Number of variables is equivalent to space

2001 ◽  
Vol 66 (3) ◽  
pp. 1217-1230 ◽  
Author(s):  
Neil Immerman ◽  
Jonathan F. Buss ◽  
David A. Mix Barrington
Keyword(s):  
Turing Machine ◽  
Descriptive Complexity ◽  
First Order ◽  
Trade Offs ◽  
Finite Set ◽  
The Universe

AbstractWe prove that the set of properties describable by a uniform sequence of first-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE[nk] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE = VAR[O[1]] [8]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier depth in descriptive complexity.

On spectra, and the negative solution of the decision problem for identities having a finite nontrivial model

1975 ◽  
Vol 40 (2) ◽  
pp. 186-196 ◽  
Author(s):  
Ralph Mckenzie
Keyword(s):  
Model Problem ◽  
Recursive Algorithm ◽  
Finite Model ◽  
Positive Direction ◽  
First Order ◽  
Sharp Form ◽  
Universal Sentence ◽  
Order Language ◽  
Finite Set ◽  
The Universe

An algorithm has been described by S. Burris [3] which decides if a finite set of identities, whose function symbols are of rank at most 1, has a finite, nontrivial model. (By “nontrivial” it is meant that the universe of the model has at least two elements.) As a consequence of some results announced in the abstracts [2] and [8], it is clear that if the restriction on the ranks of function symbols is relaxed somewhat, then this finite model problem is no longer solvable by an algorithm, or at least not by a “recursive algorithm” as the term is used today.In this paper we prove a sharp form of this negative result; showing, by the way, that Burris' result is in a sense the best possible result in the positive direction. Our main result is that in a first order language whose only function or relation symbol is a 2-place function symbol (the language of groupoids), the set of identities that have no nontrivial model, is recursively inseparable from the set of identities such that the sentence has a finite model. As a corollary, we have that each of the following problems, restricted to sentences defined in the language of groupoids, is algorithmically unsolvable: (1) to decide if an identity has a finite nontrivial model; (2) to decide if an identity has a nontrivial model; (3) to decide if a universal sentence has a finite model; (4) to decide if a universal sentence has a model. We note that the undecidability of (2) was proved earlier by McNulty [13, Theorem 3.6(i)], improving results obtained by Murskiǐ [14] and by Perkins [17]. The other parts of the corollary seem to be new.

Imaginings

2017 ◽  
Vol 9 (3) ◽  
pp. 17-30
Author(s):  
Kelly James Clark
Keyword(s):  
Religious Tradition ◽  
Second Order ◽  
Fine Tuning ◽  
Intellectual Humility ◽  
First Order ◽  
Natural Processes ◽  
Empirically Supported ◽  
Group Violence ◽  
The World ◽  
The Universe

In Branden Thornhill-Miller and Peter Millican’s challenging and provocative essay, we hear a considerably longer, more scholarly and less melodic rendition of John Lennon’s catchy tune—without religion, or at least without first-order supernaturalisms (the kinds of religion we find in the world), there’d be significantly less intra-group violence. First-order supernaturalist beliefs, as defined by Thornhill-Miller and Peter Millican (hereafter M&M), are “beliefs that claim unique authority for some particular religious tradition in preference to all others” (3). According to M&M, first-order supernaturalist beliefs are exclusivist, dogmatic, empirically unsupported, and irrational. Moreover, again according to M&M, we have perfectly natural explanations of the causes that underlie such beliefs (they seem to conceive of such natural explanations as debunking explanations). They then make a case for second-order supernaturalism, “which maintains that the universe in general, and the religious sensitivities of humanity in particular, have been formed by supernatural powers working through natural processes” (3). Second-order supernaturalism is a kind of theism, more closely akin to deism than, say, Christianity or Buddhism. It is, as such, universal (according to contemporary psychology of religion), empirically supported (according to philosophy in the form of the Fine-Tuning Argument), and beneficial (and so justified pragmatically). With respect to its pragmatic value, second-order supernaturalism, according to M&M, gets the good(s) of religion (cooperation, trust, etc) without its bad(s) (conflict and violence). Second-order supernaturalism is thus rational (and possibly true) and inconducive to violence. In this paper, I will examine just one small but important part of M&M’s argument: the claim that (first-order) religion is a primary motivator of violence and that its elimination would eliminate or curtail a great deal of violence in the world. Imagine, they say, no religion, too.Janusz Salamon offers a friendly extension or clarification of M&M’s second-order theism, one that I think, with emendations, has promise. He argues that the core of first-order religions, the belief that Ultimate Reality is the Ultimate Good (agatheism), is rational (agreeing that their particular claims are not) and, if widely conceded and endorsed by adherents of first-order religions, would reduce conflict in the world.While I favor the virtue of intellectual humility endorsed in both papers, I will argue contra M&M that (a) belief in first-order religion is not a primary motivator of conflict and violence (and so eliminating first-order religion won’t reduce violence). Second, partly contra Salamon, who I think is half right (but not half wrong), I will argue that (b) the religious resources for compassion can and should come from within both the particular (often exclusivist) and the universal (agatheistic) aspects of religious beliefs. Finally, I will argue that (c) both are guilty, as I am, of the philosopher’s obsession with belief. 

scholarly journals Electroweak baryogenesis and gravitational waves in a composite Higgs model with high dimensional fermion representations

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Ke-Pan Xie ◽  
Ligong Bian ◽  
Yongcheng Wu
Keyword(s):  
Form Factors ◽  
Higgs Doublet ◽  
Higgs Model ◽  
High Dimensional ◽  
Electroweak Phase Transition ◽  
First Order ◽  
Composite Higgs ◽  
Future Space ◽  
The Universe ◽  
Scalar Sector

Abstract We study electroweak baryogenesis in the SO(6)/SO(5) composite Higgs model with the third generation quarks being embedded in the 20′ representation of SO(6). The scalar sector contains one Higgs doublet and one real singlet, and their potential is given by the Coleman-Weinberg potential evaluated from the form factors of the lightest vector and fermion resonances. We show that the resonance masses at $$ \mathcal{O}\left(1\sim 10\kern0.5em \mathrm{TeV}\right) $$ O 1 ∼ 10 TeV can generate a potential that triggers the strong first-order electroweak phase transition (SFOEWPT). The CP violating phase arising from the dimension-6 operator in the top sector is sufficient to yield the observed baryon asymmetry of the universe. The SFOEWPT parameter space is detectable at the future space-based detectors.

scholarly journals Finite representability of semigroups with demonic refinement

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Robin Hirsch ◽  
Jaš Šemrl
Keyword(s):  
Partial Order ◽  
Turing Machines ◽  
Binary Relations ◽  
Finite Representation ◽  
First Order ◽  
Finite Structures ◽  
Finite Base ◽  
Finite Set ◽  
Finite Representability ◽  
Representation Class

AbstractThe motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ⊑ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ⊔ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) of abstract $$(\le , \circ )$$ ( ≤ , ∘ ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( ≤ , ∘ ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( ≤ , ∘ ) structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representation property holds for finite structures.

scholarly journals Testing for Negative Spillovers: Is Promoting Human Rights Really Part of the “Problem”?

2021 ◽  
Vol 75 (1) ◽  
pp. 71-102
Author(s):  
Anton Strezhnev ◽  
Judith G. Kelley ◽  
Beth A. Simmons
Keyword(s):  
Human Rights ◽  
International Law ◽  
Instrumental Variable ◽  
Burden Of Proof ◽  
Negative Consequences ◽  
Human Rights Law ◽  
International Human Rights ◽  
First Order ◽  
Trade Offs ◽  
International Human

AbstractThe international community often seeks to promote political reforms in recalcitrant states. Recently, some scholars have argued that, rather than helping, international law and advocacy create new problems because they have negative spillovers that increase rights violations. We review three mechanisms for such spillovers: backlash, trade-offs, and counteraction and concentrate on the last of these. Some researchers assert that governments sometimes “counteract” international human rights pressures by strategically substituting violations in adjacent areas that are either not targeted or are harder to monitor. However, most such research shows only that both outcomes correlate with an intervention—the targeted positively and the spillover negatively. The burden of proof, however, should be as rigorous as those for studies of first-order policy consequences. We show that these correlations by themselves are insufficient to demonstrate counteraction outside of the narrow case where the intervention is assumed to have no direct effect on the spillover, a situation akin to having a valid instrumental variable design. We revisit two prominent findings and show that the evidence for the counteraction claim is weak in both cases. The article contributes methodologically to the study of negative spillovers in general by proposing mediation and sensitivity analysis within an instrumental variables framework for assessing such arguments. It revisits important prior findings that claim negative consequences to human rights law and/or advocacy, and raises critical normative questions regarding how we empirically evaluate hypotheses about causal mechanisms.

scholarly journals Nucleation is more than critical: A case study of the electroweak phase transition in the NMSSM

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Sebastian Baum ◽  
Marcela Carena ◽  
Nausheen R. Shah ◽  
Carlos E. M. Wagner ◽  
Yikun Wang
Keyword(s):  
Phase Transition ◽  
Parameter Space ◽  
Local Minima ◽  
Critical Temperatures ◽  
Electroweak Phase Transition ◽  
First Order ◽  
Vacuum Structure ◽  
The Universe ◽  
Scalar Sector

Abstract Electroweak baryogenesis is an attractive mechanism to generate the baryon asymmetry of the Universe via a strong first order electroweak phase transition. We compare the phase transition patterns suggested by the vacuum structure at the critical temperatures, at which local minima are degenerate, with those obtained from computing the probability for nucleation via tunneling through the barrier separating local minima. Heuristically, nucleation becomes difficult if the barrier between the local minima is too high, or if the distance (in field space) between the minima is too large. As an example of a model exhibiting such behavior, we study the Next-to-Minimal Supersymmetric Standard Model, whose scalar sector contains two SU(2) doublets and one gauge singlet. We find that the calculation of the nucleation probabilities prefers different regions of parameter space for a strong first order electroweak phase transition than the calculation based solely on the critical temperatures. Our results demonstrate that analyzing only the vacuum structure via the critical temperatures can provide a misleading picture of the phase transition patterns, and, in turn, of the parameter space suitable for electroweak baryogenesis.

μ-definable sets of integers

1993 ◽  
Vol 58 (1) ◽  
pp. 291-313 ◽  
Author(s):  
Robert S. Lubarsky
Keyword(s):  
Fixed Point ◽  
Free Variable ◽  
Order Logic ◽  
First Order Logic ◽  
Inductive Definition ◽  
Natural Class ◽  
First Order ◽  
Inductive Definitions ◽  
Definable Sets ◽  
The Universe

Inductive definability has been studied for some time already. Nonetheless, there are some simple questions that seem to have been overlooked. In particular, there is the problem of the expressibility of the μ-calculus.The μ-calculus originated with Scott and DeBakker [SD] and was developed by Hitchcock and Park [HP], Park [Pa], Kozen [K], and others. It is a language for including inductive definitions with first-order logic. One can think of a formula in first-order logic (with one free variable) as defining a subset of the universe, the set of elements that make it true. Then “and” corresponds to intersection, “or” to union, and “not” to complementation. Viewing the standard connectives as operations on sets, there is no reason not to include one more: least fixed point.There are certain features of the μ-calculus coming from its being a language that make it interesting. A natural class of inductive definitions are those that are monotone: if X ⊃ Y then Γ (X) ⊃ Γ (Y) (where Γ (X) is the result of one application of the operator Γ to the set X). When studying monotonic operations in the context of a language, one would need a syntactic guarantor of monotonicity. This is provided by the notion of positivity. An occurrence of a set variable S is positive if that occurrence is in the scopes of exactly an even number of negations (the antecedent of a conditional counting as a negation). S is positive in a formula ϕ if each occurrence of S is positive. Intuitively, the formula can ask whether x ∊ S, but not whether x ∉ S. Such a ϕ can be considered an inductive definition: Γ (X) = {x ∣ ϕ(x), where the variable S is interpreted as X}. Moreover, this induction is monotone: as X gets bigger, ϕ can become only more true, by the positivity of S in ϕ. So in the μ-calculus, a formula is well formed by definition only if all of its inductive definitions are positive, in order to guarantee that all inductive definitions are monotone.

Nonrecursive tilings of the plane. I

1974 ◽  
Vol 39 (2) ◽  
pp. 283-285 ◽  
Author(s):  
William Hanf
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Positive Answer ◽  
Additional Restriction ◽  
Detailed History ◽  
Finite Set

A finite set of tiles (unit squares with colored edges) is said to tile the plane if there exists an arrangement of translated (but not rotated or reflected) copies of the squares which fill the plane in such a way that abutting edges of the squares have the same color. The problem of whether there exists a finite set of tiles which can be used to tile the plane but not in any periodic fashion was proposed by Hao Wang [9] and solved by Robert Berger [1]. Raphael Robinson [7] gives a more detailed history and a very economical solution to this and related problems; we will assume that the reader is familiar with §4 of [7]. In 1971, Dale Myers asked whether there exists a finite set of tiles which can tile the plane but not in any recursive fashion. If we make an additional restriction (called the origin constraint) that a given tile must be used at least once, then the positive answer is given by the main theorem of this paper. Using the Turing machine constructed here and a more complicated version of Berger and Robinson's construction, Myers [5] has recently solved the problem without the origin constraint.Given a finite set of tiles T1, …, Tn, we can describe a tiling of the plane by a function f of two variables ranging over the integers. f(i, j) = k specifies that the tile Tk is to be placed at the position in the plane with coordinates (i, j). The tiling will be said to be recursive if f is a recursive function.

scholarly journals On generalized Lemaitre–Tolman–Bondi metric: Fractal matter at the end of matter–antimatter recombination

2021 ◽  
pp. 2150086
Author(s):  
Sergio L. Cacciatori ◽  
Alessio Marrani ◽  
Federico Re
Keyword(s):  
Dark Matter ◽  
Relativistic Effects ◽  
Partial Explanation ◽  
First Order ◽  
Acceleration Of The Universe ◽  
Particular Solution ◽  
Ancient Times ◽  
The Universe ◽  
Recombination Epoch ◽  
Ltb Metric

Many recent researches have investigated the deviations from the Friedmannian cosmological model, as well as their consequences on unexplained cosmological phenomena, such as dark matter and the acceleration of the Universe. On one hand, a first-order perturbative study of matter inhomogeneity returned a partial explanation of dark matter and dark energy, as relativistic effects due to the retarded potentials of far objects. On the other hand, the fractal cosmology, now approximated by a Lemaitre–Tolman–Bondi (LTB) metric, results in distortions of the luminosity distances of SNe Ia, explaining the acceleration as apparent. In this work, we extend the LTB metric to ancient times. The origin of the fractal distribution of matter is explained as the matter remnant after the matter–antimatter recombination epoch. We show that the evolution of such a inhomogeneity necessarily requires a dynamical generalization of LTB, and we propose a particular solution.

scholarly journals The universe is a collection of self-driven mathematical entities; subjective consciousness is the use of mathematical models by a Turing machine

2021 ◽  
Author(s):  
Xiaoyang Yu
Keyword(s):  
Elementary Particle ◽  
Turing Machine ◽  
World Line ◽  
Conscious Experience ◽  
Physical Interaction ◽  
External Observer ◽  
Objective Reality ◽  
Technical Limitation ◽  
Subjective Reality ◽  
The Universe

Humans are limited in what they know by the technical limitation of their cortical language network. A reality is a situation model. The universe is a collection of self-driven mathematical entities. If we are happy to accept randomness, it’s obviously possible that all other so-called “worlds” in the many-worlds interpretation don’t exist objectively. The so-called “physical interaction” (aka objective interaction) among any number of elementary particles is consistent with the so-called “physical law”. From the viewpoint of an imagined external observer (who is located somewhere outside of all worlds), in all worlds, every self-driven elementary particle is changing its state to match its fated state, together form a single fated self-driven state machine; the so-called “subjective reality” (aka the so-called “subjective conscious experience”) is actually the use of a mathematical model (MM) by a Turing machine (TM). The so-called “subjective reality” shouldn’t be able to alter/impact the fated world line of any elementary particle within this world. Except one objective MM which is a fitted MM of the objective reality, every other causality is not an objective MM but a Granger causality, and is an under-fitted MM of the objective reality.

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