@article{jacobs_neural_2018,
title = {Neural {Nets} via {Forward} {State} {Transformation} and {Backward} {Loss} {Transformation}},
url = {http://arxiv.org/abs/1803.09356},
abstract = {This article studies (multilayer perceptron) neural networks with an emphasis on the transformations involved --- both forward and backward --- in order to develop a semantical/logical perspective that is in line with standard program semantics. The common two-pass neural network training algorithms make this viewpoint particularly fitting. In the forward direction, neural networks act as state transformers. In the reverse direction, however, neural networks change losses of outputs to losses of inputs, thereby acting like a (real-valued) predicate transformer. In this way, backpropagation is functorial by construction, as shown earlier in recent other work. We illustrate this perspective by training a simple instance of a neural network.},
urldate = {2019-11-21},
journal = {arXiv:1803.09356 [cs]},
author = {Jacobs, Bart and Sprunger, David},
month = mar,
year = {2018},
note = {ZSCC: 0000001
arXiv: 1803.09356},
keywords = {Categorical ML, Effectus theory, Machine learning}
}
@article{jacobs_probability_2018,
title = {From probability monads to commutative effectuses},
volume = {94},
issn = {23522208},
url = {https://linkinghub.elsevier.com/retrieve/pii/S2352220816301122},
doi = {10/gct2wr},
abstract = {Eﬀectuses have recently been introduced as categorical models for quantum computation, with probabilistic and Boolean (classical) computation as special cases. These ‘probabilistic’ models are called commutative eﬀectuses, and are the focus of attention here. The paper describes the main known ‘probability’ monads: the monad of discrete probability measures, the Giry monad, the expectation monad, the probabilistic power domain monad, the Radon monad, and the Kantorovich monad. It also introduces successive properties that a monad should satisfy so that its Kleisli category is a commutative eﬀectus. The main properties are: partial additivity, strong aﬃneness, and commutativity. It is shown that the resulting commutative eﬀectus provides a categorical model of probability theory, including a logic using eﬀect modules with parallel and sequential conjunction, predicate- and state-transformers, normalisation and conditioning of states.},
language = {en},
urldate = {2019-11-28},
journal = {Journal of Logical and Algebraic Methods in Programming},
author = {Jacobs, Bart},
month = jan,
year = {2018},
note = {ZSCC: 0000028},
keywords = {Categorical probability theory, Effectus theory},
pages = {200--237}
}
@article{jacobs_quantum_2017,
title = {Quantum effect logic in cognition},
volume = {81},
issn = {0022-2496},
url = {http://www.sciencedirect.com/science/article/pii/S0022249617300378},
doi = {10/gcnkcj},
abstract = {This paper illustrates applications of a new, modern version of quantum logic in quantum cognition. The new logic uses ‘effects’ as predicates, instead of the more restricted interpretation of predicates as projections — which is used so far in this area. Effect logic involves states and predicates, validity and conditioning, and also state and predicate transformation via channels. The main aim of this paper is to demonstrate the usefulness of this effect logic in quantum cognition, via many high-level reformulations of standard examples. The usefulness of the logic is greatly increased by its implementation in the programming language Python.},
language = {en},
urldate = {2019-11-22},
journal = {Journal of Mathematical Psychology},
author = {Jacobs, Bart},
month = dec,
year = {2017},
note = {ZSCC: 0000002},
keywords = {Categorical probability theory, Effectus theory, Psychology},
pages = {1--10}
}
@article{jacobs_predicate/state_2016,
series = {The {Thirty}-second {Conference} on the {Mathematical} {Foundations} of {Programming} {Semantics} ({MFPS} {XXXII})},
title = {A {Predicate}/{State} {Transformer} {Semantics} for {Bayesian} {Learning}},
volume = {325},
issn = {1571-0661},
url = {http://www.sciencedirect.com/science/article/pii/S1571066116300883},
doi = {10/ggdgbb},
abstract = {This paper establishes a link between Bayesian inference (learning) and predicate and state transformer operations from programming semantics and logic. Specifically, a very general definition of backward inference is given via first applying a predicate transformer and then conditioning. Analogously, forward inference involves first conditioning and then applying a state transformer. These definitions are illustrated in many examples in discrete and continuous probability theory and also in quantum theory.},
language = {en},
urldate = {2019-11-24},
journal = {Electronic Notes in Theoretical Computer Science},
author = {Jacobs, Bart and Zanasi, Fabio},
month = oct,
year = {2016},
note = {ZSCC: 0000030},
keywords = {Bayesianism, Categorical ML, Categorical probability theory, Effectus theory, Programming language theory, Semantics},
pages = {185--200}
}
@article{cho_introduction_2015,
title = {An {Introduction} to {Effectus} {Theory}},
url = {http://arxiv.org/abs/1512.05813},
abstract = {Effectus theory is a new branch of categorical logic that aims to capture the essentials of quantum logic, with probabilistic and Boolean logic as special cases. Predicates in effectus theory are not subobjects having a Heyting algebra structure, like in topos theory, but `characteristic' functions, forming effect algebras. Such effect algebras are algebraic models of quantitative logic, in which double negation holds. Effects in quantum theory and fuzzy predicates in probability theory form examples of effect algebras. This text is an account of the basics of effectus theory. It includes the fundamental duality between states and effects, with the associated Born rule for validity of an effect (predicate) in a particular state. A basic result says that effectuses can be described equivalently in both `total' and `partial' form. So-called `commutative' and `Boolean' effectuses are distinguished, for probabilistic and classical models. It is shown how these Boolean effectuses are essentially extensive categories. A large part of the theory is devoted to the logical notions of comprehension and quotient, which are described abstractly as right adjoint to truth, and as left adjoint to falisity, respectively. It is illustrated how comprehension and quotients are closely related to measurement. The paper closes with a section on `non-commutative' effectus theory, where the appropriate formalisation is not entirely clear yet.},
urldate = {2019-11-23},
journal = {arXiv:1512.05813 [quant-ph]},
author = {Cho, Kenta and Jacobs, Bart and Westerbaan, Bas and Westerbaan, Abraham},
month = dec,
year = {2015},
note = {ZSCC: 0000039
arXiv: 1512.05813},
keywords = {Effectus theory}
}
@article{jacobs_type_2015,
title = {A {Type} {Theory} for {Probabilistic} and {Bayesian} {Reasoning}},
url = {http://arxiv.org/abs/1511.09230},
abstract = {This paper introduces a novel type theory and logic for probabilistic reasoning. Its logic is quantitative, with fuzzy predicates. It includes normalisation and conditioning of states. This conditioning uses a key aspect that distinguishes our probabilistic type theory from quantum type theory, namely the bijective correspondence between predicates and side-effect free actions (called instrument, or assert, maps). The paper shows how suitable computation rules can be derived from this predicate-action correspondence, and uses these rules for calculating conditional probabilities in two well-known examples of Bayesian reasoning in (graphical) models. Our type theory may thus form the basis for a mechanisation of Bayesian inference.},
urldate = {2019-11-21},
journal = {arXiv:1511.09230 [cs, math]},
author = {Jacobs, Bart and Adams, Robin},
month = nov,
year = {2015},
note = {ZSCC: 0000013
arXiv: 1511.09230},
keywords = {Bayesianism, Effectus theory, Type theory}
}
@article{jacobs_towards_2015,
title = {Towards a {Categorical} {Account} of {Conditional} {Probability}},
volume = {195},
issn = {2075-2180},
url = {http://arxiv.org/abs/1306.0831},
doi = {10/ggdf9w},
abstract = {This paper presents a categorical account of conditional probability, covering both the classical and the quantum case. Classical conditional probabilities are expressed as a certain "triangle-fill-in" condition, connecting marginal and joint probabilities, in the Kleisli category of the distribution monad. The conditional probabilities are induced by a map together with a predicate (the condition). The latter is a predicate in the logic of effect modules on this Kleisli category. This same approach can be transferred to the category of C*-algebras (with positive unital maps), whose predicate logic is also expressed in terms of effect modules. Conditional probabilities can again be expressed via a triangle-fill-in property. In the literature, there are several proposals for what quantum conditional probability should be, and also there are extra difficulties not present in the classical case. At this stage, we only describe quantum systems with classical parametrization.},
urldate = {2019-11-21},
journal = {Electronic Proceedings in Theoretical Computer Science},
author = {Jacobs, Bart and Furber, Robert},
month = nov,
year = {2015},
note = {ZSCC: NoCitationData[s0]
arXiv: 1306.0831},
keywords = {Categorical probability theory, Effectus theory},
pages = {179--195}
}
@article{jacobs_new_2015,
title = {New {Directions} in {Categorical} {Logic}, for {Classical}, {Probabilistic} and {Quantum} {Logic}},
volume = {11},
issn = {18605974},
url = {http://arxiv.org/abs/1205.3940},
doi = {10/ggdf99},
abstract = {Intuitionistic logic, in which the double negation law not-not-P = P fails, is dominant in categorical logic, notably in topos theory. This paper follows a different direction in which double negation does hold. The algebraic notions of effect algebra/module that emerged in theoretical physics form the cornerstone. It is shown that under mild conditions on a category, its maps of the form X -{\textgreater} 1+1 carry such effect module structure, and can be used as predicates. Predicates are identified in many different situations, and capture for instance ordinary subsets, fuzzy predicates in a probabilistic setting, idempotents in a ring, and effects (positive elements below the unit) in a C*-algebra or Hilbert space. In quantum foundations the duality between states and effects plays an important role. It appears here in the form of an adjunction, where we use maps 1 -{\textgreater} X as states. For such a state s and a predicate p, the validity probability s {\textbar}= p is defined, as an abstract Born rule. It captures many forms of (Boolean or probabilistic) validity known from the literature. Measurement from quantum mechanics is formalised categorically in terms of `instruments', using L{\textbackslash}"uders rule in the quantum case. These instruments are special maps associated with predicates (more generally, with tests), which perform the act of measurement and may have a side-effect that disturbs the system under observation. This abstract description of side-effects is one of the main achievements of the current approach. It is shown that in the special case of C*-algebras, side-effect appear exclusively in the non-commutative case. Also, these instruments are used for test operators in a dynamic logic that can be used for reasoning about quantum programs/protocols. The paper describes four successive assumptions, towards a categorical axiomatisation of quantitative logic for probabilistic and quantum systems.},
number = {3},
urldate = {2019-11-23},
journal = {Logical Methods in Computer Science},
author = {Jacobs, Bart},
month = oct,
year = {2015},
note = {ZSCC: NoCitationData[s0]
arXiv: 1205.3940},
keywords = {Effectus theory},
pages = {24}
}