Primitive Recursive Research Articles

An isomorphism theorem for models of weak König’s lemma without primitive recursion

We prove that if (M,\mathcal{X}) and (M,\mathcal{Y}) are countable models of the theory \mathrm{WKL}^{*}_{0} such that \mathrm{I}\Sigma_{1}(A) fails for some A \in \mathcal{X}\cap \mathcal{Y} , then (M,\mathcal{X}) and (M,\mathcal{Y}) are isomorphic. As a consequence, the analytic hierarchy collapses to \Delta^{1}_{1} provably in \mathrm{WKL}^{*}_{0} + \neg \mathrm{I}\Sigma^{0}_{1} , and \mathrm{WKL} is the strongest \Pi^{1}_{2} statement that is \Pi^{1}_{1} -conservative over \mathrm{RCA}^{*}_{0} + \neg \mathrm{I}\Sigma^{0}_{1} . Applying our results to the \Delta^{0}_{n} -definable sets in models of \mathrm{RCA}^{*}_{0} + \mathrm{B}\Sigma^{0}_{n} + \neg \mathrm{I}\Sigma^{0}_{n} that also satisfy an appropriate relativization of weak König’s lemma, we prove that for each n \ge 1 , the set of \Pi^{1}_{2} sentences that are \Pi^{1}_{1} -conservative over \mathrm{RCA}^{*}_{0} + \mathrm{B}\Sigma^{0}_{n} + \neg \mathrm{I}\Sigma^{0}_{n} is computably enumerable. In contrast, we prove that the set of \Pi^{1}_{2} sentences that are \Pi^{1}_{1} -conservative over \mathrm{RCA}^{*}_{0} + \mathrm{B}\Sigma^{0}_{n} is \Pi_{2} -complete. This answers a question of Towsner. We also show that \mathrm{RCA}_{0} + \mathrm{RT}^{2}_{2} is \Pi^{1}_{1} -conservative over \mathrm{B}\Sigma^{0}_{2} if and only if it is conservative over \mathrm{B}\Sigma^{0}_{2} with respect to \forall \Pi^{0}_{5} sentences.

State Machines for Large Scale Computer Software and Systems

The behavior and architecture of large scale discrete state systems found in computer software and hardware can be specified and analyzed using a particular class of primitive recursive functions. This article begins with an illustration of the utility of the method via a number of small examples and then via longer specification and verification of the “Paxos” distributed consensus algorithm [ 26 ]. The “sequence maps” are then shown to provide an alternative representation of deterministic state machines and products of state machines. Distributed and composite systems, parallel and concurrent computation, and real-time behavior can all be specified naturally with these methods—which require neither extensions to the classical state machine model nor any axiomatic methods or other techniques from formal logic or other foundational methods. Compared to state diagrams or tables or the standard set-tuple-transition-maps, sequence maps are more concise and better suited to describing the behavior and compositional architecture of computer systems. Staying strictly within the boundaries of classical deterministic state machines anchors the methods to the algebraic structures of automata and makes the specifications faithful to engineering practice.

Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal

Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties.In this article, we prove that Buchholz's system of fundamental sequences for the ϑ function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the ϑ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along ϑ(εΩ+1).

Basic Predicate Calculus is Sound with Respect to a Modified Version of Strictly Primitive Recursive Realizability

Basic Predicate Calculus is Sound with Respect to a Modified Version of Strictly Primitive Recursive Realizability

Enhancing Symbolic Manipulation through Pairing Primitive Recursive String Functions and Interplay with Generalized Pairing PRSF

In earlier studies, the notion of generalized primitive recursive string functions has been presented, and their connections with abstract paring-based primitive recursive string functions have been investigated. Our study is centered around establishing a fundamental theorem that states a connection between these two distinct sorts of functions. The theorem specifically establishes that the universal definition of every generalized pairing primitive recursive string function is contingent upon its correspondence with a conventional(abstract) pairing primitive recursive string function. This article introduces innovative concept of Pairing Primitive Recursive String Functions (P-PRSF) for manipulating and interacting with word pairs. Based on the principles of primitive recursion and pairing functions, P-PRSF enables the extraction, transformation, and combination of word components. The proposed theorems validate the effectiveness of P-PRSF in capturing relationships within word pairs. Moreover, the interplay between P-PRSF and Generalized Pairing PRSF (GP-PRSF) extends the concept to involve more intricate interactions.

Certifying expressive power and algorithms of reversible primitive permutations with Lean

Reversible primitive permutations (RPP) is a class of recursive functions that models reversible computation. We present a proof, which has been verified using the proof-assistant Lean, that demonstrates RPP can encode every primitive recursive function (PRF-completeness) and that each RPP can be encoded as a primitive recursive function (PRF-soundness). Our proof of PRF-completeness is simpler and fixes some errors in the original proof, while also introducing a new reversible iteration scheme for RPP. By keeping the formalization and semi-automatic proofs simple, we are able to identify a single programming pattern that can generate a set of reversible algorithms within RPP: Cantor pairing, integer division quotient/remainder, and truncated square root. Finally, Lean source code is available for experiments on reversible computation whose properties can be certified.

On the Computability of Primitive Recursive Functions by Feedforward Artificial Neural Networks

We show that, for a primitive recursive function h(x,t), where x is a n-tuple of natural numbers and t is a natural number, there exists a feedforward artificial neural network N(x,t), such that for any n-tuple of natural numbers z and a positive natural number m, the first m+1 terms of the sequence {h(z,t)} are the same as the terms of the tuple (N(z,0),…,N(z,m)).

On Correspondences between Feedforward Artificial Neural Networks on Finite Memory Automata and Classes of Primitive Recursive Functions

When realized on computational devices with finite quantities of memory, feedforward artificial neural networks and the functions they compute cease being abstract mathematical objects and turn into executable programs generating concrete computations. To differentiate between feedforward artificial neural networks and their functions as abstract mathematical objects and the realizations of these networks and functions on finite memory devices, we introduce the categories of general and actual computabilities and show that there exist correspondences, i.e., bijections, between functions computable by trained feedforward artificial neural networks on finite memory automata and classes of primitive recursive functions.

A new perspective on completeness and finitist consistency

Abstract In this paper, we study the metamathematics of consistent arithmetical theories $T$ (containing $\textsf {I}\varSigma _{1}$); we investigate numerical properties based on proof predicates that depend on numerations of the axioms. Numeral Completeness. For every true (in $\mathbb {N}$) sentence $\vec {Q}\vec {x}.\varphi (\vec {x})$, with $\varphi (\vec {x})$ a $\varSigma _{1}(\textsf {I}\varSigma _1)$-formula, there is a numeration $\tau $ of the axioms of $T$ such that $\textsf {I}\varSigma _1\vdash \vec {Q}\vec {x}. \texttt {Pr}_{\tau }(\ulcorner \varphi (\overset {\text{.} }{\vec {x}})\urcorner )$, where $\texttt {Pr}_{\tau }$ is the provability predicate for the numeration $\tau $. Numeral Consistency. If $T$ is consistent, there is a $\varSigma _{1}(\textsf {I}\varSigma _1)$-numeration $\tau $ of the axioms of $\textsf {I}\varSigma _{1}$ such that $\textsf {I}\varSigma _1\vdash \forall\, x. \texttt {Pr}_{\tau }(\ulcorner \neg \textit {Prf}(\ulcorner \perp \urcorner , \overset {\text{.}}{x})\urcorner )$, where $\textit {Prf}(x,y)$ denotes a $\varDelta _{1}(\textsf {I}\varSigma _1)$-definition of ‘$y$ is a $T$-proof of $x$’. Finitist consistency is addressed by generalizing a result of Artemov: Partial finitism. If $T$ is consistent, there is a primitive recursive function $f$ such that, for all $n\in \mathbb {N}$, $f(n)$ is the code of an $\textsf {I}\varSigma _{1}$-proof of $\neg\, \textit{Prf}(\ulcorner \perp \urcorner ,\overline {n})$. These results are not in conflict with Gödel’s Incompleteness Theorems. Rather, they allow to extend their usual interpretation and show a deep connection to reflections in Hilbert’s last papers of 1931.

Deciding FO-rewritability of Regular Languages and Ontology-Mediated Queries in Linear Temporal Logic

Our concern is the problem of determining the data complexity of answering an ontology-mediated query (OMQ) formulated in linear temporal logic LTL over (Z,<) and deciding whether it is rewritable to an FO(<)-query, possibly with some extra predicates. First, we observe that, in line with the circuit complexity and FO-definability of regular languages, OMQ answering in AC0, ACC0 and NC1 coincides with FO(<,≡)-rewritability using unary predicates x ≡ 0 (mod n), FO(<,MOD)-rewritability, and FO(RPR)-rewritability using relational primitive recursion, respectively. We prove that, similarly to known PSᴘᴀᴄᴇ-completeness of recognising FO(<)-definability of regular languages, deciding FO(<,≡)- and FO(<,MOD)-definability is also PSᴘᴀᴄᴇ-complete (unless ACC0 = NC1). We then use this result to show that deciding FO(<)-, FO(<,≡)- and FO(<,MOD)-rewritability of LTL OMQs is ExᴘSᴘᴀᴄᴇ-complete, and that these problems become PSᴘᴀᴄᴇ-complete for OMQs with a linear Horn ontology and an atomic query, and also a positive query in the cases of FO(<)- and FO(<,≡)-rewritability. Further, we consider FO(<)-rewritability of OMQs with a binary-clause ontology and identify OMQ classes, for which deciding it is PSᴘᴀᴄᴇ-, Π2p- and coNP-complete.

Primitive recursive ordered fields and some applications

We establish primitive recursive (PR) versions of some known facts about computable ordered fields of reals and computable reals, and apply them to prove primitive recursiveness of several important problems in linear algebra and analysis. One of the central results of this paper is a partial PR analogue of Ershov–Madison’s theorem about real closures of computable ordered fields. It allows us, in particular, to obtain PR root-finding algorithms in the PR real and algebraic closures of PR fields with a certain property (PR splitting). We also relate the corresponding fields to the PR reals, as well as introduce and study the notion of a PR metric space. It enables us to derive sufficient conditions for PR computing of normal forms of matrices and solution operators of symmetric hyperbolic systems of PDEs. The methods represent a mix of symbolic and approximate algorithms.

Characterizing time computational complexity classes with polynomial differential equations

In this paper we show that several classes of languages from computational complexity theory, such as EXPTIME, can be characterized in a continuous manner by using only polynomial differential equations. This characterization applies not only to languages, but also to classes of functions, such as the classes defining the Grzegorczyk hierarchy, which implies an analog characterization of the class of elementary computable functions and the class of primitive recursive functions.

The nonarithmeticity of the predicate logic of primitive recursive realizability

The notion of primitive recursive realizability was introduced by S. Salehi as a kind of semantics for the language of basic arithmetic using primitive recursive functions. It is of interest to study the corresponding predicate logic. D. A. Viter proved that the predicate logic of primitive recursive realizability by Salehi is not arithmetical. The technically complex proof combines the methods used by the author of this article in the study of predicate logics of constructive arithmetic theories and the results of M. Ardeshir on the translation of the intuitionistic predicate logic into the basic predicate logic. The purpose of this article is to present another proof of Viter's result by directly transferring the methods used earlier in proving the nonarithmeticity of the predicate logic of recursive realizability.

Folding left and right matters: Direct style, accumulators, and continuations

Abstract The equivalence of folding left and right over Peano numbers and lists makes it possible to minimalistically inter-derive (1) structurally recursive functions in direct style, (2) structurally tail-recursive functions that use an accumulator, and (3) structurally tail-recursive functions in delimited continuation-passing style, using Ohori and Sasano’s lightweight fusion by fixed-point promotion. When the fold-left and the fold-right functions account for primitive iteration for Peano numbers, this equivalence is unconditional. When they account for primitive recursion for Peano numbers, this equivalence is modulo left permutativity of their induction-step parameter – a property which is more general than associativity and commutativity. And when they account for primitive iteration or for primitive recursion over lists, this equivalence is modulo left permutativity of their induction-step parameter if these two fold functions have the same type. Since the 1980s, however, the two fold functions for lists do not have the same type: the arguments for their induction-step parameter are swapped, a re-ordering that complicated Bird and Wadler’s duality theorems and whose history is reviewed in an appendix. Without this re-ordering, Bird and Wadler’s second duality theorem more visibly accounts for “re-bracketing,” which is a key step to make recursive programs tail recursive in the general area of program development, from Cooper in the 1960s and onwards.

First-Order Rewritability and Complexity of Two-Dimensional Temporal Ontology-Mediated Queries

Aiming at ontology-based data access to temporal data, we design two-dimensional temporal ontology and query languages by combining logics from the (extended) DL-Lite family with linear temporal logic LTL over discrete time (Z,<). Our main concern is first-order rewritability of ontology-mediated queries (OMQs) that consist of a 2D ontology and a positive temporal instance query. Our target languages for FO-rewritings are two-sorted FO(<) -- first-order logic with sorts for time instants ordered by the built-in precedence relation < and for the domain of individuals---its extension FO(<, ≡) with the standard congruence predicates t ≡ 0 (mod n), for any fixed n > 1, and FO(RPR) that admits relational primitive recursion. In terms of circuit complexity, FO(<, ≡)- and FO(RPR)-rewritability guarantee answering OMQs in uniform AC0 and NC1, respectively. We proceed in three steps. First, we define a hierarchy of 2D DL-Lite/LTL ontology languages and investigate the FO-rewritability of OMQs with atomic queries by constructing projections onto 1D LTL OMQs and employing recent results on the FO-rewritability of propositional LTL OMQs. As the projections involve deciding consistency of ontologies and data, we also consider the consistency problem for our languages. While the undecidability of consistency for 2D ontology languages with expressive Boolean role inclusions might be expected, we also show that, rather surprisingly, the restriction to Krom and Horn role inclusions leads to decidability (and ExpSpace-completeness), even if one admits full Booleans on concepts. As a final step, we lift some of the rewritability results for atomic OMQs to OMQs with expressive positive temporal instance queries. The lifting results are based on an in-depth study of the canonical models and only concern Horn ontologies.

Rogers semilattices of punctual numberings

AbstractThe paper works within the framework of punctual computability, which is focused on eliminating unbounded search from constructions in algebra and infinite combinatorics. We study punctual numberings, that is, uniform computations for families S of primitive recursive functions. The punctual reducibility between numberings is induced by primitive recursive functions. This approach gives rise to upper semilattices of degrees, which are called Rogers pr-semilattices. We show that any infinite, uniformly primitive recursive family S induces an infinite Rogers pr-semilattice R. We prove that the semilattice R does not have minimal elements, and every nontrivial interval inside R contains an infinite antichain. In addition, every non-greatest element from R is a part of an infinite antichain. We show that the $\Sigma_1$ -fragment of the theory Th(R) is decidable.

Basic Predicate Calculus is not Sound with Respect to the Strong Variant of Strictly Primitive Recursive Realizability

Basic Predicate Calculus is not Sound with Respect to the Strong Variant of Strictly Primitive Recursive Realizability

Definitional schemes for primitive recursive and computable functions

The unary primitive recursive functions can be defined in terms of a finite set of initial functions together with a finite set of unary and binary operations that are primitive recursive in their inputs. We reduce arity considerations, by show that two fixed unary operations suffice, and a single initial function can be chosen arbitrarily. The method works for many other classes of functions, including the unary partial computable functions. For this class of partial functions we also show that a single unary operation (together with any finite set of initial functions) will never suffice.

Production Assessment Using a Knowledge Transfer Framework and Evidence Theory

Operational knowledge is one of the most valuable assets in a company, as it provides a strategic advantage over competitors and ensures steady and optimal operation in machines. An (interactive) assessment system on the shop floor can optimize the process and reduce stopovers because it can provide constant valuable information regarding the machine condition to the operators. However, formalizing operational (tacit) knowledge to explicit knowledge is not an easy task. This transformation considers modeling expert knowledge, quantification of knowledge uncertainty, and validation of the acquired knowledge. This study proposes a novel approach for production assessment using a knowledge transfer framework and evidence theory to address the aforementioned challenges. The main contribution of this paper is a methodology for the formalization of tacit knowledge based on an extended failure mode and effect analysis for knowledge extraction, as well as the use of evidence theory for the uncertainty definition of knowledge. Moreover, this approach uses primitive recursive functions for knowledge modeling and proposes a validation strategy of the knowledge using machine data. These elements are integrated into an interactive recommendation system hosted on a backend that uses HoloLens as a visual interface. We demonstrate this approach using an industrial setup: a laboratory bulk good system. The results yield interesting insights, including the knowledge validation, uncertainty behavior of knowledge, and interactive troubleshooting for the machine operator.

Arithmetic Formulated in a Logic of Meaning Containment


 
 
 We assess Meyer’s formalization of arithmetic in his [21], based on the strong relevant logic R and compare this with arithmetic based on a suitable logic of meaning containment, which was developed in Brady [7]. We argue in favour of the latter as it better captures the key logical concepts of meaning and truth in arithmetic. We also contrast the two approaches to classical recapture, again favouring our approach in [7]. We then consider our previous development of Peano arithmetic including primitive recursive functions, finally extending this work to that of general recursion.