arXiv subjects

Mnacho Echenim

Publications and source records attributed to Mnacho Echenim.

At least 19 recordsLinked to original sources

arXiv2023

A formalization of the CHSH inequality and Tsirelson's upper-bound in Isabelle/HOL

We present a formalization of several fundamental notions and results from Quantum Information theory, including density matrices and projective measurements, along with the proof that the local hidden-variable hypothesis advocated by Einstein to model quantum mechanics cannot hold. The proof of the latter result is based on the so-called CHSH inequality, and it is the violation of this inequality that was experimentally evidenced by Aspect who earned the Nobel Prize in 2022 for his work. We also formalize various results related to the violation of the CHSH inequality, such as Tsirelson's bound which permits to obtain the maximum violation of this inequality in a quantum setting.

quant-ph↗

arXiv2023

Tractable and Intractable Entailment Problems in Separation Logic with Inductively Defined Predicates

We establish various complexity results for the entailment problem between formulas in Separation Logic with user-defined predicates denoting recursive data structures. The considered fragments are characterized by syntactic conditions on the inductive rules that define the semantics of the predicates. We focus on so-called P-rules, which are similar to (but simpler than) the PCE rules introduced by Iosif et al. in 2013. In particular, for a specific fragment where predicates are defined by so-called loc-deterministic inductive rules, we devise a sound and complete cyclic proof procedure running in polynomial time. Several complexity lower bounds are provided, showing that any relaxing of the provided conditions makes the problem intractable.

cs.LO↗

arXiv2022

Unbiasing and robustifying implied volatility calibration in a cryptocurrency market with large bid-ask spreads and missing quotes

We design a novel calibration procedure that is designed to handle the specific characteristics of options on cryptocurrency markets, namely large bid-ask spreads and the possibility of missing or incoherent prices in the considered data sets. We show that this calibration procedure is significantly more robust and accurate than the standard one based on trade and mid-prices.

q-fin.PR↗

arXiv2022

A Proof Procedure For Separation Logic With Inductive Definitions and Theory Reasoning

A proof procedure, in the spirit of the sequent calculus, is proposed to check the validity of entailments between Separation Logic formulas combining inductively defined predicates denoted structures of bounded tree width and theory reasoning. The calculus is sound and complete, in the sense that a sequent is valid iff it admits a (possibly infinite) proof tree. We show that the procedure terminates in the two following cases: (i) When the inductive rules that define the predicates occurring on the left-hand side of the entailment terminate, in which case the proof tree is always finite. (ii) When the theory is empty, in which case every valid sequent admits a rational proof tree, where the total number of pairwise distinct sequents occurring in the proof tree is doubly exponential w.r.t.\ the size of the end-sequent.

cs.LO↗

arXiv2022

Two Results on Separation Logic With Theory Reasoning

Two results are presented concerning the entailment problem in Separation Logic with inductively defined predicate symbols and theory reasoning. First, we show that the entailment problem is undecidable for rules with bounded tree-width, if theory reasoning is considered. The result holds for a wide class of theories, even with a very low expressive power. For instance it applies to the natural numbers with the successor function, or with the usual order. Second, we show that every entailment problem can be reduced to an entailment problem containing no equality (neither in the formulas nor in the recursive rules defining the semantics of the predicate symbols).

cs.LO↗

arXiv2021

A Superposition-Based Calculus for Quantum Diagrammatic Reasoning and Beyond

We introduce a class of rooted graphs which allows one to encode various kinds of classical or quantum circuits. We then follow a set-theoretic approach to define rewrite systems over the considered graphs and propose a new complete Superposition callculus which handles sets of formulas consisting of equations or disequations over these graphs.

cs.LO↗

arXiv2021

Quantum projective measurements and the CHSH inequality in Isabelle/HOL

We present a formalization in Isabelle/HOL of quantum projective measurements, a class of measurements involving orthogonal projectors that is frequently used in quantum computing. We also formalize the CHSH inequality, a result that holds on arbitrary probability spaces, which can used to disprove the existence of a local hidden-variable theory for quantum mechanics.

cs.LO↗

arXiv2021

Unifying Decidable Entailments in Separation Logic with Inductive Definitions

The entailment problem $φ\models ψ$ in Separation Logic \cite{IshtiaqOHearn01,Reynolds02}, between separated conjunctions of equational ($x \iseq y$ and $x \not\iseq y$), spatial ($x \mapsto (y_1,\ldots,y_\rank)$) and predicate ($p(x_1,\ldots,x_n)$) atoms, interpreted by a finite set of inductive rules, is undecidable in general. Certain restrictions on the set of inductive definitions lead to decidable classes of entailment problems. Currently, there are two such decidable classes, based on two restrictions, called \emph{establishment} \cite{IosifRogalewiczSimacek13,KatelaanMathejaZuleger19,PZ20} and \emph{restrictedness} \cite{EIP21a}, respectively. Both classes are shown to be in \twoexptime\ by the independent proofs from \cite{PZ20} and \cite{EIP21a}, respectively, and a many-one reduction of established to restricted entailment problems has been given \cite{EIP21a}. In this paper, we strictly generalize the restricted class, by distinguishing the conditions that apply only to the left- ($φ$) and the right- ($ψ$) hand side of entailments, respectively. We provide a many-one reduction of this generalized class, called \emph{safe}, to the established class. Together with the reduction of established to restricted entailment problems, this new reduction closes the loop and shows that the three classes of entailment problems (respectively established, restricted and safe) form a single, unified, \twoexptime-complete class.

cs.LO↗

arXiv2020

Decidable Entailments in Separation Logic with Inductive Definitions: Beyond Established Systems

We define a class of Separation Logic formulae, whose entailment problem: given formulae $ϕ, ψ_1, \ldots, ψ_n$, is every model of $ϕ$ a model of some $ψ_i$? is 2EXPTIME-complete. The formulae in this class are existentially quantified separating conjunctions involving predicate atoms, interpreted by the least sets of store-heap structures that satisfy a set of inductive rules, which is also part of the input to the entailment problem. Previous work consider established sets of rules, meaning that every existentially quantified variable in a rule must eventually be bound to an allocated location, i.e. from the domain of the heap. In particular, this guarantees that each structure has treewidth bounded by the size of the largest rule in the set. In contrast, here we show that establishment, although sufficient for decidability (alongside two other natural conditions), is not necessary, by providing a condition, called equational restrictedness, which applies syntactically to (dis-)equalities. The entailment problem is more general in this case, because equationally restricted rules define richer classes of structures, of unbounded treewidth. In this paper we show that (1) every established set of rules can be converted into an equationally restricted one and (2) the entailment problem is 2EXPTIME-complete in the latter case, thus matching the complexity of entailments for established sets of rules.

cs.LO↗

arXiv2020

Entailment Checking in Separation Logic with Inductive Definitions is 2-EXPTIME hard

The entailment between separation logic formulae with inductive predicates, also known as symbolic heaps, has been shown to be decidable for a large class of inductive definitions. Recently, a 2-EXPTIME algorithm was proposed and an EXPTIME-hard bound was established; however no precise lower bound is known. In this paper, we show that deciding entailment between predicate atoms is 2-EXPTIME-hard. The proof is based on a reduction from the membership problem for exponential-space bounded alternating Turing machines.

cs.LO↗

arXiv2019

Ilinva: Using Abduction to Generate Loop Invariants

We describe a system to prove properties of programs. The key feature of this approach is a method to automatically synthesize inductive invariants of the loops contained in the program. The method is generic, i.e., it applies to a large set of programming languages and application domains; and lazy, in the sense that it only generates invariants that allow one to derive the required properties. It relies on an existing system called GPiD for abductive reasoning modulo theories, and on the platform for program verification Why3. Experiments show evidence of the practical relevance of our approach.

cs.LO↗

arXiv2018

Formalizing the Cox-Ross-Rubinstein pricing of European derivatives in Isabelle/HOL

We formalize in the proof assistant Isabelle essential basic notions and results in financial mathematics. We provide generic formal definitions of concepts such as markets, portfolios, derivative products, arbitrages or fair prices, and we show that, under the usual no-arbitrage condition, the existence of a replicating portfolio for a derivative implies that the latter admits a unique fair price. Then, we provide a formalization of the Cox-Rubinstein model and we show that the market is complete in this model, i.e., that every derivative product admits a replicating portfolio. This entails that in this model, every derivative product admits a unique fair price.

cs.LO↗

arXiv2018

A Generic Framework for Implicate Generation Modulo Theories

The clausal logical consequences of a formula are called its implicates. The generation of these implicates has several applications, such as the identification of missing hypotheses in a logical specification. We present a procedure that generates the implicates of a quantifier-free formula modulo a theory. No assumption is made on the considered theory, other than the existence of a decision procedure. The algorithm has been implemented (using the solvers MiniSat, CVC4 and Z3) and experimental results show evidence of the practical relevance of the proposed approach.

cs.LO↗

arXiv2018

The Complexity of Prenex Separation Logic with One Selector

We first show that infinite satisfiability can be reduced to finite satisfiability for all prenex formulas of Separation Logic with $k\geq1$ selector fields ($\seplogk{k}$). Second, we show that this entails the decidability of the finite and infinite satisfiability problem for the class of prenex formulas of $\seplogk{1}$, by reduction to the first-order theory of one unary function symbol and unary predicate symbols. We also prove that the complexity is not elementary, by reduction from the first-order theory of one unary function symbol. Finally, we prove that the Bernays-Schönfinkel-Ramsey fragment of prenex $\seplogk{1}$ formulae with quantifier prefix in the language $\exists^*\forall^*$ is \pspace-complete. The definition of a complete (hierarchical) classification of the complexity of prenex $\seplogk{1}$, according to the quantifier alternation depth is left as an open problem.

cs.LO↗

arXiv2018

On the Expressive Completeness of Bernays-Schönfinkel-Ramsey Separation Logic

This paper investigates the satisfiability problem for Separation Logic, with unrestricted nesting of separating conjunctions and implications, for prenex formulae with quantifier prefix in the language $\exists^*\forall^*$, in the cases where the universe of possible locations is either countably infinite or finite. In analogy with first-order logic with uninterpreted predicates and equality, we call this fragment Bernays-Schönfinkel-Ramsey Separation Logic [BSR(SLk)]. We show that, unlike in first-order logic, the (in)finite satisfiability problem is undecidable for BSR(SLk) and we define two non-trivial subsets thereof, that are decidable for finite and infinite satisfiability, respectively, by controlling the occurrences of universally quantified variables within the scope of separating implications, as well as the polarity of the occurrences of the latter. The decidability results are obtained by a controlled elimination of separating connectives, described as (i) an effective translation of a prenex form Separation Logic formula into a combination of a small number of \emph{test formulae}, using only first-order connectives, followed by (ii) a translation of the latter into an equisatisfiable first-order formula.

cs.LO↗

arXiv2014

A Superposition Calculus for Abductive Reasoning

We present a modification of the superposition calculus that is meant to generate consequences of sets of first-order axioms. This approach is proven to be sound and deductive-complete in the presence of redundancy elimination rules, provided the considered consequences are built on a given finite set of ground terms, represented by constant symbols. In contrast to other approaches, most existing results about the termination of the superposition calculus can be carried over to our procedure. This ensures in particular that the calculus is terminating for many theories of interest to the SMT community.

cs.LO↗

arXiv2012

Reasoning on Schemata of Formulae

A logic is presented for reasoning on iterated sequences of formulae over some given base language. The considered sequences, or "schemata", are defined inductively, on some algebraic structure (for instance the natural numbers, the lists, the trees etc.). A proof procedure is proposed to relate the satisfiability problem for schemata to that of finite disjunctions of base formulae. It is shown that this procedure is sound, complete and terminating, hence the basic computational properties of the base language can be carried over to schemata.

cs.LO↗

arXiv2012

A Calculus for Generating Ground Explanations (Technical Report)

We present a modification of the superposition calculus that is meant to generate explanations why a set of clauses is satisfiable. This process is related to abductive reasoning, and the explanations generated are clauses constructed over so-called abductive constants. We prove the correctness and completeness of the calculus in the presence of redundancy elimination rules, and develop a sufficient condition guaranteeing its termination; this sufficient condition is then used to prove that all possible explanations can be generated infinite time for several classes of clause sets, including many of interest to the SMT community. We propose a procedure that generates a set of explanations that should be useful to a human user and conclude by suggesting several extensions to this novel approach.

cs.LO↗