We explore the formal foundations of recent studies comparing aural pattern recognition capabilities of populations of human and non-human animals. To date, these experiments have focused on the boundary between the Regular and Context-Free stringsets. We argue that experiments directed at distinguishing capabilities with respect to the Subregular Hierarchy, which subdivides the class of Regular stringsets, are likely to provide better evidence about the distinctions between the cognitive mechanisms of humans and those of other species. Moreover, the classes of the (...) Subregular Hierarchy have the advantage of fully abstract descriptive (model-theoretic) characterizations in addition to characterizations in more familiar grammar- and automata-theoretic terms. Because the descriptive characterizations make no assumptions about implementation, they provide a sound basis for drawing conclusions about potential cognitive mechanisms from the experimental results. We review the Subregular Hierarchy and provide a concrete set of principles for the design and interpretation of these experiments. (shrink)

In this paper we show how to interpret Robinson’s arithmetic Q and the theory R of Tarski, Mostowski, and Robinson as theories of cardinals in very weak theories of relations over a domain.

We define the notion of ordinal computability by generalizing standard Turing computability on tapes of length ω to computations on tapes of arbitrary ordinal length. We show that a set of ordinals is ordinal computable from a finite set of ordinal parameters if and only if it is an element of Gödel's constructible universe L. This characterization can be used to prove the generalized continuum hypothesis in L.

Let be the set of nonnegative integers. We show the two following facts about Presburger's arithmetic:1. 1. Let . If L is not definable in , + then there is an definable in , such that there is no bound on the distance between two consecutive elements of L′. and2. 2. is definable in , + if and only if every subset of which is definable in is definable in , +. These two Theorems are of independent interest but we (...) will get from them new proofs of Cobham's and Semenov's Theorems ; Semenov's Theorem is:Let k and l be multiplicatively independent . If is definable in and in then L is recognizable . Here Vm is the function which sends a nonzero natural number to the greatest power of m dividing it. (shrink)

A structure has a (finite-string) automatic presentation if the elements of its domain can be named by finite strings in such a way that the coded domain and the coded atomic operations are recognised by synchronous multitape automata. Consequently, every structure with an automatic presentation has a decidable first-order theory. The problems surveyed here include the classification of classes of structures with automatic presentations, the complexity of the isomorphism problem, and the relationship between definability and recognisability.

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)

The classical Feferman–Vaught Theorem for First Order Logic explains how to compute the truth value of a first order sentence in a generalized product of first order structures by reducing this computation to the computation of truth values of other first order sentences in the factors and evaluation of a monadic second order sentence in the index structure. This technique was later extended by Läuchli, Shelah and Gurevich to monadic second order logic. The technique has wide applications in decidability and (...) definability theory. Here we give a unified presentation, including some new results, of how to use the Feferman–Vaught Theorem, and some new variations thereof, algorithmically in the case of Monadic Second Order Logic MSOL . We then extend the technique to graph polynomials where the range of the summation of the monomials is definable in MSOL . Here the Feferman–Vaught Theorem for these polynomials generalizes well known splitting theorems for graph polynomials. Again, these can be used algorithmically. Finally, we discuss extensions of MSOL for which the Feferman–Vaught Theorem holds as well. (shrink)

We investigate the expressive power of second-order logic over finite structures, when two limitations are imposed. Let SAA ) be the set of second-order formulas such that the arity of the relation variables is bounded by k and the number of alternations of second-order quantification is bounded by n . We show that this imposes a proper hierarchy on second-order logic, i.e. for every k , n there are problems not definable in AA but definable in AA for some c (...) 1 , d 1 . The method to show this is to introduce the set AUTOSAT of formulas in F which satisfy themselves. We study the complexity of this set for various fragments of second-order loeic. For first-order logic FOL with unbounded alternation of quantifiers AUTOSAT is PSpace complete. For first-order logic FOL n with alternation of quantifiers bounded by n , AUTOSAT is definable in AA . AUTOSAT) is definable in AA for some c 1 , d 1. (shrink)

Linguistics as a science has rapidly changed during the course of a relatively short period. The mathematical foundations of the science, however, present a different story below the surface. In this paper, I argue that due to the former, the seismic shifts in theory over the past 80 years opens linguistics up to the problem of pessimistic meta-induction or radical theory change. I further argue that, due to the latter, one current solution to this problem in the philosophy of science, (...) namely structural realism :403–424, 1998; French in Proc Aristot Soc 106:167–185, 2006), should be viewed as especially enticing for linguists, as their field is a largely structural enterprise. I discuss particular historical instances of theory change in generative syntax before investigating two views on the nature of structural properties and eventually proposing an approach in terms of invariance :162–186, 2015) as a grounding for structural realism in the history and philosophy of linguistics. (shrink)

The logic L (Qu) extends first-order logic by a generalized form of counting quantifiers ("the number of elements satisfying... belongs to the set C"). This logic is investigated for structures with an injectively ω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [6]. It is shown that, as in the case of automatic structures [21], also modulo-counting quantifiers as well as infinite cardinality quantifiers ("there are χ many (...) elements satisfying...") lead to decidable theories. For a structure of bounded degree with injective ω-automatic presentation, the fragment of L(Qu) that contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold. (shrink)

The model-checking problem for a logic L on a class C of structures asks whether a given L-sentence holds in a given structure in C. In this paper, we give super-exponential lower bounds for fixed-parameter tractable model-checking problems for first-order and monadic second-order logic. We show that unless PTIME=NP, the model-checking problem for monadic second-order logic on finite words is not solvable in time f·p, for any elementary function f and any polynomial p. Here k denotes the size of the (...) input sentence and n the size of the input word. We establish a number of similar lower bounds for the model-checking problem for first-order logic, for example, on the class of all trees. (shrink)

We study the decidability problem for metric and layered temporal logics. The logics we consider are suitable to model time granularity in various contexts, and they allow one to build granular temporal models by referring to the "natural scale" in any component of the model and by properly constraining the interactions between differently-grained components. A monadic second-order language combining operators such as temporal contextualization and projection, together with the usual displacement operator of metric temporal logics, is considered, and the theory (...) of finitely-layered metric temporal structures is shown to be decidable. (shrink)

In the first part of the paper we study arithmetical properties of Pascal triangles modulo a prime power; the main result is the generalization of Lucas' theorem. Then we investigate the structure N; Bpx, where p is a prime, α is an integer greater than one, and Bpx = Rem, px); it is shown that addition is first-order definable in this structure, and that its elementary theory is decidable.

We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ₂—theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ₁—theory of multiplication and order is decidable in finite models as well (...) as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation. We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication. (shrink)

For every k ∈ ω, there is an infinite set $A_k \subseteq \omega$ and a d(k) ∈ ω such that for all $Q_0, Q_1 \subseteq A_k$ where |Q 0 | = |Q 1 or $d(k) , the structures $\langle \omega, +, Q_0\rangle$ and $\langle \omega, +, Q_1\rangle$ are indistinguishable by first-order sentences of quantifier depth k whose atomic formulas are of the form u = v, u + v = w, and Q(u), where u, v, and w are variables.

The aim of this paper is to point out the equivalence between three notions respectively issued from recursion theory, computational complexity and finite model theory. One the one hand, the rudimentary languages are known to be characterized by the linear hierarchy. On the other hand, this complexity class can be proved to correspond to monadic second‐order logic with addition. Our viewpoint sheds some new light on the close connection between these domains: We bring together the two extremal notions by providing (...) a direct logical characterization of rudimentary languages and a representation result of second‐order logic into these languages. We use natural arithmetical tools, and our proofs contain no ingredient from computational complexity. (shrink)

Presburger arithmetic is the first-order theory of the natural numbers with addition. We characterize sets that can be defined by a Presburger formula as exactly the sets whose characteristic functions can be represented by rational generating functions; a geometric characterization of such sets is also given. In addition, ifp= are a subset of the free variables in a Presburger formula, we can define a counting functiong to be the number of solutions to the formula, for a givenp. We show that (...) every counting function obtained in this way may be represented as, equivalently, either a piecewise quasi-polynomial or a rational generating function. Finally, we translate known computational complexity results into this setting and discuss open directions. (shrink)

Locally finite omega languages were introduced by Ressayre [Formal languages defined by the underlying structure of their words. J Symb Log 53(4):1009–1026, 1988]. These languages are defined by local sentences and extend ω-languages accepted by Büchi automata or defined by monadic second order sentences. We investigate their topological complexity. All locally finite ω-languages are analytic sets, the class LOC ω of locally finite ω-languages meets all finite levels of the Borel hierarchy and there exist some locally finite ω-languages which are (...) Borel sets of infinite rank or even analytic but non-Borel sets. This gives partial answers to questions of Simonnet (Automates et Théorie Descriptive, Ph.D. Thesis. Université Paris 7, March 1992) and of Duparc et al. [Computer science and the fine structure of Borel sets. Theor Comput Sci 257(1–2):85–105, 2001]. (shrink)

In this article we review some of the main results of descriptive complexity theory in order to make the reader familiar with the nature of the investigations in this area. We start by presenting the characterization of automata recognizable languages by monadic second-order logic. Afterwards we explain the characterization of various logics by fIxed-point logics. We assume familiarity with logic but try to keep knowledge of complexity theory to aminimum.

It is shown, assuming the linear case of Schinzel's Hypothesis, that the first-order theory of the structure $\langle \omega; +, P\rangle$ , where P is the set of primes, is undecidable and, in fact, that multiplication of natural numbers is first-order definable in this structure. In the other direction, it is shown, from the same hypothesis, that the monadic second-order theory of $\langle\omega; S, P\rangle$ is decidable, where S is the successor function. The latter result is proved using a general (...) result of A. L. Semenov on decidability of monadic theories, and a proof of Semenov's result is presented. (shrink)

Let θ, θ′ be two multiplicatively independent Pisot numbers, and letU,U′ be two linear numeration systems whose characteristic polynomial is the minimal polynomial of θ and θ′, respectively. For everyn≥ 1, ifA⊆ ℕnisU-andU′ -recognizable thenAis definable in 〈ℕ: + 〉.

Linear temporal logic is a widely used method for verification of model checking and expressing the system specifications. The relationship between theory of automata and logic had a great influence in the computer science. Investigation of the relationship between quantum finite automata and linear temporal logic is a natural goal. In this paper, we present a construction of quantum finite automata on finite words from linear-time temporal logic formulas. Further, the relation between quantum finite automata and linear temporal logic is (...) explored in terms of language recognition and acceptance probability. We have shown that the class of languages accepted by quantum finite automata are definable in linear temporal logic, except for measure-once one-way quantum finite automata. (shrink)

We investigate the extension of monadic second-order logic of order with cardinality quantifiers "there exists uncountably many sets such that... " and "there exists continuum many sets such that... ". We prove that over the class of countable linear orders the two quantifiers are equivalent and can be effectively and uniformly eliminated. Weaker or partial elimination results are obtained for certain wider classes of chains. In particular, we show that over the class of ordinals the uncountability quantifier can be effectively (...) and uniformly eliminated. Our argument makes use of Shelah's composition method and Ramsey-like theorem for dense linear orders. (shrink)

In every undirected graph or, more generally, in every undirected hypergraph of bounded rank, one can specify an orientation of the edges or hyperedges by monadic second-order formulas using quantifications on sets of edges or hyperedges. The proof uses an extension to hypergraphs of the classical notion of a depth-first spanning tree. Applications are given to the characterization of the classes of graphs and hypergraphs having decidable monadic theories.

It was conjectured by Halpern and Kapron (Annals of Pure and Applied Logic, vol. 69, 1994) that frame satisfiability of propositional modal formulas is incomparable in expressive power to both Σ 1 1 (Ackermann) and Σ 1 1 (Bernays-Schonfinkel). We prove this conjecture. Our results imply that Σ 1 1 (Ackermann) and Σ 1 1 (Bernays-Schonfinkel) are incomparable in expressive power, already on finite graphs. Moreover, we show that on ordered finite graphs, i.e., finite graphs with a successor, Σ 1 (...) 1 (Bernays-Schonfinkel) is strictly more expressive than Σ 1 1 (Ackermann). (shrink)

In prior papers the following question was considered: which classes of computable sets can be learned if queries about those sets can be asked by the learner? The answer depended on the query language chosen. In this paper we develop a framework for studying this question. Essentially, once we have a result for queries to [S,<]2, we can obtain the same result for many different languages. We obtain easier proofs of old results and several new results. An earlier result we (...) have an easier proof of: the set of computable sets cannot be learned with queries to the language [+,<] . A new result: the set of computable sets cannot be learned with queries to the language [+,<,POWa] where POWa is the predicate that tests if a number is a power of a. (shrink)

We design functional algebras that characterize various complexity classes of global functions. For this purpose, classical schemata from recursion theory are tailored for capturing complexity. In particular we present a functional analog of first-order logic and describe algebras of the functions computable in nondeterministic logarithmic space, deterministic and nondeterministic polynomial time, and for the functions computable by AC 1 -circuits.

John organized a state lottery and his wife won the main prize. You may feel that the event of her winning wasn’t particularly random, but how would you argue that in a fair court of law? Traditional probability theory does not even have the notion of random events. Algorithmic information theory does, but it is not applicable to real-world scenarios like the lottery one. We attempt to rectify that.

We present an elementary three-pass algorithm for computing addition in Ostrowski numeration systems. When a is quadratic, addition in the Ostrowski numeration system based on a is recognizable by a finite automaton. We deduce that a subset of X⊆Nn is definable in, where Va is the function that maps a natural number x to the smallest denominator of a convergent of a that appears in the Ostrowski representation based on a of x with a nonzero coefficient if and only if (...) the set of Ostrowski representations of elements of X is recognizable by a finite automaton. The decidability of the theory of follows. (shrink)

This paper extends a notion of local grammars in formal language theory to autosegmental representations, in order to develop a sufficiently expressive yet computationally restrictive theory of well-formedness in natural language tone patterns. More specifically, it shows how to define a class ASL\ of stringsets using local grammars over autosegmental representations and a mapping g from strings to autosegmental structures. It then defines a particular class ASL\ using autosegmental representations specific to tone and compares its expressivity to established formal language (...) grammars that have been successfully applied to other areas of phonology. (shrink)

We show in this paper that the theory of ordinal addition of any fixed ordinal ωα, with α less than ωω, admits a quantifier elimination. This in particular gives a new proof for the decidability result first established in 1965 by R. Büchi using transfinite automata. Our proof is based on the Ehrenfeucht games, and we show that quantifier elimination go through generalized power.RésuméOn montre ici que, pour tout ordinal α inférieur à ωω, la théorie additive de ωα admet une (...) élimination des quantificateurs. On obtient ainsi une nouvelle preuve de la décidabilité de ces théories . On utilise ici les jeux d'Ehrenfeucht, avec un formalisme à la Ferrante et Rackoff, et on montre que le fait d'admettre une élimination des quantificateurs se transmet par produit généralisé. (shrink)

We show here that the first order theory of the positive integers equipped with multiplication remains decidable when one adds to the language the usual order restricted to the prime numbers. We see moreover that the complexity of the latter theory is a tower of exponentials, of height O(n).

For a given numeration system U, a set X of integers is said to be U-star-free if the language of the normalized U-representations of the elements in X is star-free. Adapting a result of McNaughton and Papert, we give a first-order logical characterization of these sets for various numeration systems including integer base systems and the Fibonacci system. For k-ary systems, the problem of the base dependence of this property is also studied. Finally, the case of k-adic systems is developed.

We show that no formula of first order logic using linear ordering and the logical relation y = 2x can define the property that the size of a finite model is divisible by 3. This answers a long-standing question which may be of relevance to certain open problems in circuit complexity.