Propositional Version Research Articles

A PROOF COMPLEXITY CONJECTURE AND THE INCOMPLETENESS THEOREM

Abstract Given a sound first-order p-time theory T capable of formalizing syntax of first-order logic we define a p-time function $g_T$ that stretches all inputs by one bit and we use its properties to show that T must be incomplete. We leave it as an open problem whether for some T the range of $g_T$ intersects all infinite ${\mbox {NP}}$ sets (i.e., whether it is a proof complexity generator hard for all proof systems). A propositional version of the construction shows that at least one of the following three statements is true: 1. There is no p-optimal propositional proof system (this is equivalent to the non-existence of a time-optimal propositional proof search algorithm). 2. $E \not \subseteq P/poly$ . 3. There exists function h that stretches all inputs by one bit, is computable in sub-exponential time, and its range $Rng(h)$ intersects all infinite ${\text {NP}}$ sets.

Automated Verification of Propositional Agent Abstraction for Classical Planning via CTLK Model Checking

Abstraction has long been an effective mechanism to help find a solution in classical planning. Agent abstraction, based on the situation calculus, is a promising explainable framework for agent planning, yet its automation is still far from being tackled. In this paper, we focus on a propositional version of agent abstraction designed for finite-state systems. We investigate the automated verification of the existence of propositional agent abstraction, given a finite-state system and a mapping indicating an abstraction for it. By formalizing sound, complete and deterministic properties of abstractions in a general framework, we show that the verification task can be reduced to the task of model checking against CTLK specifications. We implemented a prototype system, and validated the viability of our approach through experimentation on several domains from classical planning.

Corrigendum to "Determinants of Normalized Bohemian Upper Hessenberg Matrices"

An amended version of Proposition 3.6 of [Fasi and Negri Porzio, Electron. J. Linear Algebra 36:352--366, 2020] is presented. The result shows that the set of possible determinants of upper Hessenberg matrices with ones on the subdiagonal and elements in the upper triangular part drawn from the set $\{-1,1\}$ is $\{ 2k \mid k \in \langle -2^{n-2} , 2^{n-2} \rangle \}$, instead of $\{ 2k \mid k \in \langle -n+1, n-1 \rangle \}$ as previously stated. This does not affect the main results of the article being corrected and shows that Conjecture 20 in the Characteristic Polynomial Database is true.

Erratum to “On asymptotic normality of certain linear rank statistics” [Statist. Probab. Lett. 145 (2019) 63–73

In our paper On asymptotic normality of certain linear rank statistics, Statistics & Probability Letters, Volume 145, 2019, Pages 63–73, https://doi.org/10.1016/j.spl.2018.08.016., we have noticed an arithmetic error in the proof of Proposition 3.3. The latter error is easy to correct and the correction strengthens current version of Proposition 3.3. The related Proposition 3.4 remains valid without significant alterations in the proofs. Some minor typos are also fixed.

Pecularities of Some Three- and Four-Valued Second Order Logics

Logics that have many truth values—more than just True and False—have been argued to be useful in the analysis of very many philosophical and linguistic puzzles (as well, sometimes, in various computational-oriented tasks). In this paper, which is a followup to (Hazen and Pelletier in K3, Ł3, LP, RM3, A3, FDE, M: How to make many-valued logics work for you. Winning paper for the Canadian Schotch-Jennings Prize, one of the prizes of the Universal Logic competition in 2018; Notre Dame J Form Log 59, 2018), we will start with a particularly well-motivated four-valued logic that has been studied mainly in its propositional and first-order versions. And we will then investigate its second-order version. This four-valued logic has two natural three-valued extensions: what is called a “gap logic” (some formulas are neither True nor False), and what is called a “glut logic” (some formulas are both True and False). We mention various results about the second-order version of these logics as well. And we then follow our earlier papers, where we had added a specific conditional connective to the three valued logics, and now add that connective to the four-valued logic under consideration. We then show that, although this addition is “conservative” in the sense that no new theorems are generated in the four-valued logic unless they employ this new conditional in their statement, nevertheless the resulting second-order versions of these logics with and without the conditional are quite different in important ways. We close with a moral for logical investigations in this realm.

A cirquent calculus system with clustering and ranking

Cirquent calculus is a new proof-theoretic and semantic approach introduced by G. Japaridze for the needs of his theory of computability logic (CoL). The earlier article “From formulas to cirquents in computability logic” by Japaridze generalized formulas in CoL to circuit-style structures termed cirquents. It showed that, through cirquents with what are termed clustering and ranking, one can capture, refine and generalize independence-friendly (IF) logic. Specifically, the approach allows us to account for independence from propositional connectives in the same spirit as IF logic accounts for independence from quantifiers. Japaridze's treatment of IF logic, however, was purely semantical, and no deductive system was proposed. The present paper syntactically constructs a cirquent calculus system with clustering and ranking, sound and complete w.r.t. the propositional fragment of cirquent-based semantics. Such a system captures the propositional version of what is called extended IF logic, thus being an axiomatization of a nontrivial fragment of that logic.

The Tarski T-Schema is a tautology (literally)

The Tarski T-Schema has a propositional version. If we use ϕ as a metavariable for formulas and use terms of the form that-ϕ to denote propositions, then the propositional version of the T-Schema is: that-ϕ is true if and only if ϕ. For example, that Cameron is Prime Minister is true if and only if Cameron is Prime Minister. If that-ϕ is represented formally as [λ ϕ], then the T-Schema can be represented as the 0-place case of λ-Conversion. If we interpret [λ…] as a truth-functional context, then using traditional logical techniques, one can prove that the propositional version of the T-Schema is a tautology, literally. Given how well-accepted these logical techniques are, we conclude that the T-Schema, in at least one of its forms, is a not just a logical truth but a tautology at that.

Mind and paradox: paradoxes depend on minds

Paradoxes are mind-dependent in a number of ways. First, by definition, paradoxes offer surprises or apparent contradictions. Since surprise and appearance rely on subjective psychological reactions, paradoxes rely on psychological events. Second, propositional versions of the liar paradox must eventually appeal to sentences if they are to achieve traction, yet sentential versions of the liar paradox rely on language and hence on mentality. Third, belief paradoxes such as B, "No one believes B", transparently hinge on the existence of mental states. (Belief paradoxes extending work by Burge, Buridan, Grim, Sainsbury, and Sorensen demonstrate a fundamental limitation on all cognitive systems: that none can hold all and only truths.) Finally, it is argued that belief paradoxes, like liar paradoxes, arise from the nature of semantic representation.

«Lamentabili sane exitu» (1907). Les documents préparatoires du Saint Office (review)

Reviewed by: «Lamentabili sane exitu» (1907). Les documents préparatoires du Saint Office C. J. T. Talar «Lamentabili sane exitu» (1907). Les documents préparatoires du Saint Office. Edited by Claus Arnold and Giacomo Losito. [Fontes Archivi Sancti Officii Romani, Vol. 6.] (Rome: Liberia Editrice Vaticana. 2011. Pp. XVI, 546. €60,00 paperback. ISBN 978-8-820-98587-5.) This volume essentially picks up the story of Vatican resistance and censure of Roman Catholic modernism where the previous one edited by Claus Arnold and Giacomo Losito, La censure d'Alfred Loisy (1903, vol. 4 in the same series) left off. From discussions surrounding the censure of five of Loisy's works in 1903 there emerged a decision to draw up a list of errors as a defense of Catholicism against a generalized rationalist menace. Over the course of the document's development, the focus narrowed to Loisy and to the French exegetical "school" of which he was perceived to be head. In a foreword Emile Poulat presents the major themes present in the final version of sixty-five propositions: critical exegesis and the ecclesial magisterium (1-9), inspiration and revelation (10-21), dogma (22-38), sacraments (52-57), and immutable truth (58-65). Those familiar with Loisy's modernist writings will readily discern the connections of these areas with those publications. The net, however, was originally cast more widely and, as the editors show, more intransigently than in the final version. In the first of the two introductory essays, Arnold provides the background for the sequencing and significance of the published documentation. From the discussions in 1903 Domenico Palmieri and Pie de Langogne were each charged with drafting a basis for discussion. The first document in the [End Page 390] series reflects Palmieri's efforts to broadly connect Loisy with Kant's errors and exhibit the dangers of the exegete's theological "evolutionism," followed by ninety-three propositions in Latin with a theological qualification attached to each. De Langogne's text (document 2) broadened its perspective beyond Loisy to encompass the "progressive" French exegetical school that included Albert Houtin and Monsignor Eudoxe Irénée Edouard Mignot. A programmatic introduction was followed by 119 propositions, without theological qualification. Both Palmieri and de Langogne, with the addition of Willem Van Rossum, were then tasked with collaborating on a third version (document 3) that was closer to de Langogne's draft and contained ninety-six propositions, sixty-six of which were judged "heretical." This joint effort then became the basis of discussion among consulters (represented in document 4) that show a moderating influence on some propositions, while discarding others. De Langogne contributed an additional document (5) examining the French hermeneutic of dogma represented by Edouard Le Roy that would figure in one of Lamentabili's propositions. Document 6 presents the results of further discussion of each proposition by the cardinals and the consulters, which was then given over to Andreas Steinhuber, who was entrusted with the redaction of the introduction to the syllabus and who eliminated a few more propositions to arrive at the final sixty-five (document 7). The editors then give the final version of each proposition, together with its sources in earlier versions, as well as the propositions that were rejected. The second introductory essay by Losito examines antimodernist activity in France and its links to the Vatican before focusing on de Langogne, who made use of this criticism in his drafts. The final version of Lamentabili is closest to the latter's vision and versions that he singled out for particular consideration. In a third section Lamentabili's impact on French public opinion is surveyed, and in the final one, the hierarchical exchanges that formed the background to Loisy's excommunication in 1908 are displayed. In Simples réflexions (Ceffonds, 1907) Loisy attempted to identify the probable source of each proposition (facilitated by the fact that most came from his own writings). The publication of the archival documentation enables scholars not only to verify the source of a proposition but also to appreciate what the framers had in mind when they deemed it censurable. Arnold and Losito also give access to the differences among those involved in the...

Response to Comment on ‘Preconditioning of matrices partitioned in 2 × 2 block form: Eigenvalue estimates and Schwarz DD for mixed FEM’

SUMMARYAs has been pointed out in a study by Cao, there is an erroneous omission in the study by Axelsson, Corollary 3.2 and Proposition 3.2, and the authors thank Zhi‐hao Cao for pointing this out.We mention, however, that these are side results and do not influence the major result of the paper ‘Preconditioning of matrices partitioned in 2 by 2 block form: Eigenvalue estimates and Schwarz DD for mixed FEM’.In this reply, we give a short background to the relations between inverses of saddle point matrices, thereby providing justified references to previous publications of the topic and present a corrected and extended version of Proposition 3.2. This gives an alternative presentation of the statements in the study by Cao, which might be useful for readers. Copyright © 2012 John Wiley & Sons, Ltd.

A tableau-based decision procedure for CTL*

AbstractWe present a sound, complete and implementable tableau method for deciding satisfiability of formulas in the propositional version of computation tree logic CTL*. This is the first such tableau. CTL* is an exceptionally important temporal logic with applications from hardware design to agent reasoning, but there is no easily automated reasoning approach to CTL*. The tableau here is a traditional tree-shaped or top-down style tableau, and affords the possibility of reasonably quick decisions on the satisfiability of medium-sized formulas and construction of small models for them. A straightforward subroutine is given for determining when looping allows successful branch termination, but much needed further development is left as future work. In particular, a more general repetition prevention mechanism is needed to speed up the task of tableau construction.

Sectoral Output Responses to Trade Openness, Oil Price and Policy Shocks in Nigeria: A CVAR Approach

This study investigated the relative effectiveness of trade and policy shocks on sectoral output growth in a small open Nigerian economy. It is a country-specific, time series study verifies whether there is difference in the effect of sectoral output response to policy shocks in Nigeria. A CVAR model was specified to assess the effects of policy shocks on real aggregate and sectoral output measures. The model included oil price shock and an interactive term of trade openness as measures of supply and external shocks to the economy. The empirical results showed that there was remarkably difference in sectoral output responses to policy distortion. The effects of monetary policy shocks were positive and significant on manufacturing, service and industrial sector while fiscal policy shock was only significant and positive on agricultural output growth. The result further showed that international oil price shock and trade openness had pronounced negative effects on both sectoral and aggregate outputs. In addition, oil and trade openness’ negative effects overwhelmed the positive effects of fiscal and monetary policy shocks. The policy implication of the finding is that the effectiveness of domestic macroeconomic policy is constrained by the external shocks from both oil price and trade openness. Thus, confirming the open economic version of policy ineffectiveness proposition of the New classical macroeconomic in Nigeria

A short note on some properties of rough groups

It is a useful method in research of group theory to construct a new group by using known groups. Lower and upper approximation operators of rough sets are applied into group theory and so the notion of a rough group has been introduced. In this paper, we first point out that there are still some incomplete propositions in [N. Kuroki, P.P. Wang, The lower and upper approximations in a fuzzy group, Inform. Sci. 90 (1996) 203–220] although some authors have showed several incorrect statements in the literature. We then present improved versions of the incomplete propositions and continue to study the image and inverse image of rough approximations of a subgroup with respect to a homomorphism between two groups.

Deontic action logic, atomic boolean algebras and fault-tolerance

We introduce a deontic action logic and its axiomatization. This logic has some useful properties ( soundness, completeness, compactness and decidability), extending the properties usually associated with such logics. Though the propositional version of the logic is quite expressive, we augment it with temporal operators, and we outline an axiomatic system for this more expressive framework. An important characteristic of this deontic action logic is that we use boolean combinators on actions, and, because of finiteness restrictions, the generated boolean algebra is atomic, which is a crucial point in proving the completeness of the axiomatic system. As our main goal is to use this logic for reasoning about fault-tolerant systems, we provide a complete example of a simple application, with an attempt at formalization of some concepts usually associated with fault-tolerance.

First Order Stålmarck

We present a proof method with a novel way of introducing universal lemmas. The method is a first order extension of Stalmarck's method, containing a branch-and-merge rule known as the dilemma rule. The dilemma rule creates two branches in a tableau-like way, but later recombines the two branches, keeping the common consequences. While the propositional version uses normal set intersection in the merges, the first order version searches for pairwise unifiable formulae in the two branches. Within branches, the system uses a special kind of variables that may not be substituted. At branch merges, these variables are replaced by universal variables, and in this way universal lemmas can be introduced. Relevant splitting formulae are found through failed unifications of variables in branches. This article presents the calculus and proof procedure, and shows soundness and completeness. Benchmarks of an implementation are also presented.

Proposition II (Book I) of Newton’s Principia

After preparing the way with comments on evanescent quantities and then Newton’s interpretation of his second law, this study of Proposition II (Book I)— Proposition II Every body that moves in some curved line described in a plane and, by a radius drawn to a point, either unmoving or moving uniformly forward with a rectilinear motion, describes areas around that point proportional to the times, is urged by a centripetal force tending toward that same point. —asks and answers the following questions: When does a version of Proposition II first appear in Newton’s work? What revisions bring that initial version to the final form in the 1726 Principia? What, exactly, does this proposition assert? In particular, what does Newton mean by the motion of a body “urged by a centripetal force”? Does it assert a true mathematical claim? If not, what revision makes it true? Does the demonstration of Proposition II persuade? Is it as convincing, for example, as the most convincing arguments of the Principia? If not, what revisions would make the demonstration more persuasive? What is the importance of Proposition II, to the physics of Book III and the mathematics of Book I?

A correction to: ``Some characterizations of space-forms"

In [1], for the application to Theorem 4.3, one needs a version of Proposition 3.1 assuming that the manifold supports an inhomogeneous Poincare-Sobolev inequality of the type (0.1) below. Indeed, an easy modification of the original argument yields the following Proposition 3.1 bis. Let (M, 〈 , 〉) be a complete manifold and assume that, for some 0 ≤ α < 1 and some non-negative function h, the inhomogeneous Sobolev– Poincare type inequality

Why the idea of framework propositions cannot contribute to an understanding of delusions

One of the tasks that recent philosophy of psychiatry has taken upon itself is to extend the range of understanding to some of those aspects of psychopathology that Jaspers deemed beyond its limits. Given the fundamental difficulties of offering a literal interpretation of the contents of primary delusions, a number of alternative strategies have been put forward including regarding them as abnormal versions of framework propositions described by Wittgenstein in On Certainty. But although framework propositions share some of the apparent epistemic features of primary delusions, their role in partially constituting the sense of inquiry rules out their role in helping to understand delusions.

Sufficient conditions for cut elimination with complexity analysis

Sufficient conditions for first-order-based sequent calculi to admit cut elimination by a Schütte–Tait style cut elimination proof are established. The worst case complexity of the cut elimination is analysed. The obtained upper bound is parameterized by a quantity related to the calculus. The conditions are general enough to be satisfied by a wide class of sequent calculi encompassing, among others, some sequent calculi presentations for the first order and the propositional versions of classical and intuitionistic logic, classical and intuitionistic modal logic S4, and classical and intuitionistic linear logic and some of its fragments. Moreover the conditions are such that there is an algorithm for checking if they are satisfied by a sequent calculus.

Using Temporal Logic to Analyse Temporal Logic: A Hierarchical Approach Based on Intervals

Temporal logic has been extensively utilized in academia and industry to formally specify and verify behavioural properties of numerous kinds of hardware and software. We present a novel way to apply temporal logic to the study of a version of itself, namely, propositional linear-time temporal logic (PTL). This involves a hierarchical framework for obtaining standard results for PTL, including a small model property, decision procedures and axiomatic completeness. A large number of the steps involved are expressed in a propositional version of Interval Temporal Logic (ITL) which is referred to as PITL. It is a natural generalization of PTL and includes operators for reasoning about periods of time and sequential composition. Versions of PTL with finite time and infinite time are both considered and one benefit of the framework is the ability to systematically reduce infinite-time reasoning to finite-time reasoning. The treatment of PTL with the operator until and past time naturally reduces to that for PTL without either one. The interval-oriented methodology differs from other analyses of PTL which typically use sets of formulas and sequences of such sets for canonical models. Instead, we represent models as time intervals expressible in PITL. The analysis furthermore relates larger intervals with smaller ones. Being an interval-based formalism, PITL is well suited for sequentially combining and decomposing the relevant formulas. Consequently, we can articulate issues of equal significance in more conventional analyses of PTL but normally only considered at the metalevel. A good example of this is the existence of bounded models with periodic suffixes for PTL formulas which are satisfiable in infinite time. We also describe decision procedures based on binary decision diagrams and exploit some links with finite-state automata. Beyond the specific issues involving PTL, the research is a significant application of ITL and interval-based reasoning and illustrates a general approach to formally reasoning about sequential and parallel behaviour in discrete linear time. The work also includes some interesting representation theorems. In addition, it has relevance to hardware description and verification since the specification languages PSL/Sugar (IEEE Standard 1850) and ‘temporal e’ (part of IEEE Standard 1647) both contain temporal constructs concerning intervals of time as does the related SystemVerilog Assertion language contained in SystemVerilog (IEEE Standard 1800), an extension of the IEEE 1364–2001 Verilog language.