## PROPOSITIONAL LOGIC (2008)

### BibTeX

@MISC{Dorais08propositionallogic,

author = {François Dorais and Timothy Goldberg and Bakhadyr Khoussainov},

title = {PROPOSITIONAL LOGIC},

year = {2008}

}

### OpenURL

### Abstract

Intuitionistic logic is an important variant of classical logic, but it is not as wellunderstood, even in the propositional case. In this thesis, we describe a faithful representation of intuitionistic propositional formulas as tree automata. This representation permits a number of consequences, including a characterization theorem for free Heyting algebras, which are the intutionistic analogue of free Boolean algebras, and a new algorithm for solving equations over intuitionistic propositional logic. BIOGRAPHICAL SKETCH

### Citations

236 |
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(Show Context)
Citation Context ...anguage is also S. We may assume that T is injective. It follows that T is already prefix-closed. We will show that the reduction of T in the sense of the Myhill-Nerode theorem for tree automata (see =-=[6]-=-) satisfies the requirements. By the Myhill-Nerode theorem for tree automata, there is an automaton T ′ which is equivalent to T and which we may also take to be injective with the following property:... |

141 |
Proofs and types, volume 7 of Cambridge Tracts
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(Show Context)
Citation Context ...s and existence proofs require witnesses. Higher-order intuitionistic systems which can express a great deal of mathematics, such as Girard’s System F and MartinLöf’s type theory (good references are =-=[9]-=- and [3]), have been developed and implemented by prominent computer scientists such as Constable, Huet and Coquand (see [2] and [1]). With all this development and with the existence of well-establis... |

28 |
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(Show Context)
Citation Context ...ely as a finite join of pairwise incomparable join-irreducibles, a property which is not shared by, for example, the countable atomless Boolean algebra, which does not have any join-irreducibles (see =-=[10]-=- for a general reference on lattice theory). Therefore if we could characterize the set Jn of join-irreducibles in Hn, we would have a characterization of Hn. Each intuitionistic-equivalence-respectin... |

17 |
Finitely generated free Heyting algebras
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(Show Context)
Citation Context ...rable elements, then there must be some i ̸= j and s ∈ S with s < s i, s j. This is used in Subsection 4.2.3 to show constructively that Hn is incomplete for n ≥ 2 (this is shown nonconstructively in =-=[4]-=-) and in Proposition 13 to show that the countable atomless Boolean algebra does not lattice-embed into any Hn. Call an element a of a lattice L join-irreducible if for all b, c ∈ L, if a = b ∨ c then... |

17 |
On formulas of one variable in intuitionistic propositional calculus
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(Show Context)
Citation Context ...16]. By a slight modification of Proposition 30, we get that the embedding of Hn into the full subalgebra of upward-closed subsets of U(n) is existentially closed. It was first determined in [15] and =-=[14]-=- that for all n, there are infinitely many intuitionistically inequivalent propositional formulas over n variables. A natural question is then to ask whether various fragments of intuitionistic propos... |

14 |
Lattices of Intermediate and Cylindric Modal Logics
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(Show Context)
Citation Context ...let the alphabet that a Kripke tree is over be An. Definition 15 (Kripke Alphabet). For n ∈ ω, let An = P(Vn), considered as an alphabet. We may also call An the Kripke alphabet over Vn. Lemma 7 (See =-=[5]-=-). A propositional formula φ ∈ Fn is a intuitionistic tautology iff for all finite Kripke models M, defined over Vn, M ⊩ φ. Proof. This follows from the fact that all finite Kripke models are p-morphi... |

11 |
Spaces, volume 3 of Cambridge studies in advanced mathematics
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(Show Context)
Citation Context ...rements: 1. 〈H, ∨, ∧, ⊥, ⊤〉 is a distributive lattice with 0 and 1. 2. For all a, b ∈ H, a → b = sup{c | c ∧ a ≤ b}. As usual, an order relation ≤ is defined on H by a ≤ b iff a ∧ b = a. Lemma 1 (See =-=[12]-=-). The class of Heyting algebras is a variety. Definition 3 (Free Heyting Algebras). The existence of free Heyting algebras is given by Lemma 1. For n ∈ ω + 1, the free Heyting algebra on n generators... |

9 |
Rules of inference with parameters for intuitionistic logic
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(Show Context)
Citation Context ...an T has. As a corollary to this, we get that the problem of deciding whether or not equations φ(x) = ⊤ in intuitionistic logic have solutions is decidable, a fact which was first shown by Rybakov in =-=[16]-=-. By a slight modification of Proposition 30, we get that the embedding of Hn into the full subalgebra of upward-closed subsets of U(n) is existentially closed. It was first determined in [15] and [14... |

8 |
Foundations of Constructive Mathematics, volume 6 of Ergebnisse der Mathematik and ihrer Grenzgebiete . Springer-Verlag
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(Show Context)
Citation Context ...istence proofs require witnesses. Higher-order intuitionistic systems which can express a great deal of mathematics, such as Girard’s System F and MartinLöf’s type theory (good references are [9] and =-=[3]-=-), have been developed and implemented by prominent computer scientists such as Constable, Huet and Coquand (see [2] and [1]). With all this development and with the existence of well-established topo... |

7 |
Sur les Algèbres de Hilbert
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(Show Context)
Citation Context ...nal logic obtained by restricting the connectives allowed are also infinite. 3The most difficult case is that where the single connective → is allowed, and this case was solved by algebraic means in =-=[8]-=-, where it is proven that for all n, the number of intuitionistically inequivalent formulas over n variables using only the connective → is finite. Subsequent proofs of this fact using semantic method... |

6 |
On the lattice theory of Brouwerian propositional logic. Acta fac. rerum
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(Show Context)
Citation Context ...akov in [16]. By a slight modification of Proposition 30, we get that the embedding of Hn into the full subalgebra of upward-closed subsets of U(n) is existentially closed. It was first determined in =-=[15]-=- and [14] that for all n, there are infinitely many intuitionistically inequivalent propositional formulas over n variables. A natural question is then to ask whether various fragments of intuitionist... |

6 |
Implicational formulas in intuitionistic logic
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- 1974
(Show Context)
Citation Context ...the number of intuitionistically inequivalent formulas over n variables using only the connective → is finite. Subsequent proofs of this fact using semantic methods (i.e., Kripke models) are given in =-=[17]-=- and [7]. An excellent analysis of all fragments of intuitionistic logic using the methods of [7] is given in [11]. In Chapter 3, I will present a purely combinatorial proof of this fact (which, in pa... |

4 |
Exact finite models for minimal propositional calculus over a finite alphabet
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(Show Context)
Citation Context ...r of intuitionistically inequivalent formulas over n variables using only the connective → is finite. Subsequent proofs of this fact using semantic methods (i.e., Kripke models) are given in [17] and =-=[7]-=-. An excellent analysis of all fragments of intuitionistic logic using the methods of [7] is given in [11]. In Chapter 3, I will present a purely combinatorial proof of this fact (which, in particular... |

1 |
Computations in propositional logic. http://staff.science.uva.nl/∼lhendrik/publications/ 55
- Hendriks
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(Show Context)
Citation Context ...sequent proofs of this fact using semantic methods (i.e., Kripke models) are given in [17] and [7]. An excellent analysis of all fragments of intuitionistic logic using the methods of [7] is given in =-=[11]-=-. In Chapter 3, I will present a purely combinatorial proof of this fact (which, in particular, does not rely on the tree automata representation given in Chapter 4) by specifying certain rewrite rule... |

1 |
Embedding of implicative lattices and superintuitionistic logics. Algebra i Logika
- Mardaev
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(Show Context)
Citation Context ...Proposition 6 and some combinatorial reasoning about U(n), we show in Theorem 4 that every Hm lattice-embeds as an interval into Hn, where m ≥ 1 and n ≥ 2, a result that was first shown by Mardaev in =-=[13]-=-. We may think of a tree automaton as assigning a state to each tree; namely, the final state of the automaton after processing the tree. We may consider all finite Kripke models to be finite labeled ... |