arXiv subjects

Santiago Figueira

Publications and source records attributed to Santiago Figueira.

At least 19 recordsLinked to original sources

arXiv2026

A Common Ancestor of PDL, Conjunctive Queries, and Unary Negation First-order Logic

We introduce and study UCPDL+, a family of expressive logics rooted in Propositional Dynamic Logic (PDL) with converse (CPDL) and universal modality (UCPDL). In terms of expressive power, UCPDL+ strictly contains PDL extended with intersection and converse (a.k.a. ICPDL), as well as Conjunctive Queries (CQ), Conjunctive Regular Path Queries (CRPQ), or some known extensions thereof (Regular Queries and CQPDL). Further, it is equivalent to the extension of the unary-negation fragment of first-order logic (UNFO) with unary transitive closure, denoted by UNTC, which in turn strictly contains a previously studied extension of UNFO with regular expressions known as UNFO$^{reg}$. We investigate the expressive power, indistinguishability via bisimulations, satisfiability, and model checking for UCPDL+ and CPDL+. We argue that natural subclasses of CPDL+ can be defined in terms of the tree-width of the underlying graphs of the formulas. We show that the class of CPDL+ formulas of tree-width 2 is equivalent to ICPDL, and that it also coincides with CPDL+ formulas of tree-width 1. However, beyond tree-width 2, incrementing the tree-width strictly increases the expressive power. We characterize the expressive power for every class of fixed tree-width formulas in terms of a bisimulation game with pebbles. Based on this characterization, we show that CPDL+ has a tree-like model property. We prove that the satisfiability problem for UCPDL+ is decidable in 2ExpTime, coinciding with the complexity of ICPDL. As a consequence, the satisfiability problem for UNTC is shown to be 2ExpTime-complete as well. We also exhibit classes for which satisfiability is reduced to ExpTime.

cs.LO↗

arXiv2026

Basic Model Theory for Path Predicate Modal Logic

Path Predicate Modal Logic (PPML) is a generalization of Basic Modal Logic, where atoms are relational predicates instead of propositional symbols. The study of PPML is motivated as a way to abstractly investigate data-aware formalisms, such as XPath or DataGL. In this paper, we investigate some basic model theoretical aspects of PPML to better characterize its expressive power. More concretely, we investigate different ways of defining Hennessy-Milner classes, and a van Benthem characterization theorem. In doing so, we discuss the main challenges of dealing with the novel features of PPML, and what are the similarities with the standard approaches.

cs.LO↗

arXiv2026

Guarded Negation Transitive Closure Logic

We study the guarded negation fragment of transitive closure logic (GNTC). We show that the satisfiability problem for GNTC is 2ExpTime-complete, by establishing the following reductions: (i) a polynomial-time reduction from the satisfiability problem for GNTC to the satisfiability problem for the unary negation fragment UNTC of GNTC, and (ii) a direct exponential-time reduction from the satisfiability problem for UNTC to the non-emptiness problem for 2-way alternating parity tree automata. Furthermore, we show that the model checking problem for GNTC is $\mathsf{P}^{\mathsf{NP}[\mathcal{O}(\log^2 n)]}$-complete in combined complexity. Our result implies $\mathsf{P}^{\mathsf{NP}[\mathcal{O}(\log^2 n)]}$-completeness for both UNTC and $\mathrm{UNFO}^{\mathrm{reg}}$, which were left open in previous works.

cs.LO↗

arXiv2026

Measuring the largest coefficients of a quantum state

We introduce a hierarchical algorithm for identifying the largest Pauli coefficients of an unknown $n$-qubit quantum state. The algorithm traverses a prefix-based tree whose nodes represent partial sums of squared Pauli coefficients, always expanding branches with the largest estimated weight and discarding the rest. Node weights are estimated using Bell sampling on two copies of the state, or alternatively via SWAP tests on subsystems. We analyze the sample complexity of each node estimation and derive bounds on the total number of nodes expanded as a function of the desired number of coefficients and the state's purity. For states admitting a sparse representation in the Pauli basis, the algorithm achieves a good reconstruction of the dominant components without requiring full state tomography. We validate the method with numerical simulations on Pauli-singleton states and random stabilizer states, showing that the algorithm's performance is competitive with other methods for structured states. Our work addresses an open problem in Pauli sampling and provides a practical tool for the targeted characterization of structured quantum states.

quant-ph↗

arXiv2025

Rauzy dimension and finite-state dimension

In 1976, Rauzy studied two complexity functions, $\underlineβ$ and $\overlineβ$, for infinite sequences over a finite alphabet. The function $\underlineβ$ achieves its maximum precisely for Borel normal sequences, while $\overlineβ$ reaches its minimum for sequences that, when added to any Borel normal sequence, result in another Borel normal sequence. We establish a connection between Rauzy's complexity functions, $\underlineβ$ and $\overlineβ$, and the notions of non-aligned block entropy, $\underline{h}$ and $\overline{h}$, by providing sharp upper and lower bounds for $\underline{h}$ in terms of $\underlineβ$, and sharp upper and lower bounds for $\overline{h}$ in terms of $\overlineβ$. We adopt a probabilistic approach by considering an infinite sequence of random variables over a finite alphabet. The proof relies on a new characterization of non-aligned block entropies, $\overline{h}$ and $\underline{h}$, in terms of Shannon's conditional entropy. The bounds imply that sequences with $\overline{h} = 0$ coincide with those for which $\overlineβ = 0$. We also show that the non-aligned block entropies, $\underline{h}$ and $\overline{h}$, are essentially subadditive.

cs.IT↗

arXiv2024

Modal Logic with Relations over Paths: a Theoretical Development through Comonadic Semantics

Game comonads provide categorical semantics for comparison games in Finite Model Theory, thus providing an abstract characterisation of logical equivalence for a wide range of logics, each one captured through a specific choice of comonad. Motivated by the goal of applying comonadic tools to the study of data-aware logics such as CoreDataXPath, in this work we introduce a generalisation of Modal Logic that allows relation symbols of arbitrary arity as atoms of the syntax, which we call Path Predicate Modal Logic or PPML. We motivate this logic as arising from a shift in perspective on a previously studied fragment of CoreDataXPath, called DataGL, and prove that PPML recovers DataGL for a specific choice of signature. We argue that this shift in perspective allows the capturing and designing of new data-aware logics. We introduce resource-bounded simulation and bisimulation games for PPML and show the Hennessy-Milner property relating bisimilarity and logical equivalence. We define the PPML comonad and prove that it captures these games. We develop the model-theoretical understanding of PPML by making systematic use of the comonadic framework. This includes results such as a tree-model property and an alternative proof of the one-way Hennessy-Milner property using a correspondence between positive PPML formulas and canonical models. We also use the comonadic perspective to establish connections with other logics, such as bounded quantifier rank and bounded variable number fragments of First Order Logic on one side and Basic Modal Logic on the other, and show how the PPML comonad induces a syntax-free characterisation of logical equivalence for DataGL, our original motivation. With respect to Basic Modal Logic, a functorial assignment from PPML unravellings into Kripke trees enables us to obtain polynomial-time reductions from PPML problems to their Basic Modal Logic counterparts.

cs.LO↗

arXiv2023

Towards exact algorithmic proofs of maximal mutually unbiased bases sets in arbitrary integer dimension

In this paper, we explore the concept of Mutually Unbiased Bases (MUBs) in discrete quantum systems. It is known that for dimensions $d$ that are powers of prime numbers, there exists a set of up to $d+1$ bases that form an MUB set. However, the maximum number of MUBs in dimensions that are not powers of prime numbers is not known. To address this issue, we introduce three algorithms based on First-Order Logic that can determine the maximum number of bases in an MUB set without numerical approximation. Our algorithms can prove this result in finite time, although the required time is impractical. Additionally, we present a heuristic approach to solve the semi-decision problem of determining if there are $k$ MUBs in a given dimension $d$. As a byproduct of our research, we demonstrate that the maximum number of MUBs in any dimension can be achieved with definable complex parameters, computable complex parameters, and other similar fields.

quant-ph↗

arXiv2023

PDL on Steroids: on Expressive Extensions of PDL with Intersection and Converse

We introduce CPDL+, a family of expressive logics rooted in Propositional Dynamic Logic (PDL). In terms of expressive power, CPDL+ strictly contains PDL extended with intersection and converse (a.k.a. ICPDL) as well as Conjunctive Queries (CQ), Conjunctive Regular Path Queries (CRPQ), or some known extensions thereof (Regular Queries and CQPDL). We investigate the expressive power, characterization of bisimulation, satisfiability, and model checking for CPDL+. We argue that natural subclasses of CPDL+ can be defined in terms of the tree-width of the underlying graphs of the formulas. We show that the class of CPDL+ formulas of tree-width 2 is equivalent to ICPDL, and that it also coincides with CPDL+ formulas of tree-width 1. However, beyond tree-width 2, incrementing the tree-width strictly increases the expressive power. We characterize the expressive power for every class of fixed tree-width formulas in terms of a bisimulation game with pebbles. Based on this characterization, we show that CPDL+ has a tree-like model property. We prove that the satisfiability problem is decidable in 2ExpTime on fixed tree-width formulas, coinciding with the complexity of ICPDL. We also exhibit classes for which satisfiability is reduced to ExpTime. Finally, we establish that the model checking problem for fixed tree-width formulas is in \ptime, contrary to the full class CPDL+.

cs.LO↗

arXiv2021

Algorithmic Pseudorandomness in Quantum Setups

The Church-Turing thesis is one of the pillars of computer science; it postulates that every classical system has equivalent computability power to the so-called Turing machine. While this thesis is crucial for our understanding of computing devices, its implications in other scientific fields have hardly been explored. Here we start this research programme in the context of quantum physics and show that computer science laws have profound implications for some of the most fundamental results of the theory. We first show how they question our knowledge on what a mixed quantum state is, as we identify situations in which ensembles of quantum states defining the same mixed state, indistinguishable according to the quantum postulates, do become distinguishable when prepared by a computer. We also show a new loophole for Bell-like experiments: if some of the parties in a Bell-like experiment use a computer to decide which measurements to make, then the computational resources of an eavesdropper have to be limited in order to have a proper observation of non-locality. Our work opens a new direction in the search for a framework unifying computer science and quantum physics.

quant-ph↗

arXiv2020

Impossibility of memory in hidden-signaling models for quantum correlations

We consider a toy model for non-local quantum correlations in which nature resorts to some form of hidden signaling (i.e., signaling between boxes but not available to the users) to generate correlations. We show that if such a model also had memory, the parties would be able to exploit the hidden-signaling and use it to send a message, achieving faster-than-light communication. Given that memory is a resource easily available for any physical system, our results add evidence against hidden signaling as the mechanism behind nature's non-local behavior

quant-ph↗

arXiv2020

Characterization of non-signaling correlations from mutual information

We present a characterization of the set of non-signaling correlations in terms of a two dimensional representation that involves the maximal value of a Bell functional and the mutual information between the parties. In particular, we apply this representation to the bipartite Bell scenario with two measurements and two outcomes. In terms of these physically meaningful quantities and through numerical optimization methods and some analytical results, we investigate the frontier between the different subsets of the non-signaling correlations, focussing on the quantum and post-quantum ones. Finally, we show that the Tsirelson bound appears as a singular point in this context without resorting to quantum mechanics.

quant-ph↗

arXiv2019

Towards a more flexible Language of Thought: Bayesian grammar updates after each concept exposure

Recent approaches to human concept learning have successfully combined the power of symbolic, infinitely productive rule systems and statistical learning to explain our ability to learn new concepts from just a few examples. The aim of most of these studies is to reveal the underlying language structuring these representations and providing a general substrate for thought. However, describing a model of thought that is fixed once trained is against the extensive literature that shows how experience shapes concept learning. Here, we ask about the plasticity of these symbolic descriptive languages. We perform a concept learning experiment that demonstrates that humans can change very rapidly the repertoire of symbols they use to identify concepts, by compiling expressions which are frequently used into new symbols of the language. The pattern of concept learning times is accurately described by a Bayesian agent that rationally updates the probability of compiling a new expression according to how useful it has been to compress concepts so far. By portraying the Language of Thought as a flexible system of rules, we also highlight the difficulties to pin it down empirically.

cs.AI↗

arXiv2018

Algorithmic identification of probabilities is hard

Suppose that we are given an infinite binary sequence which is random for a Bernoulli measure of parameter $p$. By the law of large numbers, the frequency of zeros in the sequence tends to~$p$, and thus we can get better and better approximations of $p$ as we read the sequence. We study in this paper a similar question, but from the viewpoint of inductive inference. We suppose now that $p$ is a computable real, but one asks for more: as we are reading more and more bits of our random sequence, we have to eventually guess the exact parameter $p$ (in the form of a Turing code). Can one do such a thing uniformly on all sequences that are random for computable Bernoulli measures, or even on a `large enough' fraction of them? In this paper, we give a negative answer to this question. In fact, we prove a very general negative result which extends far beyond the class of Bernoulli measures.

math.LO↗

arXiv2017

Distinguishing computable mixtures of quantum states

In this article we extend results from our previous work [Bendersky, de la Torre, Senno, Figueira and Acín, Phys. Rev. Lett. 116, 230406 (2016)] by providing a protocol to distinguish in finite time and with arbitrarily high success probability any algorithmic mixture of pure states from the maximally mixed state. Moreover, we introduce a proof-of-concept experiment consisting in a situation where two different random sequences of pure states are prepared; these sequences are indistinguishable according to quantum mechanics, but they become distinguishable when randomness is replaced with pseudorandomness within the preparation process.

quant-ph↗

arXiv2017

Axiomatizations for downward XPath on Data Trees

We give sound and complete axiomatizations for XPath with data tests by "equality" or "inequality", and containing the single "child" axis. This data-aware logic predicts over data trees, which are tree-like structures whose every node contains a label from a finite alphabet and a data value from an infinite domain. The language allows us to compare data values of two nodes but cannot access the data values themselves (i.e. there is no comparison by constants). Our axioms are in the style of equational logic, extending the axiomatization of data-oblivious XPath, by B. ten Cate, T. Litak and M. Marx. We axiomatize the full logic with tests by "equality" and "inequality", and also a simpler fragment with "equality" tests only. Our axiomatizations apply both to node expressions and path expressions. The proof of completeness relies on a novel normal form theorem for XPath with data tests.

cs.LO↗

arXiv2016

Non-signaling deterministic models for non-local correlations have to be uncomputable

Quantum mechanics postulates random outcomes. However, a model making the same output predictions but in a deterministic manner would be, in principle, experimentally indistinguishable from quantum theory. In this work we consider such models in the context of non-locality on a device independent scenario. That is, we study pairs of non-local boxes that produce their outputs deterministically. It is known that, for these boxes to be non-local, at least one of the boxes' output has to depend on the other party's input via some kind of hidden signaling. We prove that, if the deterministic mechanism is also algorithmic, there is a protocol which, with the sole knowledge of any upper bound on the time complexity of such algorithm, extracts that hidden signaling and uses it for the communication of information.

quant-ph↗

arXiv2014

Normality in non-integer bases and polynomial time randomness

It is known that if $x\in[0,1]$ is polynomial time random (i.e. no polynomial time computable martingale succeeds on the binary fractional expansion of $x$) then $x$ is normal in any integer base greater than one. We show that if $x$ is polynomial time random and $β>1$ is Pisot, then $x$ is "normal in base $β$", in the sense that the sequence $(xβ^n)_{n\in\mathbb{N}}$ is uniformly distributed modulo one. We work with the notion of "$P$-martingale", a generalization of martingales to non-uniform distributions, and show that a sequence over a finite alphabet is distributed according to an irreducible, invariant Markov measure~$P$ if an only if no $P$-martingale whose betting factors are computed by a deterministic finite automaton succeeds on it. This is a generalization of Schnorr and Stimm's characterization of normal sequences in integer bases. Our results use tools and techniques from symbolic dynamics, together with automata theory and algorithmic randomness.

math.DS↗

arXiv2013

LT^2C^2: A language of thought with Turing-computable Kolmogorov complexity

In this paper, we present a theoretical effort to connect the theory of program size to psychology by implementing a concrete language of thought with Turing-computable Kolmogorov complexity (LT^2C^2) satisfying the following requirements: 1) to be simple enough so that the complexity of any given finite binary sequence can be computed, 2) to be based on tangible operations of human reasoning (printing, repeating,...), 3) to be sufficiently powerful to generate all possible sequences but not too powerful as to identify regularities which would be invisible to humans. We first formalize LT^2C^2, giving its syntax and semantics and defining an adequate notion of program size. Our setting leads to a Kolmogorov complexity function relative to LT^2C^2 which is computable in polynomial time, and it also induces a prediction algorithm in the spirit of Solomonoff's inductive inference theory. We then prove the efficacy of this language by investigating regularities in strings produced by participants attempting to generate random strings. Participants had a profound understanding of randomness and hence avoided typical misconceptions such as exaggerating the number of alternations. We reasoned that remaining regularities would express the algorithmic nature of human thoughts, revealed in the form of specific patterns. Kolmogorov complexity relative to LT^2C^2 passed three expected tests examined here: 1) human sequences were less complex than control PRNG sequences, 2) human sequences were not stationary, showing decreasing values of complexity resulting from fatigue, 3) each individual showed traces of algorithmic stability since fitting of partial sequences was more effective to predict subsequent sequences than average fits. This work extends on previous efforts to combine notions of Kolmogorov complexity theory and algorithmic information theory to psychology, by explicitly ...

q-bio.NC↗