Sound Proof System Research Articles

Evolutionary topology optimization approach to design multiphase soundproof systems with poroelastic media

SummaryWith the constant development of cities, noise sources have become increasingly present inside and outside living environments. Consequently, soundproof systems comprised of porous materials have been widely adopted as filling fabric of closed‐space structures, such as in the components of buildings, airplanes or automobiles. However, in many situations, simply filling spaces may not be the most effective approach. In that scenario, this work introduces a multiphase acoustic topology optimization methodology to design closed‐space structures for sound attenuation. Based on the Bi‐directional Evolutionary Structural Optimization (BESO) algorithm, the proposed approach combines Biot's poroelasticity equations, expressed in the mixed /p form, and the Unified Multiphase (UMP) technique to fully describe the multiphysics involved in the acoustic, poroelastic and elastic model relations. An objective function contemplating different combinations of structural, viscous and thermal dissipated powers is maximized over multiple frequencies. Volume constraints in each material phase and a novel material interpolation scheme are also considered. The resultant topologies present enhanced dissipated power levels and manufacturability, even when compared with various baseline configurations of similar volume fractions.

Web Consequence Untangled

AbstractUnder the standard modal explication of consequence, a conclusion is a consequence of some premises just in case necessarily, if the latter are true, so is the former. Notoriously, this explication yields some results that at first glance are counter-intuitive. In particular, a necessary truth is a consequence of arbitrary premises, and premises that cannot all be true together entail arbitrary conclusions. In his paper ‘On Ground and Consequence’ (Synthese, 2021), Benjamin Schnieder introduces a novel notion of web consequence, defined on the basis of the concept of ground, which, he argues, fits our intuitive conception better. Building on his idea, the present paper examines the concept of web consequence in more detail. In particular, I provide three alternative, and simpler semantic characterizations of Schnieder’s propositional logic of web consequence, two within a form of truthmaker semantics, one within a many-valued setting. I then consider some natural variations on that logic and establish their connections to well-known subclassical logics such as FDE, $$\hbox {K}_3$$ K 3 , and LP. Finally, I provide sound and complete tableaux-based proof systems for each of the logics of web consequence so obtained.

Sound-based Two-factor Authentication: Vulnerabilities and Redesign

Reducing the level of user effort involved in traditional two-factor authentication (TFA) constitutes an important research topic. An interesting representative approach, Sound-Proof , leverages ambient sounds to detect the proximity between the second-factor device (phone) and the login terminal (browser), and it eliminates the need for the user to transfer PIN codes. In this article, we identify a weakness of the Sound-Proof system that makes it completely vulnerable to passive “environment guessing” and active “environment manipulating” remote attackers and proximity attackers. Addressing these security issues, we propose Listening-Watch , a new TFA mechanism based on a wearable device (watch/bracelet) and active browser-generated random speech sounds. As the user attempts to log in, the browser populates a short random code encoded into speech, and the login succeeds if the watch’s audio recording contains this code (decoded using speech recognition ) and is similar enough to the browser’s audio recording. The remote attacker, who has guessed/manipulated the user’s environment, will be defeated, since authentication success relies upon the presence of the random code in watch’s recordings. The proximity attacker will also be defeated unless it is extremely close (<50 cm) to the watch, since the wearable microphones are usually designed to capture only nearby sounds (e.g., voice commands).

Monotonicity and the Precision of Program Analysis

It is widely known that the precision of a program analyzer is closely related to intensional program properties, namely, properties concerning how the program is written. This explains, for instance, the interest in code obfuscation techniques, namely, tools explicitly designed to degrade the results of program analysis by operating syntactic program transformations. Less is known about a possible relation between what the program extensionally computes, namely, its input-output relation, and the precision of a program analyzer. In this paper we explore this potential connection in an effort to isolate program fragments that can be precisely analyzed by abstract interpretation, namely, programs for which there exists a complete abstract interpretation. In the field of static inference of numeric invariants, this happens for programs, or parts of programs, that manifest a monotone (either non-decreasing or non-increasing) behavior. We first formalize the notion of program monotonicity with respect to a given input and a set of numerical variables of interest. A sound proof system is then introduced with judgments specifying whether a program is monotone relatively to a set of variables and a set of inputs. The interest in monotonicity is justified because we prove that the family of monotone programs admits a complete abstract interpretation over a specific class of non-trivial numerical abstractions and inputs. This class includes all non-relational abstract domains that refine interval analysis (i.e., at least as precise as the intervals abstraction) and that satisfy a topological convexity hypothesis.

Calculational Design of [In]Correctness Transformational Program Logics by Abstract Interpretation

We study transformational program logics for correctness and incorrectness that we extend to explicitly handle both termination and nontermination. We show that the logics are abstract interpretations of the right image transformer for a natural relational semantics covering both finite and infinite executions. This understanding of logics as abstractions of a semantics facilitates their comparisons through their respective abstractions of the semantics (rather that the much more difficult comparison through their formal proof systems). More importantly, the formalization provides a calculational method for constructively designing the sound and complete formal proof system by abstraction of the semantics. As an example, we extend Hoare logic to cover all possible behaviors of nondeterministic programs and design a new precondition (in)correctness logic.

Orthologic with Axioms

We study the proof theory and algorithms for orthologic, a logical system based on ortholattices, which have shown practical relevance in simplification and normalization of verification conditions. Ortholattices weaken Boolean algebras while having polynomial-time equivalence checking that is sound with respect to Boolean algebra semantics. We generalize ortholattice reasoning and obtain an algorithm for proving a larger class of classically valid formulas. As the key result, we analyze a proof system for orthologic augmented with axioms. An important feature of the system is that it limits the number of formulas in a sequent to at most two, which makes the extension with axioms non-trivial. We show a generalized form of cut elimination for this system, which implies a sub-formula property. From there we derive a cubic-time algorithm for provability from axioms, or equivalently, for validity in finitely presented ortholattices. We further show that propositional resolution of width 5 proves all formulas provable in orthologic with axioms. We show that orthologic system subsumes resolution of width 2 and arbitrarily wide unit resolution and is complete for reasoning about generalizations of propositional Horn clauses. Moving beyond ground axioms, we introduce effectively propositional orthologic (by analogy with EPR for classical logic), presenting its semantics as well as a sound and complete proof system. Our proof system implies the decidability of effectively propositional orthologic, as well as its fixed-parameter tractability for a bounded maximal number of variables in each axiom. As a special case, we obtain a generalization of Datalog with negation and disjunction.

Morpho-logic from a topos perspective – application to symbolic AI

Modal logics have proved useful for many reasoning tasks in symbolic artificial intelligence (AI), such as belief revision, spatial reasoning, among others. On the other hand, mathematical morphology (MM) is a theory for non-linear analysis of structures, that was widely developed and applied in image analysis. Its mathematical bases rely on algebra, complete lattices, topology. Strong links have been established between MM and mathematical logics, mostly modal logics. In this paper, we propose to further develop and generalize this link between mathematical morphology and modal logic from a topos perspective, i.e. categorial structures generalizing space, and connecting logics, sets and topology. Furthermore, we rely on the internal language and logic of a topos. We define structuring elements, dilations and erosions as morphisms. Then we introduce the notion of structuring neighborhoods, and show that the dilations and erosions based on them lead to a constructive modal logic, for which a sound and complete proof system is proposed. We then show that the modal logic thus defined (called morpho-logic here), is well adapted to define concrete and efficient operators for revision, merging, and abduction of new knowledge, or even spatial reasoning.

Self-adjointness of sound-proof models for magnetic buoyancy

An ideal magneto-hydrodynamic fluid, whether fully compressible or incompressible, is a Hamiltonian system. This implies that the equations describing perturbations to any static state are self-adjoint, a fact that is useful in obtaining stability criteria. To describe weakly compressible flows, there are a number of “sound-proof” models that eliminate sound waves by making approximations to the governing equations. However, such approximations may violate the Hamiltonian structure of the system. In a recent work, we have introduced a very general sound-proof model and determined conditions under which it closely approximates the linear regime of magneto-buoyancy instability, motivated by conditions in the solar interior. In the present work, we take a complementary approach, by deriving constraints under which the linearised sound-proof system is self-adjoint. We show that there is a unique set of self-adjoint sound-proof equations that conserves the same energy as the fully compressible system.

Towards logical foundations for probabilistic computation

The overall purpose of the present work is to lay the foundations for a new approach to bridge logic and probabilistic computation. To this aim we introduce extensions of classical and intuitionistic propositional logic with counting quantifiers, that is, quantifiers that measure to which extent a formula is true. The resulting systems, called cCPL and iCPL, respectively, admit a natural semantics, based on the Borel σ-algebra of the Cantor space, together with a sound and complete proof system. Our main results consist in relating cCPL and iCPL with some central concepts in the study of probabilistic computation. On the one hand, the validity of cCPL-formulae in prenex form characterizes the corresponding level of Wagner's hierarchy of counting complexity classes, closely related to probabilistic complexity. On the other hand, proofs in iCPL correspond, in the sense of Curry and Howard, to typing derivations for a randomized extension of the λ-calculus, so that counting quantifiers reveal the probability of termination of the underlying probabilistic programs.

Logics for Epistemic Actions: Completeness, Decidability, Expressivity

We build and study dynamic versions of epistemic logic. We study languages parameterized by an action signature that allows one to express epistemic actions such as (truthful) public announcements, completely private announcements to groups of agents, and more. The language 𝓛(Σ) is modeled on dynamic logic. Its sentence-building operations include modalities for the execution of programs, and for knowledge and common knowledge. Its program-building operations include action execution, composition, repetition, and choice. We consider two fragments of 𝓛(Σ). In 𝓛1(Σ), we drop action repetition; in 𝓛0(Σ), we also drop common knowledge. We present the syntax and semantics of these languages and sound proof systems for the validities in them. We prove the strong completeness of a logical system for 𝓛0(Σ) and the weak completeness of one for 𝓛1(Σ). We show the finite model property and, hence, decidability of 𝓛1(Σ). We translate 𝓛1(Σ) into PDL, obtaining a second proof of decidability. We prove results on expressive power, comparing 𝓛1(Σ) with modal logic together with transitive closure operators. We prove that a logical language with operators for private announcements is more expressive than one for public announcements.

On counting propositional logic and Wagner's hierarchy

We introduce an extension of classical propositional logic with counting quantifiers. These forms of quantification make it possible to express that a formula is true in a certain portion of the set of all its interpretations. Beyond providing a sound and complete proof system for this logic, we show that validity problems for counting propositional logic can be used to capture counting complexity classes. More precisely, we show that the complexity of the decision problems for validity of prenex counting formulas perfectly matches the appropriate levels of Wagner's counting hierarchy.

Understanding the Relative Strength of QBF CDCL Solvers and QBF Resolution

QBF solvers implementing the QCDCL paradigm are powerful algorithms that successfully tackle many computationally complex applications. However, our theoretical understanding of the strength and limitations of these QCDCL solvers is very limited. In this paper we suggest to formally model QCDCL solvers as proof systems. We define different policies that can be used for decision heuristics and unit propagation and give rise to a number of sound and complete QBF proof systems (and hence new QCDCL algorithms). With respect to the standard policies used in practical QCDCL solving, we show that the corresponding QCDCL proof system is incomparable (via exponential separations) to Q-resolution, the classical QBF resolution system used in the literature. This is in stark contrast to the propositional setting where CDCL and resolution are known to be p-equivalent. This raises the question what formulas are hard for standard QCDCL, since Q-resolution lower bounds do not necessarily apply to QCDCL as we show here. In answer to this question we prove several lower bounds for QCDCL, including exponential lower bounds for a large class of random QBFs. We also introduce a strengthening of the decision heuristic used in classical QCDCL, which does not necessarily decide variables in order of the prefix, but still allows to learn asserting clauses. We show that with this decision policy, QCDCL can be exponentially faster on some formulas. We further exhibit a QCDCL proof system that is p-equivalent to Q-resolution. In comparison to classical QCDCL, this new QCDCL version adapts both decision and unit propagation policies.

A Completeness Proof for a Regular Predicate Logic with Undefined Truth Value

We provide a sound and complete proof system for an extension of Kleene’s ternary logic to predicates. The concept of theory is extended with, for each function symbol, a formula that specifies when the function is defined. The notion of “is defined” is extended to terms and formulas via a straightforward recursive algorithm. The “is defined” formulas are constructed so that they themselves are always defined. The completeness proof relies on the Henkin construction. For each formula, precisely one of the formula, its negation, and the negation of its “is defined” formula is true on the constructed model. Many other ternary logics in the literature can be reduced to ours. Partial functions are ubiquitous in computer science and even in (in)equation solving at schools. Our work was motivated by an attempt to precisely explain, in terms of logic, typical informal methods of reasoning in such applications.

The Quantified Argument Calculus with Two- and Three-valued Truth-valuational Semantics

We introduce a two-valued and a three-valued truth-valuational substitutional semantics for the Quantified Argument Calculus (Quarc). We then prove that the 2-valid arguments are identical to the 3-valid ones with strict-to-tolerant validity. Next, we introduce a Lemmon-style Natural Deduction system and prove the completeness of Quarc on both two- and three-valued versions, adapting Lindenbaum’s Lemma to truth-valuational semantics. We proceed to investigate the relations of three-valued Quarc and the Predicate Calculus (PC). Adding a logical predicate T to Quarc, true of all singular arguments, allows us to represent PC quantification in Quarc and translate PC into Quarc, preserving validity. Introducing a weak existential quantifier into PC allows us to translate Quarc into PC, also preserving validity. However, unlike the translated systems, neither extended system can have a sound and complete proof system with Cut, supporting the claim that these are basically different calculi.

A separation logic for negative dependence

Formal reasoning about hashing-based probabilistic data structures often requires reasoning about random variables where when one variable gets larger (such as the number of elements hashed into one bucket), the others tend to be smaller (like the number of elements hashed into the other buckets). This is an example of negative dependence , a generalization of probabilistic independence that has recently found interesting applications in algorithm design and machine learning. Despite the usefulness of negative dependence for the analyses of probabilistic data structures, existing verification methods cannot establish this property for randomized programs. To fill this gap, we design LINA, a probabilistic separation logic for reasoning about negative dependence. Following recent works on probabilistic separation logic using separating conjunction to reason about the probabilistic independence of random variables, we use separating conjunction to reason about negative dependence. Our assertion logic features two separating conjunctions, one for independence and one for negative dependence. We generalize the logic of bunched implications (BI) to support multiple separating conjunctions, and provide a sound and complete proof system. Notably, the semantics for separating conjunction relies on a non-deterministic , rather than partial, operation for combining resources. By drawing on closure properties for negative dependence, our program logic supports a Frame-like rule for negative dependence and monotone operations. We demonstrate how LINA can verify probabilistic properties of hash-based data structures and balls-into-bins processes.

Sequent Calculi for the Propositional Logic of HYPE

In this paper we discuss sequent calculi for the propositional fragment of the logic of HYPE. The logic of HYPE was recently suggested by Leitgeb (Journal of Philosophical Logic 48:305–405, 2019) as a logic for hyperintensional contexts. On the one hand we introduce a simple mathbf{G1}-system employing rules of contraposition. On the other hand we present a mathbf{G3}-system with an admissible rule of contraposition. Both systems are equivalent as well as sound and complete proof-system of HYPE. In order to provide a cut-elimination procedure, we expand the calculus by connections as introduced in Kashima and Shimura (Mathematical Logic Quarterly 40:153–172, 1994).

Implementation of a semi-Lagrangian fully-implicit time integration of the unified soundproof system of equations for numerical weather prediction

AbstractAn alternate dynamical core that employs the unified equations of A. Arakawa and C.S. Konor (2009) has been developed within Environment and Climate change Canada’s GEM (Global Environmental Multiscale) atmospheric model. As in the operational GEM dynamical core, the novel core utilizes the same fully-implicit two-time-level semi-Lagrangian scheme for time discretization while the log-pressure-based terrain-following vertical coordinate has been slightly adapted. Overall, the new dynamical core implementation required only minor changes to the existing informatics code of the GEM model and from a computational performance perspective, the new core does not incur any significant additional cost. A broad range of tests – that include both two-dimensional idealized theoretical cases and three-dimensional deterministic forecasts covering both hydrostatic and non-hydrostatic scales–have been carried out to evaluate the performance of the new dynamical core. For all the tested cases, when compared to the operational GEM model, the new dynamical core based on the unified equations has been found to produce statistically equivalent results. These results imply that the unified equations can be adopted for operational numerical weather prediction that would employ a single soundproof system of equations to produce reliable forecasts for all meteorological scales of interest with negligible changes for the computational overhead.

The logic of gossiping

The so-called gossip problem is a formal model of peer-to-peer communication. In order to perform such communication efficiently, it is important to keep track of what agents know about who holds what information at a given point in time. The knowledge that the agents possess depends strongly on the particular type of communication that is used.Here, we formally define a large number of different variants of the gossip problem, that differ in the extent to which communication is private (observable, synchronous or asynchronous), the direction of the flow of information (caller to callee, callee to caller or both) and whether the agents become aware of the exact set of information possessed by their communication partner.We consider a number of formulas that represent interesting properties that a gossip situation may or may not enjoy, and show for which variants they are valid. Additionally, we show that the model checking and validity checking problems for each variant are decidable, and we introduce sound and complete proof systems for them.

A sound and complete proof system for a unified temporal logic

Theorem proving is one of the most widely used approaches to the verification of computer systems, and its theoretical basis is generally a proof system for formal derivation of logic formulas. In this paper, we propose a sound and complete proof system for Propositional Projection Temporal Logic (PPTL) with indexed expressions, which is a unified temporal logic that subsumes the well used Linear Temporal Logic (LTL). First, the syntax, semantics and logic laws of PPTL that allows indexed expressions are introduced, and the representation of LTL constructs by PPTL formulas is illustrated. Then, the proof system for the logic is presented which consists of axioms and inference rules for the derivation of both basic constructs and indexed expressions. To show the capability of the proof system, several examples of formal proofs are provided. Finally, the soundness and completeness of the proof system are demonstrated.

Logics for Sizes with Union or Intersection

This paper presents the most basic logics for reasoning about the sizes of sets that admit either the union of terms or the intersection of terms. That is, our logics handle assertions All x y and AtLeast x y, where x and y are built up from basic terms by either unions or intersections. We present a sound, complete, and polynomial-time decidable proof system for these logics. An immediate consequence of our work is the completeness of the logic additionally permitting More x y. The logics considered here may be viewed as efficient fragments of two logics which appear in the literature: Boolean Algebra with Presburger Arithmetic and the Logic of Comparative Cardinality.