Look Ma, Backtracking without Recursion

I show how backtracking can be discovered naturally without using a recursive function (nor using a loop with an explicit stack). Rather, my approach involves a form of self application that can be elegantly expressed in an object-oriented program, and that is reminiscent of how recursion is done in lambda calculus. It also illustrates why reasoning about object-oriented programs can be hard.

Self-Verifying Pushdown and Queue Automata

We study the computational and descriptional complexity of self-verifying pushdown automata (SVPDA) and self-verifying realtime queue automata (SVRQA). A self-verifying automaton is a nondeterministic device whose nondeterminism is symmetric in the following sense. Each computation path can give one of the answers yes, no, or do not know. For every input word, at least one computation path must give either the answer yes or no, and the answers given must not be contradictory. We show that SVPDA and SVRQA are automata characterizations of so-called complementation kernels, that is, context-free or realtime nondeterministic queue automaton languages whose complement is also context free or accepted by a realtime nondeterministic queue automaton. So, the families of languages accepted by SVPDA and SVRQA are strictly between the families of deterministic and nondeterministic languages. Closure properties and various decidability problems are considered. For example, it is shown that it is not semidecidable whether a given SVPDA or SVRQA can be made self-verifying. Moreover, we study descriptional complexity aspects of these machines. It turns out that the size trade-offs between nondeterministic and self-verifying as well as between self-verifying and deterministic automata are non-recursive. That is, one can choose an arbitrarily large recursive function f, but the gain in economy of description eventually exceeds f when changing from the former system to the latter.

Counterexample-Guided Partial Bounding for Recursive Function Synthesis

Quantifier bounding is a standard approach in inductive program synthesis in dealing with unbounded domains. In this paper, we propose one such bounding method for the synthesis of recursive functions over recursive input data types. The synthesis problem is specified by an input reference (recursive) function and a recursion skeleton. The goal is to synthesize a recursive function equivalent to the input function whose recursion strategy is specified by the recursion skeleton. In this context, we illustrate that it is possible to selectively bound a subset of the (recursively typed) parameters, each by a suitable bound. The choices are guided by counterexamples. The evaluation of our strategy on a broad set of benchmarks shows that it succeeds in efficiently synthesizing non-trivial recursive functions where standard across-the-board bounding would fail.

Finitism, imperative programs and primitive recursion

Following the Crisis of Foundations Hilbert proposed to consider a finitistic form of arithmetic as mathematics’ safe core. This approach to finitism has often admitted primitive recursive function definitions as obviously finitistic, but some have advocated the inclusion of additional variants of recurrence, while others argued that, to the contrary, primitive recursion exceeds finitism. In a landmark essay, William Tait contested the finitistic nature of these extensions, due to their impredicativity, and advocated identifying finitism with primitive recursive arithmetic, a stance often referred to as Tait’s Thesis. However, a problem with Tait’s argument is that the recurrence schema has itself impredicative and non-finitistic facets, starting with an explicit reference to the functions being defined, which are after all infinite objects. It is therefore desirable to buttress Tait’s Thesis on grounds that avoid altogether any trace of concrete infinities or impredicativity. We propose here to do just that, building on the generic framework of [ 13]. We provide further evidence for Tait’s Thesis by outlining a proof of a purely finitistic version of Parsons’ theorem, whose intuitive gist is that finitistic reasoning is equivalent to finitistic computing.

Setting final target score in T-20 cricket match by the team batting first

The purpose of this paper is to develop a deterministic model for setting the target in T-20 Cricket by the team batting first. Mathematical tools were used in model development. Recursive function and secondary data statistics of T-20 cash rich cricket tournament Indian Premier League (IPL) such as runs scored in different stages, fall wickets in different stages, and type of pitch are used in developing the model. This model was tested at 120 matches held IPL 2016 and 2017. This model had been proved effective by comparing with the models developed earlier. This model can be a useful tool to the stakeholders like coach and captain of the team for adopting better strategy at any stage of the match. For future research, this model can be useful in framing a regulation work by policy makers at both national and international cricket board by deriving the target score during interruptions.

Counting to Infinity: Does learning the syntax of the count list predict knowledge that numbers are infinite?

By around the age of 5½, many children in the US judge that numbers never end, and that it is always possible to add +1 to a set. These same children also generally perform well when asked to label the quantity of a set after 1 object is added (e.g., judging that a set labeled “five” should now be “six”). These findings suggest that children have implicit knowledge of the “successor function”: every natural number, n, has a successor, n+1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language-specific counting routines (e.g., the rules in English that represent base 10 structure). We tested 4- and 5-year-old children’s knowledge of counting with three tasks, which we then related to (1) children’s belief that 1 can always be added to any number (the successor function), and (2) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge wasn’t directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as four years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end.

Concepts as Recurrent Elements of Discourse: An Approach to the History of Conceptuality in Michel Foucault’s The Archeology of Knowledge.

The article concentrates on Michel Foucault’s idea from The Archeology of Knowledge that concepts are recurrent elements of discourse; it is argued that this theme has been insufficiently appreciated and developed. The identification of concepts as a particular kind of repetitive element of discourse means that there is an unequal probability of the distribution of linguistic units in a discourse and therefore discursive regularities are evident. Concepts are also characterized by Foucault as the realization of a specific discursive temporality: their distribution and circulation indicate what is topical and urgent for a given discourse, what is retained from their own past, and what potential future is in store for them. Concepts as a recurring element and as a form of discursive temporality are correlative with the “associative field” of an utterance, i.e. with the form of correlation with the field of the “already said” which is distinctive of an utterance. Finally, recursion must be considered here particularly in the mathematical sense: the concept is a kind of value for a “variable” in a recursive function, i.e. it is the realization of a certain scheme of correlation with prior utterances, and it is dependent upon that scheme. The author considers three “subspaces” of the formation of concepts from the The Archeology of Knowledge: forms of succession, forms of coexistence, and procedures of intervention. It is argued that these subspaces should be understood primarily in terms of the “Foucault hypothesis” that runs through the four chapters of The Archeology that in turn deal with the formation of objects, of modalities, of concepts, and of strategies. The selected subspaces are independent of each other — there is no conceptual way to encompass them in any kind of unity. However, they are interrelated historically — changes in the system of one subspace are synchronized with changes at higher levels and correlates with them.