Summary Ill-Typed Programs Dont Evaluate Two-Sided Type Systems arxiv.org
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The text explores the evaluation and reasoning of ill-typed programs in two-sided type systems, covering various cases, abstractions, term construction, and providing a proof.
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Key Points
- Two-sided type systems allow for hypothetical reasoning and guarantee that well-typed programs don't go wrong and ill-typed programs don't evaluate.
- The document presents a two-sided constrained type system for a functional programming language with datatype constructors and pattern matching, showing that types can be automatically inferred and are principal.
- The system classifies functions that return a value only if a specific input is provided, and defines type disjointness as two types not having any values in common.
- The rules in the system handle let expressions, if-zero expressions, application evaluation, and the successor and predecessor functions.
- The document discusses the evaluation of ill-typed programs, the concept of getting stuck, and the construction of terms. It also explores the possibility of internalizing negation and presents a success semantics that weakens typing on the right side.
Summaries
37 word summary
Two-sided type systems allow for reasoning over program expressions and prevent ill-typed programs from evaluating. The text discusses evaluation of ill-typed programs, presenting different cases and reasoning, evaluating abstractions and constructing terms, and concludes with a proof.
83 word summary
Two-sided type systems are discussed in the document, which allow for hypothetical reasoning over compound program expressions and the refutation of typing formulas. These type systems ensure that well-typed programs do not go wrong and that ill-typed programs do not evaluate.
This text excerpt discusses the evaluation of ill-typed programs in two-sided type systems. It presents different cases and reasoning to obtain bounds, as well as the evaluation of abstractions and the construction of terms. The text concludes with the proof of Theore
1396 word summary
Two-sided type systems allow for hypothetical reasoning over the typing of compound program expressions and the refutation of typing formulas. They guarantee that well-typed programs don't go wrong and that ill-typed programs do not evaluate. This makes them suitable for incorrect
The document presents a two-sided constrained type system for a functional programming language with datatype constructors and pattern matching. It shows that types can be automatically inferred and are principal. The paper also introduces an alternative success semantics for two-sided judgements and derives a one
The types in the two-sided type system are defined by a grammar. The system classifies functions that return a value only if a specific input is provided. Type disjointness is defined as two types not having any values in common. The two-sided type
The rules (LetL1) and (LetL2) allow for reasoning about let expressions on the left. The rules (IfZL1) and (IfZL2) are used for reasoning about the if-zero expression on the left
Rules (OkApL1) and (OkApL2) express the requirement for an application to evaluate, while (OkSL) and (OkPL) express that the successor and predecessor are only defined on numerals. The (OkPr
Using (Dis), we can prove ? . pred() : Nat x Nat Nat x Nat. The semantics of the typing judgement state that : on the left means that evaluates to an , and : on the right means that evaluates to an or diverges
The excerpt discusses the corollary that states if a term is well-typed, it does not go wrong, and if it is ill-typed, it does not evaluate. It also presents two counterexamples to show that allowing higher-types on the
The document discusses a type system designed for reasoning about the shape of datatype constructions. It is a constrained type system that uses type constraints to restrict possible instantiations of type variables. The types in this system are a simplified version of the types defined in the
The types are stratified into sum types, monotypes, and type schemes. Type schemes are closed with no free type variables. The sum type is a finite set with orthogonal elements. Recursive types are captured implicitly with constructor types and type constraints. Sub
Ill-typed programs don't evaluate in two-sided type systems. The rules (CnsL), (CnsR), and (CnsK) handle constructor types. The (MchL) and (MchR) rules type pattern
Constrained type inference in two-sided type systems is similar to Hindley-Milner type inference, but with some differences. The inference algorithm requires both the left and right environments as input. The algorithm terminates because every proof tree has a maximum height determined
Syntactic soundness is achieved in the system. The authors explore the possibility of internalizing negation by adding a complement operator to types, but encounter difficulties due to asymmetry. They propose a success semantics that weakens typing on the right side
Two-sided type systems provide a complete treatment of the necessity arrow, allowing for the elimination of certain rules. By exploiting the symmetry of the typing rules, a one-sided system can be obtained by exchanging formulas between the two sides. The one-sided system is
A bespoke function type called the success typing of a function is defined. Success types are highly effective in practice because they can refute that an application has a certain type when it reduces to a value. Incorrectness logics, such as O'Hearn's
The summary is not provided.
The excerpt discusses the concept of getting stuck in ill-typed programs and presents a characterization of getting stuck. It defines different classes of values and identifies various forms of stuck terms. The excerpt also introduces the concept of reduction and presents a lemma that demonstrates the
(Beta). [ /] is of shape ( . ) and 2 = [ /]. Let ? be ? [ ? /], then ? [ /] = ( ? [ ? /]) [ /] = ( ? [ /]) [ ?
The text excerpt discusses the evaluation of ill-typed programs in two-sided type systems. The proof is presented using induction on the structure of the program. Lemmas A.5 and A.6 are introduced to support the proof. The first part of
When the shape of ? is of type ?, it is necessary to prove (i). If [ /] diverges or [ /] ? ?, then [/] diverges or there is some ? such that [/] ? ?. When the shape is ?,
The text discusses the evaluation of ill-typed programs in two-sided type systems. It presents different cases and reasoning to obtain bounds. It also discusses the evaluation of abstractions and the construction of terms. The text concludes with the proof of Theorems
Ill-typed programs don't evaluate in two-sided type systems. The summary is 10 words in length.
The section contains an algorithmic version of the constrained type system rules, along with a proof of its correctness. It also includes a type inference algorithm and its correctness proof. The theorem states the soundness of the algorithmic type assignment. The proof is
The proof shows completeness of algorithmic type assignment for ill-typed programs. It proceeds by induction on the derivation of a given algorithm. Different cases are considered, such as (GVar), (LVar), (SubL), (SubR),
By applying the inductive hypothesis, the text demonstrates that certain type systems hold for various cases such as (AppL2), (AppR), (CnsL), (CnsR), (MchL), (MchR), (
The document discusses the constrained type system inference algorithm and its correctness proof. It examines different cases, such as (CnsDL), (Var), (App), (Abs), and (Fix), and provides explanations and proofs for each case. The document
Let ? be a type variable substitution and ? be the inferred constraints. By (FixR2), we have ? I ? 2 fix. In the section of interest, the judgments provable with (CnsR2) are discussed. For
Taking := ? [ ? ] trivially satisfies the requirements. In InferL, the inferred judgment is: ? I (Ok ? ) ? : . For some , we have ? I ? 2 . : , ?, : 1 I ?
The excerpt discusses the denition of reduction and consistency in a constrained type system. It also introduces the concept of single-subject (1S) judgments and provides a lemma stating that a proof tree of a 1S conclusion only contains 1S
If the root is concluded by (CnsR), then the 1S conclusion has shape ? ? (1,...,) : (1,...,) and ? is a consistent variable environment. If the root is concluded by (MchL), then
The text discusses the soundness and completeness of algorithmic type assignment in the context of ill-typed programs. It presents different cases for reasoning about the type of expressions, including local variables, identifiers, applications, abstractions, and constructor terms. The
The excerpt discusses the soundness and completeness of algorithmic type assignment and its implications for evaluating ill-typed programs in two-sided type systems. It also presents Lemmas C.8, C.9, and C.10, which provide proofs and
Ill-typed programs don't evaluate in two-sided type systems. The proof of Theorem 5.10 (Progress) is given through induction on the derivation. The text discusses different cases, including top-level identifiers, abstractions, constructors, match
In this excerpt, the authors discuss various cases and proofs related to ill-typed programs and two-sided type systems. They analyze different scenarios and provide reasoning and conclusions based on the given assumptions and hypotheses. The proofs are organized into separate cases based on the
Ill-typed programs don't evaluate in two-sided type systems. The proof uses induction and various inference rules to reason about match expressions, fix expressions, and substitution. Lemma C.12 states that a trivial substitution holds for consistent variable environments. Lemma C
The proof is by induction on different cases. If a variable is Ok, then the desired conclusion follows. If a variable is not Ok, the reasoning is similar. If an application is present, it follows from inversion and Lemma C.12 that the
The summary of the excerpt is as follows: The result follows from the induction hypothesis and (AppR). If the context is of shape match E with (I =1 ( ?? )), it follows from inversion that there is some family and ? ?
The result follows from various rules of the system, such as (App), (OkApL1), (OkApL2), (App2), (OkL), (OkC2), (OkR), (Ok), (OkSL
The text discusses different cases and hypotheses related to the evaluation of ill-typed programs in two-sided type systems. It mentions the contradiction that arises when certain conditions are met and provides explanations based on induction hypotheses, shape of expressions, and substitution lemma. The