On the difficulty of discovering mathematical proofs

In his well-known paper [11], Post founded a general theory of recursively enumerable sets, which had its metamathematical source in questions about the decision problem for deducibility in formal systems. However, in centering attention on the notion of degree of unsolvability, Post set a course for his theory which has rarely returned to this source. Among exceptions to this tendency we may mention, as being closest to the problems considered here, the work of Kleene in [8] pp. 298–316, of Myhill in [10], and of Uspenskij in [15]. It is the purpose of this paper to make some further contributions towards bridging this gap.From a certain point of view, it may be argued that there is no real separation between metamathematics and the theory of recursively enumerable sets. For, if the notion of formal system is construed in a sufficiently wide sense, by taking as ‘axioms’ certain effectively found members of a set of ‘formal objects’ and as ‘proofs’ certain effectively found sequences of these objects, then the set of ‘provable statements’ of such a system may be identified, via Gödel's numbering technique, with a recursively enumerable set; and conversely, each recursively enumerable set is identified in this manner with some formal system (cf. [8] pp. 299–300 and 306). However, the pertinence of Post's theory is no longer clear when we turn to systems formalized within the more conventional framework of the first-order predicate calculus. It is just this restriction which serves to clarify the difference in spirit of the two disciplines.

On the number of steps in proofs

We relate various complexity measures like sensitivity, block sensitivity, certificate complexity for multi-output functions to the query complexities of such functions. Using these relations, we show that the deterministic query complexity of total search problems is at most the third power of its pseudo-deterministic query complexity. Previously, a fourth-power relation was shown by Goldreich, Goldwasser and Ron (ITCS'13). Using our proof along with a decision-tree manipulation technique, we give a simple and self-contained proof that the $$\text{SearchCNF}$$ problem on random $$\text{k}$$ -CNF has pseudo-deterministic query complexity $$\Omega(n^{1/3})$$ ; a lower bound of $$\Omega(\sqrt{n})$$ is known, due to Goldwasser, Impagliazzo, Pitassi, and Santhanam (CCC'21), but via a significantly more complex proof. We improve the known separation between pseudo-deterministic and randomized decision tree size for total search problems in two ways: (1) We exhibit an $$\text{exp}(\widetilde{\Omega}(n^{1/4}))$$ separation for the $$\text{SearchCNF}$$ relation for random $$k$$ -CNFs. This seems to be the first exponential lower bound on the pseudo-deterministic size complexity of $$\text{SearchCNF}$$ associated with random $$k$$ -CNFs. (2) We exhibit an $${\text{exp}(\Omega(n))}$$ separation for the $$\text{ApproxHW}$$ relation. The previous best known separation for any relation was $${\text{exp}(\Omega(n^{1/2}))}$$ . We also separate pseudo-determinism from randomness in $$\text{AND}$$ and $$\text{CONJ}$$ decision trees, and determinism from pseudo-determinism in $$\text{Parity}$$ decision trees. Finally, for a hypercube colouring problem, that was introduced by Goldwasswer et al. to analyze the pseudo-deterministic complexity of a complete problem in $$\text{TFNPdt}$$ , we prove that either the monotone block-sensitivity or the anti-monotone block sensitivity is $${\Omega(n^{1/3})}$$ ; Goldwasser et al. showed an $${\Omega(n^{1/2})}$$ bound for general block-sensitivity.

By the length of a proof in a formal system S we shall understand the number of formulas of which the proof consists. Further, if every natural number is represented in Sin a definite way by a symbol (a numeral) (for example, by a symbol of the form 1 + 1 + 1), then a function cp(x) will be said to be computable in S in case for each numeral m there exists a numeral n such that ¢(m) = n is provable in S. In particular, all recursively defined functions, for example, are already computable in classical arithmetic (that is, in the system S1 of the sequence defined below). Now let Si be the system of logic of the ith order, the natural numbers being taken as individuals. That is, more precisely: Si shall contain, together with the appropriate logical axioms, variables and quantifiers for natural numbers, for classes of natural numbers, for classes of classes of natural numbers, and so on, up to classes of the ith type, but no variable of a higher type. Then, as is known, there are propositions of Si that are provable in Si+1 but not in Si. If, on the other hand, we consider those formulas / that are provable in Si as well as in Si+1, then the following holds: For each function ¢ that is computable in Si there exist infinitely many formulas f such that, if k is the length of a shortest proof of f in Si and l is the length of a shortest proof of f in Si+1, then k > ¢(l). If, for example, we set ¢(n) = 106n, it follows that there are infinitely many formulas whose shortest proof in Si is more than 106 times as long as their shortest proof in Si+1. Thus, passing to the logic of the next higher order has the effect, not only of making provable certain propositions that were not provable be/ore, but also of making it possible to shorten, by an extraordinary amount, infinitely many of the proofs already available.

How does the design of technology-based service mathematics shape science students' understandings of mathematics? Reciprocally, how do the conceptions of undergraduate mathematics teachers concerning service mathematics shape their design decisions when building a technology-based course? The paper starts from the premise that simultaneous study of design and learning together may throw light on epistemological and psychological issues in learning mathematics. The corpus of data was derived from two university mathematics departments, each of which was attempting to design and implement a service course of undergraduate mathematics for science students. The findings reveal a complex relationship between design and learning, with implications for the design of instructional approaches.

Several formal systems, such as resolution and minimal model semantics, provide a framework for logic programming. In this article, we will survey the use ofstructural proof theoryas an alternative foundation. Researchers have been using this foundation for the past 35 years to elevate logic programming from its roots in first-order classical logic into higher-order versions of intuitionistic and linear logic. These more expressive logic programming languages allow for capturing stateful computations and rich forms of abstractions, including higher-order programming, modularity, and abstract data types. Term-level bindings are another kind of abstraction, and these are given an elegant and direct treatment within both proof theory and these extended logic programming languages. Logic programming has also inspired new results in proof theory, such as those involving polarity and focused proofs. These recent results provide a high-level means for presenting the differences between forward-chaining and backward-chaining style inferences. Anchoring logic programming in proof theory has also helped identify its connections and differences with functional programming, deductive databases, and model checking.

Logic of agreement: foundations, semantic system and proof theory

We are concerned with how computer theorem provers should expect users to communicate proofs to them. There are many stylistic choices that still allow the machine to generate a completely formal proof object. The most obvious choice is the amount of guidance required from the user, or from the machine perspective, the degree of automation provided. But another important consideration, which we consider particularly significant, is the bias towards a ‘procedural’ or ‘declarative’ proof style. We will explore this choice in depth, and discuss the strengths and weaknesses of declarative and procedural styles for proofs in pure mathematics and for verification applications. We conclude with a brief summary of our own experiments in trying to combine both approaches.

Union-free expressions are used in union normal form to decompose any regular language to a finite union of union-free languages. Based on the automata characterisation of the union-free languages, by restricting the 1CFPAs not to have transitions by the empty word, or to be deterministic, the n-union-free and the deterministic union-free languages are defined. Union-complexity as a measure of descriptional complexity of regular languages was introduced recently. By the minimum number of union-free/n-union-free/deterministic union-free languages needed to get a regular language as their union, its union-complexity/n-union-complexity/d-union-complexity is defined. It is already known that union-complexity and n-union-complexity are finite for every regular language, however there are regular languages with infinite d-union-complexity. Operational union-complexity, that is, to predict the union-complexity of a language obtained by a language operation from languages with known union-complexity is an important and interesting question belonging to the field of descriptional complexity of formal systems. In the present paper, the Kleene plus, the positive Kleene closure operator is studied. As the Kleene star and plus operations have very different effects on the union-free languages, it is an interesting problem to investigate how the union-complexities may change under this operation. In particular, we show that the union-complexity of a regular language is not growing when this operation is being applied on it. On the other hand, the n-union-complexity of the Kleene plus of an n-union-free language remains 1, but the n-union-complexity of the Kleene plus of other regular languages may grow. Further, the deterministic union-complexity may jump to an infinite value even if the original language had a relatively small deterministic union-complexity, e.g., 4.Keywordsunion-complexityunion-free languagesregular expressionsKleene closure

A major concern to the founders of modern logic—Frege, Peirce, Russell, and Hilbert—was to give an account of the logical structure of valid reasoning. Taking valid reasoning in mathematics as paradigmatic, these pioneers led the way in developing the accounts of logic which we teach today and that underwrite the work in model theory, proof theory, and definability theory. The resulting notions of proof, model, formal system, soundness, and completeness are things that no one claiming familiarity with logic can fail to understand, and they have also played an enormous role in the revolution known as computer science. The success of this model of inference led to an explosion of results and applications. But it also led most logicians—and those computer scientists most influenced by the logic tradition—to neglect forms of reasoning that did not fit well within this model. We are thinking, of course, of reasoning that uses devices like diagrams, graphs, charts, frames, nets, maps, and pictures. The attitude of the traditional logician to these forms of representation is evident in the quotation of Neil Tennant in Chapter I, which expresses the standard view of the role of diagrams in geometrical proofs. One aim of our work, as explained there, is to demonstrate that this dogma is misguided. We believe that many of the problems people have putting their knowledge of logic to work, whether in machines or in their own lives, stems from the logocentricity that has pervaded its study for the past hundred years. Recently, some researchers outside the logic tradition have explored uses of diagrams in knowledge representation and automated reasoning, finding inspiration in the work of Euler, Venn, and especially C. S. Peirce. This volume is a testament to this resurgence of interest in nonlinguistic representations in reasoning. While we applaud this resurgence, the aim of this chapter is to strike a cautionary note or two. Enchanted by the potential of nonlinguistic representations, it is all too easy to overreact and so to repeat the errors of the past.

My aim in this thesis is to reveal the true nature of mathematical reasoning and to show the necessity of pictorial representations in mathematics. In order to support my thesis I argue that using only the formal language, without fully integrating visual reasoning in mathematical communication, mathematics operates insufficiently and it can never reach its full potential. First of all, I provide a short historical approach to philosophy of mathematics as well as a summary of current state of visualization in mathematics. Then, I bring forth a complete picture of philosophy of mathematics and mathematical objects, which consistently allows visual reasoning, intuitions and constructions in mathematical communication. This interpretation makes it easy to see the pictorial representations as a natural part of mathematical reasoning. Moreover, I lay out the medium for visual reasoning in mathematics: Space. After providing several definitions of space I conclude that Kant’s characterization is the finest one as a base for mathematical reasoning, in particular for visual reasoning. Furthermore, I bring out the method of visual reasoning: Synthetic a priori. In order to show the link between visualization and synthetic a priori I provide hitherto definitions of visualization and argue that these lead up to Kantian characterization of constructions and synthetic a priori. Then, I provide several interpretations to show that Kant’s characterization of mathematics has elements valuable for modern mathematics that modern logic cannot capture. Finally, I offer three main arguments leading to the main thesis “pictorial repre- sentations/ diagrams are necessary parts of mathematical reasoning”: (1) There are formal systems with diagrams. Hence, not all uses of diagrams are synthetic. (2) There are necessary non-formal uses of diagrams in mathematical proofs. (3) There is a strong link between visualization and Kantian philosophy of mathematics. As a result, I claim that the real capacity of mathematics is not in turning it into a machine language but integrating and appreciating the aesthetic part of it as well. I also indicate the necessity of having a theory of diagrammatics, which should be set up as a branch of mathematics.

Certainty? Absolute certainty can be an unproductive and unwise goal, even in mathematics. The rigor aspired should be only that needed for the purposes at hand. Be ready doubt what has been proved, for some proofs depend on hidden assumptions, while others can be interpreted in various ways. Ask that a convincing argument be given for an assertion before accepting it, and continue test assertions by judging the reasonableness and usefulness of the conclusions drawn. Ask also whether the hypotheses are adequate and acceptable. Here I will use to prove in the broader sense of to test, which includes testing by pure reason, by intuition, and in practice. In this sense, the testing, or proving, of assertions is a process in which all things are proved and the good retained. Formalism and rigor have their uses. Hilbert's investigations into the foundations of geometry uncovered the hidden assumptions of classical geometry. Finding these defects in Euclidean geometry did not invalidate it, for it had already proved its worth. Hilbert's work made explicit the assumptions that should be added the list of postulates of a formal theory agreeing with Euclid's results. Before Hilbert's work, Euclid's postulates were thought be completely adequate. This was wrong, but no harm was done. The error was discovered, and matters put right. In the same way, the errors of today (if any) may be discovered and corrected. Our successors will make repairs and hold fast the good, just as we have. Hilbert used rigorous formal methods because they suited his purposes. He wished give mathematics an unshakable, formal foundation. Hence, he was concerned with axioms. His methods and axioms may not help a geometer who wants know whether there is a single tile, copies of which can cover a plane only nonperiodically-an open question in plane geometry in 1980 (see Box 1). Intuitive methods are more important for solving such problems. Barring the inaccurate reading of symbols and the failure follow rules, it is possible establish with certainty that an expression is a theorem in a certain formal system. However, despite this certainty, questions remain. Do the axioms of that system correctly reflect the properties of the objects of that theory in its intended interpretation? Do the intended mathematical objects capture important

Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Theta(√(n)) on the space needed for so-called pebbling contradictions over pyramid graphs of size n. Also, continuing the line of research initiated by (Ben-Sasson 2002) into trade-offs between different proof complexity measures, we present a simplified proof of the recent length-space trade-off result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential trade-offs in resolution.

Smullyan in [Smu61] identified the recursion theoretic essence of incompleteness results such as Gödel's first incompleteness theorem and Rosser's theorem. Smullyan (improving upon [Kle50] and [Kle52]) showed that, for sufficiently complex theories, the collection of provable formulae and the collection of refutable formulae are effectively inseparable—where formulae and their Gödel numbers are identified. This paper gives a similar treatment for proof speed-up. We say that a formal system S1is speedable over another system S0on a set of formulaeAiff, for each recursive functionh, there is a formulaαinAsuch that the length of the shortest proof ofαin S0is larger thanhof the shortest proof ofαin S1. (Here we equate the length of a proof with something like the number of characters making it up,notits number of lines.) We characterize speedability in terms of the inseparability by r.e. sets of the collection of formulae which are provable in S1but unprovable in S0from the collectionA–where again formulae and their Gödel numbers are identified. We provide precise definitions of proof length, speedability and r.e. inseparability below.We follow the terminology and notation of [Rog87] with borrowings from [Soa87]. Below,ϕis an acceptable numbering of the partial recursive functions [Rog87] andΦa (Blum) complexity measure associated withϕ[Blu67], [DW83].

The purpose of this study is to establish levels from an inductive proof to an algebraic demonstration in lower secondary school mathematics. I propose that we can establish six levels of proof in lower secondary school mathematics as steps from an inductive proof to an algebraic demonstration on the basis of three axes (contents of proof, representation of proof, and students' thinking). To reach this conclusion, I firstly examine the meaning of ‘demonstration in lower secondary school mathematics’ and ‘proof in lower secondary school mathematics’, and show the relationships between them. Secondly, I set out four basic levels of proof, as seen from two aspects (contents and representation of proof). Thirdly, I subdivide them into six levels from the third aspect of students' thinking. Finally, I illustrate my discussion with a 7th grader's activities.