A Lambda Calculus for Gödel Dummett Logic Capturing Waitfreedom
|
|
|
- Amanda Perkins
- 5 years ago
- Views:
Transcription
1 A Lambda Calculus for Gödel Dummett Logic Capturing Waitfreedom Yoichi Hirai Univ. of Tokyo, JSPS research fellow , FLOPS 2012, Kobe 1
2 2 Curry Howard Correspondence Logic Intuitionistic propositional logic + (φ ψ) (ψ φ) = Gödel Dummett Logic + φ φ = Classical propositional logic Computation Simply-typed λ each term has a unique normal form λ-gd Normal form not unique in general: utilize as distributed computation many calculi Normal form not unique in general: fight with evaluation strategy This correspondence is based on Brouwer Heyting Kolmogorov interpretation
3 Brouwer Heyting Kolmogorov Interpretation explains logical connectives in intuitionistic logic A proof of P Q is a pair x, y where x is a proof of P and y is a proof of Q A proof of P Q is a pair i, x where i = 0 and x is a proof of P, or i = 1 and x is a proof of Q A proof of P Q is a construction which permits us to transform any proof of P into a proof of Q. P Q 3
4 4 Computational Interpretation of Implication O O O O
5 5 Computational Interpretation of Implication O O O O
6 6 Computational Interpretation of Implication O O O O
7 7 Computational Interpretation of Implication O O O O
8 8 Computational Interpretation of Implication O O O O
9 Computational Interpretation of Dummett Axiom (O A) (A O)...? Gödel Dummett logic has Dummett Axiom (φ ψ) (ψ φ), which must be realized for any φ and ψ. O A or A O 9
10 10 Computational Interpretation of Dummett Axiom (O A) (A O) O or A A O
11 11 Computational Interpretation of Dummett Axiom (O A) (A O) O or A A O
12 12 Computational Interpretation of Dummett Axiom (O A) (A O) O or A A O
13 13 Computational Interpretation of Dummett Axiom (O A) (A O) O or A A O
14 14 Computational Interpretation of Dummett Axiom (O A) (A O) O or A A O
15 15 Computational Interpretation of Dummett Axiom (O A) (A O) O or A A O
16 16 Computational Interpretation of Dummett Axiom (O A) (A O) O or A A O
17 17 Computational Interpretation of Dummett Axiom (O A) (A O) O or A A O
18 (O A) (A O) is Not a Theorem of G.D. Logic But, if rendez-vous was possible, the formula is realizable O and A A O 18
19 19 (O A) (A O) is Not a Theorem of G.D. Logic But, if rendez-vous was possible, the formula is realizable O waiting... and A A O
20 (O A) (A O) is Not a Theorem of G.D. Logic But, if rendez-vous was possible, the formula is realizable O and A A O 20
21 (O A) (A O) is Not a Theorem of G.D. Logic But, if rendez-vous was possible, the formula is realizable O and A A O 21
22 22 (O A) (A O) is Not a Theorem of G.D. Logic But, if rendez-vous was possible, the formula is realizable O and A A O
23 23 Waitfree = impossible to wait for other processes (free as in tax-free and alcohol-free) A task can be waitfreely solvable or not. A task I P O P where P is processes, I is inputs and O is outputs A watifreely unsolvable task: {((x, y), (y, x)) x, y {0, 1}} A waitfreely solvable task: {((x, y), (x, x + y)), ((x, y), (x + y, y)), ((x, y), (x + y, x + y)) x, y {0, 1}} The definition of waitfreedom is long: involving a virtual machine Saks and Zaharoglou (2000)
24 24 Main Theorem A task is waitfreely solvable solvable by a typable λ-gd-term For a typed lambda calculus λ-gdfor Gödel Dummett logic.
25 25 Formulas, Sequents, Hypersequents Local Types φ ::= P φ φ φ φ φ φ Global Types φ + ::= [i]φ φ + φ + φ + φ + where i is a process Sequent S ::= Γ φ + whereγ is a finite set of global formulas Hypersequent H ::= S (S H)
26 Hyper-Natural Deduction (Based on Avron s Hypersequents) External Rules com H Γ, [i]φ H Γ, [j]ψ Inner Global Rules H Γ [i]ψ [j]φ and structural rules H Γ φ + ψ + H Γ φ + ψ + E 0 E 1 H Γ φ + H Γ ψ + and I, E, I Inner Local Rules [i]ax [i]φ, Γ [i]φ H [i]φ, Γ [i]ψ [i] I H Γ [i](φ ψ) H Γ [i](φ ψ) H Γ [i]φ [i] E H Γ [i]ψ and E, E, I, E, I 26
27 27 Example Derivation of [0](O A) [1](A O) [i] I EC com [i] I [0]O, [1]A [0]O [0]O, [1]A [1]A [0]O [0]A [1]A [1]O [0](O A) [1](A O) [0](O A) [1](A O) [0](O A) [1](A O) [0](O A) [1](A O) The conclusion as a type, represents the orange apple protocol as a formula, without modalities [i], is the Dummett axiom.
28 28 Typing Terms External Rules G 0 Γ, M : [i]φ G 1 Γ, N : [j]ψ [G 0, G 1 ] Γ l i (M) : [i]ψ l j Γ (N) : [j]φ Inner Global Rules G Γ M : φ + ψ + G Γ M : φ + ψ + G Γ π g l (M) : φ+ G Γ πr g (M) : ψ+ Inner Local Rules x : [i]φ, Γ x : [i]φ G x : [i]φ, Γ M : [i]ψ G Γ λx.m : [i](φ ψ) G 0 Γ M : [i](φ ψ) G 1 Γ N : [i]φ [G 0, G 1 ] Γ MN : [i]ψ where G 0 and G 1 has the same types.
29 29 Example Typed Term x : [0]O, y : [1]A x : [0]O x : [0]O l (x) : [0]A x : [0]O, y : [1]A y : [1]A y : [1]A l (y) : [1]O x : [0]O, y : [1]A l (x) : [0]A x : [0]O, y : [1]A l (y) : [1]O ( x : [0]O, y : [1]A inl g ) ( l (x) : φ x : [0]O, y : [1]A inr g ) l (y) : φ ( x : [0]O, y : [1]A [inl g ) ( l (x), inr g l (y) )] : (φ :=)[0]A [1]O
30 30 Example Reduction 1 From the previously typed term: write reduction ( ({ }, { }, [inl g ) ( l (x), inr g l (y) ( ({l x}, { }, [inl g ) ( l (x), inr g l (y)
31 31 Example Reduction 1 From the previously typed term: read reduction ({ }, { }, [inl g ( l (x) ), inr g ( l (y) ({l x}, { }, [inl g ( l (x) ), inr g ( l (y) ({l x}, { }, [inl g (abort ), inr g ( l (y)
32 32 Example Reduction 1 From the previously typed term: write reduction ( ({ }, { }, [inl g ) ( l (x), inr g l (y) ( ({l x}, { }, [inl g ) ( l (x), inr g l (y) ({l x}, { }, [inl g (abort ), inr g ( l (y) ({l x}, {l y}, [inl g (abort ), inr g ( l (y)
33 33 Example Reduction 1 From the previously typed term: read reduction ( ({ }, { }, [inl g ) ( l (x), inr g l (y) ( ({l x}, { }, [inl g ) ( l (x), inr g l (y) ({l x}, { }, [inl g (abort ), inr g ( l (y) ({l x}, {l y}, [inl g (abort ), inr g ( l (y) ({l x}, {l y}, [inl g (abort ), inr g (x
34 Example Reduction 1 From the previously typed term: abort propagation ( ({ }, { }, [inl g ) ( l (x), inr g l (y) ( ({l x}, { }, [inl g ) ( l (x), inr g l (y) ({l x}, { }, [inl g (abort ), inr g ( l (y) ({l x}, {l y}, [inl g (abort ), inr g ( l (y) ({l x}, {l y}, [inl g (abort ), inr g (x ({l x}, {l y}, [abort, inr g (x 34
35 Example Reduction 1 From the previously typed term: abort propagation ( ({ }, { }, [inl g ) ( l (x), inr g l (y) ( ({l x}, { }, [inl g ) ( l (x), inr g l (y) ({l x}, { }, [inl g (abort ), inr g ( l (y) ({l x}, {l y}, [inl g (abort ), inr g ( l (y) ({l x}, {l y}, [inl g (abort ), inr g (x ({l x}, {l y}, [abort, inr g (x ({l x}, {l y}, inr g (x)) 35
36 Example Reduction 2 From the same term: ({ }, { ( }, [inl g ) l (x), inr g l (y) ({ ( }, {l y}, [inl g ) l (x), inr g l (y) ({ ( }, {l y}, [inl g ) l (x), inr g (abort ( ({l x}, {l y}, [inl g ) l (x), inr g (abort ({l x}, {l y}, [inl g (y), inr g (abort ({l x}, {l y}, [inl g (y), abort ]) ({l x}, {l y}, inl g (y)) 36
37 37 Example Reduction 3 Still from the same term: ( ({ }, { }, [inl g ) ( l (x), inr g l (y) ( ({l x}, { }, [inl g ) ( l (x), inr g l (y) ( ({l x}, {l y}, [inl g ) ( l (x), inr g l (y) ({l x}, {l y}, [inl g (y), inr g ( l (y) ({l x}, {l y}, [inl g (y), inr g (x
38 Results Strong normalization No typed term can reduce infinitely often Non-abortfullness No typed term can reduce into abort Waitfree characterization A task is waitfreely solvable solvable by a typed term 38
39 History Lamport (1977) introduced what is now called waitfree computing Avron (1991) introduced a hypersequent calculus for Gödel Dummett logic and showed cut-elimination Gafni and Koutsoupias (1999) showed that it is undecidable whether a task can be solved waitfreely Herlihy and Shavit (1999), Saks and Zaharoglou (2000) gave a topological characterization of waitfree computation (Gödel Prize 2004) I found waitfreedom when I was looking at past Gödel Prizes 39
40 Avron s Question and My Answer Avron (1991): it seems to us extremely important to determine the exact computational content of [the intermediate logics with cut-elimination for hypersequents] and to develop corresponding λ-calculi. I answer: The computational content of Gödel Dummett logic is waitfreedom λ-gd is the λ-calculus for it Fermüller (2003) gave a natural deduction but not reductions Future Work: generalizing to more logics and investigating classical logic adapting to weaker shared memory consistency 40
Non-classical Logics: Theory, Applications and Tools
Non-classical Logics: Theory, Applications and Tools Agata Ciabattoni Vienna University of Technology (TUV) Joint work with (TUV): M. Baaz, P. Baldi, B. Lellmann, R. Ramanayake,... N. Galatos (US), G.
Hypersequent Calculi for some Intermediate Logics with Bounded Kripke Models
Hypersequent Calculi for some Intermediate Logics with Bounded Kripke Models Agata Ciabattoni Mauro Ferrari Abstract In this paper we define cut-free hypersequent calculi for some intermediate logics semantically
An Intuitionistic Epistemic Logic for Sequential Consistency on Shared Memory
An Intuitionistic Epistemic Logic for Sequential Consistency on Shared Memory Yoichi Hirai Abstract In the celebrated Gödel Prize winning papers, Herlihy, Shavit, Saks and Zaharoglou gave topological characterization
An Intuitionistic Epistemic Logic for Sequential Consistency on Shared Memory
An Intuitionistic Epistemic Logic for Sequential Consistency on Shared Memory Yoichi Hirai 2010-04-27, Dakar Motivation Ÿ Ÿ Treating asynchronous communication using an epistemic logic. Shared memory consistency
Intersection Synchronous Logic
UnB 2007 p. 1/2 Intersection Synchronous Logic Elaine Gouvêa Pimentel Simona Ronchi della Rocca Luca Roversi UFMG/UNITO, 2007 UnB 2007 p. 2/2 Outline Motivation UnB 2007 p. 2/2 Outline Motivation Intuitionistic
Linear Nested Sequents, 2-Sequents and Hypersequents
Linear Nested Sequents, 2-Sequents and Hypersequents Björn Lellmann TU Wien TABLEAUX 2015 Wroc law, Sep 22, 2015 Sequent systems and modal logics Sequent calculi for modal logics are well-established and
Henk Barendregt and Freek Wiedijk assisted by Andrew Polonsky. Radboud University Nijmegen. March 5, 2012
1 λ Henk Barendregt and Freek Wiedijk assisted by Andrew Polonsky Radboud University Nijmegen March 5, 2012 2 reading Femke van Raamsdonk Logical Verification Course Notes Herman Geuvers Introduction to
Gödel Logics a short survey
Gödel Logics a short survey Norbert Preining 1 Colloquium Logicum 2016 September 2016, Hamburg 1 norbert@preining.info Today s program Development of many-valued logics t-norm based logics Gödel logics
On a computational interpretation of sequent calculus for modal logic S4
On a computational interpretation of sequent calculus for modal logic S4 Yosuke Fukuda Graduate School of Informatics, Kyoto University Second Workshop on Mathematical Logic and Its Applications March
Nonclassical logics (Nichtklassische Logiken)
Nonclassical logics (Nichtklassische Logiken) VU 185.249 (lecture + exercises) http://www.logic.at/lvas/ncl/ Chris Fermüller Technische Universität Wien www.logic.at/people/chrisf/ chrisf@logic.at Winter
Rules and derivations in an elementary logic course arxiv: v1 [cs.lo] 7 Jan 2016
![Rules and derivations in an elementary logic course arxiv: v1 [cs.lo] 7 Jan 2016](/thumbs/90/101773263.jpg "Rules and derivations in an elementary logic course arxiv: v1 [cs.lo] 7 Jan 2016")
Rules and derivations in an elementary logic course arxiv:160101483v1 [cslo] 7 Jan 2016 Gilles Dowek Inria, 23 avenue d Italie, CS 81321, 75214 Paris Cedex 13, France gillesdowek@inriafr When teaching
Introduction to Intuitionistic Logic
Introduction to Intuitionistic Logic August 31, 2016 We deal exclusively with propositional intuitionistic logic. The language is defined as follows. φ := p φ ψ φ ψ φ ψ φ := φ and φ ψ := (φ ψ) (ψ φ). A
Hypersequent calculi for non classical logics
Tableaux 03 p.1/63 Hypersequent calculi for non classical logics Agata Ciabattoni Technische Universität Wien, Austria agata@logic.at Tableaux 03 p.2/63 Non classical logics Unfortunately there is not
On Urquhart s C Logic
On Urquhart s C Logic Agata Ciabattoni Dipartimento di Informatica Via Comelico, 39 20135 Milano, Italy ciabatto@dsiunimiit Abstract In this paper we investigate the basic many-valued logics introduced
A Schütte-Tait style cut-elimination proof for first-order Gödel logic
A Schütte-Tait style cut-elimination proof for first-order Gödel logic Matthias Baaz and Agata Ciabattoni Technische Universität Wien, A-1040 Vienna, Austria {agata,baaz}@logic.at Abstract. We present
Propositions as Types
Propositions as Types Martin Pfeifhofer & Felix Schett May 25, 2016 Contents 1 Introduction 2 2 Content 3 2.1 Getting Started............................ 3 2.2 Effective Computability And The Various Definitions.......
The Curry-Howard Isomorphism
The Curry-Howard Isomorphism Software Formal Verification Maria João Frade Departmento de Informática Universidade do Minho 2008/2009 Maria João Frade (DI-UM) The Curry-Howard Isomorphism MFES 2008/09
Embedding formalisms: hypersequents and two-level systems of rules
Embedding formalisms: hypersequents and two-level systems of rules Agata Ciabattoni 1 TU Wien (Vienna University of Technology) agata@logicat Francesco A Genco 1 TU Wien (Vienna University of Technology)
Structuring Logic with Sequent Calculus
Structuring Logic with Sequent Calculus Alexis Saurin ENS Paris & École Polytechnique & CMI Seminar at IIT Delhi 17th September 2004 Outline of the talk Proofs via Natural Deduction LK Sequent Calculus
From pre-models to models
From pre-models to models normalization by Heyting algebras Olivier HERMANT 18 Mars 2008 Deduction System : natural deduction (NJ) first-order logic: function and predicate symbols, logical connectors:,,,,
Constructive approach to relevant and affine term calculi
Constructive approach to relevant and affine term calculi Jelena Ivetić, University of Novi Sad, Serbia Silvia Ghilezan,University of Novi Sad, Serbia Pierre Lescanne, University of Lyon, France Silvia
Logical Preliminaries
Logical Preliminaries Johannes C. Flieger Scheme UK March 2003 Abstract Survey of intuitionistic and classical propositional logic; introduction to the computational interpretation of intuitionistic logic
Categories, Proofs and Programs
Categories, Proofs and Programs Samson Abramsky and Nikos Tzevelekos Lecture 4: Curry-Howard Correspondence and Cartesian Closed Categories In A Nutshell Logic Computation 555555555555555555 5 Categories
Preuves de logique linéaire sur machine, ENS-Lyon, Dec. 18, 2018
Université de Lorraine, LORIA, CNRS, Nancy, France Preuves de logique linéaire sur machine, ENS-Lyon, Dec. 18, 2018 Introduction Linear logic introduced by Girard both classical and intuitionistic separate
What is a Proof? Jean Gallier and Kurt W.A.J.H.Y. Reillag CIS, Upenn and Hospices de Beaune. Friday, February 22, 13
What is a Proof? Jean Gallier and Kurt W.A.J.H.Y. Reillag CIS, Upenn and Hospices de Beaune Reillag s office Another office After a bad proof! Finally, some peace! Quick History Quick History Formalizing
CHAPTER 11. Introduction to Intuitionistic Logic
CHAPTER 11 Introduction to Intuitionistic Logic Intuitionistic logic has developed as a result of certain philosophical views on the foundation of mathematics, known as intuitionism. Intuitionism was originated
Type Theory and Constructive Mathematics. Type Theory and Constructive Mathematics Thierry Coquand. University of Gothenburg
Type Theory and Constructive Mathematics Type Theory and Constructive Mathematics Thierry Coquand University of Gothenburg Content An introduction to Voevodsky s Univalent Foundations of Mathematics The
cse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018
cse371/mat371 LOGIC Professor Anita Wasilewska Fall 2018 Chapter 7 Introduction to Intuitionistic and Modal Logics CHAPTER 7 SLIDES Slides Set 1 Chapter 7 Introduction to Intuitionistic and Modal Logics
Propositional Logic Language
Propositional Logic Language A logic consists of: an alphabet A, a language L, i.e., a set of formulas, and a binary relation = between a set of formulas and a formula. An alphabet A consists of a finite
Uniform Schemata for Proof Rules
Uniform Schemata for Proof Rules Ulrich Berger and Tie Hou Department of omputer Science, Swansea University, UK {u.berger,cshou}@swansea.ac.uk Abstract. Motivated by the desire to facilitate the implementation
Logic Part II: Intuitionistic Logic and Natural Deduction
Yesterday Remember yesterday? classical logic: reasoning about truth of formulas propositional logic: atomic sentences, composed by connectives validity and satisability can be decided by truth tables
On the Construction of Analytic Sequent Calculi for Sub-classical Logics
On the Construction of Analytic Sequent Calculi for Sub-classical Logics Ori Lahav Yoni Zohar Tel Aviv University WoLLIC 2014 On the Construction of Analytic Sequent Calculi for Sub-classical Logics A
Type Systems. Lecture 9: Classical Logic. Neel Krishnaswami University of Cambridge
Type Systems Lecture 9: Classical Logic Neel Krishnaswami University of Cambridge Where We Are We have seen the Curry Howard correspondence: Intuitionistic propositional logic Simply-typed lambda calculus
Command = Value Context. Hans-Dieter Hiep. 15th of June, /17
1/17 Command = Value Context Hans-Dieter Hiep 15th of June, 2018 2/17 Example Consider a total functional programming language. swap :: A B B A swap (a, b) = (b, a) paws :: A B B A paws a = a paws b =
Proof Theoretical Reasoning Lecture 3 Modal Logic S5 and Hypersequents
Proof Theoretical Reasoning Lecture 3 Modal Logic S5 and Hypersequents Revantha Ramanayake and Björn Lellmann TU Wien TRS Reasoning School 2015 Natal, Brasil Outline Modal Logic S5 Sequents for S5 Hypersequents
Semantics for Propositional Logic
Semantics for Propositional Logic An interpretation (also truth-assignment, valuation) of a set of propositional formulas S is a function that assigns elements of {f,t} to the propositional variables in
3.2 Reduction 29. Truth. The constructor just forms the unit element,. Since there is no destructor, there is no reduction rule.
32 Reduction 29 32 Reduction In the preceding section, we have introduced the assignment of proof terms to natural deductions If proofs are programs then we need to explain how proofs are to be executed,
Sequent calculus for predicate logic
CHAPTER 13 Sequent calculus for predicate logic 1. Classical sequent calculus The axioms and rules of the classical sequent calculus are: Axioms { Γ, ϕ, ϕ for atomic ϕ Γ, Left Γ,α 1,α 2 Γ,α 1 α 2 Γ,β 1
Introduction to λ-calculus
p.1/65 Introduction to λ-calculus Ken-etsu FUJITA fujita@cs.gunma-u.ac.jp http://www.comp.cs.gunma-u.ac.jp/ fujita/ Department of Computer Science Gunma University :Church 32, 36, 40; Curry 34 1. Universal
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages
Harvard School of Engineering and Applied Sciences CS 152: Programming Languages Lecture 17 Tuesday, April 2, 2013 1 There is a strong connection between types in programming languages and propositions
FROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS.
FROM AXIOMS TO STRUCTURAL RULES, THEN ADD QUANTIFIERS. REVANTHA RAMANAYAKE We survey recent developments in the program of generating proof calculi for large classes of axiomatic extensions of a non-classical
(Algebraic) Proof Theory for Substructural Logics and Applications
(Algebraic) Proof Theory for Substructural Logics and Applications Agata Ciabattoni Vienna University of Technology agata@logic.at (Algebraic) Proof Theory for Substructural Logics and Applications p.1/59
Proofs in classical logic as programs: a generalization of lambda calculus. A. Salibra. Università Ca Foscari Venezia
Proofs in classical logic as programs: a generalization of lambda calculus A. Salibra Università Ca Foscari Venezia Curry Howard correspondence, in general Direct relationship between systems of logic
UNITARY UNIFICATION OF S5 MODAL LOGIC AND ITS EXTENSIONS
Bulletin of the Section of Logic Volume 32:1/2 (2003), pp. 19 26 Wojciech Dzik UNITARY UNIFICATION OF S5 MODAL LOGIC AND ITS EXTENSIONS Abstract It is shown that all extensions of S5 modal logic, both
A Local System for Intuitionistic Logic
A Local ystem for Intuitionistic Logic Alwen Tiu Australian ational University and ational ICT Australia Abstract. This paper presents systems for first-order intuitionistic logic and several of its extensions
Structural extensions of display calculi: a general recipe
Structural extensions of display calculi: a general recipe Agata Ciabattoni and Revantha Ramanayake Vienna University of Technology {agata,revantha}@logic.at Abstract. We present a systematic procedure
185.A09 Advanced Mathematical Logic
185.A09 Advanced Mathematical Logic www.volny.cz/behounek/logic/teaching/mathlog13 Libor Běhounek, behounek@cs.cas.cz Lecture #1, October 15, 2013 Organizational matters Study materials will be posted
Uniform Rules and Dialogue Games for Fuzzy Logics
Uniform Rules and Dialogue Games for Fuzzy Logics A. Ciabattoni, C.G. Fermüller, and G. Metcalfe Technische Universität Wien, A-1040 Vienna, Austria {agata,chrisf,metcalfe}@logic.at Abstract. We provide
Teaching Natural Deduction as a Subversive Activity
Teaching Natural Deduction as a Subversive Activity James Caldwell Department of Computer Science University of Wyoming Laramie, WY Third International Congress on Tools for Teaching Logic 3 June 2011
Evaluation Driven Proof-Search in Natural Deduction Calculi for Intuitionistic Propositional Logic
Evaluation Driven Proof-Search in Natural Deduction Calculi for Intuitionistic Propositional Logic Mauro Ferrari 1, Camillo Fiorentini 2 1 DiSTA, Univ. degli Studi dell Insubria, Varese, Italy 2 DI, Univ.
Tilburg University. Dialogues games for classical logic (short paper) Alama, J.; Knoks, A.; Uckelman, S.L. Published in: Tableaux 2011
Tilburg University Dialogues games for classical logic (short paper) Alama, J.; Knoks, A.; Uckelman, S.L. Published in: Tableaux 2011 Publication date: 2011 Link to publication Citation for published version
Focusing on Binding and Computation
Focusing on Binding and Computation Dan Licata Joint work with Noam Zeilberger and Robert Harper Carnegie Mellon University 1 Programming with Proofs Represent syntax, judgements, and proofs Reason about
Introduction to dependent type theory. CIRM, May 30
CIRM, May 30 Goals of this presentation Some history and motivations Notations used in type theory Main goal: the statement of main properties of equality type and the univalence axiom First talk P ropositions
Consequence Relations and Natural Deduction
Consequence Relations and Natural Deduction Joshua D. Guttman Worcester Polytechnic Institute September 9, 2010 Contents 1 Consequence Relations 1 2 A Derivation System for Natural Deduction 3 3 Derivations
The Extended BG-Simulation and the Characterization of t-resiliency
The Extended BG-Simulation and the Characterization of t-resiliency Eli Gafni November 17, 2008 Abstract A distributed task T on n processors is an input/output relation between a collection of processors
Labelled Calculi for Łukasiewicz Logics
Labelled Calculi for Łukasiewicz Logics D. Galmiche and Y. Salhi LORIA UHP Nancy 1 Campus Scientifique, BP 239 54 506 Vandœuvre-lès-Nancy, France Abstract. In this paper, we define new decision procedures
On Jankov-de Jongh formulas
On Jankov-de Jongh formulas Nick Bezhanishvili Institute for Logic, Language and Computation University of Amsterdam http://www.phil.uu.nl/~bezhanishvili The Heyting day dedicated to Dick de Jongh and
SEMANTICS OF INTUITIONISTIC PROPOSITIONAL LOGIC: HEYTING ALGEBRAS AND KRIPKE MODELS
SEMNTICS OF INTUITIONISTIC PROPOSITIONL LOGIC: HEYTING LGEBRS ND KRIPKE MODELS CHD E. BROWN bstract. We describe two well-known semantics for intuitionistic propositional logic: Heyting algebras and Kripke
A Brief Introduction to Justification Logic , Peking University
A Brief Introduction to Justification Logic 2015.12.08, Peking University Junhua Yu Department of Philosophy, Tsinghua University, Beijing 100084 China December 10, 2015 1 Outline History - from constructive
Outline. Overview. Syntax Semantics. Introduction Hilbert Calculus Natural Deduction. 1 Introduction. 2 Language: Syntax and Semantics
Introduction Arnd Poetzsch-Heffter Software Technology Group Fachbereich Informatik Technische Universität Kaiserslautern Sommersemester 2010 Arnd Poetzsch-Heffter ( Software Technology Group Fachbereich
Quantifiers and Functions in Intuitionistic Logic
Quantifiers and Functions in Intuitionistic Logic Association for Symbolic Logic Spring Meeting Seattle, April 12, 2017 Rosalie Iemhoff Utrecht University, the Netherlands 1 / 37 Quantifiers are complicated.
Stratifications and complexity in linear logic. Daniel Murfet
Stratifications and complexity in linear logic Daniel Murfet Curry-Howard correspondence logic programming categories formula type objects sequent input/output spec proof program morphisms cut-elimination
Inducing syntactic cut-elimination for indexed nested sequents
Inducing syntactic cut-elimination for indexed nested sequents Revantha Ramanayake Technische Universität Wien (Austria) IJCAR 2016 June 28, 2016 Revantha Ramanayake (TU Wien) Inducing syntactic cut-elimination
Using Gentzen system techniques with existing automated theorem provers and rewriting tools
Using Gentzen system techniques with existing automated theorem provers and rewriting tools Peter Jipsen Chapman Univ, Orange, CA March 6, 2008 Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen
If-then-else and other constructive and classical connectives (Or: How to derive natural deduction rules from truth tables)
If-then-else and other constructive and classical connectives (Or: How to derive natural deduction rules from truth tables) Herman Geuvers Nijmegen, NL (Joint work with Tonny Hurkens) Types for Proofs
Sequent calculi of quantum logic with strict implication
CTFM 2015 9/7 Sequent calculi of quantum logic with strict implication Tokyo Institute of Technology Graduate School of Information Science and Engineering Tomoaki Kawano About quantum logic Sequent calculi
Outline. 1 Background and Aim. 2 Main results (in the paper) 3 More results (not in the paper) 4 Conclusion
Outline 1 Background and Aim 2 Main results (in the paper) 3 More results (not in the paper) 4 Conclusion De & Omori (Konstanz & Kyoto/JSPS) Classical and Empirical Negation in SJ AiML 2016, Sept. 2, 2016
Interpolation via translations
Interpolation via translations Walter Carnielli 2,3 João Rasga 1,3 Cristina Sernadas 1,3 1 DM, IST, TU Lisbon, Portugal 2 CLE and IFCH, UNICAMP, Brazil 3 SQIG - Instituto de Telecomunicações, Portugal
The Calculus of Inductive Constructions
The Calculus of Inductive Constructions Hugo Herbelin 10th Oregon Programming Languages Summer School Eugene, Oregon, June 16-July 1, 2011 1 Outline - A bit of history, leading to the Calculus of Inductive
Sequent Calculus in the Topos of Trees
Sequent Calculus in the Topos of Trees Ranald Clouston 1 and Rajeev Goré 2 1 Department of Computer Science, Aarhus University ranald.clouston@cs.au.dk 2 Research School of Computer Science, Australian
Foundations of Computation. Ana Bove
Foundations of Computation Ana Bove Programming Logic (ProgLog) Group February 13th 2018 Outline of the talk: What we do in ProgLog Origines of computer science Courses in the area Warming-up Exercise
AN ALTERNATIVE NATURAL DEDUCTION FOR THE INTUITIONISTIC PROPOSITIONAL LOGIC
Bulletin of the Section of Logic Volume 45/1 (2016), pp 33 51 http://dxdoiorg/1018778/0138-068045103 Mirjana Ilić 1 AN ALTERNATIVE NATURAL DEDUCTION FOR THE INTUITIONISTIC PROPOSITIONAL LOGIC Abstract
A simple proof that super-consistency implies cut elimination
A simple proof that super-consistency implies cut elimination Gilles Dowek 1 and Olivier Hermant 2 1 École polytechnique and INRIA, LIX, École polytechnique, 91128 Palaiseau Cedex, France gilles.dowek@polytechnique.edu
MONADIC FRAGMENTS OF INTUITIONISTIC CONTROL LOGIC
Bulletin of the Section of Logic Volume 45:3/4 (2016), pp. 143 153 http://dx.doi.org/10.18778/0138-0680.45.3.4.01 Anna Glenszczyk MONADIC FRAGMENTS OF INTUITIONISTIC CONTROL LOGIC Abstract We investigate
Extraction from classical proofs using game models
1/16 Extraction from classical proofs using game models Valentin Blot University of Bath research funded by the UK EPSRC 2/16 The computational content of classical logic Griffin, 1990: Computational content
A Cut-Free Calculus for Second-Order Gödel Logic
Fuzzy Sets and Systems 00 (2014) 1 30 Fuzzy Sets and Systems A Cut-Free Calculus for Second-Order Gödel Logic Ori Lahav, Arnon Avron School of Computer Science, Tel Aviv University Abstract We prove that
Marie Duží
Marie Duží marie.duzi@vsb.cz 1 Formal systems, Proof calculi A proof calculus (of a theory) is given by: 1. a language 2. a set of axioms 3. a set of deduction rules ad 1. The definition of a language
Propositional Logic: Deductive Proof & Natural Deduction Part 1
Propositional Logic: Deductive Proof & Natural Deduction Part 1 CS402, Spring 2016 Shin Yoo Deductive Proof In propositional logic, a valid formula is a tautology. So far, we could show the validity of
Semantical study of intuitionistic modal logics
Semantical study of intuitionistic modal logics Department of Intelligence Science and Technology Graduate School of Informatics Kyoto University Kensuke KOJIMA January 16, 2012 Abstract We investigate
arxiv: v1 [cs.lo] 31 Aug 2014
![arxiv: v1 [cs.lo] 31 Aug 2014](/thumbs/76/74440423.jpg "arxiv: v1 [cs.lo] 31 Aug 2014")
arxiv:1409.0266v1 [cs.lo] 31 Aug 2014 Virtual Evidence: A Constructive Semantics for Classical Logics Robert L. Constable Abstract This article presents a computational semantics for classical logic using
Taming Implications in Dummett Logic
Taming Implications in Dummett Logic Guido Fiorino Dipartimento di Metodi Quantitativi per le Scienze Economiche ed Aziendali, Università di Milano-Bicocca, Piazza dell Ateneo Nuovo, 1, 20126 Milano, Italy.
Decomposing Modalities
Decomposing Modalities Frank Pfenning Department of Computer Science Carnegie Mellon University Logical Frameworks and Meta-Languages: Theory and Practice (LFMTP 15) Berlin, Germany August 1, 2015 1 /
Part III Logic. Theorems. Based on lectures by T. E. Forster Notes taken by Dexter Chua. Lent 2017
Part III Logic Theorems Based on lectures by T. E. Forster Notes taken by Dexter Chua Lent 2017 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures.
Structural completeness for discriminator varieties
Structural completeness for discriminator varieties Micha l Stronkowski Warsaw University of Technology Praha, June 2014 Deductive systems Sent - set of propositional sentences Ax - axioms ( Sent) + inference
First-Order Intuitionistic Logic with Decidable Propositional Atoms
First-Order Intuitionistic Logic with Decidable Propositional Atoms Alexander Sakharov alex@sakharov.net http://alex.sakharov.net Abstract First-order intuitionistic logic extended with the assumption
Subtractive Logic. To appear in Theoretical Computer Science. Tristan Crolard May 3, 1999
Subtractive Logic To appear in Theoretical Computer Science Tristan Crolard crolard@ufr-info-p7.jussieu.fr May 3, 1999 Abstract This paper is the first part of a work whose purpose is to investigate duality
How and when axioms can be transformed into good structural rules
How and when axioms can be transformed into good structural rules Kazushige Terui National Institute of Informatics, Tokyo Laboratoire d Informatique de Paris Nord (Joint work with Agata Ciabattoni and
Justification logic for constructive modal logic
Justification logic for constructive modal logic Sonia Marin With Roman Kuznets and Lutz Straßburger Inria, LIX, École Polytechnique IMLA 17 July 17, 2017 The big picture The big picture Justification
Gentzen-type Refutation Systems for Three-Valued Logics
Gentzen-type Refutation Systems for Three-Valued Logics Johannes Oetsch and Hans Tompits Technische Universität Wien, Institut für Informationssysteme 184/3, Favoritenstraße 9-11, A-1040 Vienna, Austria
TOWARDS A PROOF THEORY OF GÖDEL MODAL LOGICS
Logical Methods in Computer Science Vol. 7 (2:10) 2011, pp. 1 27 www.lmcs-online.org Submitted Jul. 25, 2010 Published May 17, 2011 TOWARDS A PROOF THEORY OF GÖDEL MODAL LOGICS GEORGE METCALFE a AND NICOLA
Recursion Theory. Joost J. Joosten
Recursion Theory Joost J. Joosten Institute for Logic Language and Computation University of Amsterdam Plantage Muidergracht 24 1018 TV Amsterdam Room P 3.26, +31 20 5256095 jjoosten@phil.uu.nl www.phil.uu.nl/
An overview of Structural Proof Theory and Computing
An overview of Structural Proof Theory and Computing Dale Miller INRIA-Saclay & LIX, École Polytechnique Palaiseau, France Madison, Wisconsin, 2 April 2012 Part of the Special Session in Structural Proof
Fixed-point elimination in Heyting algebras 1
1/32 Fixed-point elimination in Heyting algebras 1 Silvio Ghilardi, Università di Milano Maria João Gouveia, Universidade de Lisboa Luigi Santocanale, Aix-Marseille Université TACL@Praha, June 2017 1 See
Forcing-based cut-elimination for Gentzen-style intuitionistic sequent calculus
Forcing-based cut-elimination for Gentzen-style intuitionistic sequent calculus Hugo Herbelin 1 and Gyesik Lee 2 1 INRIA & PPS, Paris Université 7 Paris, France Hugo.Herbelin@inria.fr 2 ROSAEC center,
Automated Support for the Investigation of Paraconsistent and Other Logics
Automated Support for the Investigation of Paraconsistent and Other Logics Agata Ciabattoni 1, Ori Lahav 2, Lara Spendier 1, and Anna Zamansky 1 1 Vienna University of Technology 2 Tel Aviv University
A call-by-name lambda-calculus machine
A call-by-name lambda-calculus machine Jean-Louis Krivine University Paris VII, C.N.R.S. 2 place Jussieu 75251 Paris cedex 05 (krivine@pps.jussieu.fr) Introduction We present, in this paper, a particularly
Implicit Computational Complexity
Implicit Computational Complexity Simone Martini Dipartimento di Scienze dell Informazione Università di Bologna Italy Bertinoro International Spring School for Graduate Studies in Computer Science, 6
Simply Typed λ-calculus
Simply Typed λ-calculus Lecture 1 Jeremy Dawson The Australian National University Semester 2, 2017 Jeremy Dawson (ANU) COMP4630,Lecture 1 Semester 2, 2017 1 / 23 A Brief History of Type Theory First developed
Example. Lemma. Proof Sketch. 1 let A be a formula that expresses that node t is reachable from s
Summary Summary Last Lecture Computational Logic Π 1 Γ, x : σ M : τ Γ λxm : σ τ Γ (λxm)n : τ Π 2 Γ N : τ = Π 1 [x\π 2 ] Γ M[x := N] Georg Moser Institute of Computer Science @ UIBK Winter 2012 the proof
Taming Implications in Dummett Logic
Taming Implications in Dummett Logic Guido Fiorino Dipartimento di Metodi Quantitativi per le cienze Economiche ed Aziendali, Università di Milano-Bicocca, Piazza dell Ateneo Nuovo, 1, 20126 Milano, Italy.