Outline Introduction to Effectus Theory Background A crash course on effect algebras and effect modules TACL’17, Prague Effectuses Bart Jacobs bart@cs.ru.nl 26 June 2017 Basic results in effectus theory Effectuses for probability and classical computation Assert maps for sequential conjunction and conditioning Quotients and comprehension Tool support for effectus probability Conclusions Page 1 of 39 Jacobs 26 June 2017 Effectus Theory Page 2 of 39 Jacobs 26 June 2017 Effectus Theory Where we are, so far About this talk Overview of quantum logic research at Nijmegen ◮ ◮ Performed within context of ERC Advanced Grant Quantum Logic, Background Computation, and Security • Running period: 1 May 2013 – 1 May 2018 A crash course on effect algebras and effect modules Focus on categorical axiomatisation of the quantum world ◮ • esp. differences/similarties with probabilistic and classical Effectuses computing ◮ Key notion is effectus, a special kind of category (see later) Basic results in effectus theory Tool support for effectus probability Conclusions Page 3 of 39 Jacobs 26 June 2017 Effectus Theory Background
� � � � � � � Group picture From Boolean to intuitionistic & quantum logic both logic & probability, via indexed categories ☛ ✟ toposes Effect Algebras & via subobject logic Effect Modules ✡ ✠ allow partial ∨ ☛ ✟ ☛ ✟ Intuitionistic logic Quantum logic Heyting algebra Orthomodular lattice ✡ ✠ ✡ ✠ drop distributivity drop double negation ☛ ✟ keep double negation keep distributivity Boolean logic/algebra ✡ ✠ Page 4 of 39 Jacobs 26 June 2017 Effectus Theory Page 5 of 39 Jacobs 26 June 2017 Effectus Theory Background Background Aha-moments in categorical logic Example (without knowing yet what an effectus is) The opposite Rng op of the category of rings (with unit) is an effectus, with: predicate � 1 + 1 in Rng op R effectus theory (2010s) topos theory (1970s) = = = = = = = = = = = = = = � R Z × Z in Rng ◮ focus on characteris- ◮ focus on subobjects = = = = = = = = = = = = = = = = = = = = = = idempotent e ∈ R , so e 2 = e tic maps X → 1 + 1 A X they form an effect they form a Heyting ◮ ◮ Hence the predicates on R ∈ Rng op are its idempotents algebra algebra ◮ These idempotents e ∈ R form an effect algebra, with: orthocomplement e ⊥ = 1 − e truth 1 falsum 0 Additionally there is a partial sum e � d = e + d if ed = 0 = de . • If R is commutative, then the idempotents form a Boolean algebra! ◮ (this case is well-known/studied, eg. in sheaf theory for commutative rings) Page 6 of 39 Jacobs 26 June 2017 Effectus Theory Page 7 of 39 Jacobs 26 June 2017 Effectus Theory Background Background
Origin of ‘effectus’ Where we are, so far New Directions paper ◮ B. Jacobs, New Directions in Categorical Logic, for Classical, Probabilistic and Quantum Logic , LMCS 11(3), 2015 Background Introduces four successive assumptions (and elaborates them) ◮ A crash course on effect algebras and effect modules Intro paper Effectuses Cho, Jacobs, Westerbaan, Westerbaan, Introduction to Effectus ◮ Theory , 2015, arxiv.org/abs/1512.05813 , 150p. Basic results in effectus theory Several other papers by ERC team members, eg. Tool support for effectus probability ◮ Kenta Cho, on equivalence between ‘total’ and ‘partial’ description ◮ Robin Adams, on “effect” logic & type theory Conclusions Bas & Bram Westerbaan, on von Neumann algebra model ◮ Page 8 of 39 Jacobs 26 June 2017 Effectus Theory Background Effect algebras, definition Effect algebras, observations Effect algebras axiomatise the unit interval [ 0 , 1 ] with its (partial!) There is then a partial order, via x ≤ y iff y = x � z , for some z ◮ addition + and “negation” x ⊥ = 1 − x . ◮ Each Boolean algebra is an effect algebra, with: x ⊥ y iff x ∧ y = 0 , and then x � y = x ∨ y Definition A Partial Commutative Monoid (PCM) consists of a set M with zero In fact, each orthomodular lattice is an effect algebra (in the same way) ◮ 0 ∈ M and partial operation � : M × M → M , which is suitably Frequently occurring form: unit intervals: ◮ commutative and associative. [ 0 , 1 ] G = { x ∈ G | 0 ≤ x ≤ 1 } One writes x ⊥ y if x � y is defined. in an ordered Abelian group with order unit 1 ∈ G . x ⊥ = 1 − x • Definition • x ⊥ y iff x + y ≤ 1, and in that case x � y = x + y . An effect algebra is a PCM in which each element x has a unique ‘orthosuplement’ x ⊥ with x � x ⊥ = 1 ( = 0 ⊥ ) Additionally, x ⊥ 1 ⇒ x = 0 must hold. Page 9 of 39 Jacobs 26 June 2017 Effectus Theory Page 10 of 39 Jacobs 26 June 2017 Effectus Theory A crash course on effect algebras and effect modules A crash course on effect algebras and effect modules
Homomorphisms of effect algebras Naturality of partial sums/disjunctions in logic George Boole in 1854 thought of disjunction as a partial operation Definition “Now those laws have been de- A homomorphism of effect algebras f : X → Y satisfies: termined from the study of in- f ( 1 ) = 1 ◮ stances, in all of which it has been if x ⊥ x ′ then both f ( x ) ⊥ f ( x ′ ) and f ( x � x ′ ) = f ( x ) � f ( x ′ ) . ◮ a necessary condition, that the This yields a category EA of effect algebras. classes or things added together in thought should be mutually ex- clusive. The expression x + y Example: seems indeed uninterpretable, un- A probability measure yields a map Σ X → [ 0 , 1 ] in EA ◮ less it be assumed that the things ◮ Recall the indicator (characteristic) function 1 U : X → [ 0 , 1 ] , for a represented by x and the things subset U ⊆ X . represented by y are entirely sep- • It gives a map of effect algebras: arate; that they embrace no indi- viduals in common.” (p.66) 1 ( − ) � [ 0 , 1 ] X P ( X ) Page 11 of 39 Jacobs 26 June 2017 Effectus Theory Page 12 of 39 Jacobs 26 June 2017 Effectus Theory A crash course on effect algebras and effect modules A crash course on effect algebras and effect modules Effect modules Effect modules, main examples Effect modules are effect algebras with a scalar multiplication, with Probabilistic examples Fuzzy predicates [ 0 , 1 ] X on a set X , with scalar multiplication scalars not from R or C , but from [ 0 , 1 ] . ◮ (Or more generally from an “effect monoid”, ie. effect algebra with multiplication) def r · p = x �→ r · p ( x ) ◮ Measurable predicates Hom ( X , [ 0 , 1 ]) , for a measurable space X , Definition with the same scalar multiplication Continuous predicates Hom ( X , [ 0 , 1 ]) , for a topological space X ◮ An effect module M is a effect algebra with an action [ 0 , 1 ] × M → M that is a “bihomomorphism” Quantum examples ◮ Effects E ( H ) on a Hilbert space: operators A : H → H satisfying 0 ≤ A ≤ I , with scalar multiplication ( r , A ) �→ rA . A map of effect modules is a map of effect algebras that commutes with scalar multiplication. Effects in a C ∗ / W ∗ -algebra A : positive elements below the unit: ◮ We get a category EMod ֒ → EA . [ 0 , 1 ] A = { a ∈ A | 0 ≤ a ≤ 1 } . This one covers the previous illustrations. Page 13 of 39 Jacobs 26 June 2017 Effectus Theory Page 14 of 39 Jacobs 26 June 2017 Effectus Theory A crash course on effect algebras and effect modules A crash course on effect algebras and effect modules
� � � � � � � � � Basic adjunction, between effects and states Where we are, so far Theorem By “homming into [ 0 , 1 ] ” one gets an adjunction: Background Hom ( − , [ 0 , 1 ]) EMod op ⊤ Conv A crash course on effect algebras and effect modules Hom ( − , [ 0 , 1 ]) This adjunction restricts to an equivalence of categories between: Effectuses Banach effect modules, which have a complete norm ◮ (or equivalently, complete order unit spaces) Basic results in effectus theory ◮ convex compact Hausdorff spaces This is called Kadison duality Tool support for effectus probability Conclusions Page 15 of 39 Jacobs 26 June 2017 Effectus Theory A crash course on effect algebras and effect modules Effectus Internal logic effectus meaning An effectus is a category with finite coproducts ( 0 , +) and 1 such that ◮ these diagrams are pullbacks: objects X types id + g � id f A + X A + Y A A arrows X → Y programs 1 (final object) singleton/unit type κ 1 � κ 1 f + id � f + id ω � X id + g � B + Y id + f � A + Y A + X 1 state B + X p � 1 + 1 X predicate these arrows are jointly monic: ◮ p � 1 + 1 ω � ··· 1 validity ω | = p X ·· =[ κ 1 ,κ 2 ,κ 2 ] ω � p X + X + X � X + X � 1 + 1 ··· 1 scalar ·· =[ κ 2 ,κ 1 ,κ 2 ] f ∗ ( ω ) | = q f ∗ ( ω ) = f ◦ ω state transformation Perspective: = f ∗ ( q ) = q ◦ f predicate transformation � � � � = f ∗ ( q ) disjoint and universal disjoint ω | � � effectus coproducts coproducts Page 16 of 39 Jacobs 26 June 2017 Effectus Theory Page 17 of 39 Jacobs 26 June 2017 Effectus Theory Effectuses Effectuses
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