## Programming Languages

### BibTeX

@MISC{Wegbreit_programminglanguages,

author = {B. Wegbreit},

title = {Programming Languages},

year = {}

}

### OpenURL

### Abstract

### Citations

202 | A Basis for a Mathematical Theory of Computation - McCarthy - 1963 |

193 |
Mathematical Theory of Computation
- Manna
- 2003
(Show Context)
Citation Context ...(x)) is valid. However, • is not a tight specification: since ,I, constrains only the second component of z, it is possible to have two different z's which satisfy O; e.g. x = 2, zl = (1,1), and z2 = =-=(13,1)-=- is an assignment of values for which the left-hand side of (5.2) is true but the right-hand side is false. As ~I, is not tight, Theorem 2 does not apply. In fact, (2.2) is not valid; it reads: x _> 2... |

89 | Proving properties of programs by structural induction
- Burstall
- 1969
(Show Context)
Citation Context ...y the inductive assertion technique. The work of Basu, Misra [1], and Mills [14] has a similar rule, except that their specification is always a function rather than a general relation. Other authors =-=[3, 7, 17]-=- have presented similar ideas. Finally, subgoal induction can be viewed as a specialization of the rule of computation induCtion which was developed in the context of pure recursive functions [12]. 8.... |

84 | Correctness of a compiler for arithmetic expressions - McCarthy, Painter - 1967 |

83 |
Procedures and parameters: An axiomatic approach
- Hoare
- 1971
(Show Context)
Citation Context ... b, and c to the output z. 209 We describe a technique, subgoal induction, for proving programs correct. It may be used as an alternative or supplement to the commonly used inductive assertion method =-=[4, 7]-=-. Subgoal induction is a proof rule which matches the problem solving heuristic: "Transform a problem into a simpler one with the same general characteristics, solve the simpler problem, and use that ... |

48 | Proving theorems about LISP functions
- BOYER, MOORE
- 1975
(Show Context)
Citation Context ...alization of results in [1] and [14]. Example 7 (Under-constrained specification). An example may serve to make these considerations more concrete. Consider F(x) ~ if x _< 1 then (0, x) else (F(x-- 1)=-=[2]-=-, F(x-- 1)[1] -t- F(x-- 1)[21) 218 Communications April 1977 of Volume 20 the ACM Number 4where angle brackets denote the forming of ordered pairs and subscripts denote the decomposition of ordered p... |

39 |
A synthesizer of inductive assertions
- German, Wegbreit
- 1975
(Show Context)
Citation Context ...way, it is sometimes possible to build up a complete specification by extracting information from proofs which fail. Details of this idea and its implementation in a program verifier are discussed in =-=[5]-=-. It should be noted that such techniques do not always yield a complete specification within a reasonable number of steps [19]. They are heuristics with certain circumscribed utility. 219 Because of ... |

24 |
Inductive methods for proving properties of programs
- Manna, Ness, et al.
- 1973
(Show Context)
Citation Context ... is a strong enough specification for this to work. Here it suffices to observe a result we prove below: (2.1) and (2.2) are valid if and only if the verification conditions for computation induction =-=[12]-=- are valid. Example 1 (Subgoal induction). Let "/" denote integer division (with truncation) and define E(x) ~ if x = 0 then 1 else L(x, E(x/2)), and L(x, y) ~ if even(x) then y2 else 2. y2 The proper... |

20 |
The New Math of Computer Programming
- Mills
- 1975
(Show Context)
Citation Context ...(x). This is a tight specification and it therefore follows that the verification conditions for subgoal induction are valid. Thus this theorem can be viewed as a generalization of results in [1] and =-=[14]-=-. Example 7 (Under-constrained specification). An example may serve to make these considerations more concrete. Consider F(x) ~ if x _< 1 then (0, x) else (F(x-- 1)[2], F(x-- 1)[1] -t- F(x-- 1)[21) 21... |

19 |
An interactive program verification system
- Good, London, et al.
- 1975
(Show Context)
Citation Context ...text of pure recursive functions [12]. 8. Conclusion Currently, there are three induction methods in common use for mechanical program verification: structural induction [2, 16], inductive assertions =-=[6, 8, 18]-=-, and computation induction [15]. In proposing a fourth, subgoal induction, it is perhaps worthwhile to discuss just why it might be used in preference to one of the current methods. At a formal level... |

17 |
Automatic program verification i: a logical basis and its implementation
- Igarashi, London, et al.
- 1973
(Show Context)
Citation Context ...text of pure recursive functions [12]. 8. Conclusion Currently, there are three induction methods in common use for mechanical program verification: structural induction [2, 16], inductive assertions =-=[6, 8, 18]-=-, and computation induction [15]. In proposing a fourth, subgoal induction, it is perhaps worthwhile to discuss just why it might be used in preference to one of the current methods. At a formal level... |

11 |
Formalization of properties of functional programs
- Manna, Pnueli
- 1970
(Show Context)
Citation Context ...s to be proved are: P(x) ~ x~(x, y; H(x)) and ~P(x) A ~(N(x), y; z2) A ~(M(x), z2; Zl) A ((N(x), y) = (M(x), z2) ~ z2=z1) --)" xXt(x, y; zl) This rule is essentially the same as Manna and Pnueli's in =-=[11]-=- except for the equality "cross term" in the premise: A (3', = "r, ~z, = z,) l<s<t<m which requires that there be a functional relationship between the 3,'s and z's. This fact was implicit in the func... |

8 |
Proving loop programs
- Basu, Misra
- 1975
(Show Context)
Citation Context ... - z = G(x). This is a tight specification and it therefore follows that the verification conditions for subgoal induction are valid. Thus this theorem can be viewed as a generalization of results in =-=[1]-=- and [14]. Example 7 (Under-constrained specification). An example may serve to make these considerations more concrete. Consider F(x) ~ if x _< 1 then (0, x) else (F(x-- 1)[2], F(x-- 1)[1] -t- F(x-- ... |

8 |
Reasoning About Programs
- J, Levitt
- 1973
(Show Context)
Citation Context ...text of pure recursive functions [12]. 8. Conclusion Currently, there are three induction methods in common use for mechanical program verification: structural induction [2, 16], inductive assertions =-=[6, 8, 18]-=-, and computation induction [15]. In proposing a fourth, subgoal induction, it is perhaps worthwhile to discuss just why it might be used in preference to one of the current methods. At a formal level... |

4 |
Interactive program verification using virtual programs
- Topor
- 1975
(Show Context)
Citation Context ...y the inductive assertion technique. The work of Basu, Misra [1], and Mills [14] has a similar rule, except that their specification is always a function rather than a general relation. Other authors =-=[3, 7, 17]-=- have presented similar ideas. Finally, subgoal induction can be viewed as a specialization of the rule of computation induCtion which was developed in the context of pure recursive functions [12]. 8.... |

2 |
Mathematical theory of partial correctness
- Manna
- 1971
(Show Context)
Citation Context ... to the specification if a Q can be found which makes the formula true. By choosing Q based on the specification, this method can be viewed as a slightly weaker variant of subgoal induction. Manna in =-=[20]-=- applied the method of [11] to a flowchart, program and noted that the resulting verification condition differed from that produced by the inductive assertion technique. The work of Basu, Misra [1], a... |

1 |
Assigning meanings to programs. Moth
- Floyd
- 1967
(Show Context)
Citation Context ... b, and c to the output z. 209 We describe a technique, subgoal induction, for proving programs correct. It may be used as an alternative or supplement to the commonly used inductive assertion method =-=[4, 7]-=-. Subgoal induction is a proof rule which matches the problem solving heuristic: "Transform a problem into a simpler one with the same general characteristics, solve the simpler problem, and use that ... |

1 |
Introducing prog into the pure lisp theorem prover. CSL-74-3, Xerox Palo Alto Res
- Moore
- 1974
(Show Context)
Citation Context ...which was developed in the context of pure recursive functions [12]. 8. Conclusion Currently, there are three induction methods in common use for mechanical program verification: structural induction =-=[2, 16]-=-, inductive assertions [6, 8, 18], and computation induction [15]. In proposing a fourth, subgoal induction, it is perhaps worthwhile to discuss just why it might be used in preference to one of the c... |