In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent (...) with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument. (shrink)

Traces the development of recursive functions from their origins in the late nineteenth century to the mid-1930s, with particular emphasis on the work and influence of Kurt Gödel.

We study the computational complexity of polyadic quantifiers in natural language. This type of quantification is widely used in formal semantics to model the meaning of multi-quantifier sentences. First, we show that the standard constructions that turn simple determiners into complex quantifiers, namely Boolean operations, iteration, cumulation, and resumption, are tractable. Then, we provide an insight into branching operation yielding intractable natural language multi-quantifier expressions. Next, we focus on a linguistic case study. We use computational complexity results to investigate semantic (...) distinctions between quantified reciprocal sentences. We show a computational dichotomy<br>between different readings of reciprocity. Finally, we go more into philosophical speculation on meaning, ambiguity and computational complexity. In particular, we investigate a possibility to<br>revise the Strong Meaning Hypothesis with complexity aspects to better account for meaning shifts in the domain of multi-quantifier sentences. The paper not only contributes to the field of the formal<br>semantics but also illustrates how the tools of computational complexity theory might be successfully used in linguistics and philosophy with an eye towards cognitive science. (shrink)

Computational complexity theory is a subfield of computer science originating in computability theory and the study of algorithms for solving practical mathematical problems. Amongst its aims is classifying problems by their degree of difficulty — i.e., how hard they are to solve computationally. This paper highlights the significance of complexity theory relative to questions traditionally asked by philosophers of mathematics while also attempting to isolate some new ones — e.g., about the notion of feasibility in mathematics, the $\mathbf{P} \neq \mathbf{NP}$ (...) problem and why it has proven hard to resolve, and the role of non-classical modes of computation and proof. (shrink)

Theory of mind refers to the human capacity for reasoning about others’ mental states based on observations of their actions and unfolding events. This type of reasoning is notorious in the cognitive science literature for its presumed computational intractability. A possible reason could be that it may involve higher-order thinking. To investigate this we formalize theory of mind reasoning as updating of beliefs about beliefs using dynamic epistemic logic, as this formalism allows to parameterize ‘order of thinking.’ We prove that (...) theory of mind reasoning, so formalized, indeed is intractable. Using parameterized complexity we prove, however, that the ‘order parameter’ is not a source of intractability. We furthermore consider a set of alternative parameters and investigate which of them are sources of intractability. We discuss the implications of these results for the understanding of theory of mind. (shrink)

We overview logical and computational explanations of the notion of tractability as applied in cognitive science. We start by introducing the basics of mathematical theories of complexity: computability theory, computational complexity theory, and descriptive complexity theory. Computational philosophy of mind often identifies mental algorithms with computable functions. However, with the development of programming practice it has become apparent that for some computable problems finding effective algorithms is hardly possible. Some problems need too much computational resource, e.g., time or memory, to (...) be practically computable. Computational complexity theory is concerned with the amount of resources required for the execution of algorithms and, hence, the inherent difficulty of computational problems. An important goal of computational complexity theory is to categorize computational problems via complexity classes, and in particular, to identify efficiently solvable problems and draw a line between tractability and intractability. -/- We survey how complexity can be used to study computational plausibility of cognitive theories. We especially emphasize methodological and mathematical assumptions behind applying complexity theory in cognitive science. We pay special attention to the examples of applying logical and computational complexity toolbox in different domains of cognitive science. We focus mostly on theoretical and experimental research in psycholinguistics and social cognition. (shrink)

How many words—and which ones—are sufficient to define all other words? When dictionaries are analyzed as directed graphs with links from defining words to defined words, they reveal a latent structure. Recursively removing all words that are reachable by definition but that do not define any further words reduces the dictionary to a Kernel of about 10% of its size. This is still not the smallest number of words that can define all the rest. About 75% of the Kernel turns (...) out to be its Core, a “Strongly Connected Subset” of words with a definitional path to and from any pair of its words and no word's definition depending on a word outside the set. But the Core cannot define all the rest of the dictionary. The 25% of the Kernel surrounding the Core consists of small strongly connected subsets of words: the Satellites. The size of the smallest set of words that can define all the rest—the graph's “minimum feedback vertex set” or MinSet—is about 1% of the dictionary, about 15% of the Kernel, and part-Core/part-Satellite. But every dictionary has a huge number of MinSets. The Core words are learned earlier, more frequent, and less concrete than the Satellites, which are in turn learned earlier, more frequent, but more concrete than the rest of the Dictionary. In principle, only one MinSet's words would need to be grounded through the sensorimotor capacity to recognize and categorize their referents. In a dual-code sensorimotor/symbolic model of the mental lexicon, the symbolic code could do all the rest through recombinatory definition. (shrink)

We introduce _Interleave-Disjunction-Lock parallel multiple context-free grammars_ (IDL-PMCFG), a novel grammar formalism designed to describe the syntax of free word order languages that allow for extensive interleaving of grammatical constituents. Though interleaved constituents, and especially the so-called hyperbaton, are common in several ancient (Classical Latin and Greek, Sanskrit...) and modern (Hungarian, Finnish...) languages, these syntactic structures are often difficult to express in existing formalisms. The IDL-PMCFG formalism combines Seki et al.’s parallel multiple context-free grammars (PMCFG) with Nederhof and Satta’s IDL (...) expressions. We define the semantics of IDL-PMCFGs and study their expressivity, proving that IDL-PMCFG extends both PMCFG and IDL-CFG (context-free grammars equipped with IDL expressions) and that IDL-PMCFG parsing is \(\mathrm {NP}\) -hard. We then introduce COMPĀ, a programming language extending Ranta’s Grammatical Framework (GF) and built as a high-level front-end formalism to IDL-PMCFG for practical grammar development. We present a parsing algorithm for IDL-PMCFG inspired by earlier Earley-style PMCFG parsing algorithms and Nederhof and Satta’s IDL graphs and give a worst-case estimate of its complexity as a function of several metrics on IDL expressions, the size of the input and a new notion of the _G_-density of a language. (shrink)

We report two experiments which tested whether cognitive capacities are limited to those functions that are computationally tractable (PTIME-Cognition Hypothesis). In particular, we investigated the semantic processing of reciprocal sentences with generalized quantifiers, i.e., sentences of the form Q dots are directly connected to each other, where Q stands for a generalized quantifier, e.g. all or most. Sentences of this type are notoriously ambiguous and it has been claimed in the semantic literature that the logically strongest reading is preferred (Strongest (...) Meaning Hypothesis). Depending on the quantifier, the verification of their strongest interpretations is computationally intractable whereas the verification of the weaker readings is tractable. We conducted a picture completion experiment and a picture verification experiment to investigate whether comprehenders shift from an intractable reading to a tractable reading which should be dispreferred according to the Strongest Meaning Hypothesis. The results from the picture completion experiment suggest that intractable readings occur in language comprehension. Their verification, however, rapidly exceeds cognitive capacities in case the verification problem cannot be solved using simple heuristics. In particular, we argue that during verification, guessing strategies are used to reduce computational complexity. (shrink)

Needed for such dialogue games as dialectic are appropriate standards of fairness and rationality. The rules of procedure of dialectic must describe a game playable by actual human participants. The present paper centers on certain idealizations of the dialectician that are not allowable.

Community detection is a major issue in network analysis. This paper combines a socio-historical approach with an experimental reconstruction of programs to investigate the early automation of clique detection algorithms, which remains one of the unsolved NP-complete problems today. The research led by the archaeologist Jean-Claude Gardin from the 1950s on non-numerical information and graph analysis is retraced to demonstrate the early contributions of social sciences and humanities. The limited recognition and reception of Gardin's innovative computer application to the humanities (...) are addressed through two factors, in addition to the effects of historiography and bibliographies on the recording, discoverability, and reuse of scientific productions: (1) funding policies, evidenced by the transfer of research effort on graph applications from temporary interdisciplinary spaces to disciplinary organizations related to the then-emerging field of computer science; and (2) the erratic careers of algorithms, in which efficiency, flaws, corrections, and authors’ status, were determining factors. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

Why should moral philosophers, moral psychologists, and machine ethicists care about computational complexity? Debates on whether artificial intelligence (AI) can or should be used to solve problems in ethical domains have mainly been driven by what AI can or cannot do in terms of human capacities. In this paper, we tackle the problem from the other end by exploring what kind of moral machines are possible based on what computational systems can or cannot do. To do so, we analyze normative (...) ethics through the lens of computational complexity. First, we introduce computational complexity for the uninitiated reader and discuss how the complexity of ethical problems can be framed within Marr’s three levels of analysis. We then study a range of ethical problems based on consequentialism, deontology, and virtue ethics, with the aim of elucidating the complexity associated with the problems themselves (e.g., due to combinatorics, uncertainty, strategic dynamics), the computational methods employed (e.g., probability, logic, learning), and the available resources (e.g., time, knowledge, learning). The results indicate that most problems the normative frameworks pose lead to tractability issues in every category analyzed. Our investigation also provides several insights about the computational nature of normative ethics, including the differences between rule- and outcome-based moral strategies, and the implementation-variance with regard to moral resources. We then discuss the consequences complexity results have for the prospect of moral machines in virtue of the trade-off between optimality and efficiency. Finally, we elucidate how computational complexity can be used to inform both philosophical and cognitive-psychological research on human morality by advancing the moral tractability thesis. (shrink)

In this article we present a new kind of computing device that uses biochemical reactions networks as building blocks to implement logic gates. The architecture of a computing machine relies on these generic and composable building blocks, computation units, that can be used in multiple instances to perform complex boolean functions. Standard logical operations are implemented by biochemical networks, encapsulated and insulated within synthetic vesicles called protocells. These protocells are capable of exchanging energy and information with each other through transmembrane (...) electron transfer. In the paradigm of computation we propose, protoputing, a machine can solve only one problem and therefore has to be built specifically. Thus, the programming phase in the standard computing paradigm is represented in our approach by the set of assembly instructions that directs the wiring of the protocells that constitute the machine itself. To demonstrate the computing power of protocellular machines, we apply it to solve a NP-complete problem, known to be very demanding in computing power, the 3-SAT problem. We show how to program the assembly of a machine that can verify the satisfiability of a given boolean formula. Then we show how to use the massive parallelism of these machines to verify in less than 20 min all the valuations of the input variables and output a fluorescent signal when the formula is satisfiable or no signal at all otherwise. (shrink)

Recently, several authors have explored the connections between NP-complete problems for finite objects and the complexity of their analogs for infinite objects. In this paper, we will categorize infinite versions of several problems arising from finite complexity theory in terms of their recursion theoretic complexity and proof theoretic strength. These infinite analogs can behave in a variety of unexpected ways.

The generation of route descriptions is a fundamental task of navigation systems. A particular problem in this context is to identify routes that can easily be described and processed by users. In this work, we present a framework for representing route networks with the qualitative information necessary to evaluate and optimize route descriptions with regard to ambiguities in them. We identify different agent models that differ in how agents are assumed to process route descriptions while navigating through route networks and (...) discuss which agent models can be translated into PDL programs. Further, we analyze the computational complexity of matching route descriptions and paths in route networks in dependency of the agent model. Finally, we empirically evaluate the influence of the agent model on the optimization and the processing of route instructions. (shrink)

This paper deals with the question: what are the key requirements for a physical system to perform digital computation? Time and again cognitive scientists are quick to employ the notion of computation simpliciter when asserting basically that cognitive activities are computational. They employ this notion as if there was or is a consensus on just what it takes for a physical system to perform computation, and in particular digital computation. Some cognitive scientists in referring to digital computation simply adhere to (...) Turing’s notion of computability . Classical computability theory studies what functions on the natural numbers are computable and what mathematical problems are undecidable. Whilst a mathematical formalism of computability may perform a methodological function of evaluating computational theories of certain cognitive capacities, concrete computation in physical systems seems to be required for explaining cognition as an embodied phenomenon . There are many non-equivalent accounts of digital computation in physical systems. I examine only a handful of those in this paper: (1) Turing’s account ; (2) The triviality “account”; (3) Reconstructing Smith’s account of participatory computation ; (4) The Algorithm Execution account . My goal in this paper is twofold. First, it is to identify and clarify some of the underlying key requirements mandated by these accounts. I argue that these differing requirements justify a demand that one commits to a particular account when employing the notion of computation in regard to physical systems. Second, it is to argue that despite the informative role that mathematical formalisms of computability may play in cognitive science, they do not specify the relationship between abstract and concrete computation. (shrink)