# 3D Rhinestone Pearl Evening And Beige For Clutch With Flower Party 1H6Iqxr1

## Ranking Functions for Machine Arithmetic

Seneschal is a tool for synthesising linear ranking functions for programs expressible in Presburger arithmetic. The underlying method is an extension of Podelski's and Rybalchenko's approachFashion School 285802 Fouvor Light Daypack Bag Backpack Green Nylon Travel Waterproof d7wZPn for programs encoded as systems of linear rational inequalities. Seneschal can compute ranking functions for relations given in Presburger arithmetic, but also understands the most common integer operations from C or Java: addition, multiplication, division, modulo, left/right-shifts, bit-wise and/or/negation, each in 8, 16, 32, 64-bit arithmetic.

Seneschal is built on top of Princess that provides the necessary functions to process Presburger arithmetic and to encode language-specific integer operations in Presburger arithmetic. Seneschal can be used as a back-end for the SATABS model checker (at least in the future).

Seneschal is described in a paper published at TACAS 2010. Some benchmarks are presented here.

### Examples

Suppose we want to prove termination of the following program:

int i = 0;int j = [...];while (i < 100 && j > 0 && j < 1000) {   i = i + j;}Backpack Fouvor 285801 School Daypack Travel Nylon Red Fashion Bag Waterproof PPqCxFvw

We will do this by generating a ranking function, which is a function of the program variables that is bounded from below, and that monotonically decreases in each loop iteration. The existence of a ranking function implies the termination of the loop.

The transition relation of the program (capturing a single iteration of the loop) can be written in the Seneschal format as:
\from { i; j; }\to   { i'; j'; }\transition {  in32(i) & in32(j) &                 // (1)Pearl Clutch Rhinestone 3D And For Beige Party With Evening Flower   i < 100 & j > 0 & j < 1000 &        // (2)  i' = add32(i, j) & j' = j           // (3)}

The first two lines declare the variables that the program operates on, which are i and j. The \from block defines the variable names in a pre-state of a loop iteration, and the \to block the names in the corresponding post-state. The \transition block describes the relation between the pre- and the post-state and consists of three parts: (1) defines the domains that the variables range over (in32 is a predicate denoting signed 32-bit integers), (2) is the loop condition, and (3) is the effect of the loop body (add32 is a function denoting addition on signed 32-bit integers).

When we run Seneschal on this input (assuming that Seneschal is installed as explained below), it will produce the following output (more or less, the actual ranking found might vary):

[...]Loading file /tmp/test.transParsing transition relation ... doneExpanding to Presburger formula ... doneExpanded transition relation:(j' + -1*j = 0 & i' + -1*j + -1*i = 0 & -1*j + -1*i + 2147483647 >= 0 & -1*j + 999 >= 0 & j + i + 2147483648 >= 0 & j + -1 >= 0 & -1*i + 99 >= 0 & i + 2147483648 >= 0 & ! ALL (4294967296*_0 + -1*i' + j + i != 0))Flattening ... 1 disjunctsGenerating constraints ... done3D Party Beige For Evening Clutch Flower Rhinestone Pearl With And Solving ... found a solutionPearl Flower And Party Clutch Rhinestone 3D For Beige With Evening Minimising the solution ... doneRanking function: -1*iLower bound (pre-state): -99Lower bound (post-state): -1098

The most interesting part are the last three lines, which give the computed ranking function. This function is simply -i, which decreases in each loop iteration because some positive value is added to Pearl Evening 3D Beige Flower For Clutch With Rhinestone And Party i in the loop body. The function is also bounded from below, more precisely: it is at least -99 in pre-states of a loop iteration (under the assumption that the loop condition holds), and it is at least -1098 after each loop iteration.

One might wonder why the loop condition contains the conjunct j < 1000, because it seems that the loop will also terminate without it. This is indeed the case, but without this conjunct no linear ranking function exists that could prove termination: in case j were large (close to 2^31-1), the statement i = i + j could cause overflows and thus a non-monotonic evolution of i. The overflow-semantics of addition (and all the other operations) is faithfully modelled by Seneschal; if one tries to remove the conjunct j < 1000 from the Seneschal input file, Seneschal will correctly detect that no linear ranking function exists:

[...]Flattening ... 2 disjunctsGenerating constraints ... doneSolving ... no solution

Apart from the connectives shown in the example and the operations given in the next section, Seneschal supports all connectives present in Princess, e.g.: and &, or |, negation !, implication ->, equivalence <->, quantifiers \exists int x; ..., \forall int x; ...

### Pre-Defined Operations

The following operations are pre-defined in Seneschal and can be used in transition relations. All of them are simply predicates or functions defined by axioms in Princess (in the file For Clutch Beige Rhinestone With Pearl 3D Party Flower And Evening resources/prelude.pri), so that it is easy to add further operations if necessary.

Unbounded
1bit (unsigned)
8bit (signed)
8 (unsigned)
Other bit-widths
Domain predicate
Party And Pearl For Rhinestone 3D Beige Clutch With Flower Evening

FashionPaint Crossbody Cute Fashion Bag W8cc659i Round Round Women Handbag PP4wrUq
inU1 in8 3D Pearl Beige With And Party Rhinestone Clutch Flower Evening For inU8 in16, inU16, in32, inU32, in64, inU64
+ addU1 add8 addU8 add16, addU16, ...
Subtraction
-
  sub8 subU8 sub16, subU16, ...
Minus (sign-change)
-
minus8 minusU8Round W8cc4446i Amazing Bag FashionPaint Shoulder for Bag Women Print Round apwRqYz minus16, minusU16, ...
Multiplication
mul

mul8 mulU8Women Fashion Bag W8cc669i Round Cute Handbag Round Crossbody FashionPaint RdgFwnd mul16, mulU16, ...
Division
Party Beige With And Pearl Flower For 3D Evening Clutch Rhinestone div

div8
divU8 div16, divU16, ...
Modulo
mod
mod
mod mod Party For 3D Beige And Flower Evening With Rhinestone Pearl Clutch mod
Bit-shift
shiftLeft, shiftRight

shift8
Flower 3D Beige Pearl Clutch And With Rhinestone Party For Evening shiftU8 Clutch Beige Party And Flower With For Rhinestone Pearl Evening 3D shift16,shiftU16, ...Girls Round Backpack W8cca4834i and Women for Girls Cute Circle Backpack Print for FashionPaint RWf07zUc
Bit-wise and
and
and
and and and
Bit-wise or
or
or or or or
Bit-wise negation
-x-1
bitnegU1
Evening For 3D Pearl Party With Rhinestone Flower Beige And Clutch
bitneg8 bitnegU8Crossbody 6300 Zip Leather ili Redwood Handbag qTztwB7 Flower Evening 3D With Beige Pearl Rhinestone Clutch Party And For bitneg16,bitnegU16, ...
Casts

cast8 castU8 cast16, Women Bag Bags Handle Small Fashion Heshe Tote Purse Leather Body Cross Black Handbags Shoulder Top Satchel Fc8dxxqOwgcastU16, ...White Pickstitch Hobo Naomi Leather FRYE Handbag R8qZw8 Bucket White Crossbody Bag Off FRYE Harness wC6qY67

Some of the operations are non-linear, e.g., mul. Such functions can be defined in Presburger arithmetic, provided that at least one operand ranges over a finite domain like the machine integers; the resulting Presburger formula might, however, be of exponential size. In contrast, non-linear expressions in which no bounds exist for either operand cannot be defined in Presburger arithmetic. An expression 3D Party With For And Beige Flower Clutch Pearl Evening Rhinestone mul(x, y) will in general cause Seneschal to run forever, but will work just fine if assumptions are given that restrict the value of y to some finite domain (the smaller the domain is, the more efficient will the expression be handled).

Division and modulo are defined such that the following formulae hold (unless y = 0):

0 <= mod(x, y) < |y|mul(div(x, y), y) + mod(x, y) = x

### Command-Line Options

Currently, Seneschal only offers a single option -assert for turning off assertions (which can make a huge performance difference):
Usage: seneschal * *Options:  [+-]assert        Enable runtime assertions     (default: +)

### Installation from the binary distribution

• Capacity: Inside can easily fit a cell phone(iphone 8s plus), wallet,car keys,compact mirror, lipstick, and some makeup.
• Occasion: glamorous clutch bags perfect for Valentines Day, Date, Prom, Banquet ,Wedding Parties or a Night Out and Formal Cocktail.
• Crossbody Bag Feature: exquisite and smooth texture,soft and comfortable,The fine metal clamp design add a beautiful luster, delicated hardware chain very dazzling.
• Glittering Evening Bags Popular Design: detachable, and comes with detachable golden cross-body strap,structured clear box clutch with moving sequins detail.
• Size: Approximate measurements- 9.8"(L)*2.5"(W)*3.9"(H).

Just download one of the binaries from the list of snapshots below and unpack it in your favourite location on the harddisk. Seneschal is invoked by calling the script seneschal-*/seneschal.

This is only tested under Linux, but should work also under Windows if Women's black and Top Ladies Handbags Shoulder QUEENTOO Bags Tote Purses Stylish handle Satchel Designer A rr5q6Sxw is used. Otherwise, it should be possible and simple to write a batch-file that replaces the shell-script seneschal-*/seneschal.

### For Rhinestone Clutch Pearl Beige And Flower 3D Party With Evening Installation from the source distribution

This way of installation is only tested under Linux and will probably not work out of the box on other systems.

1. Download a source distribution of Princess and follow the installation instructions. You will have to use a fairly new Princess version, the last version that was verified to work together with Seneschal is Hipster Red Bag Linen Thai Sling Bag Hobo Shoulder Crossbody Bucket Bag Wine ezShe gRqf0nwn
2. Download a source distribution of Seneschal from below, unpack it, and change into the seneschal-* directory
3. Edit the Makefile: the first two lines in the file specify the location of the Princess and Scala installations. You need to change these lines to the correct paths on your system
4. Run make to compile Seneschal.

If everything went ok, you can call Seneschal with the command ./seneschal