Step In Proof Research Articles

Scaffolding as a strategy to help student difficulties in proving group problems

This study aims to describe the use of scaffolding as a teaching strategy in helping students difficulties in proving group problems. To achieve that goal, learning is done with scaffolding strategies on group topics. Then the problem proving test is given on a group topic, then the interview is based on the student’s proof answers. For this reason, this research is qualitative descriptive research. The subjects were students of Unirow Tuban Mathematics education who were taking Algebra Structure Courses. Scaffolding strategy design refers to the mapping of proof indicators, and the level of scaffolding developed by [1] explaining, reviewing and restructuring, and developing conceptual thinking, then associated with the scaffolding strategy developed by [2], including offering explanations, inviting student participation, verifying and clarifying student understanding, modeling desired behavior, and inviting students to contribute instructions. To explore the ability to prove, interview guidelines refer to indicators of construction proof. The results showed that the ability of students in the initial steps of proof, proof of flow, using related concepts, the argument of proof, and the language of proof had a very good increase. So the ability of students to prove after using a strategy to pursue scaffolding has increased. The implication of scaffolding as a strategy becomes an alternative teaching choice in abstract algebra subjects.

On Proof Complexities Relations in Some Systems of Propositional Calculus

The number of linear proofs steps for some sets of formulas is compared in the folowing systems of propositional calculus: PK – seguent system with cut rule, PK— - the same system without cut rule, SPK – the same system with substitution rule, QPK – the same system with quantifier rules. The number of steps of tree-like proofs in the same systems for some considered set of formulas is compared from Alessandra Carbone in [1] and some distinctive property of the system QPK is revealed: QPK has an exponential speed-up over the systems SPK and PK, which, in their turn, have an exponential speed-up over the system PK—. This result drew the heavy interest for the study of the system QPK. In this work for linear proofs steps in the same systems the other relations are received: it is showed that the system QPK has no preference over the system SPK, it is showed also that for the considered formula sets the system PK has no preference over the system PK—, which, in its turn, has no preference over the monotone system PMon. It is proved also, that the same results are reliable for some other sets of formulas and for other systems as well.

Proof searching and prediction in HOL4 with evolutionary/heuristic and deep learning techniques

Interactive theorem provers (ITPs), also known as proof assistants, allow human users to write and verify formal proofs. The proof development process in ITPs can be a cumbersome and time-consuming activity due to manual user interactions. This makes proof guidance and proof automation the two most desired features for ITPs. In this paper, we first provide two evolutionary and heuristic-based proof searching approaches for the HOL4 proof assistant, where a genetic algorithm (GA) and simulated annealing (SA) is used to search and optimize the proofs in different HOL theories. In both approaches, random proof sequences are first generated from a population of frequently occurring HOL4 proof steps that are discovered with sequential pattern mining. Generated proof sequences are then evolved using GA operators (crossover and mutation) and by applying the annealing process of SA till their fitness match the fitness of the target proof sequences. Experiments were done to compare the performance of SA with that of GA. Results have shown that the two proof searching approaches can be used to efficiently evolve the random sequences to obtain the target sequences. However, these approaches lack the ability to learn the proof process, that is important for the prediction of new proof sequences. For this purpose, we propose to use a deep learning technique known as long short-term memory (LSTM). LSTM is trained on various HOL4 theories for proof learning and prediction. Experimental results suggest that combining evolutionary/heuristic and deep learning techniques with proof assistants can greatly facilitate proof finding/optimization and proof prediction.

Machine Learning Guidance for Connection Tableaux

Connection calculi allow for very compact implementations of goal-directed proof search. We give an overview of our work related to connection tableaux calculi: first, we show optimised functional implementations of connection tableaux proof search, including a consistent Skolemisation procedure for machine learning. Then, we show two guidance methods based on machine learning, namely reordering of proof steps with Naive Bayesian probabilities, and expansion of a proof search tree with Monte Carlo Tree Search.

Kemampuan Mengkonstruksi Bukti Geometri Mahasiswa Calon Guru Matematika Pada Perkuliahan Geometri

This study aims to analyze and describe the ability to construct proofs of perspective teacher mathematics students in basic geometrical lectures on the concepts of alignment, triangles, and concordance of two triangles. This research uses a descriptive qualitative method involving 35 prospective mathematics students at the Universitas PGRI Semarang. This study uses a qualitative descriptive method involving 35 students. The results of this study show that: (1) 28% of students sketched diagrams and used geometric labels appropriately on the constructed evidence; (2) 28.57% of students have the correct initial step of proof; (3) 28.57% of students can determine the exact conjecture that leads to the correct proof; (4) 25.71% of students are correct in compiling proof arguments by the correct postulate and theorem; (5) 25.71% of the thought flow process used by coherent students leads to valid evidence; and (6) 22.86% of students mastered theorems and concepts used in compiling constructing proof.

Optimal Multiplexed Erasure Codes for Streaming Messages With Different Decoding Delays

This paper considers multiplexing two sequences of messages with two different decoding delays over a packet erasure channel. In each time slot, the source constructs a packet based on the current and previous messages and transmits the packet, which may be erased when the packet travels from the source to the destination. The destination must perfectly recover every source message in the first sequence subject to a decoding delay T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , and every source message in the second sequence subject to a shorter decoding delay T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> ≤ T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> ,. We assume that the channel loss model introduces a burst erasure of a fixed length B on the discrete timeline. Under this channel loss assumption, the capacity region for the case where T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , ≤ T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> +B was previously solved. In this paper, we fully characterize the capacity region for the remaining case T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</sub> , ) T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> +B. The key step in the achievability proof is achieving the non-trivial corner point of the capacity region through using a multiplexed streaming code constructed by superimposing two single-stream codes. The main idea in the converse proof is obtaining a genie-aided bound when the channel is subject to a periodic erasure pattern where each period consists of a length-B burst erasure followed by a length-T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</sub> noiseless duration.

Improving Proof Ability of Prospective Teachers with a Contextual Model on Trigonometry

Proof ability of prospective teachers on Trigonometry material is still lacking. It can be seen when they carry out trigonometry proof that does not meet the proof indicators to conduct the research. This study aimed at improving proof ability of prospective teachers with a contextual model on Trigonometry materials. The research method used was classroom action research with the subjects of 30 students of Mathematics Education Study Program, State University of Malang, 2019. The research was conducted in two cycles by providing scaffolding action regarding proof steps, trigonometry identity scaffolding, and conflict cognitive of proof results. The research results showed that an increase of material indicator was 4%, an increase of completeness indicator was 24%, an increase of reason indicator was 59%, an increase of clarity indicator was 31%, an increase of conclusion indicator was 2%, and an increase of application capability indicator was 4% with a total increase of student’s ability was 23%.

Extremal and Ramsey results on graph blowups

Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite Ramsey numbers. For graphs $G$ and $H$, say $G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. Then the blowup Ramsey number of $G,H,r,$ and $t$ is defined as the minimum $n$ such that $G[n] \overset{r}{\longrightarrow} H[t]$. Souza proved upper and lower bounds on $n$ that are exponential in $t$, and conjectured that the exponential constant does not depend on $G$. We prove that the dependence on $G$ in the exponential constant is indeed unnecessary, but conjecture that some dependence on $G$ is unavoidable. An important step in both Souza's proof and ours is a theorem of Nikiforov, which says that if a graph contains a constant fraction of the possible copies of $H$, then it contains a blowup of $H$ of logarithmic size. We also provide a new proof of this theorem with a better quantitative dependence.

البرهان غیر المباشر ومشکلة التفاف البراهین ومعیاریتها فی الاستنباط الطبیعى للمنطق الکلاسیکى

الملخص تعد مشکلة التفاف(1)detour البراهين في أنسقة الاستنباط الطبيعى مشکلة في غاية الأهمية، وهى مشکلة لا نجد مثيلا لها في الأنسقة الأکسيوماتيکية. وهى تعنى، على نحو مبسط، تکرار خطوات البرهان، وهو أمر معيب من ناحية اشتقاق البراهين. لقد أدرک جنتسن G. Gentzen صاحب أول نسق في الاستنباط الطبيعى تلک المشکلة، وهو قد استطاع حلها بالنسبة لأنسقة الاستنباط الطبيعى الحدسية دون الکلاسيکية، من خلال مبرهنته على معيارية Normalization / Haupstaz البراهين الحدسية، أى البرهنة على إمکانية حذف التفاف البراهين. ويعود الفضل إلى داج برافيتس D. Prawitz فى وضع أول برهان على معيارية براهين منطق الاستنباط الطبيعى الکلاسيکى من خلال استخدام قاعدة البرهان غير المباشر. في هذا البحث سوف نتساءل عن قدرة تلک القاعدة على ذلک حقا، وإمکانية وجود قاعدة بديلة الکلمات المفتاحية: برهان غير مباشر- التفاف البراهين- استنباط طبيعى-منطق حدسى- قانون بيرس [Abstract: The problem of the detour of proofs in natural deduction systems is a serious problem. We can not find a counterpart for this problem in axiomatic systems. It means, roughly speaking, repetition to the lines or steps of proof. This is a fault from a derivative point of view. G. Gentzen, the first logician to put a calculus for natural deduction , knew this problem very well, he even could solve it in respect to intuitive natural deductions systems but not to classical ones. He solved that problem by his Haupstaz or Normalization theorem for intuitive proofs, i.e. proving the possibility of eliminating the detours in proofs. It was Dag Prawitz who put the first proof of the normalization of proofs in natural deduction for classical systems by indirect proof rule. In this paper, I shall scrutinize the ability of that rule of doing what wanted from it and if there is an alternative one for it.]

Proof Learning in PVS With Utility Pattern Mining

Interactive theorem provers (ITPs) are software tools that allow human users to write and verify formal proofs. In recent years, an emerging research area in ITPs is proof mining, which consists of identifying interesting proof patterns that can be used to guide the interactive proof process in ITPs. In previous studies, some data mining techniques, such as frequent pattern mining, have been used to analyze proofs to find frequent proof steps. Though useful, such models ignore the facts that not all proof steps are equally important. To address this issue, this paper proposes a novel proof mining approach based on finding not only frequent patterns but also high utility patterns to find proof steps of high importance (utility). A proof process learning approach is proposed based on high utility itemset mining (HUIM) for the PVS (Prototype Verification System) proof assistant. Proofs in PVS theories are first abstracted to a computer-processable corpus, where each line represents a proof sequence and proof commands in proof sequences are associated with utilities representing their weightage (importance). HUIM techniques are then applied on the corpus to discover frequent proof steps/high utility patterns and their relationships with each other. Experimental results suggest that combining frequent pattern mining techniques, such as sequential pattern mining and high utility itemset mining, with proof assistants, such as PVS, is useful to learn and guide the proof development process.

В.М. Глушков і автоматизація пошуку доведень теорем в Україні: алгоритм очевидності та системи САД І SAD

Fifty years ago, in 1970, Academician V.M. Glushkov published a paper, in which he, along with a discussion of some problems of artificial intelligence, formulated a research program called Evidence Algorithm (EA) describing his vision of the problem of a computer support of human activity in looking for a proof of a particular theorem. V.M. Glushkov proposed to focus attention on the construction of an automated theorem-proving system performing simultaneous investigations in: creating formal natural languages for writing mathematical texts in a form accustomed to a human, constructing a procedure for a proof search based on the evolutionary developing of the machine notion of an evidence of a computer-made proof step, using the knowledge gained by the system during its operation and providing a user with the opportunity to assist to the system in its proof search process. Since the inception of EA, two serious attempts have been made to implement this program. The first led to the emergence in 1978 of a Russian-language automated theorem proving and the second led to the appearance in 2002 of its English-language modification named System for Automated Deduction (SAD). And if the development and trial operation of the first system were discontinued in 1992 after the output from service of the ES-line computers, on which it was realized, the SAD system, being placed on the website “nevidal.org”, is now still available in online mode. That is, at the current time, it is possible to carry out different experiments with the SAD system and to solve various problems that require rigorous mathematical reasoning. This work is devoted to a chronological description of studies on the implementation of the EA program for the entire period of its existence and to the highlighting of peculiarities of both the systems, as well as of their common features and distinguishes. Some possible ways of the further development of the SAD system are given.

On the Numbers of Minimal Tautologies and Properties of Their Proofs in Classical and Nonclassical Logic

It is proved in this paper that the number of minimal tautologies for any given logic tautology of size п can be an exponential function in п, and it is also proved that for every tautology of the given logic there is some minimal tautology such that the number of its sequential form proof steps is equal to minimal steps in the proof of sequential form for the given tautology in cut-free sequent systems for classical, intuitionistic, Joganssons and monotone logics.

Existence for (p,q) critical systems in the Heisenberg group

AbstractThis paper deals with the existence of entire nontrivial solutions for critical quasilinear systems (𝓢) in the Heisenberg group ℍn, driven by general (p,q) elliptic operators of Marcellini types. The study of (𝓢) requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p,q) systems.

Die Zeit der Einbildungskraft - Die Rolle des Schematismus in Kants Erkenntnistheorie

Abstract In this paper, I focus on Kant’s doctrine of figurative synthesis. Figurative synthesis is the result of the activity of productive transcendental imagination. This is the chief problem of the so-called “second proof step” in Kant’s deduction of the categories according to the second edition of the Critique of Pure Reason. The pure original synthetic apperception forms in the inner and outer sense - i. e. in time and space - by self-affection structures of order that make it possible to cognize empirical objects. The order of space and time through figurative syntheses (formal intuitions) must be distinguished on the one hand from space and time as forms of intuition and on the other hand from the order of the manifold given in space and time (intuition of particular contents). This clarifies the differences and relations between the constitutive noetic faculties of our knowledge apparatus.

Learning from Tactic Steps in Formal Proofs

Learning from Tactic Steps in Formal Proofs

A calculus for dynamic architectures

• A set of 35 rules to reason about component behavior in a dynamic context is provided. • The rules are implemented in Isabelle/HOL and their soundness is verified mechanically in 3500 lines of Isabelle/HOL code. • The approach is evaluated by means of four case studies resulting in 3800 additional lines of Isabelle/HOL code. With the emergence of mobile and adaptive computing, dynamic architectures have become increasingly important. In such architectures, components may appear or disappear, and connections between them may change over time. Dynamic architectures are usually specified in terms of two separate specifications: a specification of component behavior and a specification of component activation and reconfiguration. To verify them, both specifications are first interpreted over a common model for dynamic architectures and verified against a property specified over the same model. Interpreting the specifications over the model, however, introduce repetitive proof steps, which increase the length of proofs, and thus the effort to develop and maintain them. To address this problem, we developed a calculus for dynamic architectures providing rules to reason about component behavior in a dynamic environment. We proved soundness and relative completeness of the rules, implemented them in the interactive theorem prover Isabelle, and mechanized the corresponding soundness proofs. The calculus can be used to support the abstract verification of dynamic architectures in Isabelle. This is demonstrated by means of a running example and evaluated in terms of four case studies. Our results suggest that the calculus has the potential to reduce the length of proofs for the verification of dynamic architectures, thus reducing the effort to develop and maintain verification results.

Investigation and Visual Explanation in Dynamic Geometry Environment

Dynamic geometry environment allows us to investigate geometric relationships and problems more efficiently than the pencil-and-paper method. Continuous variation of geometric configurations shows the relationship from different points of view. This paper aims to provide some indications of how a dynamic geometry environment, especially GeoGebra software, can be used to offer insight and support understanding of geometry problem solving and proofs through investigation and experimentation. Our case studies show in particular how prospective teachers of mathematics deal with difficult geometry problems, designated initially to mathematically gifted students for competition, using investigation in GeoGebra. We attempt to reveal the prospective teachers' consideration systematically when they are solving a geometric problem. The prepared visual explanation and visual proof steps serve as clues or scaffolding to facilitate geometric problem solving, and they can be useful also for mathematics teachers in preparing their students for mathematics competitions.

Well-posedness to 3D Burgers’ equation in critical Gevrey Sobolev spaces

We prove that the three-dimensional periodic Burgers’ equation has a unique global in time solution in a critical Gevrey–Sobolev space. Comparatively to Navier–Stokes equations, the main difficulty is the lack of an incompressibility condition. In our proof of existence, we overcome the bootstrapping argument, which was a technical step in a precedent proof in Sololev spaces. This makes our proof shorter and gives sense of considering the Gevrey class for a mathematical study to Burgers’ equation. To prove that the unique solution is global in time, we use the maximum principle. Energy methods, Sobolev product laws, compactness methods, and Fourier analysis are the main tools.

The ability of senior high school students in comprehending mathematical proofs

Mathematical proving ability is an ability which has function to improve mathematical high order thinking and reasoning skill. This ability consists of two sub-abilities, which are mathematical proof comprehension and mathematical proof construction. Both of the sub-abilities are related each other. To construct mathematical proofs, students need to have good mathematical proof comprehension. Due to the importance of mathematical proof comprehension, a qualitative descriptive study was conducted to describe senior high school students’ ability in mathematical proof comprehension. A test was given to 10 students of 11th grade in one of senior high schools at Bandung, West Java, Indonesia. The students’ works were analyzed for each proof comprehension indicator. Overall, the research found four results, the students: (i) were good in giving a reason for each step of proofs; (ii) got difficulties in validating a step of proof; (iii) were good in applying the steps of a proof to prove similar statement; and (iv) failed in determining and proving a notion based on pattern from the given statements. This study concludes that the senior high school students’ ability in comprehending mathematical proofs has not been satisfied. In several parts, it needs to be improved during mathematics learning process.

Generalisation of proof simulation procedures for Frege systems by M.L. Bonet and S.R. Buss

ABSTRACTIn this paper, we present a generalisation of proof simulation procedures for Frege systems by Bonet and Buss to some logics for which the deduction theorem does not hold. In particular, we study the case of finite-valued Łukasiewicz logics. To this end, we provide proof systems and which augment Avron's Frege system HŁuk with nested and general versions of the disjunction elimination rule, respectively. For these systems, we provide upper bounds on speed-ups w.r.t. both the number of steps in proofs and the length of proofs. We also consider Tamminga's natural deduction and Avron's hypersequent calculus GŁuk for 3-valued Łukasiewicz logic and generalise our results considering the disjunction elimination rule to all finite-valued Łukasiewicz logics.