First-order Signature Research Articles

Proximate deconfined quantum critical point in SrCu2(BO3)2.

The deconfined quantum critical point (DQCP) represents a paradigm shift in quantum matter studies, presenting a "beyond Landau" scenario for order-order transitions. Its experimental realization, however, has remained elusive. Using high-pressure 11B nuclear magnetic resonance measurements on the quantum magnet SrCu2(BO3)2, we here demonstrate a magnetic field-induced plaquette singlet to antiferromagnetic transition above 1.8 gigapascals at a notably low temperature, Tc ≃ 0.07 kelvin. First-order signatures of the transition weaken with increasing pressure, and we observe quantum critical scaling at the highest pressure, 2.4 gigapascals. Supported by model calculations, we suggest that these observations can be explained by a proximate DQCP inducing critical quantum fluctuations and emergent O(3) symmetry of the order parameters. Our findings offer a concrete experimental platform for investigation of the DQCP.

An efficient NPN Boolean matching algorithm based on structural signature and Shannon expansion

An efficient pairwise Boolean matching algorithm for solving the problem of matching single-output specified Boolean functions under input negation and/or input permutation and/or output negation (NPN) is proposed in this paper. We present the structural signature (SS) vector, which comprises a first-order signature value, two symmetry marks, and a group mark. As a necessary condition for NPN Boolean matching, the SS is more effective than the traditional signature. A symmetry mark can distinguish symmetric variables and asymmetric variables and be used to search for multiple variable mappings in a single variable-mapping search operation, which reduces the search space significantly. Updating the SS vector via Shannon decomposition provides benefits in distinguishing unidentified variables, and the group mark and phase collision check can be used to discover incorrect variable mappings quickly, which also speeds up the NPN Boolean matching process. Using the algorithm proposed in this paper, we test both equivalent and non-equivalent matching speeds on the MCNC benchmark circuit sets and random circuit sets. In the experiment, our algorithm is shown to be 4.2 times faster than competitors when testing equivalent circuits and 172 times faster, on average, when testing non-equivalent circuits.

First-order transitions and thermodynamic properties in the 2D Blume-Capel model: the transfer-matrix method revisited

We investigate the first-order transition in the spin-1 two-dimensional Blume-Capel model in square lattices by revisiting the transfer-matrix method. With large strip widths increased up to the size of 18 sites, we construct the detailed phase coexistence curve which shows excellent quantitative agreement with the recent advanced Monte Carlo results. In the deep first-order area, we observe the exponential system-size scaling of the spectral gap of the transfer matrix from which linearly increasing interfacial tension is deduced with decreasing temperature. We find that the first-order signature at low temperatures is strongly pronounced with much suppressed finite-size influence in the examined thermodynamic properties of entropy, non-zero spin population, and specific heat. It turns out that the jump at the transition becomes increasingly sharp as it goes deep into the first-order area, which is in contrast to the Wang-Landau results where finite-size smoothing gets more severe at lower temperatures.

Describing finite groups by short first-order sentences

We say that a class of finite structures for a finite first-order signature is r-compressible for an unbounded function r: N → N+ if each structure G in the class has a first-order description of size at most O(r(|G|)). We show that the class of finite simple groups is log-compressible, and the class of all finite groups is log3-compressible. As a corollary we obtain that the class of all finite transitive permutation groups is log3-compressible. The results rely on the classification of finite simple groups, the bi-interpretability of the twisted Ree groups with finite difference fields, the existence of profinite presentations with few relators for finite groups, and group cohomology. We also indicate why the results are close to optimal.

First-order character of the displacive structural transition in BaWO4

Nearly all displacive transitions have been considered to be continuous or second order, and the rigid unit mode (RUM) provides a natural candidate for the soft mode. However, in-situ X-ray diffraction and Raman measurements show clearly the first-order evidences for the scheelite-to-fergusonite displacive transition in BaWO4: a 1.6% volume collapse, coexistence of phases, and hysteresis on release of pressure. Such first-order signatures are found to be the same as the soft modes in BaWO4, which indicates the scheelite-to-fergusonite displacive phase transition hides a deeper physical mechanism. By the refinement of atomic displacement parameters, we further show that the first-order character of this phase transition stems from a coupling of large compression of soft BaO8 polyhedrons to the small displacive distortion of rigid WO4 tetrahedrons. Such a coupling will lead to a deeper physical insight in the phase transition of the common scheelite-structured compounds.

A Formalization of the Theorem of Existence of First-Order Most General Unifiers

This work presents a formalization of the theorem of existence of most general unifiers in first-order signatures in the higher-order proof assistant PVS. The distinguishing feature of this formalization is that it remains close to the textbook proofs that are based on proving the correctness of the well-known Robinson's first-order unification algorithm. The formalization was applied inside a PVS development for term rewriting systems that provides a complete formalization of the Knuth-Bendix Critical Pair theorem, among other relevant theorems of the theory of rewriting. In addition, the formalization methodology has been proved of practical use in order to verify the correctness of unification algorithms in the style of the original Robinson's unification algorithm.

Derivatives of knots and second-order signatures

We define a set of "second-order" L^(2)-signature invariants for any algebraically slice knot. These obstruct a knot's being a slice knot and generalize Casson-Gordon invariants, which we consider to be "first-order signatures". As one application we prove: If K is a genus one slice knot then, on any genus one Seifert surface, there exists a homologically essential simple closed curve of self-linking zero, which has vanishing zero-th order signature and a vanishing first-order signature. This extends theorems of Cooper and Gilmer. We introduce a geometric notion, that of a derivative of a knot with respect to a metabolizer. We also introduce a new equivalence relation, generalizing homology cobordism, called null-bordism.

Evidence for first-order nature of the ferromagnetic transition in Ni, Fe, Co, andCoFe2O4

Nearly all ferromagnetic transitions have been considered to be continuous or second order, and the most typical examples are the ferromagnetic transitions in Ni, Fe, Co, and ${\text{CoFe}}_{2}{\text{O}}_{4}$. However, by precise measurement with electrical resistivity or impedance and differential scanning calorimetry, we show clear evidence for the first-order nature of these ``second-order transitions''---a small thermal hysteresis and latent heat. Such first-order signatures are found to be the same as those for the well-recognized first-order transitions in the ferroelectric ${\text{BaTiO}}_{3}$ and ferroelastic ${\text{Ti}}_{50}{\text{Ni}}_{47}{\text{Fe}}_{3}$. These results question the existence of genuine second-order transition in ferromagnetic systems. By a phenomenological approach, we further show that the first-order nature of ferromagnetic transition may stem from a coupling of magnetic moment to other order parameter(s) like strain. Such a coupling may provide insight into developing highly magnetoresponsive materials.

A duality between proof systems for cyclic term graphs

This paper presents a proof-theoretic observation about two kinds of proof systems for bisimilarity between cyclic term graphs.First we consider proof systems for demonstrating that μ term specifications of cyclic term graphs have the same tree unwinding. We establish a close connection between adaptations for μ terms over a general first-order signature of the coinductive axiomatisation of recursive type equivalence by Brandt and Henglein (Brandt and Henglein 1998) and of a proof system by Ariola and Klop (Ariola and Klop 1995) for consistency checking. We show that there exists a simple duality by mirroring between derivations in the former system and formalised consistency checks, which are called ‘consistency unfoldings', in the latter. This result sheds additional light on the axiomatisation of Brandt and Henglein: it provides an alternative soundness proof for the adaptation considered here.We then outline an analogous duality result that holds for a pair of similar proof systems for proving that equational specifications of cyclic term graphs are bisimilar.

Explicit substitutions and higher-order syntax

Recently there has been a great deal of interest in higher-order syntax which seeks to extend standard initial algebra semantics to cover languages with variable binding. The canonical example studied in the literature is that of the untyped ?-calculus which is handled as an instance of the general theory of binding algebras, cf. Fiore et al. [13]. Another important syntactic construction is that of explicit substitutions which are used to model local definitions and to implement reduction in the ?-calculus. The syntax of a language with explicit substitutions does not form a binding algebra as an explicit substitution may bind an arbitrary number of variables. Thus explicit substitutions are a natural test case for the further development of the theory and applications of syntax with variable binding. This paper shows that a language containing explicit substitutions and a first-order signature ? is naturally modelled as the initial algebra of the Id + F ??_ +_ ? _ endofunctor. We derive a similar formula for adding explicit substitutions to the untyped ?-calculus and then show these initial algebras provide useful datatypes for manipulating abstract syntax by implementing two reduction machines. We also comment on the apparent lack of modularity in syntax with variable binding as compared to first-order languages.

Remote sensing methods and techniques as a tool for the implementation of environmental legislation. The Greek Forest Law case study

Implementation of environmental legislation can be supported by the production of maps with two information levels, one which identifies areas where a legal rule is enforced and another which identifies areas where there is an infringement of that rule. This paper investigates the efficiency of remote sensing methods and techniques for the production of such a map regarding a specific law case, the Greek Forest Law. To do this, we identified, with the use of KVR-1000 very high resolution imagery, potential illegal buildings in forests and forest areas, which were mapped at a scale of 1:5000. For this purpose, an image processing method was developed which integrates spectral and first-order texture signatures, detects disturbances in forest environment, and classifies them in order for the buildings to be identified among other disturbances. The texture signatures were selected after an investigation of the performance of first- and second-order statistical texture methods in detecting disturbances in forest environment. By the method developed, buildings are mapped by segmented parts of their edges. Based on a sample of 163 buildings, the accuracy of the method was calculated at approximately 89%, i.e. 11% of the buildings included in the sample were not detected. The map produced would probably not be acceptable in court, however it can adequately support administrative procedures in ministries, forest services, etc., regarding identification of illegal building, incorporation of forest areas in an existing city plan, organization of the forest and forest area register book, etc.

Prelogical Relations

We study a weakening of the notion of logical relations, called prelogical relations , that has many of the features that make logical relations so useful as well as further algebraic properties including composability. The basic idea is simply to require the reverse implication in the definition of logical relations to hold only for pairs of functions that are expressible by the same lambda term. Prelogical relations are the minimal weakening of logical relations that gives composability for extensional structures and simultaneously the most liberal definition that gives the Basic Lemma. Prelogical predicates (i.e., unary prelogical relations) coincide with sets that are invariant under Kripke logical relations with varying arity as introduced by Jung and Tiuryn, and prelogical relations are the closure under projection and intersection of logical relations. These conceptually independent characterizations of prelogical relations suggest that the concept is rather intrinsic and robust. The use of prelogical relations gives an improved version of Mitchell's representation independence theorem which characterizes observational equivalence for all signatures rather than just for first-order signatures. Prelogical relations can be used in place of logical relations to give an account of data refinement where the fact that prelogical relations compose explains why stepwise refinement is sound.

Prelogical Relations

We study a weakening of the notion of logical relations, called prelogical relations, that has many of the features that make logical relations so useful as well as further algebraic properties including composability. The basic idea is simply to require the reverse implication in the definition of logical relations to hold only for pairs of functions that are expressible by the same lambda term. Prelogical relations are the minimal weakening of logical relations that gives composability for extensional structures and simultaneously the most liberal definition that gives the Basic Lemma. Prelogical predicates (i.e., unary prelogical relations) coincide with sets that are invariant under Kripke logical relations with varying arity as introduced by Jung and Tiuryn, and prelogical relations are the closure under projection and intersection of logical relations. These conceptually independent characterizations of prelogical relations suggest that the concept is rather intrinsic and robust. The use of prelogical relations gives an improved version of Mitchell's representation independence theorem which characterizes observational equivalence for all signatures rather than just for first-order signatures. Prelogical relations can be used in place of logical relations to give an account of data refinement where the fact that prelogical relations compose explains why stepwise refinement is sound.

Polymorphic syntax definition

Context-free grammars are used in several algebraic specification formalisms instead of first-order signatures for the definition of the structure of algebras, because grammars provide better notation than signatures. The rigidity of these first-order structures enforces a choice between strongly typed structures with little genericity or generic operations over untyped structures. In two-level signatures level 1 defines the algebra of types used at level 0 providing the possibility to define polymorphic abstract data types. Two-level grammars are the grammatical counterpart of two-level signatures. This paper discusses the correspondence between context-free grammars and first-order signatures, the extension of this correspondence to two-level grammars and signatures, examples of the usage of two-level grammars for polymorphic syntax definition, a restriction of the class of two-level grammars for which the parsing problem is decidable, a parsing algorithm that yields a minimal and finite set of most general parse trees for this class of grammars, and a proof of its correctness.

EXPRESSIBILITY OF THE SEMANTICS OF SEQUENTIAL PROGRAMS IN FIRST-ORDER LOGIC

The notion of an expressive interpretation was originally introduced by Cook in order to formulate completeness results for Hoare-style proof systems. An interpretation is called expressive for a programming language if the input/output relation of every program can be expressed by a first-order formula. Various necessary or sufficient conditions for an interpretation to be expressive are known. In this paper a characterization is given which improves on these results. It holds for a wide spectrum of programming languages. An interpretation is expressive iff it is either 1. uniformly locally finite or 2. weakly arithmetic and it has evaluation predicates (which return the truth value of a boolean expression, given its godel number and the values of its free variables). Besides this characterization, it is investigated which first-order signatures allow nontrivial (i.e. non uniformly locally finite) expressive interpretations. Signatures which have (besides constants) only one unary function symbol and only unary relation symbols, have only trivial expressive interpretations. All other signatures allow nontrivial expressive interpretations as well.

Incompleteness of first-order temporal logic with until

The results presented in this paper concern the axiomatizability problem of first-order temporal logic with linear and discrete time. We show that the logic is incomplete, i.e., it cannot be provided with a finitistic and complete proof system. We show two incompleteness theorems. Although the first one is weaker (it assumes some first-order signature), we decided to present it, for its proof is much simpler and contains an interesting fact that finite sets are characterizable by means of temporal formulas. The second theorem shows that the logic is incomplete independently of any particular signature.