Logical Strength Research Articles

On the logical strengths of partial solutions to mathematical problems

We use the framework of reverse mathematics to address the question of, given a mathematical problem, whether or not it is easier to find an infinite partial solution than it is to find a complete solution. Following Flood, we say that a Ramsey-type variant of a problem is the problem with the same instances but whose solutions are the infinite partial solutions to the original problem. We study Ramsey-type variants of problems related to K\"onig's lemma, such as restrictions of K\"onig's lemma, Boolean satisfiability problems, and graph coloring problems. We find that sometimes the Ramsey-type variant of a problem is strictly easier than the original problem (as Flood showed with weak K\"onig's lemma) and that sometimes the Ramsey-type variant of a problem is equivalent to the original problem. We show that the Ramsey-type variant of weak K\"onig's lemma is robust in the sense of Montalban: it is equivalent to several perturbations of itself. We also clarify the relationship between Ramsey-type weak K\"onig's lemma and algorithmic randomness by showing that Ramsey-type weak weak K\"onig's lemma is equivalent to the problem of finding diagonally non-recursive functions and that these problems are strictly easier than Ramsey-type weak K\"onig's lemma. This answers a question of Flood.

The strength of infinitary Ramseyan principles can be accessed by their densities

In this article, we conduct a model-theoretic investigation of three infinitary Ramseyan statements: the Infinite Ramsey Theorem for pairs and two colours (RT22), the Canonical Ramsey Theorem for pairs (CRT2) and the Regressive Ramsey Theorem for pairs (RegRT2). We approximate the logical strength of these principles by the strength of their first-order iterated versions, known as density principles. We then investigate their logical strength using strong initial segments of models of Peano Arithmetic, in the spirit of the classical article by Paris and Kirby, hereby re-proving old results model-theoretically. The article is concluded by a discussion of two further outreaches of densities. One is a further investigation of the strength of the Ramsey Theorem for pairs. The other deals with the asymptotics of the standard Ramsey function R22.

Reverse mathematics and marriage problems with finitely many solutions

We analyze the logical strength of theorems on marriage problems with a fixed finite number of solutions via the techniques of reverse mathematics. We show that if a marriage problem has k solutions, then there is a finite set of boys such that the marriage problem restricted to this set has exactly k solutions, each of which extend uniquely to a solution of the original marriage problem. The strength of this assertion depends on whether or not the marriage problem has a bounding function. We also answer three questions from our previous work on marriage problems with unique solutions.

Fragments of Kripke–Platek set theory and the metamathematics of $$\alpha $$ α -recursion theory

The foundation scheme in set theory asserts that every nonempty class has an $$\in $$ź-minimal element. In this paper, we investigate the logical strength of the foundation principle in basic set theory and $$\alpha $$ź-recursion theory. We take KP set theory without foundation (called KP$$^-$$-) as the base theory. We show that KP$$^-$$- + $$\Pi _1$$ź1-Foundation + $$V=L$$V=L is enough to carry out finite injury arguments in $$\alpha $$ź-recursion theory, proving both the Friedberg-Muchnik theorem and the Sacks splitting theorem in this theory. In addition, we compare the strengths of some fragments of KP.

Software Reliability Prediction using Fuzzy Inference System: Early Stage Perspective

paper presents a reliability prediction model that predicts the reliability of the developing software using fuzzy inference system. The focus of the study is on the reliability prediction prior to the coding phase so that the developers use this information for optimally performing resource planning and quality assessment of the software under development. Requirements and object-oriented design level product measures have participated for early reliability prediction. The paper has also utilized the strengths of fuzzy logic to deal with the uncertainties and vagueness involved in the early stage measures. The model has also been statistically validated through the data set obtained through twenty real software projects. The values of the Pearson's correlation coefficient along with the predictive accuracy measures are quite encouraging, and support that the developed model is a better and improved reliability prediction model.

Logical Probability and the Strength of Mathematical Conjectures

Mathematicians often speak of the evidence for unproved conjectures, such as the Riemann Hypothesis. It is argued that such evidence should be seen in terms of logical probability in Keynes's sense: a strictly logical degree of partial implication. That is essentially the same as objective Bayesianism. Examples are given and explained in terms of the objective logical strength of evidence.

Labelled Interpolation Systems for Hyper-Resolution, Clausal, and Local Proofs.

Craig’s interpolation theorem has numerous applications in model checking, automated reasoning, and synthesis. There is a variety of interpolation systems which derive interpolants from refutation proofs; these systems are ad-hoc and rigid in the sense that they provide exactly one interpolant for a given proof. In previous work, we introduced a parametrised interpolation system which subsumes existing interpolation methods for propositional resolution proofs and enables the systematic variation of the logical strength and the elimination of non-essential variables in interpolants. In this paper, we generalise this system to propositional hyper-resolution proofs as well as clausal proofs. The latter are generated by contemporary SAT solvers. Finally, we show that, when applied to local (or split) proofs, our extension generalises two existing interpolation systems for first-order logic and relates them in logical strength.

Logical strength of complexity theory and a formalization of the PCP theorem in bounded arithmetic

We present several known formalizations of theorems from computational complexity in bounded arithmetic and formalize the PCP theorem in the theory PV1 (no formalization of this theorem was known). This includes a formalization of the existence and of some properties of the (n,d,{\lambda})-graphs in PV1.

A new hybrid artificial intelligence approach to predicting global thermal comfort of stretch knitted fabrics

Today numerous consumers consider thermal comfort to be one of the most significant attributes when purchasing textile and apparel products, so there is a need to develop a model able to simulate objectively the consumers’ perception. The global thermal comfort of stretch knitted fabrics is a multi-criteria phenomenon that requires the satisfaction of several properties at the same time. In this paper, we used the desirability functions to evaluate the satisfaction degree of global thermal comfort. Statistical method was used to investigate the interrelationship among knit thermo-physical properties, and group them into factors. Two models of artificial neural network (general and special) have been set up to predict the global thermal comfort from structural parameters (inputs) of knitted fabrics made from pure yarn cotton (cellulose) and viscose (regenerated cellulose) fibers and plated knitted with elasthane (Lycra) fibers. A virtual leave one out approach dealing with over fitting phenomenon and allowing the selection of the optimal neural network architecture was used. By combining the strengths of statistics and fuzzy logic (data reduction and information summation) also a neural network (self-learning ability), hybrid model was developed to simulate the consumer thermal comfort perception. After that, ANN model is inverted. With a required output value and some input parameters it is possible to calculate the unknown optimum input parameter. Finally, this forecasting can help industrials to anticipate the consumer’s taste. Thus, they can adjust the knitting production parameter to reach the desired global thermal comfort to satisfy this consumer.

On uniform relationships between combinatorial problems

The enterprise of comparing mathematical theorems according to their logical strength is an active area in mathematical logic, with one of the most common frameworks for doing so being reverse mathematics. In this setting, one investigates which theorems provably imply which others in a weak formal theory roughly corresponding to computable mathematics. Since the proofs of such implications take place in classical logic, they may in principle involve appeals to multiple applications of a particular theorem, or to non-uniform decisions about how to proceed in a given construction. In practice, however, if a theorem Q \mathsf {Q} implies a theorem P \mathsf {P} , it is usually because there is a direct uniform translation of the problems represented by P \mathsf {P} into the problems represented by Q \mathsf {Q} , in a precise sense formalized by Weihrauch reducibility. We study this notion of uniform reducibility in the context of several natural combinatorial problems, and compare and contrast it with the traditional notion of implication in reverse mathematics. We show, for instance, that for all n , j , k ≥ 1 n,j,k \geq 1 , if j > k j > k , then Ramsey’s theorem for n n -tuples and k k many colors is not uniformly, or Weihrauch, reducible to Ramsey’s theorem for n n -tuples and j j many colors. The two theorems are classically equivalent, so our analysis gives a genuinely finer metric by which to gauge the relative strength of mathematical propositions. We also study Weak König’s Lemma, the Thin Set Theorem, and the Rainbow Ramsey’s Theorem, along with a number of their variants investigated in the literature. Weihrauch reducibility turns out to be connected with sequential forms of mathematical principles, where one wishes to solve infinitely many instances of a particular problem simultaneously. We exploit this connection to uncover new points of difference between combinatorial problems previously thought to be more closely related.

What Do Reciprocals Mean?

Research on reciprocals has uncovered a variety of semantic contributions that the reciprocal can make, creating problems for proposals that the reciprocal unambiguously means something weak (e.g., Langendoen 1978). However, there is no real evidence that reciprocals are ambiguous, despite previous claims to the contrary (e.g., Fiengo and Lasnik 1973). First, we classify the apparently heterogeneous list of meanings proposed in previous research into a natural taxonomy, showing how they arise from a small stock of logical operations and predicates. Second, we exhibit a partial ordering of the various reciprocal meanings according to logical strength, which we make crucial use of in determining what reciprocals mean in each specific context where they appear. Third, we hypothesize that a reciprocal statement expresses the strongest candidate meaning that is consistent with known properties of the relation expressed by the scope of the reciprocal. This hypothesis is supported by analysis of a large collection of examples we have gathered from various corpora.

CONSISTENCY AND THE THEORY OF TRUTH

AbstractWhat is the logical strength of theories of truth? That is: If you take a theory${\cal T}$and add a theory of truth to it, how strong is the resulting theory, as compared to${\cal T}$? Once the question has been properly formulated, the answer turns out to be about as elegant as one could want: At least when${\cal T}$is finitely axiomatized theory, theories of truth act more or less as a kind of abstract consistency statement. To prove this result, however, we have to formulate truth-theories somewhat differently from how they have been and instead follow Tarski in ‘disentangling’ syntactic theories from object theories.

Coordinated Power and Performance Guarantee with Fuzzy MIMO Control in Virtualized Server Clusters

It is important but challenging to assure the performance of multi-tier Internet applications with the power consumption cap of virtualized server clusters mainly due to system complexity of shared infrastructure and dynamic and bursty nature of workloads. This paper presents PERFUME, a system that simultaneously guarantees power and performance targets with flexible tradeoffs and service differentiation among co-hosted applications while assuring control accuracy and system stability. Based on the proposed fuzzy MIMO control technique, it effectively controls both the throughput and percentile-based response time of multi-tier applications due to its novel self-adaptive fuzzy modeling that integrates the strengths of fuzzy logic, MIMO control and artificial neural network. Furthermore, we address an important challenge of pro-actively avoiding violations of power and performance targets in anticipation of future workload changes. We implement PERFUME in a testbed of virtualized blade servers hosting multi-tier RUBiS applications. Performance evaluation based on synthetic and real-world Web workloads demonstrates its control accuracy, flexibility in selecting tradeoffs between conflicting targets, service differentiation capability and robustness against highly dynamic and bursty workloads. It outperforms a representative utility based approach in providing guarantee of the system throughput, percentile-based response time and power budget.

Reverse mathematics and marriage problems with unique solutions

We analyze the logical strength of theorems on marriage problems with unique solutions using the techniques of reverse mathematics, restricting our attention to problems in which each boy knows only finitely many girls. In general, these marriage theorems assert that if a marriage problem has a unique solution then there is a way to enumerate the boys so that for every m, the first m boys know exactly m girls. The strength of each theorem depends on whether the underlying marriage problem is finite, infinite, or bounded.

Reverse mathematics and Isbell's zig‐zag theorem

The paper explores the logical strength of Isbell's zig‐zag theorem using the framework of reverse mathematics. Working in , we show that is equivalent to Isbell's zig‐zag theorem for countable monoids: If B is a monoid extension of A, then is dominated by A if and only if b has a zig‐zag over A. Our proof of Isbell's zig‐zag theorem avoids use of strong comprehension axioms common in traditional proofs. We also analyze the strength of theorems concerning binary relations.

Improved Separations of Regular Resolution from Clause Learning Proof Systems

This paper studies the relationship between resolution and conflict driven clause learning (CDCL) without restarts, and refutes some conjectured possible separations. We prove that the guarded, xor-ified pebbling tautology clauses, which Urquhart proved are hard for regular resolution, as well as the guarded graph tautology clauses of Alekhnovich, Johannsen, Pitassi, and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses. For the latter set of clauses, we extend this to prove that a CDCL search without restarts can refute these clauses in polynomial time, provided it makes the right choices for decision literals and clause learning. This holds even if the CDCL search is required to greedily process conflicts arising from unit propagation. This refutes the conjecture that the guarded graph tautology clauses or the guarded xor-ified pebbling tautology clauses can be used to separate CDCL without restarts from general resolution. Together with subsequent results by Buss and Kolodziejczyk, this means we lack any good conjectures about how to establish the exact logical strength of conflict-driven clause learning without restarts.

Slow consistency

The fact that “natural” theories, i.e. theories which have something like an “idea” to them, are almost always linearly ordered with regard to logical strength has been called one of the great mysteries of the foundation of mathematics. However, one easily establishes the existence of theories with incomparable logical strengths using self-reference (Rosser-style). As a result, PA+Con(PA) is not the least theory whose strength is greater than that of PA. But still we can ask: is there a sense in which PA+Con(PA) is the least “natural” theory whose strength is greater than that of PA? In this paper we exhibit natural theories in strength strictly between PA and PA+Con(PA) by introducing a notion of slow consistency.

IndentifyingMaqasid al-Qur’an: A Critical Analysis Of Rashid Riaz’sviews

This articleexaminesMuÍammadRashid Rida’s (1865-1935 C.E.) views of Maqasid al-Qur’an.Rida’s work on this subject is one of a few rear pieces that comprehensively deal with this emerging distinctive concept of understanding the Qur’Énic discourse. Thus, it is significant to view RiÌÉ’s insight and contribution to this subject.This article presents a critical analysis of the ten main objectives of the Qur’an that he dealt with in his book al-WaÍy al-MuÍammadÊ as well as TafsÊr al-ManÉr. This study mainly explores the methodological and logical strength of RiÌÉ’s identification of objectives of the Qur’an.

Unconfusing Merely Confused Supposition in Albert of Saxony

Abstract In this essay I argue that Albert would reject the need for a separate fourth mode of common personal supposition, and that his view of merely confused supposition has not been fully explicated by modern scholars. I first examine the various examples of conjunct descent given by modern scholars from his Perutilis logica, and show that Albert clearly adopts it in resolving the sophistic examples involved. Second, I explicate the view of merely confused supposition that Albert defends in his Sophismata, and then attempt to answer the question: which view of merely confused supposition was his final view, the view articulated in the Perutilis logica or the view in the Sophismata? I conclude that based upon his Sophismata view of merely confused supposition, Albert came to realize the logical strength his revised theory of personal supposition afforded, and consequently, that he is one of the earliest 14th-century logicians to adopt conjunct terminal descent to resolve various sophisms, a move which gave his theory of personal supposition a logical symmetry having two sorts of propositional descents to singulars, and two sorts of terminal descents to singulars.

Reverse mathematics, trichotomy, and dichotomy

Using the techniques of reverse mathematics, we analyze the logical strength of statements similar to trichotomy and dichotomy for sequences of reals. Capitalizing on the connection between sequential statements and constructivity, we find computable restrictions of the statements for sequences and constructive restrictions of the original principles. 2010 Mathematics Subject Classification 03B30, 03F35 (primary); 03F60, 30F50 (secondary)