arXiv subjects

Antonio E. Porreca

Publications and source records attributed to Antonio E. Porreca.

14 recordsLinked to original sources

arXiv2026

Injective and pseudo-injective polynomial equations: From permutations to dynamical systems

We study the computational complexity of decomposing finite discrete dynamical systems (FDDSs) in terms of the semiring operations of alternative and synchronous execution, which is useful for the analysis of discrete phenomena in science and engineering. More specifically, we investigate univariate polynomials of the form $P(X) = B$, that is with a constant side, first over the subsemiring of permutations and then over general FDDSs. We find a characterization of injective polynomials $P$ and efficient algorithms for solving the associated equations. Then, we introduce the more general notion of pseudo-injective polynomial, which is based on a condition on the lengths of the limit cycles of its coefficients, and prove that the corresponding equations are also solvable efficiently. These results also apply even when permutations are encoded in an exponentially more compact way.

cs.DM↗

arXiv2025

Solving "pseudo-injective" polynomial equations over finite dynamical systems

We consider the semiring of abstract finite dynamical systems up to isomorphism, with the operations of alternative and synchronous execution. We continue searching for efficient algorithms for solving polynomial equations of the form $P(X) = B$, with a constant side B, with the goal of decomposing complex behaviors into simpler systems. Taking inspiration from the characterization of injective polynomials P over dynamical systems, which is based on a condition on the lengths of limit cycles of their coefficients, we introduce a more general notion of pseudo-injectivity by relaxing this constraint. We prove that the associated equations can be solved efficiently, even in certain cases where the input is encoded in an exponentially more compact way.

cs.DM↗

arXiv2025

Roots in the semiring of finite deterministic dynamical systems

Finite discrete-time dynamical systems (FDDS) model phenomena that evolve deterministically in discrete time. It is possible to define sum and product operations on these systems (disjoint union and direct product, respectively) giving a commutative semiring. This algebraic structure led to several works employing polynomial equations to model hypotheses on phenomena modelled using FDDS. To solve these equations, algorithms for performing the division and computing $k$-th roots are needed. In this paper, we propose two polynomial algorithms for these tasks, under the condition that the result is a connected FDDS. This ultimately leads to an efficient solution to equations of the type $AX^k=B$ for connected $X$. These results are some of the important final steps for solving more general polynomial equations on FDDS.

cs.DM↗

arXiv2025

Injectivity of polynomials over finite discrete dynamical systems

The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering. The decomposition of dynamics into simpler subsystems allows us to simplify this analysis (or design). Here we focus on an algebraic approach to decomposition, based on alternative and synchronous execution as the sum and product operations; this gives rise to polynomial equations (with a constant side). In this article we focus on univariate, injective polynomials, giving a characterization in terms of the form of their coefficients and a polynomial-time algorithm for solving the associated equations.

cs.DM↗

arXiv2024

Unconventional complexity classes in unconventional computing (extended abstract)

Many unconventional computing models, including some that appear to be quite different from traditional ones such as Turing machines, happen to characterise either the complexity class P or PSPACE when working in deterministic polynomial time (and in the maximally parallel way, where this applies). We discuss variants of cellular automata and membrane systems that escape this dichotomy and characterise intermediate complexity classes, usually defined in terms of Turing machines with oracles, as well as some possible reasons why this happens.

cs.CC↗

arXiv2024

Polynomial-delay generation of functional digraphs up to isomorphism

We describe a procedure for the generation of functional digraphs up to isomorphism; these are digraphs with uniform outdegree 1, also called mapping patterns, finite endofunctions, or finite discrete-time dynamical systems. This procedure is based on a reverse search algorithm for the generation of connected functional digraphs, which is then applied as a subroutine for the generation of arbitrary ones. Both algorithms output solutions with $O(n^2)$ delay and require linear space with respect to the number $n$ of vertices.

cs.DS↗

arXiv2024

Decomposition and factorisation of transients in Functional Graphs

Functional graphs (FGs) model the graph structures used to analyse the behaviour of functions from a discrete set to itself. In turn, such functions are used to study real complex phenomena evolving in time. As the systems involved can be quite large, it is interesting to decompose and factorise them into several subgraphs acting together. Polynomial equations over functional graphs provide a formal way to represent this decomposition and factorisation mechanism, and solving them validates or invalidates hypotheses on their decomposability. The current solution method breaks down a single equation into a series of basic equations of the form AxX = B (with A, X, and B being FGs) to identify the possible solutions. However, it is able to consider just FGs made of cycles only. This work proposes an algorithm for solving these basic equations for general connected FGs. By exploiting a connection with the cancellation problem, we prove that the upper bound to the number of solutions is closely related to the size of the cycle in the coefficient A of the equation. The cancellation problem is also involved in the main algorithms provided by the paper. We introduce a polynomial-time semi-decision algorithm able to provide constraints that a potential solution will have to satisfy if it exists. Then, exploiting the ideas introduced in the first algorithm, we introduce a second exponential-time algorithm capable of finding all solutions by integrating several 'hacks' that try to keep the exponential as tight as possible.

math.DS↗

arXiv2022

Profiles of dynamical systems and their algebra

The commutative semiring $\mathbf{D}$ of finite, discrete-time dynamical systems was introduced in order to study their (de)composition from an algebraic point of view. However, many decision problems related to solving polynomial equations over $\mathbf{D}$ are intractable (or conjectured to be so), and sometimes even undecidable. In order to take a more abstract look at those problems, we introduce the notion of "topographic" profile of a dynamical system $(A,f)$ with state transition function $f \colon A \to A$ as the sequence $\mathop{\mathrm{prof}} A = (|A|_i)_{i \in \mathbb{N}}$, where $|A|_i$ is the number of states having distance $i$, in terms of number of applications of $f$, from a limit cycle of $(A,f)$. We prove that the set of profiles is also a commutative semiring $(\mathbf{P},+,\times)$ with respect to operations compatible with those of $\mathbf{D}$ (namely, disjoint union and tensor product), and investigate its algebraic properties, such as its irreducible elements and factorisations, as well as the computability and complexity of solving polynomial equations over $\mathbf{P}$.

cs.DM↗

arXiv2019

The many roads to the simulation of reaction systems

Reaction systems are a computational model inspired by the bio-chemical reactions that happen inside biological cells. They have been and currently are studied for their many nice theoretical properties. They are also a useful modeling tool for biochemical systems, but in order to be able to employ them effectively in the field the presence of efficient and widely available simulators is essential. Here we explore three different algorithms and implementations of the simulation, comparing them to the current state of the art. We also show that we can obtain performances comparable to GPU-based simulations on real-world systems by using a carefully tuned CPU-based simulator.

cs.FL↗

arXiv2019

Complexity of the dynamics of reaction systems

Reaction systems are discrete dynamical systems inspired by bio-chemical processes, whose dynamical behaviour is expressed by set-theoretic operations on finite sets. Reaction systems thus provide a description of bio-chemical phenomena that complements the more traditional approaches, for instance those based on differential equations. A comprehensive list of decision problems about the dynamical behavior of reaction systems (such as cycles and fixed/periodic points, attractors, and reachability) is provided along with the corresponding computational complexity, which ranges from tractable problems to PSPACE-complete problems.

cs.FL↗

arXiv2019

Characterizing PSPACE with shallow non-confluent P systems

In P systems with active membranes, the question of understanding the power of non-confluence within a polynomial time bound is still an open problem. It is known that, for shallow P systems, that is, with only one level of nesting, non-confluence allows them to solve conjecturally harder problems than confluent P systems, thus reaching PSPACE. Here we show that PSPACE is not only a bound, but actually an exact characterization. Therefore, the power endowed by non-confluence to shallow P systems is equal to the power gained by confluent P systems when non-elementary membrane division and polynomial depth are allowed, thus suggesting a connection between the roles of non-confluence and nesting depth.

cs.CC↗

arXiv2019

On the dynamical behaviour of linear higher-order cellular automata and its decidability

Higher-order cellular automata (HOCA) are a variant of cellular automata (CA) used in many applications (ranging, for instance, from the design of secret sharing schemes to data compression and image processing), and in which the global state of the system at time $t$ depends not only on the state at time $t-1$, as in the original model, but also on the states at time $t-2, \ldots, t-n$, where $n$ is the memory size of the HOCA. We provide decidable characterizations of two important dynamical properties, namely, sensitivity to the initial conditions and equicontinuity, for linear HOCA over the alphabet $\mathbb{Z}_m$. Such characterizations extend the ones shown in [23] for linear CA (LCA) over the alphabet $\mathbb{Z}^{n}_m$ in the case $n=1$. We also prove that linear HOCA of size memory $n$ over $\mathbb{Z}_m$ form a class that is indistinguishable from a specific subclass of LCA over $\mathbb{Z}_m^n$. This enables to decide injectivity and surjectivity for linear HOCA of size memory $n$ over $\mathbb{Z}_m$ using the decidable characterization provided in [2] and [19] for injectivity and surjectivity of LCA over $\mathbb{Z}^n_m$. Finally, we prove an equivalence between LCA over $\mathbb{Z}_m^n$ and an important class of non-uniform CA, another variant of CA used in many applications.

cs.FL↗

arXiv2019

Solving QSAT in sublinear depth

Among $\mathbf{PSPACE}$-complete problems, QSAT, or quantified SAT, is one of the most used to show that the class of problems solvable in polynomial time by families of a given variant of P systems includes the whole $\mathbf{PSPACE}$. However, most solutions require a membrane nesting depth that is linear with respect to the number of variables of the QSAT instance under consideration. While a system of a certain depth is needed, since depth 1 systems only allows to solve problems in $\mathbf{P^{\#P}}$, it was until now unclear if a linear depth was, in fact, necessary. Here we use P systems with active membranes with charges, and we provide a construction that proves that QSAT can be solved with a sublinear nesting depth of order $\frac{n}{\log n}$, where $n$ is the number of variables in the quantified formula given as input.

cs.CC↗

arXiv2019

A Turing machine simulation by P systems without charges

It is well known that the kind of P systems involved in the definition of the P conjecture is able to solve problems in the complexity class $\mathbf{P}$ by leveraging the uniformity condition. Here we show that these systems are indeed able to simulate deterministic Turing machines working in polynomial time with a weaker uniformity condition and using only one level of membrane nesting. This allows us to embed this construction into more complex membrane structures, possibly showing that constructions similar to the one performed in [1] for P systems with charges can be carried out also in this case.

cs.CC↗