scholarly journals Subformula Linking for Intuitionistic Logic with Application to Type Theory

2021 ◽  
pp. 200-216
Author(s):  
Kaustuv Chaudhuri
Keyword(s):  
Theorem Proving ◽  
Intuitionistic Logic ◽  
Type Theory ◽  
Linear Logic ◽  
Interactive Theorem Proving ◽  
Order Logic ◽  
First Order Logic ◽  
Direct Manipulation ◽  
First Order

AbstractSubformula linking is an interactive theorem proving technique that was initially proposed for (classical) linear logic. It is based on truth and context preserving rewrites of a conjecture that are triggered by a user indicating links between subformulas, which can be done by direct manipulation, without the need of tactics or proof languages. The system guarantees that a true conjecture can always be rewritten to a known, usually trivial, theorem. In this work, we extend subformula linking to intuitionistic first-order logic with simply typed lambda-terms as the term language of this logic. We then use a well known embedding of intuitionistic type theory into this logic to demonstrate one way to extend linking to type theory.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

scholarly journals Formalization of Linear Space Theory in the Higher-Order Logic Proving System

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Jie Zhang ◽  
Danwen Mao ◽  
Yong Guan
Keyword(s):  
Linear Space ◽  
Theorem Proving ◽  
Predicate Logic ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Space Theory ◽  
Higher Order Logic ◽  
First Order ◽  
Important Approach

Theorem proving is an important approach in formal verification. Higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and stronger semantics. Higher-order logic is more expressive. This paper presents the formalization of the linear space theory in HOL4. A set of properties is characterized in HOL4. This result is used to build the underpinnings for the application of higher-order logic in a wider spectrum of engineering applications.

scholarly journals Checkable Proofs for First-Order Theorem Proving

10.29007/s6d1 ◽  
2018 ◽  
Author(s):  
Giles Reger ◽  
Martin Suda
Keyword(s):  
Theorem Proving ◽  
Critical Mass ◽  
Current Situation ◽  
Boolean Satisfiability ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Theorem Provers ◽  
Interactive Theorem Provers ◽  
Theory Reasoning

Inspired by the success of the DRAT proof format for certification of boolean satisfiability (SAT),we argue that a similar goal of having unified automatically checkable proofs should be soughtby the developers of automated first-order theorem provers (ATPs). This would not onlyhelp to further increase assurance about the correctness of prover results,but would also be indispensable for tools which rely on ATPs,such as ``hammers'' employed within interactive theorem provers.The current situation, represented by the TSTP format is unsatisfactory,because this format does not have a standardised semantics and thus cannot be checked automatically.Providing such semantics, however, is a challenging endeavour. One would ideallylike to have a proof format which covers only-satisfiability-preserving operations such as Skolemisationand is versatile enough to encompass various proving methods (i.e. not just superposition)or is perhaps even open ended towards yet to be conceived methods or at least easily extendable in principle.Going beyond pure first-order logic to theory reasoning in the style of SMT orbeyond proofs to certification of satisfiability are further interesting challenges.Although several projects have already provided partial solutions in this direction,we would like to use the opportunity of ARCADE to further promote the idea andgather critical mass needed for its satisfactory realisation.

scholarly journals Different Proofs are Good Proofs

10.29007/kwk9 ◽  
2018 ◽  
Author(s):  
Geoff Sutcliffe ◽  
Cynthia Chang ◽  
Li Ding ◽  
Deborah McGuinness ◽  
Paulo Pinheiro da Silva
Keyword(s):  
Theorem Proving ◽  
Automated Theorem Proving ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Problem Library

In order to compare the quality of proofs, it is necessary to measure artifacts of the proofs, and evaluate the measurements to determine differences between the proofs. This paper discounts the approach of ranking measurements of proof artifacts, and takes the position that different proofs are good proofs. The position is based on proofs in the TSTP solution library, which are generated by Automated Theorem Proving (ATP) systems applied to first-order logic problems in the TPTP problem library.

scholarly journals Completeness theorems for first-order logic analysed in constructive type theory

2021 ◽  
Vol 31 (1) ◽  
pp. 112-151
Author(s):  
Yannick Forster ◽  
Dominik Kirst ◽  
Dominik Wehr
Keyword(s):  
Type Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
The Standard Model ◽  
Coq Proof Assistant ◽  
Game Theoretic ◽  
Completeness Theorems

Abstract We study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov’s Principle and Weak K̋nig’s Lemma and discuss non-standard semantics admitting assumption-free completeness. We contribute a reusable Coq library for first-order logic containing all results covered in this paper.

Logic in the 1930s: Type Theory and Model Theory

2013 ◽  
Vol 19 (4) ◽  
pp. 433-472 ◽  
Author(s):  
Georg Schiemer ◽  
Erich H. Reck
Keyword(s):  
Twentieth Century ◽  
Model Theory ◽  
Type Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Gradual Transformation ◽  
Standard Framework ◽  
Theory And Model ◽  
Made In

AbstractIn historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style of Principia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics of axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition (Carnap and Tarski) and two proponents of the first-order tradition (Gödel and Hilbert), we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory.

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