In All But Finitely Many Possible Worlds: Model-Theoretic Investigations on ‘<Emphasis Type="Italic">Overwhelming Majority</Emphasis>’ Default Conditionals

Defeasible conditionals are statements of the form ‘if A then normally B’. One plausible interpretation introduced in nonmonotonic reasoning dictates that (\(A\Rightarrow B\)) is true iff B is true in ‘most’ A-worlds. In this paper, we investigate defeasible conditionals constructed upon a notion of ‘overwhelming majority’, defined as ‘truth in a cofinite subset of \(\omega \)’, the first infinite ordinal. One approach employs the modal logic of the frame \((\omega , <)\), used in the temporal logic of discrete linear time. We introduce and investigate conditionals, defined modally over \((\omega , <)\); several modal definitions of the conditional connective are examined, with an emphasis on the nonmonotonic ones. An alternative interpretation of ‘majority’ as sets cofinal (in \(\omega \)) rather than cofinite (subsets of \(\omega \)) is examined. For these modal approaches over \((\omega , <)\), a decision procedure readily emerges, as the modal logic \({\mathbf {K4DLZ}}\) of this frame is well-known and a translation of the conditional sentences can be mechanically checked for validity; this allows also for a quick proof of \(\mathsf {NP}\)-completeness of the satisfiability problem for these logics. A second approach employs the conditional version of Scott-Montague semantics, in the form of \(\omega \)-many possible worlds, endowed with neighborhoods populated by collections of cofinite subsets of \(\omega \). This approach gives rise to weak conditional logics, as expected. The relative strength of the conditionals introduced is compared to (the conditional logic ‘equivalent’ of) KLM logics and other conditional logics in the literature.     

Optimisation Validation

We introduce the idea of optimisation validation, which is to formally establish that an instance of an optimising transformation indeed improves with respect to some resource measure. This is related to, but in contrast with, translation validation, which aims to establish that a particular instance of a transformation undertaken by an optimising compiler is semantics preserving. Our main setting is a program logic for a subset of Java bytecode, which is sound and complete for a resource-annotated operational semantics. The latter employs resource algebras for measuring dynamic costs such as time, space and more elaborate examples. We describe examples of optimisation validation that we have formally verified in Isabelle/HOL using the logic. We also introduce a type and effect system for measuring static costs such as code size, which is proved consistent with the operational semantics.     

Non-determinism in Gödel’s System <Emphasis Type="Italic">T</Emphasis>

The calculus T^? is a successor-free version of Gödel’s T. It is well known that a number of important complexity classes, like e.g. the classes logspace, \(\textsc{p}\), \(\textsc{linspace}\), \(\textsc{etime}\) and \(\textsc{pspace}\), are captured by natural fragments of T^? and related calculi. We introduce the calculus T^∽, which is a non-deterministic variant of T^?, and compare the computational power of T^∽ and T^?. First, we provide a denotational semantics for T^∽ and prove this semantics to be adequate. Furthermore, we prove that \(\textsc{linspace}\subseteq \mathcal {G}^{\backsim }_{0} \subseteq \textsc{linspace}\) and \(\textsc{etime}\subseteq \mathcal {G}^{\backsim }_{1} \subseteq \textsc{pspace}\) where \(\mathcal {G}^{\backsim }_{0}\) and \(\mathcal {G}^{\backsim }_{1}\) are classes of problems decidable by certain fragments of T^∽. (It is proved elsewhere that the corresponding fragments of T^? equal respectively \(\textsc{linspace}\) and \(\textsc{etime}\).) Finally, we show a way to interpret T^∽ in T^?.     

Quantum codes from cyclic codes over $$F_q+uF_q+vF_q+uvF_q$$

In this paper, we study quantum codes over \(F_q\) from cyclic codes over \(F_q+uF_q+vF_q+uvF_q,\) where \(u^2=u,~v^2=v,~uv=vu,~q=p^m\), and p is an odd prime. We give the structure of cyclic codes over \(F_q+uF_q+vF_q+uvF_q\) and obtain self-orthogonal codes over \(F_q\) as Gray images of linear and cyclic codes over \(F_q+uF_q+vF_q+uvF_q\). In particular, we decompose a cyclic code over \(F_q+uF_q+vF_q+uvF_q\) into four cyclic codes over \(F_q\) to determine the parameters of the corresponding quantum code.     

New q-ary quantum MDS codes with distances bigger than $$\frac{ q}{2}$$

The construction of quantum MDS codes has been studied by many authors. We refer to the table in page 1482 of (IEEE Trans Inf Theory 61(3):1474–1484, 2015) for known constructions. However, there have been constructed only a few q-ary quantum MDS \([[n,n-2d+2,d]]_q\) codes with minimum distances \(d>\frac{q}{2}\) for sparse lengths \(n>q+1\). In the case \(n=\frac{q^2-1}{m}\) where \(m|q+1\) or \(m|q-1\) there are complete results. In the case \(n=\frac{q^2-1}{m}\) while \(m|q^2-1\) is neither a factor of \(q-1\) nor \(q+1\), no q-ary quantum MDS code with \(d> \frac{q}{2}\) has been constructed. In this paper we propose a direct approach to construct Hermitian self-orthogonal codes over \(\mathbf{F}_{q^2}\). Then we give some new q-ary quantum codes in this case. Moreover many new q-ary quantum MDS codes with lengths of the form \(\frac{w(q^2-1)}{u}\) and minimum distances \(d > \frac{q}{2}\) are presented.     

Efficient parallelizable hashing using small non-compressing primitives

A well-established method of constructing hash functions is to base them on non-compressing primitives, such as one-way functions or permutations. In this work, we present \(S^r\), an \(rn\)-to-\(n\)-bit compression function (for \(r\ge 1\)) making \(2r-1\) calls to \(n\)-to-\(n\)-bit primitives (random functions or permutations). \(S^r\) compresses its inputs at a rate (the amount of message blocks per primitive call) up to almost 1/2, and it outperforms all existing schemes with respect to rate and/or the size of underlying primitives. For instance, instantiated with the \(1600\)-bit permutation of NIST’s SHA-3 hash function standard, it offers about \(800\)-bit security at a rate of almost 1/2, while SHA-3-512 itself achieves only \(512\)-bit security at a rate of about \(1/3\). We prove that \(S^r\) achieves asymptotically optimal collision security against semi-adaptive adversaries up to almost \(2^{n/2}\) queries and that it can be made preimage secure up to \(2^n\) queries using a simple tweak.     

The remote set problem on lattices

We initiate studying the Remote Set Problem (\({\mathsf{RSP}}\)) on lattices, which given a lattice asks to find a set of points containing a point which is far from the lattice. We show a polynomial-time deterministic algorithm that on rank n lattice \({\mathcal{L}}\) outputs a set of points, at least one of which is \({\sqrt{\log n / n} \cdot \rho(\mathcal{L})}\) -far from \({\mathcal{L}}\) , where \({\rho(\mathcal{L})}\) stands for the covering radius of \({\mathcal{L}}\) (i.e., the maximum possible distance of a point in space from \({\mathcal{L}}\)). As an application, we show that the covering radius problem with approximation factor \({\sqrt{n / \log n}}\) lies in the complexity class \({\mathsf{NP}}\) , improving a result of Guruswami et al. (Comput Complex 14(2): 90–121, 2005) by a factor of \({\sqrt{\log n}}\) .Our results apply to any \({\ell_p}\) norm for \({2 \leq p \leq \infty}\) with the same approximation factors (except a loss of \({\sqrt{\log \log n}}\) for \({p = \infty}\)). In addition, we show that the output of our algorithm for \({\mathsf{RSP}}\) contains a point whose \({\ell_2}\) distance from \({\mathcal{L}}\) is at least \({(\log n/n)^{1/p} \cdot \rho^{(p)}(\mathcal{L})}\) , where \({\rho^{(p)}(\mathcal{L})}\) is the covering radius of \({\mathcal{L}}\) measured with respect to the \({\ell_p}\) norm. The proof technique involves a theorem on balancing vectors due to Banaszczyk (Random Struct Algorithms 12(4):351–360, 1998) and the “six standard deviations” theorem of Spencer (Trans Am Math Soc 289(2):679–706, 1985).     

Unconditional Superconvergence Analysis for Nonlinear Parabolic Equation with $${\textit{EQ}}_1^{rot}$$ Nonconforming Finite Element

Nonlinear parabolic equation is studied with a linearized Galerkin finite element method. First of all, a time-discrete system is established to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, a rigorous analysis for the regularity of the time-discrete system is presented based on the proof of the temporal error skillfully. On the other hand, the spatial error is derived \(\tau \)-independently with the above achievements. Then, the superclose result of order \(O(h^2+\tau ^2)\) in broken \(H^1\)-norm is deduced without any restriction of \(\tau \). The two typical characters of the \({\textit{EQ}}_1^{rot}\) nonconforming FE (see Lemma 1 below) play an important role in the procedure of proof. At last, numerical results are provided in the last section to confirm the theoretical analysis. Here, h is the subdivision parameter, and \(\tau \), the time step.     

Generic construction of an $$\mathrm {}$$-secure key exchange protocol in the standard model

LaMacchia, Lauter and Mityagin presented a strong security model for authenticated key agreement, namely the \(\mathrm {eCK}\) model. They also constructed a protocol, namely the NAXOS protocol, that enjoys a simple security proof in the \(\mathrm {eCK}\) model. However, the NAXOS protocol uses a random oracle-based technique to combine the long-term secret key and the per session randomness, so-called NAXOS trick, in order to achieve the \(\mathrm {eCK}\) security definition. For NAXOS trick-based protocols, the leakage of per session randomness modeled in the \(\mathrm {eCK}\) model is somewhat unnatural, because the \(\mathrm {eCK}\) model leaks per session randomness, while the output of the NAXOS trick computation remains safe. In this work, we present a standard model \(\mathrm {eCK}\)-secure protocol construction, eliminating the NAXOS trick. Moreover, our protocol is a generic construction, which can be instantiated with arbitrary suitable cryptographic primitives. Thus, we present a generic \(\mathrm {eCK}\)-secure, NAXOS-free, standard model key exchange protocol. To the best of our knowledge this is the first paper on generic transformation of a \(\mathrm {CCA2}\)-secure public-key encryption scheme to an \(\mathrm {eCK}\)-secure key exchange protocol in the standard model.     

New asymmetric quantum codes over $$F_{q}$$

Two families of new asymmetric quantum codes are constructed in this paper. The first family is the asymmetric quantum codes with length \(n=q^{m}-1\) over \(F_{q}\), where \(q\ge 5\) is a prime power. The second one is the asymmetric quantum codes with length \(n=3^{m}-1\). These asymmetric quantum codes are derived from the CSS construction and pairs of nested BCH codes. Moreover, let the defining set \(T_{1}=T_{2}^{-q}\), then the real Z-distance of our asymmetric quantum codes are much larger than \(\delta _\mathrm{max}+1\), where \(\delta _\mathrm{max}\) is the maximal designed distance of dual-containing narrow-sense BCH code, and the parameters presented here have better than the ones available in the literature.     

Mutually unbiased maximally entangled bases in $$\mathbb {C}^d\otimes \mathbb {C}^d$$

We study mutually unbiased maximally entangled bases (MUMEB’s) in bipartite system \(\mathbb {C}^d\otimes \mathbb {C}^d (d \ge 3)\). We generalize the method to construct MUMEB’s given in Tao et al. (Quantum Inf Process 14:2291–2300, 2015), by using any commutative ring R with d elements and generic character of \((R,+)\) instead of \(\mathbb {Z}_d=\mathbb {Z}/d\mathbb {Z}\). Particularly, if \(d=p_1^{a_1}p_2^{a_2}\ldots p_s^{a_s}\) where \(p_1, \ldots , p_s\) are distinct primes and \(3\le p_1^{a_1}\le \cdots \le p_s^{a_s}\), we present \(p_1^{a_1}-1\) MUMEB’s in \(\mathbb {C}^d\otimes \mathbb {C}^d\) by taking \(R=\mathbb {F}_{p_1^{a_1}}\oplus \cdots \oplus \mathbb {F}_{p_s^{a_s}}\), direct sum of finite fields (Theorem 3.3).     

Verified bytecode verification and type-certifying compilation

This article presents a type certifying compiler for a subset of Java and proves the type correctness of the bytecode it generates in the proof assistant Isabelle. The proof is performed by defining a type compiler that emits a type certificate and by showing a correspondence between bytecode and the certificate which entails well-typing. The basis for this work is an extensive formalization of the Java bytecode type system, which is first presented in an abstract, lattice-theoretic setting and then instantiated to Java types.     

Slicing from formal semantics: Chisel—a tool for generic program slicing

We describe Chisel, a tool that synthesizes a program slicer directly from a given algebraic specification of a programming language operational semantics \(\mathcal {S}\). \(\mathcal {S}\) is assumed to be a rewriting logic specification, given in Maude, while the program is a ground term of this specification. Chisel takes \(\mathcal {S}\) and synthesizes language constructs, i.e., instructions, that produce features relevant for slicing, e.g., data dependency. We implement syntheses adjusted to each feature as model checking properties over an abstract representation of \(\mathcal {S}\). The syntheses results are used by a traditional interprocedural slicing algorithm that we parameterize by the synthesized language features. We present the tool on two language paradigms: high-level, imperative and low-level, assembly languages. Computing program slices for these languages allows for extracting traceability properties in standard compilation chains and makes our tool fitting for the validation of embedded system designs. Chisel’s slicing benchmark evaluation is based on benchmarks used in avionics.     

Counter machines and crystallographic structures

One way to depict a crystallographic structure is by a periodic (di)graph, i.e., a graph whose group of automorphisms has a translational subgroup of finite index acting freely on the structure. We establish a relationship between periodic graphs representing crystallographic structures and an infinite hierarchy of intersection languages \(\mathcal {DCL}_d,\,d=0,1,2,\ldots \), within the intersection classes of deterministic context-free languages. We introduce a class of counter machines that accept these languages, where the machines with d counters recognize the class \(\mathcal {DCL}_d\). An intersection of d languages in \(\mathcal {DCL}_1\) defines \(\mathcal {DCL}_d\). We prove that there is a one-to-one correspondence between sets of walks starting and ending in the same unit of a d-dimensional periodic (di)graph and the class of languages in \(\mathcal {DCL}_d\). The proof uses the following result: given a digraph \(\Delta \) and a group G, there is a unique digraph \(\Gamma \) such that \(G\le \mathrm{Aut}\,\Gamma ,\,G\) acts freely on the structure, and \(\Gamma /G \cong \Delta \).     

An improved low-cost yoking proof protocol based on Kazahaya’s flaws

Peris-Lopez et al. (J Netw Comput Appl 34:833–845, 2011) recently provided some guidelines that should be followed to design a secure yoking proof protocol. In addition, conforming to those guidelines and EPC C1-G2, they presented a yoking proof for medical systems based on low-cost RFID tags, named Kazahaya. In this paper, we compromise its security and show how a passive adversary can retrieve secret parameters of a patient’s tag in cost of \(O(2^{16})\) off-line PRNG evaluations. Nevertheless, to show other weaknesses of the protocol and rule out any possible improvement by increasing the length of the used PRNG, we present a forgery attack that proves that a generated proof at time \(t_n\) can be used to forge a valid proof for any desired time \(t_j\). The success probability of this attack is ‘1’ and the complexity is negligible. In addition, we present a new lightweight protocol based on 128-bit PRNG function to solve the problems of Kazahaya protocol. In terms of security, we evaluate the new protocol based on formal and informal methods and prove that the improved protocol is not vulnerable to RFID attacks.     

Unconditionally Optimal Error Analysis of Crank–Nicolson Galerkin FEMs for a Strongly Nonlinear Parabolic System

In this paper, we present unconditionally optimal error estimates of linearized Crank–Nicolson Galerkin finite element methods for a strongly nonlinear parabolic system in \(\mathbb {R}^d\ (d=2,3)\). However, all previous works required certain time-step conditions that were dependent on the spatial mesh size. In order to overcome several entitative difficulties caused by the strong nonlinearity of the system, the proof takes two steps. First, by using a temporal-spatial error splitting argument and a new technique, optimal \(L^2\) error estimates of the numerical schemes can be obtained under the condition \(\tau \ge h\), where \(\tau \) denotes the time-step size and h is the spatial mesh size. Second, we obtain the boundedness of numerical solutions by mathematical induction and inverse inequality when \(\tau \le h\). Then, optimal \(L^2\) and \(H^1\) error estimates are proved in a different way for such case. Numerical results are given to illustrate our theoretical analyses.     

Equivalential Structures for Binary and Ternary Syllogistics

The aim of this paper is to provide a contribution to the natural logic program which explores logics in natural language. The paper offers two logics called \( \mathcal {R}(\forall ,\exists ) \) and \( \mathcal {G}(\forall ,\exists ) \) for dealing with inference involving simple sentences with transitive verbs and ditransitive verbs and quantified noun phrases in subject and object position. With this purpose, the relational logics (without Boolean connectives) are introduced and a model-theoretic proof of decidability for they are presented. In the present paper we develop algebraic semantics (bounded meet semi-lattice) of the logics using congruence theory.     

Exploring Dynamic Impact of Foreign Direct Investment on China’s CO$$_{}$$ Emissions Using Markov-Switching Vector Error Correction Model

The impact of foreign direct investment (FDI) on China’s CO\(_{2}\) emissions is an important index to evaluate the effect of foreign investment policy. This paper uses the monthly data of CO\(_{2}\) emissions and FDI from January 1997 to December 2013 to analyze the regime states, switching probability and regime correlation between FDI and CO\(_{2}\) emissions with the help of nonlinear Markov-switching vector error correction model (MS-VECM), The results indicate that the influence of FDI on CO\(_{2}\) emissions shows the two-regime dynamic characteristics, FDI has played a stimulating role in promoting China’s CO\(_{2}\) emissions during the period from January 1997 to October 2003, while played an inhibiting role during the period from November 2003 to December 2013. The duration of the inhibiting effect of FDI on CO\(_{2}\) emissions is longer, and the frequency is higher than that of the stimulating effect. Therefore, the overall influence of FDI on CO\(_{2}\) emissions during the period from January 1997 to December 2013 is inhibitive, which means FDI has contributed to CO\(_{2}\) emissions reduction. The innovation points of this study are mainly reflected in the following two aspects: first, nonlinear MS-VECM is introduced to dynamically study the relationship between FDI and CO\(_{2}\) emissions in contrast to prior studies that simply use static analysis method; second, the effect of China’s foreign investment policies on CO\(_{2}\) emissions is evaluated in each period according to the empirical results of MS-VECM.     

Tight bounds for parallel randomized load balancing

Given a distributed system of \(n\) balls and \(n\) bins, how evenly can we distribute the balls to the bins, minimizing communication? The fastest non-adaptive and symmetric algorithm achieving a constant maximum bin load requires \(\varTheta (\log \log n)\) rounds, and any such algorithm running for \(r\in {\mathcal {O}}(1)\) rounds incurs a bin load of \(\varOmega ((\log n/\log \log n)^{1/r})\). In this work, we explore the fundamental limits of the general problem. We present a simple adaptive symmetric algorithm that achieves a bin load of 2 in \(\log ^* n+{\mathcal {O}}(1)\) communication rounds using \({\mathcal {O}}(n)\) messages in total. Our main result, however, is a matching lower bound of \((1-o(1))\log ^* n\) on the time complexity of symmetric algorithms that guarantee small bin loads. The essential preconditions of the proof are (i) a limit of \({\mathcal {O}}(n)\) on the total number of messages sent by the algorithm and (ii) anonymity of bins, i.e., the port numberings of balls need not be globally consistent. In order to show that our technique yields indeed tight bounds, we provide for each assumption an algorithm violating it, in turn achieving a constant maximum bin load in constant time.     

Realization of a quantum gate using gravitational search algorithm by perturbing three-dimensional harmonic oscillator with an electromagnetic field

The aim of this paper is to design a current source obtained as a representation of p information symbols \(\{I_k\}\) so that the electromagnetic (EM) field generated interacts with a quantum atomic system producing after a fixed duration T a unitary gate U(T) that is as close as possible to a given unitary gate \(U_g\). The design procedure involves calculating the EM field produced by \(\{I_k\}\) and hence the perturbing Hamiltonian produced by \(\{I_k\}\) finally resulting in the evolution operator produced by \(\{I_k\}\) up to cubic order based on the Dyson series expansion. The gate error energy is thus obtained as a cubic polynomial in \(\{I_k\}\) which is minimized using gravitational search algorithm. The signal to noise ratio (SNR) in the designed gate is higher as compared to that using quadratic Dyson series expansion. The SNR is calculated as the ratio of the Frobenius norm square of the desired gate to that of the desired gate error.