Recursively Defined Research Articles

The combinatorics of Farey words and their traces

We introduce a family of 3 -variable ‘Farey polynomials’ that are closely connected with the geometry and topology of 3 -manifolds and orbifolds as they can be used to produce concrete realisations of the boundaries and local coordinates for one-dimensional (over \mathbb{C} ) deformation spaces of Kleinian groups. As such, this family of polynomials has a number of quite remarkable properties. We study these polynomials from an abstract combinatorial viewpoint, including a recursive definition extending that which is known in the literature for the special case of manifolds, even beyond what the geometry predicts. We also present some intriguing examples and conjectures which we would like to bring to the attention of researchers interested in algebraic combinatorics and hypergeometric functions. The results in this paper additionally provide a practical approach to various classification problems for rank 2 subgroups of \operatorname{PSL}(2,\mathbb{C}) since they, together with other recent work of the authors, make it possible to provide certificates that certain groups are discrete and free, and effective ways to identify relators.

Addition and Differentiation of ZX-diagrams

The ZX-calculus is a powerful framework for reasoning in quantum computing. It provides in particular a compact representation of matrices of interests. A peculiar property of the ZX-calculus is the absence of a formal sum allowing the linear combinations of arbitrary ZX-diagrams. The universality of the formalism guarantees however that for any two ZX-diagrams, the sum of their interpretations can be represented by a ZX-diagram. We introduce a general, inductive definition of the addition of ZX-diagrams, relying on the construction of controlled diagrams. Based on this addition technique, we provide an inductive differentiation of ZX-diagrams. Indeed, given a ZX-diagram with variables in the description of its angles, one can differentiate the diagram according to one of these variables. Differentiation is ubiquitous in quantum mechanics and quantum computing (e.g. for solving optimization problems). Technically, differentiation of ZX-diagrams is strongly related to summation as witnessed by the product rules. We also introduce an alternative, non inductive, differentiation technique rather based on the isolation of the variables. Finally, we apply our results to deduce a diagram for an Ising Hamiltonian.

Large-W limit of the knapsack problem.

We formulate the knapsack problem (KP) as a statistical physics system and compute the corresponding partition function as an integral in the complex plane. The introduced formalism allows us to derive three statistical-physics-based algorithms for the KP: one based on the recursive definition of the exact partition function, another based on the large weight limit of that partition function, and a final one based on the zero-temperature limit of the second. Comparing the performances of the algorithms, we find that they do not consistently outperform (in terms of runtime and accuracy) dynamic programming, annealing, or standard greedy algorithms. However, the exact partition function is shown to reproduce the dynamic programming solution to the KP, and the zero-temperature algorithm is shown to produce a greedy solution. Therefore, although dynamic programming and greedy solutions to the KP are conceptually distinct, a statistical physics formalism introduced reveals that the large weight-constraint limit of the former leads to the latter. We conclude by discussing how to extend this formalism in order to obtain more accurate versions of the introduced algorithms and other similar combinatorial optimization problems.

Reifying dynamical algebra: Maximal ideals in countable rings, constructively

The existence of a maximal ideal in a general nontrivial commutative ring is tied together with the axiom of choice. Following Berardi, Valentini and thus Krivine but using the relative interpretation of negation (that is, as “implies 0 = 1”) we show, in constructive set theory with minimal logic, how for countable rings one can do without any kind of choice and without the usual decidability assumption that the ring is strongly discrete (membership in finitely generated ideals is decidable). By a functional recursive definition we obtain a maximal ideal in the sense that the quotient ring is a residue field (every noninvertible element is zero), and with strong discreteness even a geometric field (every element is either invertible or else zero). Krull’s lemma for the related notion of prime ideal follows by passing to rings of fractions. By employing a construction variant of set-theoretic forcing due to Joyal and Tierney, we expand our treatment to arbitrary rings and establish a connection with dynamical algebra: We recover the dynamical approach to maximal ideals as a parametrized version of the celebrated double negation translation. This connection allows us to give formal a priori criteria elucidating the scope of the dynamical method. Along the way we do a case study for proofs in algebra with minimal logic, and generalize the construction to arbitrary inconsistency predicates. A partial Agda formalization is available at an accompanying repository.11 See https://github.com/iblech/constructive-maximal-ideals/. This text is a revised and extended version of the conference paper (In Revolutions and Revelations in Computability. 18th Conference on Computability in Europe (2022) Springer). The conference paper only briefly sketched the connection with dynamical algebra; did not compare this connection with other flavors of set-theoretic forcing; and sticked to the case of commutative algebra only, passing on the generalization to inconsistency predicates and well-orders.

Computability and non-monotone induction

Non-monotone inductive definitions were studied in the late 1960’s and early 1970’s with the aim of understanding connections between the complexity of the formulas defining the induction steps and the ordinals measuring the rank of the inductive process. In general, any type 2 functional will generate an inductive process, and in this paper we will view non-monotone induction as a functional of type 3. We investigate the associated computation theory of this functional inherited from the Kleene schemes and we investigate the associated companion of sets with codes computable in the functional of non-monotone induction. The interest in this functional is motivated from observing that constructions via non-monotone induction appear to be natural in classical analysis. There are two groups of results: We establish strong closure properties of the least ordinal without a code computable in the functional of non-monotone induction, and we provide a characterisation of the class of functionals of type 3 computable from this functional, a characterisation in terms of sequential operators working in transfinite time. We will also see that the full computational power of non-monotone induction is required when this principle is used to construct functionals witnessing the compactness of the Cantor space and of closed, bounded intervals.

Compact Circuits for Efficient Möbius Transform

The Möbius transform is a linear circuit used to compute the evaluations of a Boolean function over all points on its input domain. The operation is very useful in finding the solution of a system of polynomial equations over GF(2) for obvious reasons. However the operation, although linear, needs exponential number of logic operations (around n · 2n−1 bit xors) for an n-variable Boolean function. As such, the only known hardware circuit to efficiently compute the Möbius Transform requires silicon area that is exponential in n. For Boolean functions whose algebraic degree is bound by some parameter d, recursive definitions of the Möbius Transform exist that requires only O(nd+1) space in software. However converting the mathematical definition of this space-efficient algorithm into a hardware architecture is a non-trivial task, primarily because the recursion calls notionally lead to a depth-first search in a transition graph that requires context switches at each recursion call for which straightforward mapping to hardware is difficult. In this paper we look to overcome these very challenges in an engineering sense. We propose a space efficient sequential hardware circuit for the Möbius Transform that requires only polynomial circuit area (i.e. O(nd+1)) provided the algebraic degree of the Boolean function is limited to d. We show how this circuit can be used as a component to efficiently solve polynomial equations of degree at most d by using fast exhaustive search. We propose three different circuit architectures for this, each of which uses the Möbius Transform circuit as a core component. We show that asymptotically, all the solutions of a system of m polynomials in n unknowns and algebraic degree d over GF(2) can be found using a circuit of silicon area proportional to m · nd+1 and circuit depth proportional to 2 · log2(n − d).In the second part of the paper we introduce a fourth hardware solver that additionally aims to achieve energy efficiency. The main idea is to reduce the solution space to a small enough value by parallel application of Möbius Transform circuits over the first few equations of the system. This is done so that one can check individually whether the vectors of this reduced solution space satisfy each of the remaining equations of the system using lower power consumption. The new circuit has area also bound by m · nd+1 and has circuit depth proportional to d · log2 n. We also show that further optimizations with respect to energy consumption may be obtained by using depth-bound Möbius circuits that exponentially decrease run time at the cost of additional logic area and depth.

Base-extension semantics for modal logic

Abstract In proof-theoretic semantics, meaning is based on inference. It may seen as the mathematical expression of the inferentialist interpretation of logic. Much recent work has focused on base-extension semantics, in which the validity of formulas is given by an inductive definition generated by provability in a ‘base’ of atomic rules. Base-extension semantics for classical and intuitionistic propositional logic have been explored by several authors. In this paper, we develop base-extension semantics for the classical propositional modal systems $K$, $KT$, $K4$ and $S4$, with $\square $ as the primary modal operator. We establish appropriate soundness and completeness theorems and establish the duality between $\square $ and a natural presentation of $\lozenge $. We also show that our semantics is in its current form not complete with respect to euclidean modal logics. Our formulation makes essential use of relational structures on bases.

A cross-sectional study on incidence and indications for induction of labor and the factors associated with failure of induction

Background: Induction fails in about 20% of induced pregnancies and has no universally accepted definition. Failed induction is the failure to establish labor or inability to achieve the active stage of labor, considering the definition of induction of labor (IOL). It needs to be differentiated from the failure of progress of labor. There is no evidence to recommend treatment for failed induction, but usually, delivery by cesarean section will be required. Aims and Objectives: The study was designed to start induction for an indication that justifies operative delivery in cases of failure. Materials and Methods: A prospective observational study was done over a period of 1 year among antenatal women confirmed to be at term with cephalic presentation in the absence of active labor with unscarred uterus. Results: The incidence of IOL was found to be 25.97% and failure was observed in 50 (21.83%) of those included in the study. Among those with failure, failed pre-induction ripening was seen in 26 (11.3%) and failed induction in 24 (10.4%). Although many variables were found to be associated with the outcome of labor, a statistically significant association was observed with premature rupture of membranes, pre-induction Bishop score, and birth weight of the baby. Conclusion: The study helps in better clinical judgment of patients who will have a successful induction or a failed induction and the patients could be counseled accordingly. This judgment will also help to avoid unnecessary inductions in those who have the risk factors for failed induction unless the induction is absolutely necessary.

An Infinite Needle in a Finite Haystack: Finding Infinite Counter-Models in Deductive Verification

First-order logic, and quantifiers in particular, are widely used in deductive verification of programs and systems. Quantifiers are essential for describing systems with unbounded domains, but prove difficult for automated solvers. Significant effort has been dedicated to finding quantifier instantiations that establish unsatisfiability of quantified formulas, thus ensuring validity of a system’s verification conditions. However, in many cases the formulas are satisfiable—this is often the case in intermediate steps of the verification process, e.g., when an invariant is not yet inductive. For such cases, existing tools are limited to finding finite models as counterexamples. Yet, some quantified formulas are satisfiable but only have infinite models, which current solvers are unable to find. Such infinite counter-models are especially typical when first-order logic is used to approximate the natural numbers, the integers, or other inductive definitions such as linked lists, which is common in deductive verification. The inability of solvers to find infinite models makes them diverge in these cases, providing little feedback to the user as they try to make progress in their verification attempts. In this paper, we tackle the problem of finding such infinite models, specifically, finite representations thereof that can be presented to the user of a deductive verification tool. These models give insight into the verification failure, and allow the user to identify and fix bugs in the modeling of the system and its properties. Our approach consists of three parts. First, we introduce symbolic structures as a way to represent certain infinite models, and show they admit an efficient model checking procedure. Second, we describe an effective model finding procedure that symbolically explores a given (possibly infinite) family of symbolic structures in search of an infinite model for a given formula. Finally, we identify a new decidable fragment of first-order logic that extends and subsumes the many-sorted variant of EPR, where satisfiable formulas always have a model representable by a symbolic structure within a known family, making our model finding procedure a decision procedure for that fragment. We evaluate our approach on examples from the domains of distributed consensus protocols and of heap-manipulating programs (specifically, linked lists). Our implementation quickly finds infinite counter-models that demonstrate the source of verification failures in a simple way, while state-of-the-art SMT solvers and theorem provers such as Z3, cvc5, and Vampire diverge or return “unknown”.

Unboxed Data Constructors: Or, How cpp Decides a Halting Problem

We propose a new language feature for ML-family languages, the ability to selectively unbox certain data constructors, so that their runtime representation gets compiled away to just the identity on their argument. Unboxing must be statically rejected when it could introduce confusion, that is, distinct values with the same representation. We discuss the use-case of big numbers, where unboxing allows to write code that is both efficient and safe, replacing either a safe but slow version or a fast but unsafe version. We explain the static analysis necessary to reject incorrect unboxing requests. We present our prototype implementation of this feature for the OCaml programming language, discuss several design choices and the interaction with advanced features such as Guarded Algebraic Datatypes. Our static analysis requires expanding type definitions in type expressions, which is not necessarily normalizing in presence of recursive type definitions. In other words, we must decide normalization of terms in the first-order λ-calculus with recursion. We provide an algorithm to detect non-termination on-the-fly during reduction, with proofs of correctness and completeness. Our algorithm turns out to be closely related to the normalization strategy for macro expansion in the cpp preprocessor.

LB4TL: A Smooth Semantics for Temporal Logic to Train Neural Feedback Controllers

This paper presents a framework for training neural network (NN)-based feedback controllers for autonomous agents with deterministic nonlinear dynamics to satisfy task objectives and safety constraints expressed in discrete-time Signal Temporal Logic (DT-STL). Control synthesis that uses the robustness semantics of DT-STL poses challenges due to its non-convexity, non-differentiability, and recursive definition, in particular when it is used to train NN-based controllers. We introduce a smooth neuro-symbolic computation graph to encode DT-STL robustness to represent a smooth approximation of the robustness, enabling the use of powerful stochastic gradient descent and backpropagation-based optimization for training. Our approximation guarantees that it lower bounds the robustness value of a given DT-STL formula, and shows orders of magnitude improvement over existing smooth approximations when applied to control synthesis. We demonstrate our approach on planning to satisfy complex spatio-temporal and sequential tasks, and show scalability with formula complexity.

Information models of tactical and operational planning levels in energy construction

The strategic, tactical and operational levels of energy construction planning are considered. The purpose of the article is to present information models for solving multi-criteria tasks of forming schedules of various types. For the first information model based on vector representations of aggregations, a multilevel classification of criteria and ranking algorithms for multi-criteria tasks is proposed. A structural analysis of criteria and ranking algorithms for multi-criteria tasks is carried out. Schemes of hierarchies of multiple inclusion of criteria and ranking algorithms for information planning models are presented. Recursive definitions of criteria and ranking algorithms are obtained. For the second information model, a heuristic paradigm of centralized scheduling is described for various types of presentation of source data, namely: sets of independent applications, sets of application vectors and sets of hierarchical or network structures of applications. The scheduling paradigm is based on the use of two schemes for generating schedules and priority rules. Using the first constructive scheme, the initial schedule is cyclically formed.

W1+∞ and [formula omitted] algebras, and Ward identities

It was demonstrated recently that the W1+∞ algebra contains commutative subalgebras associated with all integer slope rays (including the vertical one). In this paper, we realize that every element of such a ray is associated with a generalized W˜ algebra. In particular, the simplest commutative subalgebra associated with the rational Calogero Hamiltonians is associated with the W˜ algebras studied earlier. We suggest a definition of the generalized W˜ algebra as differential operators in variables pk basing on the matrix realization of the W1+∞ algebra, and also suggest an unambiguous recursive definition, which, however, involves more elements of the W1+∞ algebra than is contained in its commutative subalgebras. The positive integer rays are associated with W˜ algebras that form sets of Ward identities for the WLZZ matrix models, while the vertical ray associated with the trigonometric Calogero-Sutherland model describes the hypergeometric τ-functions corresponding to the completed cycles.

Inductive definitions in logic versus programs of real-time cellular automata

Descriptive complexity provides intrinsic, i.e. machine-independent, characterizations of the main complexity classes. On the other hand, logic can be useful for designing programs in a natural declarative way. This is especially important for parallel computation models such as cellular automata, since designing parallel programs is considered a difficult task.This paper establishes three logical characterizations of the three classical complexity classes modeling minimal time, called real-time, of one-dimensional cellular automata according to their canonical variations: unidirectional or bidirectional communication, input word given in a parallel or sequential way.Our three logics are natural restrictions of existential second-order Horn logic with built-in successor and predecessor functions. These logics correspond exactly to the three ways of deciding a language on a square grid circuit of side n according to one of the three natural locations of an input word of length n: along a side of the grid, on the diagonal that contains the output cell – placed on the vertex (n,n) of the square grid–, or on the diagonal opposite to the output cell.The key ingredient to our results is a normalization method that transforms a formula from one of our three logics into an equivalent normalized formula that closely mimics a grid circuit.Then, we extend our logics by allowing a limited use of negation on hypotheses like in Stratified Datalog. By revisiting in detail a number of representative classical problems - recognition of the set of primes by Fisher's algorithm, Dyck language recognition, Firing Squad Synchronization problem, etc. - we show that this extension makes easier programming and we prove that it does not change the real-time complexity of our logics.Finally, based on our experience in expressing these representative problems in logic, we argue that our logics are high-level programming languages: they make it possible to express in a natural, complete and synthetic way the algorithms of the literature, based on signals – and even to design new inductive algorithms –, and to translate them automatically into cellular automata of the same complexity.

The failure of cut-elimination in cyclic proof for first-order logic with inductive definitions

Abstract A cyclic proof system is a proof system whose proof figure is a tree with cycles. The cut-elimination in a proof system is fundamental. It is conjectured that the cut-elimination in the cyclic proof system for first-order logic with inductive definitions does not hold. This paper shows that the conjecture is correct by giving a sequent not provable without the cut rule but provable in the cyclic proof system.

The geometry of discrete L-algebras

Abstract The relationship of discrete L-algebras to projective geometry is deepened and made explicit in several ways. Firstly, a geometric lattice is associated to any discrete L-algebra. Monoids of I-type are obtained as a special case where the perspectivity relation is trivial. Secondly, the structure group of a non-degenerate discrete L-algebra X is determined and shown to be a complete invariant. It is proved that X ∖ {1} is a projective space with an orthogonality relation. A new definition of non-symmetric quantum sets, extending the recursive definition of symmetric quantum sets, is provided and shown to be equivalent to the former one. Quantum sets are characterized as complete projective spaces with an anisotropic duality, and they are also characterized in terms of their complete lattice of closed subspaces, which is one-sided orthomodular and semimodular. For quantum sets of finite cardinality n > 3, a representation as a projective space with duality over a skew-field is given. Quantum sets of cardinality 2 are classified, and the structure group of their associated L-algebra is determined.

A Proof Procedure for Separation Logic with Inductive Definitions and Data

A Proof Procedure for Separation Logic with Inductive Definitions and Data

More Fixpoints! (Functional Pearl)

Haskell’s laziness allows the programmer to solve some problems naturally and declaratively via recursive equations. Unfortunately, if the input is “too recursive”, these very elegant idioms can fall into the dreaded black hole, and the programmer has to resort to more pedestrian approaches. It does not have to be that way: We built variants of common pure data structures (Booleans, sets) where recursive definitions are productive. Internally, the infamous unsafePerformIO is at work, but the user only sees a beautiful and pure API, and their pretty recursive idioms – magically – work again.

POSIX Lexing with Derivatives of Regular Expressions

Brzozowski introduced the notion of derivatives for regular expressions. They can be used for a very simple regular expression matching algorithm. Sulzmann and Lu cleverly extended this algorithm in order to deal with POSIX matching, which is the underlying disambiguation strategy for regular expressions needed in lexers. Their algorithm generates POSIX values which encode the information of how a regular expression matches a string—that is, which part of the string is matched by which part of the regular expression. In this paper we give our inductive definition of what a POSIX value is and show that Sulzmann and Lu’s algorithm always generates such a value. We also show that our inductive definition of a POSIX value is equivalent to an alternative definition by Okui and Suzuki which identifies POSIX values as least elements according to an ordering of values.

Computing with Infinite Objects: the Gray Code Case

Infinite Gray code has been introduced by Tsuiki as a redundancy-free representation of the reals. In applications the signed digit representation is mostly used which has maximal redundancy. Tsuiki presented a functional program converting signed digit code into infinite Gray code. Moreover, he showed that infinite Gray code can effectively be converted into signed digit code, but the program needs to have some non-deterministic features (see also H. Tsuiki, K. Sugihara, "Streams with a bottom in functional languages"). Berger and Tsuiki reproved the result in a system of formal first-order intuitionistic logic extended by inductive and co-inductive definitions, as well as some new logical connectives capturing concurrent behaviour. The programs extracted from the proofs are exactly the ones given by Tsuiki. In order to do so, co-inductive predicates $\bS$ and $\bG$ are defined and the inclusion $\bS \subseteq \bG$ is derived. For the converse inclusion the new logical connectives are used to introduce a concurrent version $\S_{2}$ of $S$ and $\bG \subseteq \bS_{2}$ is shown. What one is looking for, however, is an equivalence proof of the involved concepts. One of the main aims of the present paper is to close the gap. A concurrent version $\bG^{*}$ of $\bG$ and a modification $\bS^{*}$ of $\bS_{2}$ are presented such that $\bS^{*} = \bG^{*}$. A crucial tool in U. Berger, H. Tsuiki, "Intuitionistic fixed point logic" is a formulation of the Archimedean property of the real numbers as an induction principle. We introduce a concurrent version of this principle which allows us to prove that $\bS^{*}$ and $\bG^{*}$ coincide. A further central contribution is the extension of the above results to the hyperspace of non-empty compact subsets of the reals.