The complexity of disjunction in intuitionistic logic

2020 ◽  
Vol 30 (1) ◽  
pp. 421-445
Author(s):  
R Ramanujam ◽  
Vaishnavi Sundararajan ◽  
S P Suresh
Keyword(s):  
Intuitionistic Logic ◽  
Linear Time ◽  
Proof Rules ◽  
Core Fragment ◽  
Better Than

Abstract We study procedures for the derivability problem of fragments of intuitionistic logic. Intuitionistic logic is known to be PSPACE-complete, with implication being one of the main contributors to this complexity. In fact, with just implication alone, we still have a PSPACE-complete logic. We study fragments of intuitionistic logic with restricted implication and develop algorithms for these fragments which are based on the proof rules. We identify a core fragment whose derivability is solvable in linear time. Adding disjunction elimination to this core gives a logic which is solvable in co-NP. These sub-procedures are applicable to a wide variety of logics with rules of a similar flavour. We also show that we cannot do better than co-NP whenever disjunction elimination interacts with other rules.

FAULT TOLERANT ROUTING IN HYPERCUBES AND STAR GRAPHS

1996 ◽  
Vol 06 (01) ◽  
pp. 127-136 ◽  
Author(s):  
QIAN-PING GU ◽  
SHIETUNG PENG
Keyword(s):  
Free Path ◽  
Fault Tolerant ◽  
Linear Time ◽  
Time Efficiency ◽  
Routing Problem ◽  
Star Graphs ◽  
Linear Time Algorithms ◽  
Better Than

In this paper, we give two linear time algorithms for node-to-node fault tolerant routing problem in n-dimensional hypercubes Hn and star graphs Gn. The first algorithm, given at most n−1 arbitrary fault nodes and two non-fault nodes s and t in Hn, finds a fault-free path s→t of length at most [Formula: see text] in O(n) time, where d(s, t) is the distance between s and t. Our second algorithm, given at most n−2 fault nodes and two non-fault nodes s and t in Gn, finds a fault-free path s→t of length at most d(Gn)+3 in O(n) time, where [Formula: see text] is the diameter of Gn. When the time efficiency of finding the routing path is more important than the length of the path, the algorithms in this paper are better than the previous ones.

How to Keep an Eye on Small Things

2021 ◽  
pp. 1-24
Author(s):  
Bengt J. Nilsson ◽  
Paweł Żyliński
Keyword(s):  
Linear Time ◽  
Simple Polygon ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Approximation Factor ◽  
Fpt Algorithm ◽  
Optimal Linear ◽  
Small Things ◽  
Set Of Points ◽  
Better Than

We present new results on two types of guarding problems for polygons. For the first problem, we present an optimal linear time algorithm for computing a smallest set of points that guard a given shortest path in a simple polygon having [Formula: see text] edges. We also prove that in polygons with holes, there is a constant [Formula: see text] such that no polynomial-time algorithm can solve the problem within an approximation factor of [Formula: see text], unless P=NP. For the second problem, we present a [Formula: see text]-FPT algorithm for computing a shortest tour that sees [Formula: see text] specified points in a polygon with [Formula: see text] holes. We also present a [Formula: see text]-FPT approximation algorithm for this problem having approximation factor [Formula: see text]. In addition, we prove that the general problem cannot be polynomially approximated better than by a factor of [Formula: see text], for some constant [Formula: see text], unless P [Formula: see text]NP.

Linear mixed model better than repeated measures analysis

2019 ◽  
Vol 30 (6) ◽  
pp. NP1-NP2 ◽  
Author(s):  
Işıl Kutluturk Karagoz ◽  
Berhan Keskin ◽  
Flora Özkalaycı ◽  
Ali Karagöz
Keyword(s):  
Mixed Models ◽  
Repeated Measures ◽  
Mixed Model ◽  
Linear Mixed Model ◽  
Linear Time ◽  
Case Series ◽  
Model Fit ◽  
Repeated Measures Analysis ◽  
Technical Issues ◽  
Better Than

We have some criticism regarding some technical issues. Mixed models have begun to play a pivotal role in statistical analyses and offer many advantages over more conventional analyses regarding repeated variance analyses. First, they allow to avoid conducting multiple t-tests; second, they can accommodate for within-patient correlation; third, they allow to incorporate not only a random coefficient, but also a random slope, typically ‘linear’ time in longitudinal case series when there are enough data and patients’ trajectories vary a lot and improving model fit.

Embedding theorems for LTL and its variants

2014 ◽  
Vol 25 (1) ◽  
pp. 83-134 ◽  
Author(s):  
NORIHIRO KAMIDE
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Linear Time ◽  
Cut Elimination ◽  
Embedding Theorems ◽  
Infinitary Logic ◽  
Spatial Logic ◽  
Linear Time Temporal Logic ◽  
Np Complete ◽  
Dynamic Topological Logic

In this paper, we prove some embedding theorems for LTL (linear-time temporal logic) and its variants:viz. some generalisations, extensions and fragments of LTL. Using these embedding theorems, we give uniform proofs of the completeness, cut-elimination and/or decidability theorems for LTL and its variants. The proposed embedding theorems clarify the relationships between some LTL-variations (for example, LTL, a dynamic topological logic, a fixpoint logic, a spatial logic, Prior's logic, Davies' logic and an NP-complete LTL) and some traditional logics (for example, classical logic, intuitionistic logic and infinitary logic).

scholarly journals Acyclic Coloring of Graphs of Maximum Degree $\Delta$

2005 ◽  
Vol DMTCS Proceedings vol. AE,... (Proceedings) ◽  
Author(s):  
Guillaume Fertin ◽  
André Raspaud
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Maximum Degree ◽  
Linear Time ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Acyclic Coloring ◽  
International Audience ◽  
Better Than

International audience An acyclic coloring of a graph $G$ is a coloring of its vertices such that: (i) no two neighbors in $G$ are assigned the same color and (ii) no bicolored cycle can exist in $G$. The acyclic chromatic number of $G$ is the least number of colors necessary to acyclically color $G$, and is denoted by $a(G)$. We show that any graph of maximum degree $\Delta$ has acyclic chromatic number at most $\frac{\Delta (\Delta -1) }{ 2}$ for any $\Delta \geq 5$, and we give an $O(n \Delta^2)$ algorithm to acyclically color any graph of maximum degree $\Delta$ with the above mentioned number of colors. This result is roughly two times better than the best general upper bound known so far, yielding $a(G) \leq \Delta (\Delta -1) +2$. By a deeper study of the case $\Delta =5$, we also show that any graph of maximum degree $5$ can be acyclically colored with at most $9$ colors, and give a linear time algorithm to achieve this bound.

On storage operators

1998 ◽  
Author(s):  
Karim Nour
Keyword(s):  
Intuitionistic Logic ◽  
Classical Logic ◽  
Input Data ◽  
Type System ◽  
Second Order ◽  
Data Type ◽  
Simple Type ◽  
Output Data ◽  
Major Drawback ◽  
Proof Rules

λ-calculus as such is not a computational model. A reduction strategy is needed. In this paper, we consider λ-calculus with the left reduction. This strategy has many advantages: it always terminates when applied to a normalizable λ-term and it seems more economic since we compute λ-term only when we need it. But the major drawback of this strategy is that a function must compute its argument every time it uses it. This is the reason why this strategy is not really used. In 1990 Krivine (1990b) introduced the notion of storage operators in order to avoid this problem and to simulate call-by-value when necessary. The AF2 type system is a way of interpreting the proof rules for second-order intuitionistic logic plus equational reasoning as construction rules for terms. Krivine (1990b) has shown that, by using Gödel translation from classical to intuitionistic logic (denoted byg), we can find in system AF2 a very simple type for storage operators. Historically the type was discovered before the notion of storage operator itself. Krivine (1990a) proved that as far as totality of functions is concerned second-order classical logic is conservative over second-order intuitionistic logic. To prove this, Krivine introduced the following notions: A[x] is an input (resp. output) data type if one can prove intuitionistically A[x] → Ag[x] (resp. Ag[x] → ⇁⇁A[x]). Then if A[x] is an input data type and B[x] is an output data type, then if one can prove A[x] → B[x] classically one can prove it intuitionistically. The notion of storage operator was discovered by investigating the property of all λ-terms of type Ng[x] → ⇁⇁N[x] where N[x] is the type of integers. Parigot (1992) and Krivine (1994) have extended the AF2 system to classical logic. The method of Krivine is very simple: it consists of adding a new constant, denoted by C, with the declaration С: ∀X{⇁⇁ X → X} which axiomatizes classical logic over intuitionistic logic. For the constant C, he adds a new reduction rule which is a particular case of a rule given by Felleisen (1987) for control operator.

scholarly journals Prefix Block-Interchanges on Binary and Ternary Strings

2019 ◽  
Author(s):  
Md. Khaledur Rahman ◽  
M. Sohel Rahman
Keyword(s):  
Upper Bound ◽  
Genome Rearrangement ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Linear Time Algorithm ◽  
Minimum Number ◽  
Binary Strings ◽  
Better Than

AbstractThe genome rearrangement problem computes the minimum number of operations that are required to sort all elements of a permutation. A block-interchange operation exchanges two blocks of a permutation which are not necessarily adjacent and in a prefix block-interchange, one block is always the prefix of that permutation. In this paper, we focus on applying prefix block-interchanges on binary and ternary strings. We present upper bounds to group and sort a given binary/ternary string. We also provide upper bounds for a different version of the block-interchange operation which we refer to as the ‘restricted prefix block-interchange’. We observe that our obtained upper bound for restricted prefix block-interchange operations on binary strings is better than that of other genome rearrangement operations to group fully normalized binary strings. Consequently, we provide a linear-time algorithm to solve the problem of grouping binary normalized strings by restricted prefix block-interchanges. We also provide a polynomial time algorithm to group normalized ternary strings by prefix block-interchange operations. Finally, we provide a classification for ternary strings based on the required number of prefix block-interchange operations.

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