some class F. This fundamental notion can be used to model applications arising in a wide variety of contexts, ranging from machine learning to formal language theory. In this work, we study functional decomposition by leveraging on the notion of functional reconfiguration. In this setting, constraints are imposed not only on the factor functions f_1,...,f_k but also on the intermediate functions arising during the composition process. We introduce a symbolic framework to address functional reconfiguration and decomposition problems. In our framework, functions arising during the reconfiguration process are represented symbolically, using ordered binary decision diagrams (OBDDs). The function g used to specify the reconfiguration process is represented by a Boolean circuit C. Finally, the function class F is represented by a second-order finite automaton A. Our main result states that functional reconfiguration, and hence functional decomposition, can be solved in fixed-parameter linear time when parameterized by the width of the input OBDD, by structural parameters associated with the reconfiguration circuit C, and by the size of the second-order finite automaton A.

One of the most fundamental problems in computational learning theory is the problem of learning a finite automaton A consistent with a finite set P of positive examples and with a finite set N of negative examples. By consistency, we mean that A accepts all strings in P and rejects all strings in N. It is well known that this problem is NP-complete. In the literature, it is stated that NP-hardness holds even in the case of a binary alphabet. As a standard reference for this theorem, the work of Gold from 1978 is either cited or adapted. Nevertheless, the results in Gold's work are stated in terms of Mealy machines, and not in terms of deterministic finite automata (DFAs) as most commonly defined. As Mealy machines are equipped with an output function, they can be more compact than DFAs which accept the same language. We show that the adaptations of Gold's construction for Mealy machines stated in the literature have some issues, and provide a correct proof for the fact that the DFA-consistency problem for binary alphabets is NP-complete.

A central computational problem in the realm of automata theory is the problem of determining whether a finite automaton A has a synchronizing word. This problem has found applications in a variety of subfields of artificial intelligence, including planning, robotics, and multi-agent systems. In this work, we study this problem within the framework of diversity of solutions, an up-and-coming trend in the field of artificial intelligence where the goal is to compute a set of solutions that are sufficiently distinct from one another. We define a notion of diversity of solutions that is suitable for contexts were solutions are strings that may have distinct lengths. Using our notion of diversity, we show that for each fixed r ∈ N, each fixed finite automaton A, and each finite automaton B given at the input, the problem of determining the existence of a diverse set {w1,w2, . . . ,wr} ⊆ L(B) of words that are synchronizing for A can be solved in polynomial time. Finally, we generalize this result to the realm of conformant planning, where the goal is to devise plans that achieve a goal irrespectively of initial conditions and of nondeterminism that may occur during their execution.

In the field of parameterized complexity theory, the study of graph width measures has been intimately connected with the development of width-based model checking algorithms for combinatorial properties on graphs. In this work, we introduce a general framework to convert a large class of width-based model-checking algorithms into algorithms that can be used to test the validity of graph-theoretic conjectures on classes of graphs of bounded width. Our framework is modular and can be applied with respect to several well-studied width measures for graphs, including treewidth and cliquewidth. As a quantitative application of our framework, we prove analytically that for several long-standing graph-theoretic conjectures, there exists an algorithm that takes a number k as input and correctly determines in time double-exponential in a polynomial of k whether the conjecture is valid on all graphs of treewidth at most k. These upper bounds, which may be regarded as upper-bounds on the size of proofs/disproofs for these conjectures on the class of graphs of treewidth at most k, improve significantly on theoretical upper bounds obtained using previously available techniques.

Abstract Petri nets are one of the most prominent system-level formalisms for the specification of causality in concurrent, distributed, or multi-agent systems. This formalism is abstract enough to be analyzed using theoretical tools, and at the same time, concrete enough to eliminate ambiguities that would arise at implementation level. One interesting feature of Petri nets is that they can be studied from the point of view of true concurrency, where causal scenarios are specified using partial orders, instead of approaches based on interleaving. On the other hand, message sequence chart (MSC) languages, are a standard formalism for the specification of causality from a purely behavioral perspective. In other words, this formalism specifies a set of causal scenarios between actions of a system, without providing any implementation-level details about the system. In this work, we establish several new connections between MSC languages and Petri nets, and show that several computational problems involving these formalisms are decidable. Our results fill some gaps in the literature that had been open for several years. To obtain our results we develop new techniques in the realm of slice automata theory, a framework introduced one decade ago in the study of the partial order behavior of bounded Petri nets. These techniques can also be applied to establish connections between Petri nets and other well studied behavioral formalisms, such as the notion of Mazurkiewicz trace languages.

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 (). Two closely related open problems are point-to-point and edge-to-edge reachability for .

Abstract Traditionally, finite automata theory has been used as a framework for the representation of possibly infinite sets of strings. In this work, we introduce the notion of second-order finite automata, a formalism that combines finite automata with ordered decision diagrams, with the aim of representing possibly infinitesets of setsof strings. Our main result states that second-order finite automata can be canonized with respect to the second-order languages they represent. Using this canonization result, we show that sets of sets of strings represented by second-order finite automata are closed under the usual Boolean operations, such as union, intersection, difference and even under a suitable notion of complementation. Additionally, emptiness of intersection and inclusion are decidable. We provide two algorithmic applications for second-order automata. First, we show that several width/size minimization problems for deterministic and nondeterministic ODDs are solvable in fixed-parameter tractable time when parameterized by the width of the input ODD. In particular, our results imply FPT algorithms for corresponding width/size minimization problems for ordered binary decision diagrams (OBDDs) with a fixed variable ordering. Previously, only algorithms that take exponential time in the size of the input OBDD were known for width minimization, even for OBDDs of constant width. Second, we show that for eachkandwone can count the number of distinct functions computable by ODDs of width at mostwand lengthkin timeh(|Σ|,w) ⋅kO(1), for a suitable $h:\mathbb {N}\times \mathbb {N}\rightarrow \mathbb {N}$ h:ℕ×ℕ→ℕ . This improves exponentially on the time necessary to explicitly enumerate all such functions, which is exponential in both the width parameterwand in the lengthkof the ODDs.

In the Intersection Non-Emptiness problem, we are given a list of finite automata $A_1,A_2,\dots,A_m$ over a common alphabet $Σ$ as input, and the goal is to determine whether some string $w\in Σ^*$ 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 zero or one half and already PSPACE-hard when all automata accept a language from the level one. Conversely, we identify a tetrachotomy in the Straubing-Thérien hierarchy. More precisely, we show that the problem is in AC$^0$ when restricted to level zero; complete for LOGSPACE or NLOGSPACE, depending on the input representation, when restricted to languages in the level one half; NP-complete when the input is given as DFAs accepting a language in from level one or three half; and finally, PSPACE-complete when the input automata accept languages in level two or higher. Moreover, we show that the proof technique used to show containment in NP for DFAs accepting languages in the Straubing-Thérien hierarchy levels one ore three half does not generalize to the context of NFAs. To prove this, we identify a family of languages that provide an exponential separation between the state complexity of general NFAs and that of partially ordered NFAs. To the best of our knowledge, this is the first superpolynomial separation between these two models of computation.

Bounded width branching programs are a formalism that can be used to capture the notion of non-uniform constant-space computation. In this work, we study a generalized version of bounded width branching programs where instructions are defined by unitary matrices of bounded dimension. We introduce a new learning framework for these branching programs that leverages on a combination of local search techniques with gradient descent over Riemannian manifolds. We also show that gapped, read-once branching programs of bounded dimension can be learned with a polynomial number of queries in the presence of a teacher. Finally, we provide explicit near-quadratic size lower-bounds for bounded-dimension unitary branching programs, and exponential size lower-bounds for bounded-dimension read-once gapped unitary branching programs. The first lower bound is proven using a combination of Neciporuk’s lower bound technique with classic results from algebraic geometry. The second lower bound is proven within the framework of communication complexity theory.

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 that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is 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 impact of this combination in the field of Kemeny Rank Aggregation, a well-studied class of problems lying in the intersection of order theory and social choice theory and also in the field of order theory itself. In particular, we show that KRA is fixed-parameter tractable with respect to natural parameters providing natural formalizations of the notions of diversity and of the notion of a sufficiently good solution. Our main results work both when considering the traditional setting of aggregation over linearly ordered votes, and in the more general setting where votes are partially ordered.

In this work, we consider the following order reconfiguration problem: Given a graph G together with linear orders ω and ω' of the vertices of G, can one transform ω into ω' by a sequence of swaps of adjacent elements in such a way that at each time step the resulting linear order has cutwidth (pathwidth) at most k? We show that this problem always has an affirmative answer when the input linear orders ω and ω' have cutwidth (pathwidth) at most k/2. Using this result, we establish a connection between two apparently unrelated problems: the reachability problem for two-letter string rewriting systems and the graph isomorphism problem for graphs of bounded cutwidth. This opens an avenue for the study of the famous graph isomorphism problem using techniques from term rewriting theory.

Clique-width and treewidth are two of the most important and useful graph parameters, and several problems can be solved efficiently when restricted to graphs of bounded clique-width or treewidth. Bounded treewidth implies bounded clique-width, but not vice versa. Problems like Longest Cycle, Longest Path, MaxCut, Edge Dominating Set, and Graph Coloring are fixed-parameter tractable when parameterized by the treewidth, but they cannot be solved in FPT time when parameterized by the clique-width unless FPT = W[1], as shown by Fomin, Golovach, Lokshtanov, and Saurabh [SIAM J. Comput. 2010, SIAM J. Comput. 2014]. For a given problem that is fixed-parameter tractable when parameterized by treewidth, but intractable when parameterized by clique-width, there may exist infinite families of instances of bounded clique-width and unbounded treewidth where the problem can be solved efficiently. In this work, we initiate a systematic study of the parameters co-treewidth (the treewidth of the complement of the input graph) and co-degeneracy (the degeneracy of the complement of the input graph). We show that Longest Cycle, Longest Path, and Edge Dominating Set are FPT when parameterized by co-degeneracy. On the other hand, Graph Coloring is para-NP-complete when parameterized by co-degeneracy but FPT when parameterized by the co-treewidth. Concerning MaxCut, we give an FPT algorithm parameterized by co-treewidth, while we leave open the complexity of the problem parameterized by co-degeneracy. Additionally, we show that Precoloring Extension is fixed-parameter tractable when parameterized by co-treewidth, while this problem is known to be W[1]-hard when parameterized by treewidth. These results give evidence that co-treewidth is a useful width parameter for handling dense instances of problems for which an FPT algorithm for clique-width is unlikely to exist. Finally, we develop an algorithmic framework for co-degeneracy based on the notion of Bondy-Chvátal closure.

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 obtained from real-world applications or from randomly generated data sets.

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 $\mathfrak{A}$ that takes as input a CMSO sentence $\varphi$, a positive integer $t$, and a connected graph $G$ of maximum degree at most $\Delta$, and determines, in time $f(|\varphi|,t)\cdot 2^{O(\Delta \cdot t)}\cdot |G|^{O(t)}$, whether $G$ has a supergraph $G'$ of treewidth at most $t$ such that $G'\models \varphi$. 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 $f(d)\cdot 2^{O(\Delta \cdot d)}\cdot |G|^{O(d)}$, whether a connected graph of maximum degree $\Delta$ has a planar supergraph of diameter at most $d$. Additionally, we show that for each fixed $k$, the problem of determining whether $G$ has an $k$-outerplanar supergraph of diameter at most $d$ is strongly uniformly fixed parameter tractable with respect to the parameter $d$. This result can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter $\mathbf{p}$. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth.

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.

When modeling an application of practical relevance as an instance of a combinatorial problem X, we are often interested not merely in finding one optimal solution for that instance, but in finding a sufficiently diverse collection of good solutions. In this work we initiate a systematic study of diversity from the point of view of fixed-parameter tractability theory. We consider an intuitive notion of diversity of a collection of solutions which suits a large variety of combinatorial problems of practical interest. Our main contribution is an algorithmic framework which --automatically-- converts a tree-decomposition-based dynamic programming algorithm for a given combinatorial problem X into a dynamic programming algorithm for the diverse version of X. Surprisingly, our algorithm has a polynomial dependence on the diversity parameter.

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 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 Turing machines working in nondeterministic linear space. For such problems, our main result immediately yields an algorithm 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.

It can be shown that each permutation group G ⊑ 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⊑ 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 ⊑ 𝕊_n that can be embedded into a connected graph with m vertices, treewidth k, and maximum degree Δ, can also be generated by a context-free grammar of size 2^{O(kΔlogΔ)}⋅ m^{O(k)}. By combining our upper bound with a connection established by Pesant, Quimper, Rousseau and Sellmann [Gilles Pesant et al., 2009] between the extension complexity of a permutation group and the grammar complexity of a formal language, we also get that these permutation groups can be represented by polytopes of extension complexity 2^{O(kΔlogΔ)}⋅ m^{O(k)}. The above upper bounds can be used to provide trade-offs between the index of permutation groups, and the number of vertices, treewidth and maximum degree of connected graphs embedding these groups. In particular, by combining our main result with a celebrated 2^{Ω(n)} lower bound on the grammar complexity of the symmetric group 𝕊_n due to Glaister and Shallit [Glaister and Shallit, 1996] we have that connected graphs of treewidth o(n/log n) and maximum degree o(n/log n) embedding subgroups of 𝕊_n of index 2^{cn} for some small constant c must have n^{ω(1)} vertices. This lower bound can be improved to exponential on graphs of treewidth n^{ε} for ε < 1 and maximum degree o(n/log n).

In the embedded planar diameter improvement problem (EPDI) we are given a graph G embedded in the plane and a positive integer d. The goal is to determine whether one can add edges to the planar embedding of G in such a way that planarity is preserved and in such a way that the resulting graph has diameter at most d. Using non-constructive techniques derived from Robertson and Seymour's graph minor theory, together with the effectivization by self-reduction technique introduced by Fellows and Langston, one can show that EPDI can be solved in time f(d)* |V(G)|^{O(1)} for some function f(d). The caveat is that this algorithm is not strongly uniform in the sense that the function f(d) is not known to be computable. On the other hand, even the problem of determining whether EPDI can be solved in time f_1(d)* |V(G)|^{f_2(d)} for computable functions f_1 and f_2 has been open for more than two decades [Cohen at. al. Journal of Computer and System Sciences, 2017]. In this work we settle this later problem by showing that EPDI can be solved in time f(d)* |V(G)|^{O(d)} for some computable function f. Our techniques can also be used to show that the embedded k-outerplanar diameter improvement problem (k-EOPDI), a variant of EPDI where the resulting graph is required to be k-outerplanar instead of planar, can be solved in time f(d)* |V(G)|^{O(k)} for some computable function f. This shows that for each fixed k, the problem k-EOPDI is strongly uniformly fixed parameter tractable with respect to the diameter parameter d.

We show that any quantum circuit of treewidth t, built from r-qubit gates, requires at least gates to compute the element distinctness function. Our result generalizes a near-quadratic lower bound for quantum formula size obtained by Roychowdhury and Vatan [SIAM J. on Computing, 2001]. The proof of our lower bound follows by an extension of Nečiporuk's method to the context of quantum circuits of constant treewidth. This extension is made via a combination of techniques from structural graph theory, tensor-network theory, and the connected-component counting method, which is a classic tool in algebraic geometry. In particular, an essential key to proving our lower bound is the development of a new algorithm for tensor network contraction which may be of independent interest.

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.

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 Max-Duo PSM in time 4^k * n^{O(1)}, and a deterministic algorithm that solves this problem in time 6.855^k * n^{O(1)}. The previous best known (deterministic) algorithm for this problem has running time (8e)^{2k+o(k)} * n^{O(1)} [Beretta et al., Theor. Comput. Sci. 2016]. We also show that Max-Duo PSM admits a problem kernel of size O(k^3), improving upon the previous best known problem kernel of size O(k^6).

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 can be generalized in two directions. First, the diameter parameter can be replaced by any contraction-closed effectively CMSO-definable parameter p. Examples of such parameters are vertex-cover number, dominating number, and many other contraction-bidimensional parameters. In the second direction, the planarity requirement can be relaxed to bounded genus, and more generally, to bounded local treewidth.

The ground term reachability problem consists in determining whether a given variable-free term t can be transformed into a given variable-free term t' by the application of rules from a term rewriting system R. The joinability problem, on the other hand, consists in determining whether there exists a variable-free term t'' which is reachable both from t and from t'. Both problems have proven to be of fundamental importance for several subfields of computer science. Nevertheless, these problems are undecidable even when restricted to linear term rewriting systems. In this work, we approach reachability and joinability in linear term rewriting systems from the perspective of parameterized complexity theory, and show that these problems are fixed parameter tractable with respect to the depth of derivations. More precisely, we consider a notion of parallel rewriting, in which an unbounded number of rules can be applied simultaneously to a term as long as these rules do not interfere with each other. A term t_1 can reach a term t_2 in depth d if t_2 can be obtained from t_1 by the application of d parallel rewriting steps. Our main result states that for some function f(R,d), and for any linear term rewriting system R, one can determine in time f(R,d)*|t_1|*|t_2| whether a ground term t_2 can be reached from a ground term t_1 in depth at most d by the application of rules from R. Additionally, one can determine in time f(R,d)^2*|t_1|*|t_2| whether there exists a ground term u, such that u can be reached from both t_1 and t_2 in depth at most d. Our algorithms improve exponentially on exhaustive search, which terminates in time 2^{|t_1|*2^{O(d)}}*|t_2|, and can be applied with regard to any linear term rewriting system, irrespective of whether the rewriting system in question is terminating or confluent.

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 minor must have size at least Omega(n^2/log^{4}(n)). For certain well studied functions, such as the triangle-freeness function, this last lower bound can be improved to Omega(n^2/log^2(n)).

In [18] Lorenz and Juhás raised the question of whether there exists a suitable formalism for the representation of infinite families of partial orders generated by Petri nets. Restricting ourselves to bounded p/t-nets, we propose Hasse diagram gener