1 of 18

## What's hot

Graph theory 1
Graph theory 1Tech_MX

Lie Convexity for Super-Standard Arrow
Lie Convexity for Super-Standard Arrowjorgerodriguessimao

5.vector geometry Further Mathematics Zimbabwe Zimsec Cambridge
5.vector geometry Further Mathematics Zimbabwe Zimsec Cambridgealproelearning

2 linear independence
2 linear independenceAmanSaeed11

Math Conference Poster
Math Conference Postermary41679

The Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphsMarco Benini

Graph: Euler path and Euler circuit
Graph: Euler path and Euler circuitLiwayway Memije-Cruz

Unitary spaces
Unitary spacesHuni Malik

Interpreting Multiple Regression via an Ellipse Inscribed in a Square Extensi...
Interpreting Multiple Regression via an Ellipse Inscribed in a Square Extensi...Toshiyuki Shimono

### What's hot(16)

Graph theory 1
Graph theory 1

Lecture 1
Lecture 1

Lie Convexity for Super-Standard Arrow
Lie Convexity for Super-Standard Arrow

nossi ch 6
nossi ch 6

Seminar on Complex Geometry
Seminar on Complex Geometry

Lecture 3
Lecture 3

5.vector geometry Further Mathematics Zimbabwe Zimsec Cambridge
5.vector geometry Further Mathematics Zimbabwe Zimsec Cambridge

Vector[1]
Vector[1]

2 linear independence
2 linear independence

Math Conference Poster
Math Conference Poster

The Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphs

Limits BY ATC
Limits BY ATC

Limit
Limit

Graph: Euler path and Euler circuit
Graph: Euler path and Euler circuit

Unitary spaces
Unitary spaces

Interpreting Multiple Regression via an Ellipse Inscribed in a Square Extensi...
Interpreting Multiple Regression via an Ellipse Inscribed in a Square Extensi...

## Viewers also liked

Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard ...
Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard ...Marco Benini

CORCON2014: Does programming really need data structures?
CORCON2014: Does programming really need data structures?Marco Benini

Fondazione point-free della matematica
Fondazione point-free della matematicaMarco Benini

Промышленность Германии
Промышленность ГерманииIvan Matyash

Dealing with negative results
Dealing with negative resultsMarco Benini

L'occhio del biologo: elementi di fotografia
L'occhio del biologo: elementi di fotografiaMarco Benini

Variations on the Higman's Lemma
Variations on the Higman's LemmaMarco Benini

### Viewers also liked(9)

Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard ...
Intuitionistic First-Order Logic: Categorical semantics via the Curry-Howard ...

CORCON2014: Does programming really need data structures?
CORCON2014: Does programming really need data structures?

Fondazione point-free della matematica
Fondazione point-free della matematica

Промышленность Германии
Промышленность Германии

Dealing with negative results
Dealing with negative results

L'occhio del biologo: elementi di fotografia
L'occhio del biologo: elementi di fotografia

Variations on the Higman's Lemma
Variations on the Higman's Lemma

## Similar to June 22nd 2014: Seminar at JAIST

Point-free foundation of Mathematics
Point-free foundation of MathematicsMarco Benini

Proof-Theoretic Semantics: Point-free meaninig of first-order systems
Proof-Theoretic Semantics: Point-free meaninig of first-order systemsMarco Benini

Point-free semantics of dependent type theories
Point-free semantics of dependent type theoriesMarco Benini

Theory of Relational Calculus and its Formalization
Theory of Relational Calculus and its FormalizationYoshihiro Mizoguchi

Dialectica and Kolmogorov Problems
Dialectica and Kolmogorov ProblemsValeria de Paiva

alexbeloi_thesis_082715_final
alexbeloi_thesis_082715_finalAlex Beloi

Commutative algebra
Commutative algebraSpringer

Class 10 mathematics compendium
Class 10 mathematics compendiumAPEX INSTITUTE

Analysis and algebra on differentiable manifolds
Analysis and algebra on differentiable manifoldsSpringer

Lectures on Analytic Geometry
Lectures on Analytic GeometryAnatol Alizar

Information geometry: Dualistic manifold structures and their uses
Information geometry: Dualistic manifold structures and their usesFrank Nielsen

Ireducible core and equal remaining obligations rule for mcst games
Ireducible core and equal remaining obligations rule for mcst gamesvinnief

Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...Antonio Lieto

The Khalimsky Line Topology- Countability and Connectedness
The Khalimsky Line Topology- Countability and ConnectednessAI Publications

Proportional and decentralized rule mcst games
Proportional and decentralized rule mcst gamesvinnief

An elementary introduction to information geometry
An elementary introduction to information geometryFrank Nielsen

A Coq Library for the Theory of Relational Calculus
A Coq Library for the Theory of Relational CalculusYoshihiro Mizoguchi

Programming modulo representations
Programming modulo representationsMarco Benini

### Similar to June 22nd 2014: Seminar at JAIST(20)

Point-free foundation of Mathematics
Point-free foundation of Mathematics

Proof-Theoretic Semantics: Point-free meaninig of first-order systems
Proof-Theoretic Semantics: Point-free meaninig of first-order systems

Point-free semantics of dependent type theories
Point-free semantics of dependent type theories

Theory of Relational Calculus and its Formalization
Theory of Relational Calculus and its Formalization

Dialectica and Kolmogorov Problems
Dialectica and Kolmogorov Problems

alexbeloi_thesis_082715_final
alexbeloi_thesis_082715_final

Commutative algebra
Commutative algebra

Class 10 mathematics compendium
Class 10 mathematics compendium

Analysis and algebra on differentiable manifolds
Analysis and algebra on differentiable manifolds

1411.4232v1
1411.4232v1

Lectures on Analytic Geometry
Lectures on Analytic Geometry

Information geometry: Dualistic manifold structures and their uses
Information geometry: Dualistic manifold structures and their uses

Linear algebra havard university
Linear algebra havard university

Ireducible core and equal remaining obligations rule for mcst games
Ireducible core and equal remaining obligations rule for mcst games

Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...
Conceptual Spaces for Cognitive Architectures: A Lingua Franca for Different ...

The Khalimsky Line Topology- Countability and Connectedness
The Khalimsky Line Topology- Countability and Connectedness

Proportional and decentralized rule mcst games
Proportional and decentralized rule mcst games

An elementary introduction to information geometry
An elementary introduction to information geometry

A Coq Library for the Theory of Relational Calculus
A Coq Library for the Theory of Relational Calculus

Programming modulo representations
Programming modulo representations

## More from Marco Benini

The Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphsMarco Benini

Explaining the Kruskal Tree Theore
Explaining the Kruskal Tree TheoreMarco Benini

Dealing with negative results
Dealing with negative resultsMarco Benini

Well Quasi Orders in a Categorical Setting
Well Quasi Orders in a Categorical SettingMarco Benini

Numerical Analysis and Epistemology of Information
Numerical Analysis and Epistemology of InformationMarco Benini

Marie Skłodowska Curie Intra-European Fellowship
Marie Skłodowska Curie Intra-European FellowshipMarco Benini

Algorithms and Their Explanations
Algorithms and Their ExplanationsMarco Benini

Programming modulo representations
Programming modulo representationsMarco Benini

Fondazione point-free della matematica
Fondazione point-free della matematicaMarco Benini

### More from Marco Benini(9)

The Graph Minor Theorem: a walk on the wild side of graphs
The Graph Minor Theorem: a walk on the wild side of graphs

Explaining the Kruskal Tree Theore
Explaining the Kruskal Tree Theore

Dealing with negative results
Dealing with negative results

Well Quasi Orders in a Categorical Setting
Well Quasi Orders in a Categorical Setting

Numerical Analysis and Epistemology of Information
Numerical Analysis and Epistemology of Information

Marie Skłodowska Curie Intra-European Fellowship
Marie Skłodowska Curie Intra-European Fellowship

Algorithms and Their Explanations
Algorithms and Their Explanations

Programming modulo representations
Programming modulo representations

Fondazione point-free della matematica
Fondazione point-free della matematica

CHROMATOGRAPHY PALLAVI RAWAT.pptx
CHROMATOGRAPHY PALLAVI RAWAT.pptxpallavirawat456

DECOMPOSITION PATHWAYS of TM-alkyl complexes.pdf
DECOMPOSITION PATHWAYS of TM-alkyl complexes.pdfDivyaK787011

Environmental acoustics- noise criteria.pptx
Environmental acoustics- noise criteria.pptxpriyankatabhane

Immunoblott technique for protein detection.ppt
Immunoblott technique for protein detection.pptAmirRaziq1

KDIGO-2023-CKD-Guideline-Public-Review-Draft_5-July-2023.pdf
KDIGO-2023-CKD-Guideline-Public-Review-Draft_5-July-2023.pdfGABYFIORELAMALPARTID1

Observational constraints on mergers creating magnetism in massive stars
Observational constraints on mergers creating magnetism in massive starsSérgio Sacani

projectile motion, impulse and moment
projectile motion, impulse and momentdonamiaquintan2

GLYCOSIDES Classification Of GLYCOSIDES Chemical Tests Glycosides
GLYCOSIDES Classification Of GLYCOSIDES Chemical Tests GlycosidesNandakishor Bhaurao Deshmukh

Gas-ExchangeS-in-Plants-and-Animals.pptx

linear Regression, multiple Regression and Annova
linear Regression, multiple Regression and AnnovaMansi Rastogi

Abnormal LFTs rate of deco and NAFLD.pptx
Abnormal LFTs rate of deco and NAFLD.pptxzeus70441

DOG BITE management in pediatrics # for Pediatric pgs# topic presentation # f...
DOG BITE management in pediatrics # for Pediatric pgs# topic presentation # f...HafsaHussainp

Oxo-Acids of Halogens and their Salts.pptx
Oxo-Acids of Halogens and their Salts.pptxfarhanvvdk

Replisome-Cohesin Interfacing A Molecular Perspective.pdf
Replisome-Cohesin Interfacing A Molecular Perspective.pdfAtiaGohar1

WEEK 4 PHYSICAL SCIENCE QUARTER 3 FOR G11
WEEK 4 PHYSICAL SCIENCE QUARTER 3 FOR G11GelineAvendao

Fertilization: Sperm and the egg—collectively called the gametes—fuse togethe...
Fertilization: Sperm and the egg—collectively called the gametes—fuse togethe...D. B. S. College Kanpur

DNA isolation molecular biology practical.pptx
DNA isolation molecular biology practical.pptxGiDMOh

CHROMATOGRAPHY PALLAVI RAWAT.pptx
CHROMATOGRAPHY PALLAVI RAWAT.pptx

DECOMPOSITION PATHWAYS of TM-alkyl complexes.pdf
DECOMPOSITION PATHWAYS of TM-alkyl complexes.pdf

AZOTOBACTER AS BIOFERILIZER.PPTX
AZOTOBACTER AS BIOFERILIZER.PPTX

PLASMODIUM. PPTX
PLASMODIUM. PPTX

Environmental acoustics- noise criteria.pptx
Environmental acoustics- noise criteria.pptx

Immunoblott technique for protein detection.ppt
Immunoblott technique for protein detection.ppt

KDIGO-2023-CKD-Guideline-Public-Review-Draft_5-July-2023.pdf
KDIGO-2023-CKD-Guideline-Public-Review-Draft_5-July-2023.pdf

Observational constraints on mergers creating magnetism in massive stars
Observational constraints on mergers creating magnetism in massive stars

projectile motion, impulse and moment
projectile motion, impulse and moment

GLYCOSIDES Classification Of GLYCOSIDES Chemical Tests Glycosides
GLYCOSIDES Classification Of GLYCOSIDES Chemical Tests Glycosides

Gas-ExchangeS-in-Plants-and-Animals.pptx
Gas-ExchangeS-in-Plants-and-Animals.pptx

linear Regression, multiple Regression and Annova
linear Regression, multiple Regression and Annova

Abnormal LFTs rate of deco and NAFLD.pptx
Abnormal LFTs rate of deco and NAFLD.pptx

DOG BITE management in pediatrics # for Pediatric pgs# topic presentation # f...
DOG BITE management in pediatrics # for Pediatric pgs# topic presentation # f...

Oxo-Acids of Halogens and their Salts.pptx
Oxo-Acids of Halogens and their Salts.pptx

Replisome-Cohesin Interfacing A Molecular Perspective.pdf
Replisome-Cohesin Interfacing A Molecular Perspective.pdf

WEEK 4 PHYSICAL SCIENCE QUARTER 3 FOR G11
WEEK 4 PHYSICAL SCIENCE QUARTER 3 FOR G11

Fertilization: Sperm and the egg—collectively called the gametes—fuse togethe...
Fertilization: Sperm and the egg—collectively called the gametes—fuse togethe...

DNA isolation molecular biology practical.pptx
DNA isolation molecular biology practical.pptx

### June 22nd 2014: Seminar at JAIST

• 1. Point-free foundation of Mathematics Dr M Benini Università degli Studi dell’Insubria Correctness by Construction project visiting JAIST until 16th June 2014 marco.benini@uninsubria.it April 22nd, 2014
• 2. Introduction This seminar aims at introducing an alternative foundation of Mathematics. Is it possible to deﬁne logical theories without assuming the existence of elements? This talk will positively answer to the above question by providing a sound and complete semantics for multi-sorted, ﬁrst-order, intuitionistic-based logical theories. (2 of 18)
• 3. Introduction Actually, there is more: the semantics does not interpret terms as elements of some universe, but rather as the glue which keeps together the meaning of formulae; the semantics allows to directly interpret the Curry-Howard isomorphism so that each theory is naturally equipped with a computational meaning; the semantics allows for a classifying model, that is, a model such that every other model of the theory can be obtained by applying a suitable functor to it; most other semantics for these logical systems can be mapped to the one presented: Heyting categories, elementary toposes, Kripke models, hyperdoctrines, and Grothendieck toposes. In this talk, we will focus on the ﬁrst aspect only. (3 of 18)
• 4. Introduction Most of this talk is devoted to introduce a single deﬁnition, logically distributive categories, which identiﬁes the models of our logical systems. These models are suitable categories, equipped with an interpretation of formulae and a number of requirements on their structure. Although the propositional part has already been studied by, e.g., Paul Taylor, the ﬁrst-order extension is novel. (4 of 18)
• 5. Logically distributive categories Let Σ = 〈S,F,R〉 be a ﬁrst-order signature, with S the set of sort symbols, F the set of function symbols, of the form f : s1 ×···×sn → s0, with si ∈ S for all 0 ≤ i ≤ n, R the set of relation symbols, of the form r : s1 ×···×sn, with si ∈ S for all 1 ≤ i ≤ n. Also, let T be a theory on Σ, i.e., a collection of axioms. A logically distributive category is a pair 〈C,M〉 where C is a category and M a map from formulae on Σ to ObjC, satisfying seven structural conditions, indicated as (C1) to (C7). Informally, objects of C will denote formulae while arrows will denote proofs where the domain is the theorem and the co-domain is the assumption(s). (5 of 18)
• 6. Logically distributive categories The ﬁrst four conditions allows to interpret propositional intuitionistic logic, as shown in P. Taylor, Practical foundations of mathematics, Cambridge University Press, 1999. (C1) C has ﬁnite products; (C2) C has ﬁnite co-products; (C3) C has exponentiation; (C4) C is distributive, i.e., for every A,B,C ∈ ObjC the arrow ∆ ≡ [1A ×ι1,1A ×ι2] : (A×B)+(A×C) → A×(B +C) has an inverse. Here [_,_] is the co-universal arrow of the (A×B)+(A×C) co-product, 1A is the identity on A, and ι1 : B → B +C, ι2 : C → B +C are the injections of the B +C co-product, _×_ is the product arrow. (6 of 18)
• 7. Logically distributive categories To express the other conditions, we need additional notation. For every s ∈ S, A formula, and x :s variable, let ΣA(x :s) be the functor from the discrete category of terms of sort s to C, deﬁned by t :s → M(A[t/x]). Also, let C∀x :s.A be the subcategory of C whose objects are the vertexes of the cones on ΣA(x :s) such that each vertex is of the form MB for some formula B with x :s ∈ FVB. The arrows of C∀x :s.A, apart identities, are all the arrows in the category of cones over ΣA(x :s) whose co-domain lies in C∀x :s.A and whose domain is M(∀x :s.A). (7 of 18)
• 8. Logically distributive categories Dually, C∃x :s.A is the subcategory of C whose objects are the vertexes of the co-cones on ΣA(x :s) such that each vertex is of the form MB for some formula B with x :s ∈ FVB. The arrows of C∃x :s.A, apart identities, are all the arrows in the category of cones over ΣA(x :s) whose domain lies in C∃x :s.A and whose co-domain is M(∃x :s.A). We require that (C5) All the subcategories C∀x :s.A have a terminal object, and all the subcategories C∃x :s.A have an initial object. Evidently, M(∀x :s.A) is the terminal object in C∀x :s.A, and M(∃x :s.A) is the initial object in C∃x :s.A. More important for us, from each object MB in C∀x :s.A there is a unique arrow to M(∀x :s.A), and dually, to each object MB in C∃x :s.A there is a unique arrow from M(∃x :s.A). (8 of 18)
• 9. Logically distributive categories (C6) We constrain the map M to be as follows: M( ) = 1C, the terminal object of C, M(⊥) = 0C, the initial object of C, M(A∧B) = MA×MB, the binary product in C, M(A∨B) = MA+MB, the binary co-product in C, M(A ⊃ B) = MBMA, the exponential object in C, M(∀x : s.A) = 1C∀x : s.A , the terminal object in C∀x : s.A, M(∃x : s.A) = 0C∃x : s.A , the initial object in C∃x : s.A. Since M is given, the deﬁnition is not circular. But, evidently, it is impredicative. (9 of 18)
• 10. Logically distributive categories For each variable x :s, A,B formulae with x :s ∈ FVA, it is easy to see that MA×M(∃x :s.B) is an object of C∃x :s.A∧B. Thus, there is a unique arrow δ: M(∃x :s.A∧B) → M(A∧(∃x :s.B)) in C∃x :s.A∧B by (C5). Our last condition is that (C7) the δ arrow above has an inverse in C. (10 of 18)
• 11. Semantics Given a theory T over a signature Σ and a logically distributive category 〈C,M〉, we interpret each formula on Σ as MA. Given a proof π: Γ T B with Γ = {x1 :A1,...,xn :An}, where assumptions are named x1,...,xn, π will become an arrow x1 :A1,...,xn :An.π:B : A1 ×···×An → B . To lighten notation, the context will be written as x, and A ≡ A1 ×···×An. (11 of 18)
• 12. Semantics A model for T is a logically distributive category together with a map MAx from T to ObjC such that each axiom A is mapped in an arrow a: 1C → MA. Assuming the standard rules of natural deduction by Prawitz, we inductively interpret proofs as follows: a proof by assumption becomes a projection from the context; a proof by axiom a:B becomes the universal arrow from the context to the terminal object composed by the arrow given by MAx; conjunction eliminations become the projections of the binary product, while conjunction introduction becomes the universal arrow; disjunction elimination become the injections of the binary co-product, while disjunction introduction is reduced to the co-universal arrow; false elimination and truth introduction become the co-universal arrow of the initial object and the universal arrow of the terminal object, respectively; (12 of 18)
• 13. Semantics universal elimination becomes the projection M(∀x :s.C) → M(A[t/x]) in the unique cone over ΣC (x :s) having M(∀x :s.C) as vertex; universal introduction becomes the universal arrow to the terminal object in the C∀x :s.C subcategory; existential introduction becomes the injection M(A[t/x]) → M(∃x :s.C) in the unique co-cone over ΣC (x :s) having M(∃x :s.C) as vertex; existential elimination becomes the co-universal arrow from the initial object of C∃x :s.A∧C . Actually, the precise deﬁnition is a bit more complex, to take into account the context. Also, some additional properties of the existential and universal subcategories are needed. But this is just technique. . . (13 of 18)
• 14. Semantics A formula A is valid in the model 〈C,M,MAx〉 when there exists an arrow 1C → MA. A formula A is a logical consequence of B1,...,Bn in the model when there exists an arrow M(B1 ∧...∧Bn) → MA in C. A formula A is a logical consequence of B1,...,Bn when it is so in any model for the theory. (14 of 18)
• 15. Soundness and completeness Theorem 1 A formula A is a logical consequence of B1,...,Bn in the theory T if and only if there is a proof of A from the hypotheses B1,...,Bn. The proof is long and complex: it can be found in M. Benini, Intuitionistic First-Order Logic: Categorical Semantics via the Curry-Howard Isomorphism, http://arxiv.org/abs/1307.0108, 2013. As a side eﬀect of the completeness proof, it follows that the syntactic category forms a classifying model with respect to the class of functors preserving the logically distributive structure. (15 of 18)
• 16. The role of terms How terms get interpreted? Variables are used to identify the required subcategories C∀x : s.A and C∃x : s.A; variables are also used to construct the substitution functor ΣA(x :s); all terms contribute to the substitution process, which induces the structure used by the semantics. Thus, it is really the substitution process, formalised in the ΣA(x :s) functors, that matters: terms are just the glue that enable us to construct the C∀x : s.A and C∃x : s.A subcategories. It is clear the topological inspiration of the whole construction. In particular, it is evident that terms are not interpreted in some universe, and their role is limited to link together formulae in subcategories that control how quantiﬁers are interpreted. (16 of 18)
• 17. Inconsistent theories A theory T is inconsistent when it allows to derive falsity. However, in our semantics, T has a model as well. A closer look to each model of T reveals that they are categorically “trivial” in the sense that the initial and the terminal objects are isomorphic. This provides a way to show that a theory is consistent. However, this is not ultimately easier or diﬀerent than ﬁnding an internal contradiction in the theory. Actually, it does make sense in a purely computational view that an inconsistent theory has a model: it means that, although the speciﬁcation of a program is ultimately wrong, there are pieces of code which are perfectly sound. (17 of 18)
• 18. Conclusion Much more than this has been done on logically distributive categories. But, still, I am in the beginning of the exploration of this foundation setting. So, any question, comment, suggestion is welcome! Questions? (18 of 18)
Current LanguageEnglish
Español
Portugues
Français
Deutsche