Generating certified code from formal proofs: a case study in homological algebra

Modern theorem provers such as Vampire utilise premise selection algorithms to control the proof search explosion. Premise selection heuristics often employ an array of continuous and discrete parameters. The quality of recommended premises varies depending on the parameter assignment. In this work, we introduce a principled probabilistic framework for optimisation of a premise selection algorithm. We present results using Sumo Inference Engine (SInE) and the Archive of Formal Proofs (AFP) as a case study. Our approach can be used to optimise heuristics on large theories in minimum number of steps.

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NATURAL FORMALIZATION: DERIVING THE CANTOR-BERNSTEIN THEOREM IN ZF

The Review of Symbolic Logic ◽

10.1017/s175502031900056x ◽

2019 ◽

pp. 1-35

Author(s):

WILFRIED SIEG ◽

PATRICK WALSH

Keyword(s):

Proof Theory ◽

Mathematical Proof ◽

Formal Proofs ◽

Bernstein Theorem ◽

Search System ◽

Inference Mechanism ◽

Mathematical Proofs ◽

Computational Work ◽

Definitional Extension

Abstract Natural Formalization proposes a concrete way of expanding proof theory from the meta-mathematical investigation of formal theories to an examination of “the concept of the specifically mathematical proof.” Formal proofs play a role for this examination in as much as they reflect the essential structure and systematic construction of mathematical proofs. We emphasize three crucial features of our formal inference mechanism: (1) the underlying logical calculus is built for reasoning with gaps and for providing strategic directions, (2) the mathematical frame is a definitional extension of Zermelo–Fraenkel set theory and has a hierarchically organized structure of concepts and operations, and (3) the construction of formal proofs is deeply connected to the frame through rules for definitions and lemmas. To bring these general ideas to life, we examine, as a case study, proofs of the Cantor–Bernstein Theorem that do not appeal to the principle of choice. A thorough analysis of the multitude of “different” informal proofs seems to reduce them to exactly one. The natural formalization confirms that there is one proof, but that it comes in two variants due to Dedekind and Zermelo, respectively. In this way it enhances the conceptual understanding of the represented informal proofs. The formal, computational work is carried out with the proof search system AProS that serves as a proof assistant and implements the above inference mechanism; it can be fully inspected at http://www.phil.cmu.edu/legacy/Proof_Site/. We must—that is my conviction—take the concept of the specifically mathematical proof as an object of investigation. Hilbert 1918

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Bayesian Optimisation for Premise Selection in Automated Theorem Proving (Student Abstract)

Proceedings of the AAAI Conference on Artificial Intelligence ◽

10.1609/aaai.v34i10.7232 ◽

2020 ◽

Vol 34 (10) ◽

pp. 13919-13920

Author(s):

Agnieszka Słowik ◽

Chaitanya Mangla ◽

Mateja Jamnik ◽

Sean B. Holden ◽

Lawrence C. Paulson

Keyword(s):

Parameter Space ◽

Theorem Proving ◽

Automated Theorem Proving ◽

Search Space ◽

Large Set ◽

Probabilistic Framework ◽

Formal Proofs ◽

Dimensional Parameter ◽

Theorem Provers

Modern theorem provers utilise a wide array of heuristics to control the search space explosion, thereby requiring optimisation of a large set of parameters. An exhaustive search in this multi-dimensional parameter space is intractable in most cases, yet the performance of the provers is highly dependent on the parameter assignment. In this work, we introduce a principled probabilistic framework for heuristic optimisation in theorem provers. We present results using a heuristic for premise selection and the Archive of Formal Proofs (AFP) as a case study.

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The Genesis and Internal Dynamics of El Salvador's People's Revolutionary Army, 1970–1976

Behavioral and Brain Sciences ◽

10.1017/s0140525x13009850 ◽

2014 ◽

Vol 38 (01) ◽

pp. 102-129

Author(s):

ALBERTO MARTÍN ÁLVAREZ ◽

EUDALD CORTINA ORERO

Keyword(s):

Latin America ◽

El Salvador ◽

Early Years ◽

Left Wing ◽

Internal Dynamics ◽

Internal Coherence ◽

The 1960S

AbstractUsing interviews with former militants and previously unpublished documents, this article traces the genesis and internal dynamics of the Ejército Revolucionario del Pueblo (People's Revolutionary Army, ERP) in El Salvador during the early years of its existence (1970–6). This period was marked by the inability of the ERP to maintain internal coherence or any consensus on revolutionary strategy, which led to a series of splits and internal fights over control of the organisation. The evidence marshalled in this case study sheds new light on the origins of the armed Salvadorean Left and thus contributes to a wider understanding of the processes of formation and internal dynamics of armed left-wing groups that emerged from the 1960s onwards in Latin America.

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Culture and the plasticity of perception

Behavioral and Brain Sciences ◽

10.1017/s0140525x19002887 ◽

2020 ◽

Vol 43 ◽

Author(s):

Michael Lifshitz ◽

T. M. Luhrmann

Keyword(s):

Cultural Context ◽

Sensory Experience ◽

Cultural Learning ◽

The World

Abstract Culture shapes our basic sensory experience of the world. This is particularly striking in the study of religion and psychosis, where we and others have shown that cultural context determines both the structure and content of hallucination-like events. The cultural shaping of hallucinations may provide a rich case-study for linking cultural learning with emerging prediction-based models of perception.

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Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

Behavioral and Brain Sciences ◽

10.1017/s0140525x19000451 ◽

2019 ◽

Vol 42 ◽

Author(s):

Daniel J. Povinelli ◽

Gabrielle C. Glorioso ◽

Shannon L. Kuznar ◽

Mateja Pavlic

Keyword(s):

Temporal Reasoning ◽

Higher Order ◽

Important Case ◽

Human Cognition ◽

Relational Reasoning ◽

Dual Systems ◽

First Order ◽

Reasoning System ◽

Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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Sociocultural memory development research drives new directions in gadgetry science

Behavioral and Brain Sciences ◽

10.1017/s0140525x19000979 ◽

2019 ◽

Vol 42 ◽

Author(s):

Penny Van Bergen ◽

John Sutton

Keyword(s):

Episodic Memory ◽

Autobiographical Memory ◽

Developmental Psychology ◽

Cross Cultural ◽

Memory Development ◽

Development Research ◽

Cultural Research ◽

New Directions ◽

Cross Cultural Research

Abstract Sociocultural developmental psychology can drive new directions in gadgetry science. We use autobiographical memory, a compound capacity incorporating episodic memory, as a case study. Autobiographical memory emerges late in development, supported by interactions with parents. Intervention research highlights the causal influence of these interactions, whereas cross-cultural research demonstrates culturally determined diversity. Different patterns of inheritance are discussed.

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Construction of plastic contact deformation maps on ceramics: a case study on aluminum nitride

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100165641 ◽

1996 ◽

Vol 54 ◽

pp. 636-637

Author(s):

D. L. Callahan

Keyword(s):

Plastic Deformation ◽

Aluminum Nitride ◽

Mechanical Response ◽

Three Dimensional ◽

Bright Field Image ◽

Perfectly Plastic ◽

Contrast Analysis ◽

Deformation Patterns

Modern polishing, precision machining and microindentation techniques allow the processing and mechanical characterization of ceramics at nanometric scales and within entirely plastic deformation regimes. The mechanical response of most ceramics to such highly constrained contact is not predictable from macroscopic properties and the microstructural deformation patterns have proven difficult to characterize by the application of any individual technique. In this study, TEM techniques of contrast analysis and CBED are combined with stereographic analysis to construct a three-dimensional microstructure deformation map of the surface of a perfectly plastic microindentation on macroscopically brittle aluminum nitride.The bright field image in Figure 1 shows a lg Vickers microindentation contained within a single AlN grain far from any boundaries. High densities of dislocations are evident, particularly near facet edges but are not individually resolvable. The prominent bend contours also indicate the severity of plastic deformation. Figure 2 is a selected area diffraction pattern covering the entire indentation area.

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