Reasoning with Computer Code: a new Mathematical Logic


A logic is a mathematical model of knowledge used to study how we reason, how we describe the world, and how we infer the conclusions that determine our behavior. The logic presented here is natural. It has been experimentally observed, not designed. It represents knowledge as a causal set, includes a new type of inference based on the minimization of an action functional, and generates its own semantics, making it unnecessary to prescribe one. This logic is suitable for high-level reasoning with computer code, including tasks such as self-programming, objectoriented analysis, refactoring, systems integration, code reuse, and automated programming from sensor-acquired data.

A strong theoretical foundation exists for the new logic. The inference derives laws of conservation from the permutation symmetry of the causal set, and calculates the corresponding conserved quantities. The association between symmetries and conservation laws is a fundamental and well-known law of nature and a general principle in modern theoretical Physics. The conserved quantities take the form of a nested hierarchy of invariant partitions of the given set. The logic associates elements of the set and binds them together to form the levels of the hierarchy. It is conjectured that the hierarchy corresponds to the invariant representations that the brain is known to generate. The hierarchies also represent fully object-oriented, self-generated code, that can be directly compiled and executed (when a compiler becomes available), or translated to a suitable programming language.

The approach is constructivist because all entities are constructed bottom-up, with the fundamental principles of nature being at the bottom, and their existence is proved by construction.

The new logic is mathematically introduced and later discussed in the context of transformations of algorithms and computer programs. We discuss what a full self-programming capability would really mean. We argue that self-programming and the fundamental question about the origin of algorithms are inextricably linked. We discuss previously published, fully automated applications to self-programming, and present a virtual machine that supports the logic, an algorithm that allows for the virtual machine to be simulated on a digital computer, and a fully explained neural network implementation of the algorithm.

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  • Bolognesi, T. 2010. Causal Sets from Simple Models of Computation. arXiv 1004(3128):1–33. arXiv:1004.3128 Available electronically from

  • Caspard, N.; Leclerc, B.; and Monjardet, B. 2012. Finite Ordered Sets. New York: Cambridge University Press.

  • Cuntz, H.; Mathy, A.; and H¨ausser, M. June 2012. A scaling law derived from optimal dendritic wiring. PNAS, 2012, DOI: 10.1073/pnas.1200430109 1–5. Available electronically from

  • Hawkins, J. 2004. On Intelligence. New york: Times Books.

  • Hofstadter, D. R. 1985. Metamagical Themas: Questing for the Essence of Mind and Pattern. New York: Basic Books, Inc.

  • Lin, L.; Osan, R.; and Tsien, J. Z. 2006. Organizing principles of real-time memory encoding: neural clique assemblies and universal neural codes. Trends in Neuroscience 29(1):48–57. Available electronically from

  • Noether, E. 1918. Invariant Variation Problems. Nachr. d. K¨onig. Gesellsch. d. Wiss. zu G¨ottingen Math-phys 1918:235–257. English translation: arXiv:physics/0503066v1 Available electronically from

  • Opdyke, W. F. 1992. Refactoring Object-Oriented Frameworks. Ph.D. Dissertation, Dep. of Computer Science, Univ. of Illinois, Urbana-Champaign, Illinois, USA. Available electronically from gibson/Teaching/CSC7302/ReadingMaterial/Opdyke92.pdf.

  • Pissanetzky, S. 1984. Sparse Matrix Technology. London: Academic Press.

  • Pissanetzky, S. 2009. A new Universal Model of Computation and its Contribution to Learning, Intelligence, Parallelism, Ontologies, Refactoring, and the Sharing of Resources. Int. J. of Information and Mathematical Sciences 5:143–173. Available electronically from

  • Pissanetzky, S. 2010. Coupled Dynamics in Host-Guest Complex Systems Duplicates Emergent Behavior in the Brain. World Academy of Science, Engineering, and Technology 68:1–9. Available electronically from

  • Pissanetzky, S. 2011a. Emergence and Self-organization in Partially Ordered Sets. Complexity 17(2):19–38.

  • Pissanetzky, S. 2011b. Emergent inference and the future of NASA. Workshop, NASA, NASA Gilruth Center, Johnson Space Center, Clear Lake, TX. Available electronically at

  • Pissanetzky, S. 2011c. Structural Emergence in Partially Ordered Sets is the Key to Intelligence. In Artificial General Intelligence, 92–101. Available electronically from

  • Pissanetzky, S. 2012a. A case study: the European Example. Available electronically at

  • Pissanetzky, S. 2012b. The Detailed Dynamics of Dynamical Systems. Available electronically at

  • Pissanetzky, S. 2012c. Overview of Previous Work on Causal Logic. Available electronically at

  • Pissanetzky, S. 2012d. Separating points. Available electronically at

  • Pissanetzky, S. 2012e. Symmetry, structure, and causets in discrete quantum gravity. Bulletin of the American Physical Society 57(2):H1.0005. Available electronically from

  • Pissanetzky, S. 2012f. Verification of the Theory of Detailed Dynamics. Available electronically at

  • Schröder, B. S. W. 2002. Ordered sets. Boston, USA: Birkh¨auser.

  • Shafer, G. 1998. Causal Logic. Available electronically from

  • Wedeen, V. J.; Rosene, D. L.; Wang, R.; Dai, G.; Mortazavi, F.; Hagmann, P.; Kaas, J. H.; and Tseng, W. I. March 2012. The geometric structure of the brain fiber pathways. Science DOI: 10.1126/science.1215280:1628–1634. Available electronically from


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