## Process Algebra with Five-Valued Conditions (1999)

Citations: | 3 - 1 self |

### BibTeX

@MISC{Bergstra99processalgebra,

author = {Jan A. Bergstra and Alban Ponse},

title = {Process Algebra with Five-Valued Conditions},

year = {1999}

}

### OpenURL

### Abstract

. We propose a ve-valued logic that can be motivated from an algorithmic point of view and from a logical perspective. This logic is combined with process algebra. For process algebra with ve-valued logic we present an operational semantics in SOS-style and a completeness result. Finally, we discuss some generalizations. Key words & Phrases: Concurrency, process algebra, many-valued logic, conditional guard construct, conditional composition. 1991 CR Categories: F.3, F.4.3, I.1. 1 Introduction Assume P is some simple program or algorithm. Then the initial behaviour of if then P else P depends on evaluation of the condition : either it yields an immediate error, or it starts performing P , or it diverges in evaluation of . Note that the second possibility only requires that is either true or false. The following three nonclassical truth values accommodate these intuitions: Meaningless. Typical examples are errors that are detectable during execution such as a type-clash or...

### Citations

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- McCarthy, Hayes
- 1969
(Show Context)
Citation Context ...fined in Σ5 by x / y . z = ((y ∨ D) ∧b (x ∨b G)) / C . ((¬y ∨ D) ∧b (z ∧b H)), where x / C . y is given above, G = (y ∧b x)∨(¬y ∧b z), andH = (¬y ∨b x)∧(y ∨b z). Fluents. Following McCarthy and Hayes =-=[23]-=-, let f, g, ... be names for fluents, i.e., objects that in any state (i.e., at each instance of time) may take a deterministic value, thus a value in {M,T, F,D}. We write f : DetFluent to express thi... |

394 |
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- Kleene
- 1956
(Show Context)
Citation Context ...n-classical logics in process theory is interesting in its own right. Expressivity can be enlarged by involving recursive ingredients. For process description we propose the (binary) Kleene star (see =-=[19]-=-), which in process algebra is defined by X∗Y = X · (X∗Y ) + Y (see also [7]). In a more general setting, one can define X∗φY = X · (X∗φY )+ φ Y and write X∗Y for X∗CY . Examples with recursively defi... |

384 | Process algebra for synchronous communication
- Bergstra, Klop
- 1984
(Show Context)
Citation Context ...Σ5. The identity stated in the following lemma is used later on, and can be easily proved. Lemma 2.2. Σ5(P4) |= φ ∨ D = φ ∨b D. 3 ACP with Five-Valued Conditions The axiom system ACP(A, γ) (see e.g., =-=[9, 10, 6]-=-) is parameterized with a set A of constants a, b, c, ... denoting atomic actions (atoms), i.e., processes that are not subject to further division, and that execute in finite time. In ACP(A, γ) there... |

363 |
Cooperating sequential processes
- Dijkstra
- 1965
(Show Context)
Citation Context ...itional composition, and that provides a parameterized version of the parallel composition operations. Next we provide an algebraic setting for the Cooperating Sequential Processes (CpSP) of Dijkstra =-=[15]-=-. We can do this for all logics that contain C. We define the following operations, where A is the set of atomic actions, Pr is the sort of processes, and L is the particular logic involved. — Constan... |

212 | A basis for a mathematical theory of computation
- McCarthy
- 1963
(Show Context)
Citation Context ... connectives and conditional composition, and we shall incorporate fluents to represent “deterministic conditions”. Asymmetric connectives. With ∧b we denote McCarthy’s left to right conjunction (cf. =-=[21]-=-), adopting the asymmetric notation from [8]. First the left argument is evaluated, and if necessary the right argument. From [8] and the intuitions provided for M,C, and D it follows that c ∧b x = c ... |

120 |
Transition system specifications with negative premises
- Groote
- 1993
(Show Context)
Citation Context ...cuted (if anything, otherwise √ symbolizes successful termination). The axioms and rules in Tables 4 and 5 yield a structured operational semantics (SOS) with negative premises in the style of Groote =-=[16]-=-. Moreover, they satisfy the so called panth-format defined by Verhoef [24] and define the following notion of bisimulation equivalence: Definition 4.1. Let B ⊆ ACPC,µ(A, γ,P4) × ACPC,µ(A, γ,P4). Then... |

114 |
Process Algebra. Cambridge Tracts in Theoretical Computer Science 18
- Baeten, Weijland
- 1990
(Show Context)
Citation Context ...Σ5. The identity stated in the following lemma is used later on, and can be easily proved. Lemma 2.2. Σ5(P4) |= φ ∨ D = φ ∨b D. 3 ACP with Five-Valued Conditions The axiom system ACP(A, γ) (see e.g., =-=[9, 10, 6]-=-) is parameterized with a set A of constants a, b, c, ... denoting atomic actions (atoms), i.e., processes that are not subject to further division, and that execute in finite time. In ACP(A, γ) there... |

114 | A congruence theorem for structured operational semantics with predicates
- Baeten, Verhoef
- 1993
(Show Context)
Citation Context ...oms and rules in Tables 4 and 5 yield a structured operational semantics (SOS) with negative premises in the style of Groote [16]. Moreover, they satisfy the so called panth-format defined by Verhoef =-=[24]-=- and define the following notion of bisimulation equivalence: Definition 4.1. Let B ⊆ ACPC,µ(A, γ,P4) × ACPC,µ(A, γ,P4). Then B is a bisimulation if for all P, Q with PBQ the following conditions hold... |

95 | Laws of programming
- Hoare, Hayes, et al.
- 1987
(Show Context)
Citation Context ...and ∧b : HH H @ @@ T F D M C (The outer rhombus represents the original lattice from [8, 13], without C.) Conditional composition. The expression x / y . z, of which the notation stems from =-=[17]-=-, denotes if y then x else z. Sequential connectives provide a useful intuition if conditional composition is introduced in the logic: y ∧b x = x / y . F. This is plausible because it provides the ver... |

94 |
On the Notation of Ordinal Numbers
- Kleene
- 1938
(Show Context)
Citation Context ...ed by the axioms in Table 1, where T stands for “true”, F for “false”, and ∗ denotes a “third truth value”: Kleene’s three-valued logic K3. This three-valued logic, which we call K3, is introduced in =-=[18]-=- to model propositional combination of partial predicates. K3 is defined by the following truth tables: x ¬x T F F T ∗ ∗ ∧ T F ∗ T T F ∗ F F F F ∗ ∗ F ∗ ∨ T F ∗ T T T T F T F ∗ ∗ T ∗ ∗ and is characte... |

76 | Formalizing Common Sense: Papers by John McCarthy - McCarthy - 1990 |

70 | Ten years of Hoare’s logic, a survey, part I
- Apt
- 1981
(Show Context)
Citation Context ...er to combine process algebra and five-valued logic, we finally introduce the ‘rule of equivalence’ (ROE) |= φ = ψ ` φ :→ X = ψ :→ X This rule reflects the ‘rule of consequence’ in Hoare’s Logic (cf. =-=[1]-=-). We write ACPC,µ(A, γ,P4) + ROE5 ` X = Y, or shortly ` X = Y , if X = Y follows from the axioms of ACPC,µ(A, γ,P4), the axioms and rules for Σ5(P4), and the appropriate rule of equivalence (ROE5) Σ5... |

68 |
Axiomatizing probabilistic processes: ACP with generative probabilities
- Baeten, Bergstra, et al.
- 1995
(Show Context)
Citation Context ...ion (either processes or again conditions). Therefore we introduce the following alternative notation X+ φ Y = X / φ . Y, which has been borrowed from the conventions in probabilistic process algebra =-=[5]-=-. We use association to the right. The above term then reads as a1 ·X1+φ1a2 ·X2+φ2a3 ·X3+φ3a4 ·X4, which is easier to grasp. A condition in Σ5(P4) is called deterministic if it does not contain C. The... |

64 |
Klop, The algebra of recursively defined processes and the algebra of regular processes
- Bergstra, W
- 1984
(Show Context)
Citation Context ...Σ5. The identity stated in the following lemma is used later on, and can be easily proved. Lemma 2.2. Σ5(P4) |= φ ∨ D = φ ∨b D. 3 ACP with Five-Valued Conditions The axiom system ACP(A, γ) (see e.g., =-=[9, 10, 6]-=-) is parameterized with a set A of constants a, b, c, ... denoting atomic actions (atoms), i.e., processes that are not subject to further division, and that execute in finite time. In ACP(A, γ) there... |

51 |
Process algebra with iteration and nesting
- Bergstra, Bethke, et al.
- 1994
(Show Context)
Citation Context ...in [8], which also comprises an account of McCarthy’s asymmetric connectives. Table 1. Axioms for three-valued logic. (1) ¬T = F (5) x ∧ y = y ∧ x (2) ¬∗ = ∗ (6) x ∧ (y ∧ z) = (x ∧ y) ∧ z (3) ¬¬x = x =-=(7)-=- T ∧ x = x (4) ¬(x ∧ y) = ¬x ∨ ¬y (8) x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z) We observe that two different intuitions for Kleene’s non-classical truth value can be distinguished: choice or undetermined, furt... |

42 |
Global renaming operators in concrete process algebra
- Baeten, Bergstra
- 1988
(Show Context)
Citation Context ...P-process which involves the interpretation of fluents we consider a data-state space S ⊆ T × W for some further unspecified set T and the set W of interpretations. We use a state operator λs( ) (see =-=[2]-=-) to model how the execution of actions affects interpretations. Typically, process aX in data-state s is represented as λs(aX) and satisfies λs(aX) = a′ · λs′(X) where a′ is the action (or δ or µ) th... |

32 |
Process Algebra with Signals and Conditions
- Baeten, Bergstra
- 1992
(Show Context)
Citation Context ...e of process algebra is conditional composition X / φ . Y, which represents if φ then X else Y where X,Y range over processes and φ is a condition. Its introduction in process algebra is described in =-=[3]-=-. In [11– 13] we have extended the scope of the condition in conditional composition to various many-valued logics as described in [8], with the intention to model and analyze the occurrence of error-... |

26 |
Process algebra with propositional signals
- Baeten, Bergstra
- 1997
(Show Context)
Citation Context ...logics containing T, F, with process algebra yields an equational completeness result (adopting our restriction on the interpretation of fluents, discarding C, and using Lemma 4.3). This follows from =-=[4, 12]-=-. Adding M (meaningless) to K4 yields a five-valued logic, which we extended with McCarthy’s asymmetric connectives to provide a useful combination with process algebra. We presented a non-equational ... |

23 | Representation of events in in nerve nets and automata - Kleene - 1956 |

20 | A propositional logic with 4 values: true, false, divergent and meaningless
- Bergstra, Bethke, et al.
- 1995
(Show Context)
Citation Context ...rifying a completeness result (on K4). 2 Five-Valued Logic First we shortly consider the incorporation of each of the previously mentioned non-classical truth values in classical two-valued logic. In =-=[8]-=- it is established that there are only two three-valued logics that satisfy the (nice) algebraic properties defined by the axioms in Table 1, where T stands for “true”, F for “false”, and ∗ denotes a ... |

15 | Transition system speci with negative premises - Groote - 1990 |

8 |
Process algebra with four-valued logic
- Bergstra, Ponse
- 2000
(Show Context)
Citation Context ...rranged in the following partial ordering, reflecting information order and (argumentwise) monotony of ∧ and ∧b : HH H @ @@ T F D M C (The outer rhombus represents the original lattice from =-=[8, 13]-=-, without C.) Conditional composition. The expression x / y . z, of which the notation stems from [17], denotes if y then x else z. Sequential connectives provide a useful intuition if conditional com... |

6 | Bochvar-McCarthy logic and process algebra
- Bergstra, Ponse
- 1997
(Show Context)
Citation Context ...ch (roughly) expresses if φ then X. In Table 3 axioms are given for combining ACP(A, γ) with five-valued conditions. Here the constant µ represents the operational contents of M and was introduced in =-=[11, 13]-=-. Furthermore, the φ in the conditional guard construct ranges over Σ5(P4), so φ :→ is considered as a unary operation and related to conditional composition by axiom (Cond). Later on we show that φ :... |

5 | The algebra of recursively de processes and the algebra of regular processes - Bergstra, Klop - 1984 |

5 | Kleene’s three-valued logic and process algebra
- Bergstra, Ponse
- 1998
(Show Context)
Citation Context ...logics containing T, F, with process algebra yields an equational completeness result (adopting our restriction on the interpretation of fluents, discarding C, and using Lemma 4.3). This follows from =-=[4, 12]-=-. Adding M (meaningless) to K4 yields a five-valued logic, which we extended with McCarthy’s asymmetric connectives to provide a useful combination with process algebra. We presented a non-equational ... |

3 | On a 3-valued logical calculus and its application to the analysis of contradictions (in Russian). Matématic̆eskij sbornik - Bochvar - 1939 |