Intuitionistic ancestral logic

2019 ◽  
Vol 29 (4) ◽  
pp. 469-486 ◽  
Author(s):  
Liron Cohen ◽  
Robert L Constable
Keyword(s):  
Programming Language ◽  
Type Theory ◽  
Transitive Closure ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Constructive Type Theory ◽  
Programming Logic ◽  
Natural Type ◽  
Definition Of

Abstract In this article we define pure intuitionistic Ancestral Logic ( iAL ), extending pure intuitionistic First-Order Logic ( iFOL ). This logic is a dependently typed abstract programming language with computational functionality beyond iFOL given by its realizer for the transitive closure, TC . We derive this operator from the natural type theoretic definition of TC using intersection. We show that provable formulas in iAL are uniformly realizable, thus iAL is sound with respect to constructive type theory. We further show that iAL subsumes Kleene Algebras with tests and thus serves as a natural programming logic for proving properties of program schemes. We also extract schemes from proofs that iAL specifications are solvable.

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.

scholarly journals Formula size games for modal logic and μ-calculus

2019 ◽  
Vol 29 (8) ◽  
pp. 1311-1344 ◽  
Author(s):  
Lauri T Hella ◽  
Miikka S Vilander
Keyword(s):  
Modal Logic ◽  
Optimal Strategy ◽  
Order Logic ◽  
First Order Logic ◽  
Kripke Models ◽  
First Order ◽  
Modal Operators ◽  
Crucial Difference ◽  
Definition Of ◽  
Person Game

Abstract We propose a new version of formula size game for modal logic. The game characterizes the equivalence of pointed Kripke models up to formulas of given numbers of modal operators and binary connectives. Our game is similar to the well-known Adler–Immerman game. However, due to a crucial difference in the definition of positions of the game, its winning condition is simpler, and the second player does not have a trivial optimal strategy. Thus, unlike the Adler–Immerman game, our game is a genuine two-person game. We illustrate the use of the game by proving a non-elementary succinctness gap between bisimulation invariant first-order logic $\textrm{FO}$ and (basic) modal logic $\textrm{ML}$. We also present a version of the game for the modal $\mu $-calculus $\textrm{L}_\mu $ and show that $\textrm{FO}$ is also non-elementarily more succinct than $\textrm{L}_\mu $.

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.).

Church’s theorem and the decision problem

2018 ◽  
Author(s):  
Rohit Parikh
Keyword(s):  
Decision Problem ◽  
Combinatorial Problem ◽  
Order Logic ◽  
First Order Logic ◽  
Negative Solution ◽  
Great Contribution ◽  
Solvable Problem ◽  
First Order ◽  
Mathematical Definition ◽  
Definition Of

Church’s theorem, published in 1936, states that the set of valid formulas of first-order logic is not effectively decidable: there is no method or algorithm for deciding which formulas of first-order logic are valid. Church’s paper exhibited an undecidable combinatorial problem P and showed that P was representable in first-order logic. If first-order logic were decidable, P would also be decidable. Since P is undecidable, first-order logic must also be undecidable. Church’s theorem is a negative solution to the decision problem (Entscheidungsproblem), the problem of finding a method for deciding whether a given formula of first-order logic is valid, or satisfiable, or neither. The great contribution of Church (and, independently, Turing) was not merely to prove that there is no method but also to propose a mathematical definition of the notion of ‘effectively solvable problem’, that is, a problem solvable by means of a method or algorithm.

A proof-theoretic treatment of λ-reduction with cut-elimination: λ-calculus as a logic programming language

2011 ◽  
Vol 76 (2) ◽  
pp. 673-699 ◽  
Author(s):  
Michael Gabbay
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Functional Programming ◽  
Order Logic ◽  
First Order Logic ◽  
Cut Elimination ◽  
Logic Programming Language ◽  
First Order ◽  
Functional Programming Language ◽  
Sequent System

AbstractWe build on an existing a term-sequent logic for the λ-calculus. We formulate a general sequent system that fully integrates αβη-reductions between untyped λ-terms into first order logic.We prove a cut-elimination result and then offer an application of cut-elimination by giving a notion of uniform proof for λ-terms. We suggest how this allows us to view the calculus of untyped αβ-reductions as a logic programming language (as well as a functional programming language, as it is traditionally seen).

scholarly journals VERIFYING PROBABILISTIC PROGRAMS USING A HOARE LIKE LOGIC

2002 ◽  
Vol 13 (03) ◽  
pp. 315-340 ◽  
Author(s):  
J. I. DEN HARTOG ◽  
E. P. DE VINK
Keyword(s):  
Programming Language ◽  
Denotational Semantics ◽  
Order Logic ◽  
First Order Logic ◽  
Proof System ◽  
Probabilistic Choice ◽  
First Order ◽  
Probabilistic Programs

Probability, be it inherent or explicitly introduced, has become an important issue in the verification of programs. In this paper we study a formalism which allows reasoning about programs which can act probabilistically. To describe probabilistic programs, a basic programming language with an operator for probabilistic choice is introduced and a denotational semantics is given for this language. To specify propertics of probabilistic programs, standard first order logic predicates are insufficient, so a notion of probabilistic predicates is introduced. A Hoare-style proof system to check properties of probabilistic programs is given. The proof system for a sublanguage is shown to be sound and complete; the properties that can be derived are exactly the valid properties. Finally some typical examples illustrate the use of the probabilistic predicates and the proof system.

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