Call-by-name, call-by-value and the λ-calculus
This paper examines the old question of the relationship between ISWIM and the λ-calculus, using the distinction between call-by-value and call-by-name. It is held that the relationship should be mediated by a standardisation theorem. Since this leads to difficulties, a new λ-calculus is introduced whose standardisation theorem gives a good correspondence with ISWIM as given by the SECD machine, but without the letrec feature. Next a call-by-name variant of ISWIM is introduced which is in an analogous correspondence withthe usual λ-calculus. The relation between call-by-value and call-by-name is then studied by giving simulations of each language by the other and interpretations of each calculus in the other. These are obtained as another application of the continuation technique. Some emphasis is placed throughout on the notion of operational equality (or contextual equality). If terms can be proved equal in a calculus they are operationally equal in the corresponding language. Unfortunately, operational equality is not preserved by either of the simulations.
All elementary functions from a single operator
A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin\sin, cos\cos, \sqrt{\phantom{x}}, and log\log has always required multiple distinct operations. Here we show that a single binary operator,
A modal analysis of staged computation | Journal of the ACM
We show that a type system based on the intuitionistic modal logic S4 provides an expressive framework for specifying and analyzing computation stages in the context of typed λ-calculi and functional languages. We directly demonstrate the sense in which ...
A formulation of the simple theory of types
The purpose of the present paper is to give a formulation of the simple theory of types which incorporates certain features of the calculus of λ-conversion. A complete incorporation of the calculus of λ-conversion into the theory of types is impossible if we require that λx and juxtaposition shall retain their respective meanings as an abstraction operator and as denoting the application of function to argument. But the present partial incorporation has certain advantages from the point of view of type theory and is offered as being of interest on this basis (whatever may be thought of the finally satisfactory character of the theory of types as a foundation for logic and mathematics).For features of the formulation which are not immediately connected with the incorporation of λ-conversion, we are heavily indebted to Whitehead and Russell, Hilbert and Ackermann, Hilbert and Bernays, and to forerunners of these, as the reader familiar with the works in question will recognize.The class of type symbols is described by the rules that ı and o are each type symbols and that if α and β are type symbols then (αβ) is a type symbol: it is the least class of symbols which contains the symbols ı and o and is closed under the operation of forming the symbol (αβ) from the symbols α and β.
All elementary functions from a single binary operator
A single two-input gate suffices for all of Boolean logic in digital hardware. No comparable primitive has been known for continuous mathematics: computing elementary functions such as sin, cos, sqrt, and log has always required multiple distinct operations. Here I show that a single binary operator, eml(x,y)=exp(x)-ln(y), together with the constant 1, generates the standard repertoire of a scientific calculator. This includes constants such as e, pi, and i; arithmetic operations including addition, subtraction, multiplication, division, and exponentiation as well as the usual transcendental and algebraic functions. For example, exp(x)=eml(x,1), ln(x)=eml(1,eml(eml(1,x),1)), and likewise for all other operations. That such an operator exists was not anticipated; I found it by systematic exhaustive search and established constructively that it suffices for the concrete scientific-calculator basis. In EML (Exp-Minus-Log) form, every such expression becomes a binary tree of identical nodes, yielding a grammar as simple as S -> 1 | eml(S,S). This uniform structure also enables gradient-based symbolic regression: using EML trees as trainable circuits with standard optimizers (Adam), I demonstrate the feasibility of exact recovery of closed-form elementary functions from numerical data at shallow tree depths up to 4. The same architecture can fit arbitrary data, but when the generating law is elementary, it may recover the exact formula.
Putting Paradoxes to Work: Contextuality in Measurement-Based Quantum Computation
We describe a joint cohomological framework for measurement-based quantum computation (MBQC) and the corresponding contextuality proofs. The central object in this framework is an element $$[\beta _\Psi ]$$[βΨ]in the second cohomology group of the chain complex describing a given MBQC. $$[\beta _\Psi ]$$[βΨ]contains the function computed therein up to gauge equivalence, and at the same time is a contextuality witness. The present cohomological description only applies to temporally flat MBQCs, and we outline an approach for extending it to the temporally ordered case.
Functional programming with bananas, lenses, envelopes and barbed wire
We develop a calculus for lazy functional programming based on recursion operators associated with data type definitions. For these operators we derive various algebraic laws that are useful in deriving and manipulating programs. We shall show that all example functions in Bird and Wadler's “Introduction to Functional Programming” can be expressed using these operators.
Cognitive Integrity Framework: Formal Foundations for Multiagent Security
Multiagent AI systems introduce cognitive attack surfaces absent in single-model inference. When agents delegate to agents, forming beliefs about beliefs through recursive trust hierarchies, manipulation of reason- ing processes—rather than mere data corruption—becomes a primary security concern. This paper presents the Cognitive Integrity Framework (CIF), providing formal foundations for cognitive security in multiagent operators. We develop four interconnected theoretical contributions: a Trust Calculus with bounded delegation (exponential 𝛿𝑑 decay) that prevents trust amplification through delegation chains; a Defense Com- position Algebra with series and parallel composition theorems establishing multiplicative detection bounds; Information-Theoretic Limits relating stealth constraints to maximum attack impact through a fundamental stealth-impact tradeoff; and a formal Adversary Hierarchy (Ω1–Ω5) characterizing external, peripheral, agent-level, coordination, and systemic threats with increasing capability and decreasing detectability. The framework provides complete coverage of the OWASP Top 10 for Agentic Applications through formal threat models grounded in cognitive state manipulation rather than traditional input/output filtering. CIF bridges classical security concepts with the cognitive requirements of agentic systems. We extend Byzantine fault tolerance to cognitive manipulation—agents that appear functional but hold corrupted beliefs—and adapt trust management systems to continuous trust evolution with provable decay bounds. The framework formalizes five architectural defense mechanisms (cognitive firewalls, belief sandboxing, behavioral tripwires, provenance tracking, Byzantine consensus) with composition rules enabling formal reasoning about layered security. Technical foundations include: operational semantics for message passing and trust updates; invariants for belief integrity, goal preservation, and trust boundedness; model checking configurations for safety property verification; and a complete notation system for attack parameterization, defense specification, and cognitive state representation. This is Part 1 of a three-part series: Part 1 (this paper, DOI: 10.5281/zenodo.18364119) presents formal foundations and theoretical analysis; Part 2 (DOI: 10.5281/zenodo.18364128) provides computational validation and implementation; Part 3 (DOI: 10.5281/zenodo.18364130) offers practical deployment guidance. The framework will continue to be developed and versioned at https://github.com/docxology/cognitive_integrity/ .
Probabilistic compositional semantics, purely
We provide a general framework for the integration of formal semantics with probabilistic reasoning. This framework is conservative, in the sense that it relies only on typed λ-calculus and is thus co
Protocol: How Control Exists after Decentralization by …
How Control Exists after Decentralization by Galloway, …
Compiling Higher-Order Specifications to SMT Solvers: How to Deal with Rejection Constructively | Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs
We present a decision procedure that combines reasoning about datatypes and codatatypes. The dual of the acyclicity rule for datatypes is a uniqueness rule that identifies observationally equal codatatype values, including cyclic values. The procedure ...
Structured reasoning about actor systems | Proceedings of the 2013 workshop on Programming based on actors, agents, and decentralized control
Successfully attaining consensus in the absence of a centralized coordinator is a fundamental problem in distributed multi-agent systems. We analyze progress in the Synod consensus protocol—which does not assume a unique leader—under the ...
Structured reasoning about actor systems | Proceedings of the 2013 workshop on Programming based on actors, agents, and decentralized control
Successfully attaining consensus in the absence of a centralized coordinator is a fundamental problem in distributed multi-agent systems. We analyze progress in the Synod consensus protocol—which does not assume a unique leader—under the ...