Univariate fractions can be transformed to mixed fractions in the equational theory of meadows of... more Univariate fractions can be transformed to mixed fractions in the equational theory of meadows of characteristic zero.
In an arithmetical structure one can make division a total function by defining 1/0 to be an elem... more In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element such as infinity ∞ or error element ⊥. A wheel is an algebra in which division is totalised by setting 1/0 = ∞ but which also contains an error element ⊥ to help control its use. We construct the wheel of rational numbers as an abstract data type Qw and give it an equational specification without auxiliary operators under initial algebra semantics.
Division by zero is a controversial theme. Why is division by zero a relevant issue and how can t... more Division by zero is a controversial theme. Why is division by zero a relevant issue and how can this issue be addressed from different perspectives? Fracterm is used as an abbreviation for fractional expression. Three types of occurrence of the division symbol in a fracterm are distinguished: prospective occurrence, retrospective occurrence, and formal occurrence. Mathematics mostly features retrospective occurrences of division, computer programming gives rise to prospective occurrences, and so does automated proof checking. The use of division in an axiom system may indicate the presence of formal occurrences of division symbols.
In an arithmetical structure one can make division a total function by defining 1/0 to be an elem... more In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to its opposite, and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics.
The strong, intermediate, and weak Turing impossibility properties are introduced. Some facts con... more The strong, intermediate, and weak Turing impossibility properties are introduced. Some facts concerning Turing impossibility for stack machine programming are trivially adapted from previous work. Several intriguing questions are raised about the Turing impossibility properties concerning different method interfaces for stack machine programming.
We will examine totalising a partial operation in a general algebra by using an absorbtive elemen... more We will examine totalising a partial operation in a general algebra by using an absorbtive element, bottom, such as an error flag. We then focus on the simplest example of a partial operation, namely subtraction on the natural numbers: n - m is undefined whenever n < m. We examine the use of bottom in algebraic structures for the natural numbers, especially semigroups and semirings. We axiomatise this totalisation process and introduce the algebraic concept of a team, being an additive cancellative semigroup with totalised subtraction. Also, with the natural numbers in mind, we introduce the property of being generated by an iterative function, which we call a splinter. We prove a number of theorems about the algebraic specification of datatypes of natural numbers.
An inversive meadow is a commutative ring with identity equipped with a multiplicative inverse op... more An inversive meadow is a commutative ring with identity equipped with a multiplicative inverse operation made total by choosing 0 as its value at 0. Previously, inversive meadows were shortly called meadows. A divisive meadow is an inversive meadows with the multiplicative inverse operation replaced by a division operation. In the spirit of Peacock's arithmetical algebra, we introduce variants of inversive and divisive meadows without an additive identity element and an additive inverse operation. We give equational axiomatizations of several classes of such variants of inversive and divisive meadows as well as of several instances of them.
For each function on bit strings, its restriction to bit strings of any given length can be compu... more For each function on bit strings, its restriction to bit strings of any given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. Backward jump instructions are not necessary for this, but instruction sequences can be significantly shorter with them. We take the function on bit strings that models the multiplication of natural numbers on their representation in the binary number system to demonstrate this by means of a concrete example. The example is reason to discuss points concerning the halting problem and the concept of an algorithm.
Four options for assigning a meaning to Islamic Logic are surveyed including a new proposal for a... more Four options for assigning a meaning to Islamic Logic are surveyed including a new proposal for an option named Real Islamic Logic (RIL). That approach to Islamic Logic should serve modern Islamic objectives in a way comparable to the functionality of Islamic Finance. The prospective role of RIL is analyzed from several perspectives: (i) parallel distributed systems design, (ii) reception by a community structured audience, (iii) informal logic and applied non-classical logics, and (iv) (in)tractability and artificial intelligence.
We give a rough sketch of the Judaic, Greek, Islamic and Christian positions in the matter of int... more We give a rough sketch of the Judaic, Greek, Islamic and Christian positions in the matter of interest prohibition during the last few millennia and discuss the way in which interest prohibition is dealt with in Islamic finance, the problems with authority-based arguments for interest prohibition, and the prospects of interest prohibition with the advent of electronic money.
We use the notion of a promise to define local trust between agents possessing autonomous decisio... more We use the notion of a promise to define local trust between agents possessing autonomous decision-making. An agent is trustworthy if it is expected that it will keep a promise. This definition satisfies most commonplace meanings of trust. Reputation is then an estimation of this expectation value that is passed on from agent to agent. Our definition distinguishes types of trust, for different behaviours, and decouples the concept of agent reliability from the behaviour on which the judgement is based. We show, however, that trust is fundamentally heuristic, as it provides insufficient information for agents to make a rational judgement. A global trustworthiness, or community trust can be defined by a proportional, self-consistent voting process, as a weighted eigenvectorcentrality function of the promise theoretical graph.
Decision taking can be performed as a service to other parties and it is amenable to outtasking r... more Decision taking can be performed as a service to other parties and it is amenable to outtasking rather than to outsourcing. Outtasking decision taking is compatible with selfsourcing of decision making activities carried out in preparation of decision taking. Decision taking as a service (DTaaS) is viewed as an instance of so-called decision casting. Preconditions for service casting are examined, and compliance of decision taking with these preconditions is confirmed. Potential advantages and disadvantages of using decision taking as a service are considered.
Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities.... more Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) a role which can be played by a number. Sumterm splitting operators are introduced and it is argued that without further precautions the presence of these operators gives rise to an instance of the so-called sum splitting paradox. A survey of solutions to the sum splitting paradox is given.
Interaction with services provided by an execution environment forms part of the behaviours exhib... more Interaction with services provided by an execution environment forms part of the behaviours exhibited by instruction sequences under execution. Mechanisms related to the kind of interaction in question have been proposed in the setting of thread algebra. Like thread, service is an abstract behavioural concept. The concept of a functional unit is similar to the concept of a service, but more concrete. A state space is inherent in the concept of a functional unit, whereas it is not inherent in the concept of a service. In this paper, we establish the existence of a universal computable functional unit for natural numbers and related results.
We review the exposition of division by zero and the definition of total arithmetical functions i... more We review the exposition of division by zero and the definition of total arithmetical functions in ``Introduction to Logic" by Patrick Suppes, 1957, and provide a hyperlink to the archived text. This book is a pedagogical introduction to first-order predicate calculus with logical, mathematical, physical and philosophical examples, some presented in exercises. It is notable for (i) presenting division by zero as a problem worthy of contemplation, (ii) considering five totalisations of real arithmetic, and (iii) making the observation that each of these solutions to ``the problem of division by zero" has both advantages and disadvantages -- none of the proposals being fully satisfactory. We classify totalisations by the number of non-real symbols they introduce, called their Extension Type. We compare Suppes' proposals for division by zero to more recent proposals. We find that all totalisations of Extension Type 0 are arbitrary, hence all non-arbitrary totalisations ar...
For money-like informational commodities the notions of architectural adequacy and evolutionary a... more For money-like informational commodities the notions of architectural adequacy and evolutionary adequacy are proposed as the first two stages of a moneyness maturity hierarchy. Then three classes of informational commodities are distinguished: exclusively informational commodities, strictly informational commodities, and ownable informational commodities. For each class money-like instances of that commodity class, as well as monies of that class may exist. With the help of these classifications and making use of previous assessments of Bitcoin, it is argued that at this stage Bitcoin is unlikely ever to evolve into a money. Assessing the evolutionary adequacy of Bitcoin is perceived in terms of a search through its design hull for superior design alternatives. An extensive comparison is made between the search for superior design alternatives to Bitcoin and the search for design alternatives to a specific and unconventional view on the definition of fractions.
Recent Trends in Algebraic Development Techniques, 2017
The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A co... more The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability theory are restated in this framework.
The number of instructions of an instruction sequence is taken for its logical SLOC, and is abbre... more The number of instructions of an instruction sequence is taken for its logical SLOC, and is abbreviated with LLOC. A notion of quantitative expressiveness is based on LLOC and in the special case of operation over a family of single bit registers a collection of elementary properties are established. A dedicated notion of interface is developed and is used for stating relevant properties of classes of instruction sequences
Fracterms are introduced as a proxy for fractions. A precise definition of fracterms is formulate... more Fracterms are introduced as a proxy for fractions. A precise definition of fracterms is formulated and on that basis reasonably precise definitions of various classes of fracterms are given. In the context of the meadow of rational numbers viewing fractions as fracterms provides an adequate theory of fractions. A very different view on fractions is that fractions are values, i.e. rational numbers. Fracterms are used to provide a range of intermediate definitions between these two definitions of fractions
Univariate fractions can be transformed to mixed fractions in the equational theory of meadows of... more Univariate fractions can be transformed to mixed fractions in the equational theory of meadows of characteristic zero.
In an arithmetical structure one can make division a total function by defining 1/0 to be an elem... more In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element such as infinity ∞ or error element ⊥. A wheel is an algebra in which division is totalised by setting 1/0 = ∞ but which also contains an error element ⊥ to help control its use. We construct the wheel of rational numbers as an abstract data type Qw and give it an equational specification without auxiliary operators under initial algebra semantics.
Division by zero is a controversial theme. Why is division by zero a relevant issue and how can t... more Division by zero is a controversial theme. Why is division by zero a relevant issue and how can this issue be addressed from different perspectives? Fracterm is used as an abbreviation for fractional expression. Three types of occurrence of the division symbol in a fracterm are distinguished: prospective occurrence, retrospective occurrence, and formal occurrence. Mathematics mostly features retrospective occurrences of division, computer programming gives rise to prospective occurrences, and so does automated proof checking. The use of division in an axiom system may indicate the presence of formal occurrences of division symbols.
In an arithmetical structure one can make division a total function by defining 1/0 to be an elem... more In an arithmetical structure one can make division a total function by defining 1/0 to be an element of the structure, or by adding a new element, such as an error element also denoted with a new constant symbol, an unsigned infinity or one or both signed infinities, one positive and one negative. We define an enlargement of a field to a transfield, in which division is totalised by setting 1/0 equal to the positive infinite value and -1/0 equal to its opposite, and which also contains an error element to help control their effects. We construct the transrational numbers as a transfield of the field of rational numbers and consider it as an abstract data type. We give it an equational specification under initial algebra semantics.
The strong, intermediate, and weak Turing impossibility properties are introduced. Some facts con... more The strong, intermediate, and weak Turing impossibility properties are introduced. Some facts concerning Turing impossibility for stack machine programming are trivially adapted from previous work. Several intriguing questions are raised about the Turing impossibility properties concerning different method interfaces for stack machine programming.
We will examine totalising a partial operation in a general algebra by using an absorbtive elemen... more We will examine totalising a partial operation in a general algebra by using an absorbtive element, bottom, such as an error flag. We then focus on the simplest example of a partial operation, namely subtraction on the natural numbers: n - m is undefined whenever n < m. We examine the use of bottom in algebraic structures for the natural numbers, especially semigroups and semirings. We axiomatise this totalisation process and introduce the algebraic concept of a team, being an additive cancellative semigroup with totalised subtraction. Also, with the natural numbers in mind, we introduce the property of being generated by an iterative function, which we call a splinter. We prove a number of theorems about the algebraic specification of datatypes of natural numbers.
An inversive meadow is a commutative ring with identity equipped with a multiplicative inverse op... more An inversive meadow is a commutative ring with identity equipped with a multiplicative inverse operation made total by choosing 0 as its value at 0. Previously, inversive meadows were shortly called meadows. A divisive meadow is an inversive meadows with the multiplicative inverse operation replaced by a division operation. In the spirit of Peacock's arithmetical algebra, we introduce variants of inversive and divisive meadows without an additive identity element and an additive inverse operation. We give equational axiomatizations of several classes of such variants of inversive and divisive meadows as well as of several instances of them.
For each function on bit strings, its restriction to bit strings of any given length can be compu... more For each function on bit strings, its restriction to bit strings of any given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. Backward jump instructions are not necessary for this, but instruction sequences can be significantly shorter with them. We take the function on bit strings that models the multiplication of natural numbers on their representation in the binary number system to demonstrate this by means of a concrete example. The example is reason to discuss points concerning the halting problem and the concept of an algorithm.
Four options for assigning a meaning to Islamic Logic are surveyed including a new proposal for a... more Four options for assigning a meaning to Islamic Logic are surveyed including a new proposal for an option named Real Islamic Logic (RIL). That approach to Islamic Logic should serve modern Islamic objectives in a way comparable to the functionality of Islamic Finance. The prospective role of RIL is analyzed from several perspectives: (i) parallel distributed systems design, (ii) reception by a community structured audience, (iii) informal logic and applied non-classical logics, and (iv) (in)tractability and artificial intelligence.
We give a rough sketch of the Judaic, Greek, Islamic and Christian positions in the matter of int... more We give a rough sketch of the Judaic, Greek, Islamic and Christian positions in the matter of interest prohibition during the last few millennia and discuss the way in which interest prohibition is dealt with in Islamic finance, the problems with authority-based arguments for interest prohibition, and the prospects of interest prohibition with the advent of electronic money.
We use the notion of a promise to define local trust between agents possessing autonomous decisio... more We use the notion of a promise to define local trust between agents possessing autonomous decision-making. An agent is trustworthy if it is expected that it will keep a promise. This definition satisfies most commonplace meanings of trust. Reputation is then an estimation of this expectation value that is passed on from agent to agent. Our definition distinguishes types of trust, for different behaviours, and decouples the concept of agent reliability from the behaviour on which the judgement is based. We show, however, that trust is fundamentally heuristic, as it provides insufficient information for agents to make a rational judgement. A global trustworthiness, or community trust can be defined by a proportional, self-consistent voting process, as a weighted eigenvectorcentrality function of the promise theoretical graph.
Decision taking can be performed as a service to other parties and it is amenable to outtasking r... more Decision taking can be performed as a service to other parties and it is amenable to outtasking rather than to outsourcing. Outtasking decision taking is compatible with selfsourcing of decision making activities carried out in preparation of decision taking. Decision taking as a service (DTaaS) is viewed as an instance of so-called decision casting. Preconditions for service casting are examined, and compliance of decision taking with these preconditions is confirmed. Potential advantages and disadvantages of using decision taking as a service are considered.
Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities.... more Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) a role which can be played by a number. Sumterm splitting operators are introduced and it is argued that without further precautions the presence of these operators gives rise to an instance of the so-called sum splitting paradox. A survey of solutions to the sum splitting paradox is given.
Interaction with services provided by an execution environment forms part of the behaviours exhib... more Interaction with services provided by an execution environment forms part of the behaviours exhibited by instruction sequences under execution. Mechanisms related to the kind of interaction in question have been proposed in the setting of thread algebra. Like thread, service is an abstract behavioural concept. The concept of a functional unit is similar to the concept of a service, but more concrete. A state space is inherent in the concept of a functional unit, whereas it is not inherent in the concept of a service. In this paper, we establish the existence of a universal computable functional unit for natural numbers and related results.
We review the exposition of division by zero and the definition of total arithmetical functions i... more We review the exposition of division by zero and the definition of total arithmetical functions in ``Introduction to Logic" by Patrick Suppes, 1957, and provide a hyperlink to the archived text. This book is a pedagogical introduction to first-order predicate calculus with logical, mathematical, physical and philosophical examples, some presented in exercises. It is notable for (i) presenting division by zero as a problem worthy of contemplation, (ii) considering five totalisations of real arithmetic, and (iii) making the observation that each of these solutions to ``the problem of division by zero" has both advantages and disadvantages -- none of the proposals being fully satisfactory. We classify totalisations by the number of non-real symbols they introduce, called their Extension Type. We compare Suppes' proposals for division by zero to more recent proposals. We find that all totalisations of Extension Type 0 are arbitrary, hence all non-arbitrary totalisations ar...
For money-like informational commodities the notions of architectural adequacy and evolutionary a... more For money-like informational commodities the notions of architectural adequacy and evolutionary adequacy are proposed as the first two stages of a moneyness maturity hierarchy. Then three classes of informational commodities are distinguished: exclusively informational commodities, strictly informational commodities, and ownable informational commodities. For each class money-like instances of that commodity class, as well as monies of that class may exist. With the help of these classifications and making use of previous assessments of Bitcoin, it is argued that at this stage Bitcoin is unlikely ever to evolve into a money. Assessing the evolutionary adequacy of Bitcoin is perceived in terms of a search through its design hull for superior design alternatives. An extensive comparison is made between the search for superior design alternatives to Bitcoin and the search for design alternatives to a specific and unconventional view on the definition of fractions.
Recent Trends in Algebraic Development Techniques, 2017
The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A co... more The Kolmogorov axioms for probability functions are placed in the context of signed meadows. A completeness theorem is stated and proven for the resulting equational theory of probability calculus. Elementary definitions of probability theory are restated in this framework.
The number of instructions of an instruction sequence is taken for its logical SLOC, and is abbre... more The number of instructions of an instruction sequence is taken for its logical SLOC, and is abbreviated with LLOC. A notion of quantitative expressiveness is based on LLOC and in the special case of operation over a family of single bit registers a collection of elementary properties are established. A dedicated notion of interface is developed and is used for stating relevant properties of classes of instruction sequences
Fracterms are introduced as a proxy for fractions. A precise definition of fracterms is formulate... more Fracterms are introduced as a proxy for fractions. A precise definition of fracterms is formulated and on that basis reasonably precise definitions of various classes of fracterms are given. In the context of the meadow of rational numbers viewing fractions as fracterms provides an adequate theory of fractions. A very different view on fractions is that fractions are values, i.e. rational numbers. Fracterms are used to provide a range of intermediate definitions between these two definitions of fractions
Accusations play a pivotal role in human communication, including in contemporary political affai... more Accusations play a pivotal role in human communication, including in contemporary political affairs. However, there is a lack of a philosophical conceptualisation that is necessary for appropriate descriptions of accusations and further philosophical scrutiny. Accusation Theory is proposed as such a theoretical framework. This paper (1) aims to present some general tools for the description of accusations as speech acts, (2) tries to understand fundamental features of accusations, (3) analyses some elements of the practice of accusing, (4) presents a short overview of relevant literature and (5) outlines some possible avenues of further research.
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Papers by Jan Bergstra