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Type Compatibility

Type compatibility in TypeScript is based on structural subtyping. Structural typing is a way of relating types based solely on their members. This is in contrast with nominal typing. Consider the following code:

In nominally-typed languages like C# or Java, the equivalent code would be an error because the Dog class does not explicitly describe itself as being an implementer of the Pet interface.

TypeScript’s structural type system was designed based on how JavaScript code is typically written. Because JavaScript widely uses anonymous objects like function expressions and object literals, it’s much more natural to represent the kinds of relationships found in JavaScript libraries with a structural type system instead of a nominal one.

A Note on Soundness

TypeScript’s type system allows certain operations that can’t be known at compile-time to be safe. When a type system has this property, it is said to not be “sound”. The places where TypeScript allows unsound behavior were carefully considered, and throughout this document we’ll explain where these happen and the motivating scenarios behind them.

Starting out

The basic rule for TypeScript’s structural type system is that x is compatible with y if y has at least the same members as x . For example consider the following code involving an interface named Pet which has a name property:

To check whether dog can be assigned to pet , the compiler checks each property of pet to find a corresponding compatible property in dog . In this case, dog must have a member called name that is a string. It does, so the assignment is allowed.

The same rule for assignment is used when checking function call arguments:

Note that dog has an extra owner property, but this does not create an error. Only members of the target type ( Pet in this case) are considered when checking for compatibility. This comparison process proceeds recursively, exploring the type of each member and sub-member.

Be aware, however, that object literals may only specify known properties . For example, because we have explicitly specified that dog is of type Pet , the following code is invalid:

Comparing two functions

While comparing primitive types and object types is relatively straightforward, the question of what kinds of functions should be considered compatible is a bit more involved. Let’s start with a basic example of two functions that differ only in their parameter lists:

To check if x is assignable to y , we first look at the parameter list. Each parameter in x must have a corresponding parameter in y with a compatible type. Note that the names of the parameters are not considered, only their types. In this case, every parameter of x has a corresponding compatible parameter in y , so the assignment is allowed.

The second assignment is an error, because y has a required second parameter that x does not have, so the assignment is disallowed.

You may be wondering why we allow ‘discarding’ parameters like in the example y = x . The reason for this assignment to be allowed is that ignoring extra function parameters is actually quite common in JavaScript. For example, Array#forEach provides three parameters to the callback function: the array element, its index, and the containing array. Nevertheless, it’s very useful to provide a callback that only uses the first parameter:

Now let’s look at how return types are treated, using two functions that differ only by their return type:

The type system enforces that the source function’s return type be a subtype of the target type’s return type.

Function Parameter Bivariance

When comparing the types of function parameters, assignment succeeds if either the source parameter is assignable to the target parameter, or vice versa. This is unsound because a caller might end up being given a function that takes a more specialized type, but invokes the function with a less specialized type. In practice, this sort of error is rare, and allowing this enables many common JavaScript patterns. A brief example:

You can have TypeScript raise errors when this happens via the compiler flag strictFunctionTypes .

Optional Parameters and Rest Parameters

When comparing functions for compatibility, optional and required parameters are interchangeable. Extra optional parameters of the source type are not an error, and optional parameters of the target type without corresponding parameters in the source type are not an error.

When a function has a rest parameter, it is treated as if it were an infinite series of optional parameters.

This is unsound from a type system perspective, but from a runtime point of view the idea of an optional parameter is generally not well-enforced since passing undefined in that position is equivalent for most functions.

The motivating example is the common pattern of a function that takes a callback and invokes it with some predictable (to the programmer) but unknown (to the type system) number of arguments:

Functions with overloads

When a function has overloads, each overload in the target type must be matched by a compatible signature on the source type. This ensures that the source function can be called in all the same cases as the target function.

Enums are compatible with numbers, and numbers are compatible with enums. Enum values from different enum types are considered incompatible. For example,

Classes work similarly to object literal types and interfaces with one exception: they have both a static and an instance type. When comparing two objects of a class type, only members of the instance are compared. Static members and constructors do not affect compatibility.

Private and protected members in classes

Private and protected members in a class affect their compatibility. When an instance of a class is checked for compatibility, if the target type contains a private member, then the source type must also contain a private member that originated from the same class. Likewise, the same applies for an instance with a protected member. This allows a class to be assignment compatible with its super class, but not with classes from a different inheritance hierarchy which otherwise have the same shape.

Because TypeScript is a structural type system, type parameters only affect the resulting type when consumed as part of the type of a member. For example,

In the above, x and y are compatible because their structures do not use the type argument in a differentiating way. Changing this example by adding a member to Empty<T> shows how this works:

In this way, a generic type that has its type arguments specified acts just like a non-generic type.

For generic types that do not have their type arguments specified, compatibility is checked by specifying any in place of all unspecified type arguments. The resulting types are then checked for compatibility, just as in the non-generic case.

For example,

Advanced Topics

Subtype vs assignment.

So far, we’ve used “compatible”, which is not a term defined in the language spec. In TypeScript, there are two kinds of compatibility: subtype and assignment. These differ only in that assignment extends subtype compatibility with rules to allow assignment to and from any , and to and from enum with corresponding numeric values.

Different places in the language use one of the two compatibility mechanisms, depending on the situation. For practical purposes, type compatibility is dictated by assignment compatibility, even in the cases of the implements and extends clauses.

any , unknown , object , void , undefined , null , and never assignability

The following table summarizes assignability between some abstract types. Rows indicate what each is assignable to, columns indicate what is assignable to them. A ” ✓ ” indicates a combination that is compatible only when strictNullChecks is off.

Reiterating The Basics :

  • Everything is assignable to itself.
  • any and unknown are the same in terms of what is assignable to them, different in that unknown is not assignable to anything except any .
  • unknown and never are like inverses of each other. Everything is assignable to unknown , never is assignable to everything. Nothing is assignable to never , unknown is not assignable to anything (except any ).
  • void is not assignable to or from anything, with the following exceptions: any , unknown , never , undefined , and null (if strictNullChecks is off, see table for details).
  • When strictNullChecks is off, null and undefined are similar to never : assignable to most types, most types are not assignable to them. They are assignable to each other.
  • When strictNullChecks is on, null and undefined behave more like void : not assignable to or from anything, except for any , unknown , never , and void ( undefined is always assignable to void ).

The TypeScript docs are an open source project. Help us improve these pages by sending a Pull Request ❤

Ryan Cavanaugh  (51)

Last updated: Feb 08, 2024  

Type systems

An interpreter for free, type system goals, static semantics, structural induction, proving type safety.

PDF version of this page.

Recall from last week that we had a big issue with our lambda calculus. We were able to construct programs with undefined behavior, i.e. ones that would evaluate to a stuck state, by having free variables in our expressions, like

Moreover, we had no way to easily enforce higher-level constraints about our functions. For example, let’s say we had a function that would apply an argument twice to a function.

We could accidentally give this a function that only takes one argument 1 , e.g.

The desired properties above are all examples of invariants , or program properties that should always hold true. Invariants are things like:

  • In my ATM program, customers should not withdraw more money than they have in their account.
  • In my TCP implementation, neither party should exchange data until the initial handshake is complete.
  • In the driver for my mouse, the output coordinates for the mouse should always be within the bounds of my screen.

There are three main considerations in the design of invariants:

  • Structure. What is the “language” of invariants? How can we write down a particular invariant?
  • Inference. Which invariants can be inferred from the program, and which need to be provided by the programmer?
  • Time of check. When, in the course of a program’s execution, is an invariant checked? Before the program is run?

For example, consider the humble assert statement. This is usually a built-in function that takes as input a boolean expression from the host language, and raises an error if the expression evaluates to false. In Python:

For these kinds of asserts, the language of invariants is the same as the host language, i.e. a Python expression. This is quite powerful! You can do arbitrary computations while checking your invariants. These invariants are never inferred—you have to write the assert statements yourself 2 . And lastly, assert statements are checked at runtime, when the interpreter reaches the assert. Nothing guarantees that an assert is checked before code relying on its invariant is executed (e.g. accidentally dividing and then asserting in the case above).

By contrast, now consider the traditional notion of a “type system” as you know it from today’s popular programming languages. For most type systems, the language of invariants is quite restricted—types specify that a variable is a “kind” of thing, e.g. n is an int , but cannot specify further that n != 0 . Most stone-age programming languages require the programmer to explicitly provide type annotations (e.g. int n = 0 ), but modern languages increasingly use type inference to deduce types automatically (e.g. let n = 0 ). Lastly, types can be checked either ahead of time (“statically”, e.g. C, Java) or during program execution (“dynamically”, e.g. Python, Javascript) 3 .

Key idea: type systems and runtime assertions derive from the same conceptual framework of enforcing invariants, just with different decisions on when and how to do the checks.

In this course, our focus is going to be on static analysis: what invariants can we describe, infer, and enforce before ever executing the program? And by enforcement, I mean iron law. We don’t want our type systems to waffle around with “well, you know, I bet this n is going to be an integer, but I’m only like, 75% sure.” We expect Robocop type systems that tell us: I HAVE PROVED TO 100% MATHEMATICAL CERTAINTY THAT IN THE INFINITE METAVERSE OF BOUNDLESS POSSIBILITIES, THIS “n” IS ALWAYS AN INTEGER.

This is the core impetus behind most modern research in PL theory. Advances in refinement types , dependent types , generalized algebraic data types , module systems , effect systems , traits , concurrency models , and theorem provers have pushed the boundaries of static program analysis. Today, we can prove more complex invariants than ever before. While cutting-edge PL research is mostly beyond the scope of this course, you will be equipped with the necessary fundamentals to continue exploring this space.

Typed lambda calculus

To understand the formal concept of a type system, we’re going to extend our lambda calculus from last week (henceforth the “untyped” lambda calculus) with a notion of types (the “simply typed” lambda calculus). Here’s the essentials of the language:

Indeed, our operational semantics are just the lambda calculus plus arithmetic. Zero change from before.

A brief aside: the main reason we’re using OCaml in this course (as opposed to, say, Haskell or Scala) is that feels quite similar to the typed lambda calculus. In fact, if we change a few keywords, we can use OCaml to execute exactly the language described above. (See the OCaml setup guide to follow along). If we wanted to transcribe the two examples above:

Of course, OCaml can do much more than this—it has strings, exceptions, if statements, modules, and so on. We’ll get there, all in due time. I point this out to show you that by learning the lambda calculus, you are actually learning the principles of real programming languages, not just highfalutin theory. When you go to assignment 2 and start on your first OCaml program, the language will feel more familiar than you may expect!

Before we dive into the type system, it’s worth asking the motivational question: what invariants of our language do we want to statically check? One way to answer this is by thinking of edge cases we want to avoid.

This is an important exercise, since it gives us an intuition for where errors might arise. However, even if we had a method for completely eliminating the edge cases we thought of, how can we know we caught all the cases? What if we just didn’t think of a possible error?

Remember that all of these issues fundamentally boil down to stuck states, or undefined behavior. We specified our operational semantics over “well-defined” programs, but that doesn’t prevent us from writing invalid programs. As before, the goal is to take a program and step it to a value. This leads us to a safety goal: if a program is well-defined, it should never enter a stuck state after each step. If we can formally prove that this safety goal holds for our language, then that means there are no missing edge cases!

Lastly, we want to prove the “type safety” of our language with two theorems:

Intuitively, progress says: if an expression is well-typed and is not a value, then we should be able to step the expression (it is not in a stuck state). However, this isn’t enough to prove our safety goal, since we also need preservation: if an expression is well-typed, when it steps, its type is preserved. For example, if we have an expression of integer type, it shouldn’t turn into a function after being stepped.

In this conceptual framework of type systems, the first thing we need to do is define how we determine the type of an expression. In our grammar, we defined a type language , but now we need a type semantics (or “static semantics”). First, we’ll define the judgments for numbers:

Note a subtlety here: \tau_1 is given to us from the program, while we have to compute \tau_2 . Our typed lambda calculus mixes types that are explicitly annotated and implicitly inferred.

At this point, you should understand the mechanics of our type system: how we define our typing rules, and how they can be used to construct proofs about the types of expressions. But it’s not sufficient just to have a type system, we need a good type system! Remember, we want to demonstrate that if a program is well-typed, then it will never enter a stuck state. To do that, we have to prove the progress and preservation theorems, i.e. verify that our language is actually “type safe.” This is an example of a metatheoretical property of our programming language. We use the type system to prove that expressions has certain types, and on the next level up we prove that our type system (or proof system) has certain properties.

For proofs with programming languages, we generalize the idea of mathematical induction to structural induction . Until this point, we’ve done proofs about individual programs, e.g. above showing that a particular concrete expression has a particular type. As you’ve seen, these have an inductive flavor—to prove a statement, you recursively prove statements about its sub-components until you reach a base case.

Note: the progress and preservation theorems are defined with respect to “closed terms”, i.e. expressions which don’t need a type context at the top level to prove their type. Or put another way, expressions with no free variables.

Proof. By rule induction on the static semantics.

Third, we case on the three ways in which a binary operation can step:

This is vacuously true, since there is no rule to step a function value.

Third, we case on the two ways an application can step:

Hence, preservation holds in either case.

Lastly, let’s prove progress.

In each case, the expression steps, so progress holds.

In the lambda calculus, all functions technically take one argument, so when I say “a function that takes one argument”, I mean as opposed to a function that returns another function.  ↩

Except where built into the language, of course. In the div example, both of the asserted invariants (int types and nonzero) will be checked by the division operator in the language runtime.  ↩

The options provided do not strictly form a dichotomy. “Gradual” or “hybrid” invariant enforcement that mixes static/dynamic checks is an active area of research, e.g. gradual typing .  ↩

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A Type Assignment System For The Game Semantics

Profile image of Gianluca Franco

this paper an alternative description of the game semantics for the untyped lambda calculus is given. More precisely, we introduce a finitary description of lambda terms. This description turns out to be equivalent to a particular game denotational semantics of the lambda calculus. Introduction

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Pietro Di Gianantonio

type assignment system

Gianluca Franco

Abstract. We study extensional models of the untyped lambda calculus in the setting of the game semantics introduced by Abramsky, Hyland et alii. In particular we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in a standard category of games, induce the same λ-theory. This is H∗, the maximal theory induced already by the classical C.P.O. model D∞, introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction.

Furio Honsell

. We study extensional models of the untyped lambda calculus in the setting of game semantics. In particular, we show that, somewhat unexpectedly and contrary to what happens in ordinary categories of domains, all reflexive objects in the category of games G, introduced by Abramsky, Jagadeesan and Malacaria, induce the same -theory. This is H , the maximal theory induced already by the classical CPO model D1 , introduced by Scott in 1969. This results indicates that the current notion of game carries a very specific bias towards head reduction. Introduction -theories are congruences over -terms, which extend pure fi-conversion. Their interest lies in the fact that they correspond to the possible operational (obser- vational) semantics of -calculus. Although researchers have mainly focused on only three such operational semantics, namely those given by head reduction, head lazy reduction or call-by-value reduction, the class of -theories is, in effect, unfathomly rich, see e.g. [6...

Electronic Proceedings in Theoretical Computer Science

Ugo de'Liguoro

We propose an intersection type system for an imperative λ -calculus based on a state monad and equipped with algebraic operations to read and write to the store. The system is derived by solving a suitable domain equation in the category of ω -algebraic lattices; the solution consists of a filter-model generalizing the well known construction for ordinary λ -calculus. Then the type system is obtained out of the term interpretations into the filter-model itself. The so obtained type system satisfies the “type-semantics” property, and it is sound and complete by construction.

Zhaohui Luo

Samson Abramsky

The aim of this chapter is to give an introduction to some recent work on the application of game semantics to the study of programming languages. An initial success for game semantics was its use in giving the first syntax-free descriptions of the fully abstract model for the functional programming language PCF [1,16,29]. One goal of semantics is to characterize the "universe of discourse" implicit in a programming language or a logic.

Lambda Calculus with Types

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Information and Computation/information and Control

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Combination of Multiplex PCRs for Staphylococcal Cassette Chromosome mec Type Assignment: Rapid Identification System for mec , ccr , and Major Differences in Junkyard Regions ▿

Juntendo University, Graduate School of Medicine, Department of Infection Control Science, Tokyo, Japan, 1 Juntendo University, Department of Bacteriology, Tokyo, Japan, 2 Public Health Research Institute, Newark, New Jersey, 3 Faculté de Médecine Laennec, Centre National de Référence des Staphylocoques, IFR62, INSERM E02030, 7 rue guillaume Paradin, 69008 Lyon, France 4

Xiao Xue Ma

Shinya watanabe, barry n. kreiswirth, jerome etienne, keiichi hiramatsu.

Staphylococcal cassette chromosome mec (SCC mec ) typing, in combination with genotyping of the Staphylococcus aureus chromosome, has become essential for defining methicillin-resistant S. aureus (MRSA) clones in epidemiological studies. We have developed a convenient system for SCC mec type assignment. The system consists of six multiplex PCRs (M-PCRs) for identifying the ccr gene complex ( ccr ), the mec gene complex ( mec ), and specific structures in the junkyard (J) regions: M-PCR with primer set 1 (M-PCR 1) identified five types of ccr genes; M-PCR 2 identified class A to class C mec ; M-PCRs 3 and 4 identified specific open reading frames in the J1 regions of type I and IV and of type II, III, and V SCC mec elements, respectively; M-PCR 5 identified the transposons Tn 554 and ΨTn 554 integrated into the J2 regions of type II and III SCC mec elements; and M-PCR 6 identified plasmids pT181 and pUB110 integrated into J3 regions. The system was validated with 99 MRSA strains carrying SCC mec elements of different types. The SCC mec types of 93 out of the 99 MRSA strains could be assigned. The SCC mec type assignments were identical to those made with a PCR system that uses numerous primer pairs to identify genes or gene alleles. Our system of six M-PCRs is thus a convenient and reliable method for typing SCC mec elements.

Methicillin-resistant Staphylococcus aureus (MRSA) strains have become prevalent in health care facilities and in the community worldwide ( 3 , 4 ). MRSA strains produce penicillin binding protein 2′ or 2a, which is poorly acylated by β-lactam antibiotics ( 5 , 22 , 25 ). The mecA gene, encoding PBP2a, is carried on a peculiar type of mobile genetic element inserted into the staphylococcal chromosome, designated staphylococcal cassette chromosome mec (SCC mec ) elements ( 12 , 14 , 24 ).

SCC mec elements typically share four characteristics: first, they carry the mec gene complex ( mec ) consisting of the methicillin resistance determinant mecA and its regulatory genes and insertion sequences; second, they carry the ccr gene complex ( ccr ) consisting of ccr genes that are responsible for the mobility of the element and its surrounding sequences; third, they have characteristic directly repeated nucleotide sequences and inverted complementary sequences at both ends; and last, they integrate into the 3′ end of an open reading frame (ORF), orfX .

Despite these similarities, the structures of SCC mec elements are rather divergent. Allotypic differences that are used for SCC mec type definitions have been identified in both ccr and mec . Five types of ccr and four classes of mec have been reported. ccr types 1 to 4 carry the ccrA and ccrB genes, which share approximately 80% identity with each other, and the type 5 ccr carries the ccrC gene ( 10 , 11 , 17 , 19 ). Four classes of the mec gene complexes have been identified among methicillin-resistant staphylococcal strains of various species: class A mec , consisting of IS 431mec-mecA-mecR1-mecI ; class B mec , consisting of IS 431mec-mecA -Δ mecR1 -IS 1272 ; class C mec , consisting of IS 431mec-mecA -Δ mecR1- IS 431 ; and class D mec , consisting of IS 431mec-mecA -Δ mecR1 with no insertion sequences downstream of Δ mecR1 identified by PCR as of yet ( 13 ). In S. aureus strains, mec classes A, B, and C have been identified. Insertion sequences have sometimes been found to be integrated in or around the class A mec . A class A mec carrying IS 431 downstream of mecI was found in Staphylococcus haemolyticus ( 13 ). Recently, Shore et al. identified MRSA strains carrying class A mec with an insertion of IS 1182 in and around the mecI gene and designated them classes A3 and A4 ( 23 ).

The SCC mec element type has been defined by the combination of ccr type and mec class. In MRSA strains, six types of SCC mec elements, that is, six combinations of ccr and mec , have been reported (Table ​ (Table1). 1 ). These six SCC mec elements have been further classified by differences in regions other than ccr and mec , which are designated junkyard (J) regions. The J regions comprise three parts: J1 (the region between ccr and the right-flanking chromosomal region), J2 (the region between mec and ccr ), and J3 (the region between orfX and mec ). The J regions are not always specific to each SCC mec type, but certain J regions are commonly shared among certain types of SCC mec elements. Of the three regions, we regard J1 as being the most fundamental, because we presume that it reflects the original form of SCC into which a mec gene complex integrated. Moreover, several different J1 regions have been identified in type II and type IV SCC mec elements ( 7 , 15 , 17 , 20 , 23 ). The presence or absence of integrated plasmids encoding drug resistance genes in the J3 regions of SCC mec elements can also be used as markers to classify SCC mec elements further in epidemiological studies ( 1 , 19 ).

List of SCC mec elements and strains used for standard strains

In this study, we describe a convenient and reliable method for SCC mec typing based on a set of multiplex PCRs (M-PCRs). In this system, M-PCRs 1 and 2 are used for SCC mec type assignment; M-PCR 3 or M-PCR 4 is used for J1 region difference-based subtyping, and M-PCRs 5 and 6 are used for the identification of integrated copies of transposons (Tn 554 or ΨTn 554 ) and plasmids (pUB110 or pT181).

MATERIALS AND METHODS

Bacterial strains..

Twelve MRSA strains were used as standard strains for SCC mec typing (Table ​ (Table1). 1 ). In addition, 99 MRSA strains were used to validate the multiplex PCR, including 78 strains isolated in the United States, Canada, Ireland, England, and Egypt from 1960 through 1993 (including 8 strains with no record of their origin) provided by B. N. Kreiswirth (note that these strains were included in the study by Pfaller et al. [ 21 ]); 10 strains isolated in France in 1996, provided by J. Etienne; and 11 strains isolated in the United States and Australia, as reported previously by Okuma et al. ( 18 ).

Nomenclature of SCC mec elements.

In this paper, we use our proposed nomenclature for SCC mec elements ( 2 ). As shown in Table ​ Table1, 1 , the SCC mec element type (defined by the combination of ccr and mec allotypes) is indicated with roman numerals, while the SCC mec element subtype (defined by differences in the junkyard regions) is indicated with Arabic numbers separated by a period, where each number indicates the structure of the J regions (J1, J2, and J3, respectively) according to the chronological priority of the description. It should be noted that the description of the J regions has been modified from our original proposal according to the suggestion of Kunyan Zhang (Calgary University, Canada).

Chromosomal DNA was extracted from MRSA strains by using the small-scale phenol extraction method and was used as a template ( 8 ).

The primer pairs used for PCR experiments are listed in Table ​ Table2. 2 . M-PCR 1 for ccr type assignment contained two primers to identify mecA and eight primers used for the identification of five ccr genes: four primers including a common forward primer (common to ccrB1-3 ) and three reverse primers specific for ccrA1 , ccrA2 , and ccrA3 for identifying ccr1-3 based on the differences in ccrA genes; two primers for identifying ccr4 ; and two primers for identifying ccr5 . M-PCR 2 for mec class assignment contained four primers to identify the gene lineages of mecA-mecI (class A mec ), mecA -IS 1272 (class B mec ), and mecA -IS 431 (class C mec ). M-PCR 3 contained five primer pairs: one pair for identifying specific ORF in the J1 region of type I SCC mec elements and four pairs for identifying specific ORFs in the J1 regions of four subtypes of type IV SCC mec elements. M-PCR 4 contained six primer pairs: four pairs for identifying specific ORFs in J1 regions of four subtypes of type II SCC mec elements, one pair for identifying specific ORFs in the J1 region of type III SCC mec elements, and one pair for identifying specific ORFs in the J1 region of type V SCC mec elements. M-PCR 5 contained three primers: one primer specific to the J2 regions of type II and type III SCC mec elements and two primers specific to ermA and cadB , respectively, to identify the J2 regions of type II or type III SCC mec elements. M-PCR 6 contained three primers: one primer specific for mecA and two primers specific for ant(4 ′ ) in plasmid pUB110 and tetK in plasmid pT181, respectively, to identify integrated plasmids in the J3 regions.

Primers used in this experiment

For M-PCR 1, reaction mixtures contained 10 ng chromosomal DNA, oligonucleotide primers (0.1 μM), 200 μM each deoxynucleotide triphosphates, Ex Taq buffer, and 2.5 U Ex Taq polymerase (Takara Bio Inc., Kyoto, Japan) in a final volume of 50 μl. The concentration of MgCl 2 was 3.2 mM. A Takara PCR thermal cycler was used for amplification with an initial denaturation step (94°C, 2 min); 30 cycles of denaturation (94°C, 2 min), annealing (57°C, 1 min), and extension (72°C, 2 min); and a final elongation step at 72°C for 2 min. For M-PCRs 2 to 5, the reaction mixtures were the same as those for M-PCR 1 except that the concentration of MgCl 2 was 2 mM and the annealing temperature was raised to 60°C for 1 min to avoid the generation of nonspecific DNA fragments. For M-PCR 6, we performed long-range PCR using the Expand High Fidelity PCR system according to the manufacturer's recommendations (Roche Diagnostics Co., Indianapolis, IN). Briefly, reactions were performed using a final reaction mixture volume of 50 μl, which contained 10 ng template DNA, oligonucleotide primers (0.3 μM), 200 μl each deoxynucleotide triphosphate, 1× Expand High Fidelity buffer, 1.5 mM MgCl 2 , and 2.6 U Expand High Fidelity PCR system enzyme mix. The PCR consisted of a denaturation step (94°C, 2 min); 10 cycles of denaturation (94°C, 15 s), annealing (50°C, 30 s), and extension (68°C, 8 min); 20 cycles of denaturation (94°C, 15 s), annealing (50°C, 30 s), and extension (68°C, 12 min); and a final elongation step (72°C, 7 min).

PCRs for the identification of SCC mercury and customary (nonmultiplexed) PCRs to identify pls were carried out with the primer pairs listed in Table ​ Table2 2 according to a previously described procedure ( 10 ). PCR products were visualized by agarose gel electrophoresis.

PCR-based identification and determination of part of the nucleotide sequence of the type II.4 SCC mec element.

DNA fragments encompassing the entire SCC mec element of strain RN7170 were amplified by long-range PCR using the Expand High Fidelity PCR system under the same conditions as those used for M-PCR 6. The primer sets used for amplifying the DNA fragments are given in Table ​ Table2. 2 . The amplicon sizes estimated by agarose gel electrophoresis are as follows: the region from orfX to mecA , amplified with primers cR1 and mA3, was 11 kb; the region from mecA to ermA in Tn 554 , amplified with primers mA2 and ermA1, was 11 kb; the region from ermA in Tn 554 to ccr , amplified with primers ermA3 and 2AJ1, was 12 kb; and the region from ccr to the right-flanking chromosomal region, amplified with primers cβ and cL4, was 15 kb. The locations of the primers are indicated in Fig. ​ Fig.1. 1 . PCR products were purified with the QIAquick PCR purification kit (QIAGEN, Hilden, Germany), and nucleotide sequences of the DNA fragments from ccr to the right-flanking chromosomal region were determined by primer walking.

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Object name is zac0010762090001.jpg

Schematic structures of representative SCC mec elements based on the nucleotide sequences deposited in the EMBL/GenBank/DDBJ database under the accession numbers listed in Table ​ Table1. 1 . Circles indicate the ccr genes that can be identified by M-PCR 1. The mec gene complexes that can be identified by M-PCR 2 are indicated by squares. Black bars indicate the locations of the J1 region-specific primers used for M-PCRs 3 and 4. The locations of primers used for the amplification of the entire type II.4 SCC mec region of strain RN7170 are indicated by red arrowheads.

Nucleotide sequence accession number.

The nucleotide sequence of the region containing J1 and the ccr gene complex of the type II.4 SCC mec of strain RN7170 has been deposited in the DDBJ/EMBL/GenBank database under accession no. {"type":"entrez-nucleotide","attrs":{"text":"AB261975","term_id":"119926920","term_text":"AB261975"}} AB261975 .

RESULTS AND DISCUSSION

Development of two m-pcrs for ccr and mec components of scc mec elements..

M-PCR 1 was developed to identify ccrC and ccr4 in addition to three ccr types that could be identified with previous systems ( 10 ). M-PCR 1 successfully amplified DNA fragments corresponding in size to each ccr gene as follows: type 1 ccr , 695 bp; type 2 ccr , 937 bp; type 3 ccr , 1,791 bp; type 4 ccr , 1,287 bp; and type 5 ccr , 518 bp (Table ​ (Table2 2 and Fig. ​ Fig.2A). 2A ). The 286-bp amplification product appearing in each lane represents mecA , the most essential gene in SCC mec , which is used as an internal amplification control. Two DNA fragments of 518 bp and 1,791 bp were amplified with chromosomal DNA of strain 85/2082, indicating that it carried both type 3 ccr and ccrC , consistent with the fact that this strain carries two SCC elements, a type III SCC mec (type 3 ccr ) and an SCC mercury ( ccrC ) (Table ​ (Table1 1 ).

An external file that holds a picture, illustration, etc.
Object name is zac0010762090002.jpg

Six multiplex PCRs for SCC mec type assignment. (A) M-PCR 1 for identification of ccr genes for the assignment of the type of ccr gene complex. Chromosomal DNAs from standard strains were used as templates. Lane 1, NCTC10442; lane 2, N315; lane 3, 85/2082; lane 4, CA05; lane 5, WIS; lane 6, HDE288. DNA fragments of the expected sizes for each ccr gene were amplified. The DNA fragment corresponding to mecA served as an internal control in each lane. (B) M-PCR 2 for identification of three gene alleles for assignment of the mec gene complex. Chromosomal DNAs from standard strains were used as templates. Lane 1, NCTC10442; lane 2, N315; lane 3, 85/2082; lane 4, CA05; lane 5, WIS. DNA fragments of the expected sizes for class A mec (lanes 2 and 3), class B mec (lanes 1 and 4), and class C mec (lane 5) were amplified. (C) M-PCR 3 for J1 region difference-based subtyping of type I and type IV SCC mec elements, which carry the class B mec . Chromosomal DNAs from standard strains were used as templates. Lane 1, NCTC10442; lane 2, CA05; lane 3, 8/6-3P; lane 4, 81/108; lane 5, JCSC4469. DNA fragments of the expected sizes for each J1 region were amplified. (D) M-PCR 4 for J1 region-based subtyping of type II and type III SCC mec elements, which carry the class A mec , and the type V SCC mec , which carries the class C mec . Chromosomal DNAs from standard strains were used as templates. Lane 1, N315; lane 2, JCSC3063; lane 3, BK351; lane 4, RN7170; lane 5, 85/2082; lane 6, WIS. (E) Identification of resistance determinants. Transposons (Tn 554 and ΨTn 554 ) were assigned by M-PCR 5 (lanes 1 to 3), and plasmids (pUB110 and pT181) were assigned by M-PCR 6 (lanes 4 and 5). SCC mercury was assigned with an M-PCR with the four sets of primers listed in Table ​ Table2 2 (lane 6). Chromosomal DNAs from standard strains were used as templates. Lanes 1 and 4, N315; lane 2, BK645; lanes 3, 5, 6, and 85/2082. MWM, molecular weight marker.

M-PCR 2, developed for assigning the mec class, contained primer pairs for identifying mecA-mecI (class A), mecA -IS 1272 (class B), and mecA -IS 431 (class C), and it can replace the three reactions used in traditional methods. The sizes of the amplified DNA fragments matched the sizes expected for each class of mec : 1,963 or 1,797 bp for mecA-mecI (class A), 2,827 bp for mecA -IS 1272 (class B), and 804 bp for mecA -IS 431 (class C) (Table ​ (Table2 2 and Fig. ​ Fig.2B). 2B ). The size of the amplified DNA fragment obtained with strain 85/2082 was shorter than that obtained with strain N315, in agreement with the fact that mecR1 of 85/2082 has a 166-bp deletion relative to that in N315 ( 10 ). This case seems to be exceptional, because we successfully amplified DNA fragments of sizes similar to those of N315 in 49 of 49 class A mec strains tested.

Development of two M-PCRs for J1 regions of SCC mec elements.

M-PCRs 3 and 4 were developed to identify specific ORFs in the J1 region of each SCC mec element: M-PCR 3 for type I and type IV SCC mec elements carrying class B mec and M-PCR 4 for type II and type III SCC mec elements carrying class A mec and type V SCC mec elements carrying class C mec .

With M-PCR 3, the J1 regions of all five SCC mec type I or IV elements (type I.1, type IV.1, type IV.2, type IV.3, and type IV.4) (Table ​ (Table1) 1 ) were identified with primer pairs specific for each subtype. The sizes of the amplified DNA fragments matched those predicted for the J1 regions of the five SCC mec elements (type I.1, 154 bp; type IV.1, 458 bp; type IV.2, 726 bp; type IV.3, 259 bp; type IV.4, 1,242 bp) (Fig. ​ (Fig.2C 2C and Table ​ Table2 2 ).

In this study, we identified a new subtype of type II SCC mec elements carried by RN7170. By amplifying entire SCC mec region and determining the nucleotide sequence of the J1 region of the element, we designated it type II.4 SCC mec . Interestingly, the J1 region carried by type IIB, IIC, IID, and IIE SCC mec elements was the same as that of the type IV.2 SCC mec element ( 23 ). Therefore, we considered these SCC mec elements to be type II.3, type II elements carrying the third identified J1 region (although it was identified previously in the type IV.2 SCC mec ). With M-PCR 4, the J1 regions of six SCC mec types, type II, III, or V (type II.1, type II.2, type II.3, type II.4, type III.1, and type V.1) (Table ​ (Table1), 1 ), were identified with primer pairs specific for each type. The sizes of the amplified DNA fragments matched those predicted for the J1 regions of all six SCC mec types (type II.1, 287 bp; type II.2, 1,518 bp; type II.3, 726 bp; type II.4, 2,003 bp; type III.1, 503 bp; type V.1, 1,159 bp) (Fig. ​ (Fig.2D 2D and Table ​ Table2 2 ).

Development of two M-PCRs to identify resistance plasmids/transposons integrated into SCC mec elements (J2 and J3 regions).

M-PCR 5 was developed to identify transposon Tn 554 or ΨTn 554 integrated into type II and type III SCC mec elements by targeting ORFs located in the J2 region flanking these elements and resistance determinants carried by the transposons ( ermA by Tn 554 and cadB by ΨTn 554 ). As expected, M-PCR 5 amplified DNA fragments of 2,756 bp (Fig. ​ (Fig.2E, 2E , lanes 1 and 2), corresponding to CN030- ermA , and 1,540 bp (lane 3), corresponding to CZ021- cadB .

M-PCR 6 was developed to identify differences in the J3 region based on the presence or absence of resistance plasmids pUB110 and pT181 by amplifying the J3 regions mecA-ant(4 ′ ) -1 for pUB110 and mecA-tetK for pT181. As expected, the DNA fragments amplified by M-PCR 6 were 4,952 bp (Fig. ​ (Fig.2E, 2E , lane 4) for mecA-ant(4′) -1 and 7,406 bp (Fig. ​ (Fig.2E, 2E , lane 5) for mecA-tetK . It should be noted that with M-PCR 6, plasmids pUB110 and pT181 could be detected if they were located downstream of mecA but could not be detected if they were located distant from mecA . In contrast, in the M-PCR developed previously by Oliveira and Lencastre ( 20 ), these plasmids were identified with primer sets specific for each plasmid, regardless of their relative position to mecA , leaving the possibility that these plasmids are located outside the SCC mec element.

Validation of six multiplex PCRs.

We evaluated our system of six M-PCRs by examining a total of 99 MRSA strains. The results obtained with the M-PCRs were identical to those obtained by traditional methods, which mostly used a set of two primers to identify each gene or different allotypes ( 7 , 10 , 18 ).

M-PCR 1 amplified a single DNA fragment belonging to one of five ccr types from the chromosomal DNAs of 83 of the 99 MRSA strains. In 10 strains, two DNA fragments were amplified, signifying that these strains carry two ccr s, while in 6 strains, no DNA fragments were amplified (Table ​ (Table3). 3 ). Among the 10 strains that carried two ccr s, 8 carried a type 3 ccr and ccrC and 2 carried a type 1 ccr and ccrC . Since ccrC is present in SCC elements carrying the mercury resistance operon (SCC mercury ) or the capsule gene cluster (SCC cap1 ) ( 2 , 16 ), we conducted an M-PCR experiment to identify the mercury resistance operon and J region in SCC mercury by using four primers (merA2, merG, mN21, and mN22) listed in Table ​ Table2. 2 . DNA fragments indicating the carriage of the mercury resistance operon were amplified from the chromosomal DNAs of all 10 strains, and DNA fragments indicating the carriage of the J region of SCC mercury were amplified from the chromosomal DNAs of 9 of them, signifying that these 9 strains carried SCC mercury . We tentatively regarded the remaining strain as a type 1 ccr strain, presuming that ccrC might be carried by other unknown mobile genetic elements, since no SCC mec element carrying the combination of ccrC (type 5 ccr ) and class A mec or class B mec has been identified yet.

SCC mec types identified in 99 MRSA strains by PCR

With M-PCR 2, the mec gene complexes of 95 of 99 tested strains were judged to belong to either class A or class B mec , and those of 4 strains were unclassifiable (Table ​ (Table3 3 ).

As such, SCC mec elements carried by 93 of the 99 tested strains were classified into one of the six known types of SCC mec elements (Table ​ (Table3 3 ).

With M-PCRs 3 and 4, these SCC mec elements could be further classified based on differences in the J1 region. Overall, we were able to classify the J1 regions of SCC mec elements carried by 92 of 93 strains that had an identified SCC mec type (Table ​ (Table3). 3 ). Interestingly, no DNA fragment from the chromosomal DNAs of the six untypeable strains was amplified by M-PCR 3 and M-PCR 4, suggesting that these strains might carry new SCC mec elements.

We used M-PCR 5 to establish the presence or absence of Tn 554 and ΨTn 554 in the J2 regions of type II and type III SCC mec elements. Tn 554 was identified in 36 of 37 type II SCC mec elements and 1 of 12 type III SCC mec elements. ΨTn 554 was identified in 11 of 12 type III SCC mec elements (Table ​ (Table3). 3 ). One type III SCC mec strain carried Tn 554 instead of ΨTn 554 at the J2 region, as previously reported ( 9 ).

We used M-PCR 6 to determine the presence or absence of plasmids pUB110 and pT181. Plasmid pUB110 was carried by 1 of 17 type I SCC mec elements, 35 of 37 type II SCC mec elements, 10 of 27 type IV SCC mec elements, and 3 of the 6 untypeable strains. Remarkably, all 10 tested gentamicin-susceptible MRSA strains (isolated in France in 1996) carried a type IV.3 SCC mec with an integrated plasmid pUB110. No type III SCC mec harbored pUB110 downstream of mecA . In contrast, 8 of 12 type III SCC mec elements harbored plasmid pT181, which was not identified in type I, type II, and type IV SCC mec elements.

Atypical and unclassifiable SCC mec elements.

We were not able to identify the ccr gene of six mecA -positive strains with M-PCR 1. Two of them carried class A mec , the J2 region of the type II SCC mec element, and the integrated plasmid pUB110, indicating that the gene lineage pUB110-IS 431-mecA-mecR1-mecI -Tn 554 , which is usually located in the type II SCC mec element, was carried by these two strains. Further studies to determine the nucleotide sequences of the region between Tn 554 and the right-flanking chromosomal region will clarify whether these strains carry a deleted type II SCC mec or a novel region with new ccr genes. Three of them carried the mecR1 gene, whereas the mecI gene was not detected by PCR testing for the respective genes ( 18 ). One strain carried neither mecR1 nor mecI . The structures of those elements will be the subject of further investigation.

Comparison to previously reported M-PCRs.

Our M-PCRs do not conflict with two previously reported M-PCRs based primarily on the identification of junkyard regions ( 20 , 26 ). The M-PCR described previously by Oliveira and Lencastre ( 20 ) has the advantage that it identifies multiple loci simultaneously (e.g., the mecA gene, the mecI gene, the J1 region of type I and type II SCC mec elements, ccrC , the dcs [downstream constant sequence] region, pT181, and pUB110). The M-PCR described previously by Zhang et al. ( 26 ) has the advantage that it identifies the J1 region of eight SCC mec elements simultaneously. M-PCRs 3 to 6, which were developed to identify specific ORFs in J regions, are based on the same concept as that of the two previously reported M-PCRs.

We first used the pls (plasmin-sensitive protein) gene to identify the J1 region of the type I.1 SCC mec , but that approach was changed since only 12 of 17 strains were positive for this gene ( 6 ). In addition, the pls gene was identified with the chromosomal DNA from a type III SCC mec strain, suggesting that it could not be a specific marker for the J1 region of type I SCC mec elements. Oliveira and Lencastre and Zhang et al. also designed primers specific to the J1 region of the type I SCC mec in regions other than the pls gene. The J region of 16 of 17 type I SCC mec strains was classified as subtype 1 with both the primers described previously by Oliveira et al. and our primers.

The identification of the J1 region of the type III SCC mec element was a bit confusing because we first reported the nucleotide sequence of the type III SCC mec element carried by strain 85/2082 as being the longest one, but this turned out to be a composite of SCC mercury and a type III SCC mec element (as indicated in Fig. ​ Fig.1, 1 , the length of the type III SCC mec element is different from that originally reported). Both Oliveira and Lencastre and Zhang et al. happened to design primers on the nucleotide sequence of SCC mercury for the identification of type III SCC mec elements. Therefore, the identification of locus E and locus F in the M-PCR described previously by Oliveira and Lencastre and the identification of the locus used by Zhang et al. indicate the likely presence of SCC mercury and are not specific for type III SCC mec elements.

The identification of the J1 regions of type IV SCC mec elements is similar with our and primers and those described previously by Zhang et al., except for a primer pair used for the identification of a type IV.3 SCC mec element designed by Zhang et al. on a locus outside the type IV.3 SCC mec , designated IE25923.

Prospects for assignment of SCC mec elements.

The typing system designed here is not final and should be developed further, since it could not identify every known difference. For example, some differences in the J3 regions of the type IV SCC mec element, such as the carriage of Tn 4001 in the type IV SCC mec of strain 81/108, the different J3 region structures in type IVE and type IVF SCC mec elements reported previously by Shore et al. ( 23 ), and the presence of dcs in the type III SCC mec element reported previously by Chongtrakool et al. ( 2 ), were not identified with our system. Although the structure of the J1 region is rather specific to each type and correlates well with the SCC mec type, we want to emphasize that the identification of the J region did not correlate exactly with the type of SCC mec element. The type II.3 MRSA strain isolated in Ireland is a good example: if only the multiplex PCR identifying the J1 region had been conducted, it would have been classified as type IV.2.

It is not easy to conduct six M-PCRs for every case. We suggest that M-PCRs 1 and 2, for identifying ccr and mec , should be conducted first to assign types of SCC mec elements, and they might be enough in most of the cases for epidemiological purposes. In cases where further typing is required, we suggest to proceed with identification of the J1 region structure with M-PCR 3 or 4 or with M-PCRs developed by Zhang et al., since the structure of J1 region might reflect the structure of an SCC in which the mec gene complex was integrated. The remaining two M-PCRs, M-PCR 5 and M-PCR 6, should be conducted, if necessary, for additional typing.

Although it might be difficult to determine all SCC mec types carried by staphylococci, the determination of as many unknown SCC mec types as possible would be of help for epidemiological studies as well as for inferring the origins of MRSA strains. Further discussion is needed in order to form a consensus among staphylococcal researchers regarding how to define SCC mec element types and how to assign SCC mec elements to cope with the ever-increasing diversity of SCC mec elements.

Acknowledgments

This work was supported by a Grant-in-Aid for 21st Century COE Research and a Grant-in-Aid for Scientific Research on Priority Areas (13226114) from the Ministry of Education, Science, Sports, Culture, and Technology of Japan.

▿ Published ahead of print on 16 October 2006.

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Say What? Chat With RTX Brings Custom Chatbot to NVIDIA RTX AI PCs

Chatbots are used by millions of people around the world every day, powered by NVIDIA GPU-based cloud servers. Now, these groundbreaking tools are coming to Windows PCs powered by NVIDIA RTX for local, fast, custom generative AI .

Chat with RTX , now free to download , is a tech demo that lets users personalize a chatbot with their own content, accelerated by a local NVIDIA GeForce RTX 30 Series GPU or higher with at least 8GB of video random access memory, or VRAM.

Ask Me Anything

Chat with RTX uses retrieval-augmented generation (RAG), NVIDIA TensorRT-LLM software and NVIDIA RTX acceleration to bring generative AI capabilities to local, GeForce-powered Windows PCs. Users can quickly, easily connect local files on a PC as a dataset to an open-source large language model like Mistral or Llama 2, enabling queries for quick, contextually relevant answers.

Rather than searching through notes or saved content, users can simply type queries. For example, one could ask, “What was the restaurant my partner recommended while in Las Vegas?” and Chat with RTX will scan local files the user points it to and provide the answer with context.

The tool supports various file formats, including .txt, .pdf, .doc/.docx and .xml. Point the application at the folder containing these files, and the tool will load them into its library in just seconds.

Users can also include information from YouTube videos and playlists. Adding a video URL to Chat with RTX allows users to integrate this knowledge into their chatbot for contextual queries. For example, ask for travel recommendations based on content from favorite influencer videos, or get quick tutorials and how-tos based on top educational resources.

type assignment system

Since Chat with RTX runs locally on Windows RTX PCs and workstations, the provided results are fast — and the user’s data stays on the device. Rather than relying on cloud-based LLM services, Chat with RTX lets users process sensitive data on a local PC without the need to share it with a third party or have an internet connection.

In addition to a GeForce RTX 30 Series GPU or higher with a minimum 8GB of VRAM, Chat with RTX requires Windows 10 or 11, and the latest NVIDIA GPU drivers.

Editor’s note: We have identified an issue in Chat with RTX that causes installation to fail when the user selects a different installation directory. This will be fixed in a future release. For the time being, users should use the default installation directory (“C:\Users\<username>\AppData\Local\NVIDIA\ChatWithRTX”).

Develop LLM-Based Applications With RTX

Chat with RTX shows the potential of accelerating LLMs with RTX GPUs. The app is built from the TensorRT-LLM RAG developer reference project, available on GitHub . Developers can use the reference project to develop and deploy their own RAG-based applications for RTX, accelerated by TensorRT-LLM. Learn more about building LLM-based applications .

Enter a generative AI-powered Windows app or plug-in to the NVIDIA Generative AI on NVIDIA RTX developer contest, running through Friday, Feb. 23, for a chance to win prizes such as a GeForce RTX 4090 GPU, a full, in-person conference pass to NVIDIA GTC and more.

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NYC fails controversial remote-learning snow day ‘test,’ public schools chancellor says

Image: Large Winter Storm Brings Snow To The Northeast

New York City's public schools chancellor said the city did not pass Tuesday's remote-learning “test” because of technical issues.

“As I said, this was a test. I don’t think that we passed this test,” David Banks said at a news briefing, adding that he felt "disappointed, frustrated and angry" as a result of the technical issues.

NYC Public Schools did a lot of work to prepare for the remote-learning day, Banks said, but shortly before 8 a.m. they were notified that parents and students were having difficulty signing on to remote learning.

Follow along for live coverage of the storm

It is the first time the school system has implemented remote learning on a snow day since it introduced the no-snow-day policy in 2022. The district serves 1.1 million students in more than 1,800 schools.

Banks blamed the technical issues on IBM, which helps facilitate the city’s remote-learning program.

“IBM was not ready for prime time,” Banks said, adding that the company was overwhelmed with the surge of people signing on for school.

IBM has since expanded its capacity, and 850,000 students and teachers are currently online, Banks said.

“We’ll work harder to do better next time,” he said, adding that there will be a deeper analysis into what went wrong.

The new system is controversial among parents who lament the end of the snow days of their childhoods, dread a return to the frustrations that remote learning caused during the pandemic and argue that online learning is a far cry from the classroom.

On Monday, Mayor Eric Adams said parents who are not willing to navigate computers for their children’s remote learning represent “a sad commentary.”

Adams defended his words Tuesday, saying they were related to a specific question he was asked about parents who do not want to sign on to remote learning.

“That is not the energy we should be showing right now. Our children have to catch up. They need to be engaged,” he said.

Adams also blamed IBM for Tuesday’s remote-learning issues, saying he hopes the company will be able to provide the product the city is paying it for.

“IBM, I’m hoping this was a teaching moment for them, as well," Adams said.

In a statement, IBM said it has been working closely with New York City Public Schools "to address this situation as quickly as possible."

An IBM spokesperson said, “The issues have been largely resolved, and we regret the inconvenience to students and parents across the city."

New York City Public Schools were the outlier in implementing remote learning during Tuesday's storm. Hundreds of districts in Boston , Connecticut , Philadelphia  and  New York were shuttered for snow days.

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  1. Assignment 0

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COMMENTS

  1. Type system

    In computer programming, a type system is a logical system comprising a set of rules that assigns a property called a type (for example, integer, floating point, string) to every term (a word, phrase, or other set of symbols).

  2. Intersection type discipline

    The Barendregt-Coppo-Dezani type assignment system extends the Coppo-Dezani type assignment system in the following three aspects: ( ⊢ BCD ) {\displaystyle (\vdash _{\text{BCD}})} introduces the universal type constant ω {\displaystyle \omega } (akin to the empty intersection) that can be assigned to any λ-term.

  3. Relation between Type Assignment system (TA) and Hindley-Milner system

    The last one is also known as Type Assignment system (TA). I'm thinking about the relations between TA and Hindley-Milner (HM), the system in languages like ML and Haskell. The book Lambda-Calculus and Combinators: An Introduction (Hindley) says that TA is polymorphic (pag. 119).

  4. A type assignment system for game semantics

    Traditionally,typeassignmentsystemsdescribethesemanticinterpretationoftermsindomain-theoreticmodels.Quitesurprisingly, the type assignment system presented in this paper is very similar to the traditional ones, the main difference being the omission of the subtyping rules. c© 2008 Elsevier B.V. All rights reserved.

  5. A type assignment system for game semantics

    Traditionally, type assignment systems describe the semantic interpretation of terms in domain-theoretic models. Quite surprisingly, the type assignment system presented in this paper is very similar to the traditional ones, the main difference being the omission of the subtyping rules.

  6. PDF Intersection Type Assignment Systems

    A disadvantage of the BCD-system is that type assignment in this system is undecidable. In recent years, some decidable restrictions have been studied. The first was the Rank2 intersection type assignment system [2], as first suggested by D. Leivant in [26], that is very close to the notion of type assignment as used in ML.

  7. PDF Quick Introduction to Type Systems

    type is a description that characterizes the expected form of the result of a computation. In particular, if e is an expression, a typing : int is an assertion that when e is evaluated, its value will be an integer. We call this assertion a typing judgment.

  8. PDF Principal Type Schemes for the Strict Type Assignment System

    The first type assignment system with intersection types was presented in [7]; the CDV-system with intersection types and ω is introduced in [9] and in [23]. The best-known intersection type assignment system is the BCD-system, as presented in [6], that is an extension of the CDV-system: there it is strengthened further by introducing a

  9. TypeScript: Documentation

    The basic rule for TypeScript's structural type system is that x is compatible with y if y has at least the same members as x. For example consider the following code involving an interface named Pet which has a name property: interface Pet { name: string; } let pet: Pet; let dog = { name: "Lassie", owner: "Rudd Weatherwax" }; pet = dog;

  10. F-semantics for type assignment systems

    In general, a type assignment system Tx is said to be complete for models, simple models, or F-models when F h- M : Q is derived in Tx iff it is satisfied by every model, simple model, or F-model, respectively. Mitchell [19] proved that Tdfl is complete for all models. For simple models, Tv# is not complete. He introduced the following rule ...

  11. A type assignment system for game semantics

    DOI: 10.1016/j.tcs.2008.01.023 Corpus ID: 146730; A type assignment system for game semantics @article{Gianantonio2008ATA, title={A type assignment system for game semantics}, author={Pietro Di Gianantonio and Furio Honsell and Marina Lenisa}, journal={Theor.

  12. A type assignment system for game semantics

    A Type Assignment System For The Game Semantics 1993 • Gianluca Franco this paper an alternative description of the game semantics for the untyped lambda calculus is given. More precisely, we introduce a finitary description of lambda terms.

  13. Principal Type Schemes for the Strict Type Assignment System

    The strict type assignment system, a restriction of the intersection type discipline, has the principal type property, and it is proved that it has all pairs deducible for M can be obtained from the principal one by these operations. We study the strict type assignment system, a restriction of the intersection type discipline (6), and prove that it has t he principal type property. We define ...

  14. A type assignment system for game semantics

    Particularly related to the topic of this paper is the system defined by De Carvalho in [10], inspired by [8], where a logical semantics of λ-calculus has been given, using a type assignment ...

  15. Introduction to generalized type systems

    Berardi (1988) and Terlouw (1988) have independently generalized the method of constructing systems in the λ-cube. Moreover, Berardi (1988, 1990) showed that the generalized type systems are flexible enough to describe many logical systems. In that way the well-known propositions-as-types interpretation obtains a nice canonical form.

  16. A Soft Type Assignment System for lambda -Calculus

    A type assignment system for the λ-calculus (STA) is designed, which assigns toλ-terms as types (a proper subset of) SLL formulas, in such a way that typable terms inherit the good complexity properties of the logical system. Soft Linear Logic (SLL) is a subsystem of second-order linear logic with restricted rules for exponentials, which is correct and complete for PTIME.

  17. PDF A Soft Type Assignment System for λ-Calculus

    polynomial complexity. A type assignment for λ-calculus correct and complete with respect to polynomial time computations has been designed by Baillot and Terui [4], using as types formulae of Light Affine Logic (LAL), a simplified ver-sionof LLL defined in [5,6]. The aim of this paper is to designa type assignment system for λ-calculus ...

  18. CS 242: Type systems

    Type systems PDF version of this page. Recall from last week that we had a big issue with our lambda calculus. We were able to construct programs with undefined behavior, i.e. ones that would evaluate to a stuck state, by having free variables in our expressions, like (λ x . x) y

  19. A Type Assignment System For The Game Semantics

    this paper an alternative description of the game semantics for the untyped lambda calculus is given. More precisely, we introduce a finitary description of lambda terms. This description turns out to be equivalent to a particular game denotational

  20. Combination of Multiplex PCRs for Staphylococcal Cassette Chromosome

    The SCC mec type assignments were identical to those made with a PCR system that uses numerous primer pairs to identify genes or gene alleles. Our system of six M-PCRs is thus a convenient and reliable method for typing SCC mec elements.

  21. A Type Assignment System For Game Semantics

    Quite surprisingly, the type assignment system presented in this paper is very similar to the traditional ones, the main difference being the omission of the subtyping rules. Key words: Lambda Calculus, Game Semantics, Type Assignment System 1 Introduction About twentyfive years ago, Mario Coppo, Mariangiola Dezani, and their group, started to ...

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    OpenAI on Thursday teased its text-to-video artificial intelligence model Sora, which can generate videos up to a minute long based prompts users type into a text box.

  23. Example of a Quality Management System (With Definition and Types)

    Types of quality management systems The type of quality management system you use can depend upon your industry. For example, food and medical products may need to use a specific system to meet health guidelines. Companies may design their own quality management systems, or pursue certification through the ISO, by meeting their standard systems.

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