scholarly journals Unifying Decidable Entailments in Separation Logic with Inductive Definitions

2021 ◽  
pp. 183-199
Author(s):  
Mnacho Echenim ◽  
Radu Iosif ◽  
Nicolas Peltier
Keyword(s):  
Separation Logic ◽  
Restricted Class ◽  
Complete Class ◽  
Inductive Definitions ◽  
Decidable Classes ◽  
Finite Set ◽  
The Right

AbstractThe entailment problem $$\upvarphi \models \uppsi $$ φ ⊧ ψ in Separation Logic [12, 15], between separated conjunctions of equational ($$x \approx y$$ x ≈ y and $$x \not \approx y$$ x ≉ y ), spatial ($$x \mapsto (y_1,\ldots ,y_\upkappa )$$ x ↦ ( y 1 , … , y κ ) ) and predicate ($$p(x_1,\ldots ,x_n)$$ p ( x 1 , … , x n ) ) atoms, interpreted by a finite set of inductive rules, is undecidable in general. Certain restrictions on the set of inductive definitions lead to decidable classes of entailment problems. Currently, there are two such decidable classes, based on two restrictions, called establishment [10, 13, 14] and restrictedness [8], respectively. Both classes are shown to be in $$\mathsf {2\text {EXPTIME}}$$ 2 EXPTIME by the independent proofs from [14] and [8], respectively, and a many-one reduction of established to restricted entailment problems has been given [8]. In this paper, we strictly generalize the restricted class, by distinguishing the conditions that apply only to the left- ($$\upvarphi $$ φ ) and the right- ($$\uppsi $$ ψ ) hand side of entailments, respectively. We provide a many-one reduction of this generalized class, called safe, to the established class. Together with the reduction of established to restricted entailment problems, this new reduction closes the loop and shows that the three classes of entailment problems (respectively established, restricted and safe) form a single, unified, $$\mathsf {2\text {EXPTIME}}$$ 2 EXPTIME -complete class.

Navigating on the Information Highway

1994 ◽  
Vol 38 (4) ◽  
pp. 271-274 ◽  
Author(s):  
Arnold M. Lund
Keyword(s):  
Design Issues ◽  
Information Highway ◽  
Human Interface ◽  
Practical Design ◽  
Finite Set ◽  
Critical Problems ◽  
The Right ◽  
Time Frames ◽  
Theoretical Issues ◽  
Design Guidance

The “information highway” is being created now, and the design task is monumental. Some of the most critical problems in designing a navigation environment have never been addressed before, and developers are demanding that either we (as human interface designers) supply answers now or they will make up their own answers. There is too little time and there are too few resources to fully explore interesting theoretical issues, and yet the results of such exploration (if on the right topics) could be critically important. This paper will review some of the practical design issues that Ameritech has identified as being important in defining a navigation environment, and approaches we have taken to those design issues. These approaches have been shaped by a lack of design guidance in some areas, and a finite set of resources (especially time). The paper will also identify directions where research would be useful, and time frames during which the results will still be valuable.

Implicational formulas in intuitionistic logic

1974 ◽  
Vol 39 (4) ◽  
pp. 661-664 ◽  
Author(s):  
Alasdair Urquhart
Keyword(s):  
Propositional Calculus ◽  
Modus Ponens ◽  
Equivalence Classes ◽  
Alternative Proof ◽  
Hilbert Algebra ◽  
Hilbert Algebras ◽  
Free Generators ◽  
Finite Set ◽  
Image Position ◽  
The Right

In [1] Diego showed that there are only finitely many nonequivalent formulas in n variables in the positive implicational propositional calculus P. He also gave a recursive construction of the corresponding algebra of formulas, the free Hilbert algebra In on n free generators. In the present paper we give an alternative proof of the finiteness of In, and another construction of free Hilbert algebras, yielding a normal form for implicational formulas. The main new result is that In is built up from n copies of a finite Boolean algebra. The proofs use Kripke models [2] rather than the algebraic techniques of [1].Let V be a finite set of propositional variables, and let F(V) be the set of all formulas built up from V ⋃ {t} using → alone. The algebra defined on the equivalence classes , by settingis a free Hilbert algebra I(V) on the free generators . A set T ⊆ F(V) is a theory if ⊦pA implies A ∈ T, and T is closed under modus ponens. For T a theory, T[A] is the theory {B ∣ A → B ∈ T}. A theory T is p-prime, where p ∈ V, if p ∉ T and, for any A ∈ F(V), A ∈ T or A → p ∈ T. A theory is prime if it is p-prime for some p. Pp(V) denotes the set of p-prime theories in F(V), P(V) the set of prime theories. T ∈ P(V) is minimal if there is no theory in P(V) strictly contained in T. Where X = {A1, …, An} is a finite set of formulas, let X → B be A1 →····→·An → B (ϕ → B is B). A formula A is a p-formula if p is the right-most variable occurring in A, i.e. if A is of the form X → p.

An Algorithmic Solution for a Word Problem in Group Theory

1964 ◽  
Vol 16 ◽  
pp. 509-516 ◽  
Author(s):  
N. S. Mendelsohn
Keyword(s):  
Group Theory ◽  
Finite Number ◽  
Word Problem ◽  
Finite Index ◽  
Unit Element ◽  
Systematic Procedure ◽  
Algorithmic Solution ◽  
Finite Set ◽  
The Right

This paper describes a systematic procedure which yields in a finite number of steps a solution to the following problem. Let G be a group generated by a finite set of generators g1, g2, g3, . . . , gr and defined by a finite set of relations R1 = R2 = . . . = Rk = I, where I is the unit element of G and R1R2, . . . , Rk are words in the gi and gi-1. Let H be a subgroup of G, known to be of finite index, and generated by a finite set of words, W1, W2, . . . , Wt. Let W be any word in G. Our problem is the following. Can we find a new set of generators for H, together with a set of representatives h1 = 1, h2, . . . , hu of the right cosets of H (i.e. G = H1 + Hh2 + . . . + Hhu) such that W can be expressed in the form W = Uhp, where U is a word in .

Exponentially Convergent Parallel Discretization Methods for the First Order Evolution Equations

2001 ◽  
Vol 1 (4) ◽  
pp. 333-355 ◽  
Author(s):  
Ivan Gavrilyuk ◽  
Vladimir L. Makarov
Keyword(s):  
Evolution Equations ◽  
Operator Coefficient ◽  
Cauchy Integral ◽  
Initial Value ◽  
Discretization Methods ◽  
First Order ◽  
Finite Set ◽  
The Right ◽  
Sinc Quadrature ◽  
Complex Coefficients

AbstractWe propose a new discretization of an initial value problem for differen- tial equations of the first order in a Banach space with a strongly P-positive operator coefficient. Using the strong positiveness, we represent the solution as a Dunford- Cauchy integral along a parabola in the right half of the complex plane, then transform it into real integrals over (−∞,∞), and finally apply an exponentially convergent Sinc quadrature formula to this integral. The integrand values are the solutions of a finite set of elliptic problems with complex coefficients, which are independent and may be solved in parallel.

scholarly journals Attack on Kayawood protocol: uncloaking private keys

2020 ◽  
Vol 15 (1) ◽  
pp. 237-249
Author(s):  
Matvei Kotov ◽  
Anton Menshov ◽  
Alexander Ushakov
Keyword(s):  
Heuristic Algorithm ◽  
Braid Group ◽  
Key Agreement ◽  
Private Key ◽  
The Core ◽  
Key Agreement Protocol ◽  
Finite Set ◽  
Security Properties ◽  
The Right ◽  
Parameter Values

AbstractWe analyze security properties of a two-party key-agreement protocol recently proposed by I. Anshel, D. Atkins, D. Goldfeld, and P. Gunnels, called Kayawood protocol. At the core of the protocol is an action (called E-multiplication) of a braid group on some finite set. The protocol assigns a secret element of a braid group to each party (private key). To disguise those elements, the protocol uses a so-called cloaking method that multiplies private keys on the left and on the right by specially designed elements (stabilizers for E-multiplication).We present a heuristic algorithm that allows a passive eavesdropper to recover Alice’s private key by removing cloaking elements. Our attack has 100% success rate on randomly generated instances of the protocol for the originally proposed parameter values and for recent proposals that suggest to insert many cloaking elements at random positions of the private key. Implementation of the attack is available on GitHub.

scholarly journals Entailment Checking in Separation Logic with Inductive Definitions is 2-EXPTIME hard

10.29007/f5wh ◽  
2020 ◽  
Author(s):  
Mnacho Echenim ◽  
Radu Iosif ◽  
Nicolas Peltier
Keyword(s):  
Lower Bound ◽  
Large Class ◽  
Separation Logic ◽  
Membership Problem ◽  
Turing Machines ◽  
Inductive Definitions ◽  
Alternating Turing Machines ◽  
Exponential Space

The entailment between separation logic formulæ with inductive predicates, also known as sym- bolic heaps, has been shown to be decidable for a large class of inductive definitions [7]. Recently, a 2-EXPTIME algorithm was proposed [10, 14] and an EXPTIME-hard bound was established in [8]; however no precise lower bound is known. In this paper, we show that deciding entailment between predicate atoms is 2-EXPTIME-hard. The proof is based on a reduction from the membership problem for exponential-space bounded alternating Turing machines [5].

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